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# Introduction to turbulence/Stationarity and homogeneity

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|                                       | == Processes statistically stationary in time ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |                                                                                                                                               | == Processes statistically stationary in time ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
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|                                       | An alternative way of looking at ''stationarity'' is to note that ''the statistics of the process are independent of the origin in time''. It is obvious from the above, for example, that if the statistics of a process are time independent, then <math> \left\langle  u^{n} \left( t \right) \right\rangle = \left\langle u^{n} \left( t + T \right) \right\rangle </math> , etc., where <math> T </math> is some arbitrary translation of the origin in time. Less obvious, but equally true, is that the product <math> \left\langle u \left( t \right) u \left( t' \right) \right\rangle </math> depends only on time difference <math> t'-t </math> and not on <math> t </math> (or <math> t' </math> ) directly. This consequence of stationarity can be extended to any product moment. For example <math> \left\langle u \left( t \right) v \left( t' \right) \right\rangle </math> can depend only on the time difference <math> t'-t </math>. And <math> \left\langle u \left( t \right) v \left( t' \right) w \left( t'' \right)\right\rangle </math> can depend only on the two time differences <math> t'- t </math> and <math> t'' - t </math> (or <math> t'' - t' </math> ) and not <math> t </math> , <math> t' </math> or <math> t'' </math> directly. |                                                                                                                                               | An alternative way of looking at ''stationarity'' is to note that ''the statistics of the process are independent of the origin in time''. It is obvious from the above, for example, that if the statistics of a process are time independent, then <math> \left\langle  u^{n} \left( t \right) \right\rangle = \left\langle u^{n} \left( t + T \right) \right\rangle </math> , etc., where <math> T </math> is some arbitrary translation of the origin in time. Less obvious, but equally true, is that the product <math> \left\langle u \left( t \right) u \left( t' \right) \right\rangle </math> depends only on time difference <math> t'-t </math> and not on <math> t </math> (or <math> t' </math> ) directly. This consequence of stationarity can be extended to any product moment. For example <math> \left\langle u \left( t \right) v \left( t' \right) \right\rangle </math> can depend only on the time difference <math> t'-t </math>. And <math> \left\langle u \left( t \right) v \left( t' \right) w \left( t'' \right)\right\rangle </math> can depend only on the two time differences <math> t'- t </math> and <math> t'' - t </math> (or <math> t'' - t' </math> ) and not <math> t </math> , <math> t' </math> or <math> t'' </math> directly. |
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| -                                     | == The autocorrelation ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                                                                                                             | == Autocorrelation ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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|                                       | One of the most useful statistical moments in the study of stationary random processes (and turbulence, in particular) is the '''autocorrelation''' defined as the average of the product of the random variable evaluated at two times, i.e. <math> \left\langle u \left( t \right) u \left( t' \right)\right\rangle </math>. Since the process is assumed stationary, this product can depend only on the time difference <math> \tau = t' - t </math>. Therefore the autocorrelation can be written as:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                                                                                                               | One of the most useful statistical moments in the study of stationary random processes (and turbulence, in particular) is the '''autocorrelation''' defined as the average of the product of the random variable evaluated at two times, i.e. <math> \left\langle u \left( t \right) u \left( t' \right)\right\rangle </math>. Since the process is assumed stationary, this product can depend only on the time difference <math> \tau = t' - t </math>. Therefore the autocorrelation can be written as:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
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|                                       | </td><td width="5%">(4)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |                                                                                                                                               | </td><td width="5%">(4)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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| -                                     | == The autocorrelation coefficient ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                                                                                                             | == Autocorrelation coefficient ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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|                                       | It is convenient to define the ''autocorrelation coefficient'' as:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                                                                                                               | It is convenient to define the ''autocorrelation coefficient'' as:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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|                                       | for all values of <math> \tau </math> .                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                                                                                                               | for all values of <math> \tau </math> .                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
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| -                                     | == The integral scale ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                                                                                                             | == Integral scale ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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|                                       | One of the most useful measures of the length of a time a process is correlated with itself is the integral scale defined by                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                                                                                                               | One of the most useful measures of the length of a time a process is correlated with itself is the integral scale defined by                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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|                                       | It is easy to see why this works by looking at Figure 5.2. In effect we have replaced the area under the correlation coefficient by a rectangle of height unity and width <math> T_{int} </math> .                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                                                                                                               | It is easy to see why this works by looking at Figure 5.2. In effect we have replaced the area under the correlation coefficient by a rectangle of height unity and width <math> T_{int} </math> .                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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| -                                     | == The temporal Taylor microscale ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                                                                                                             | == Temporal Taylor microscale ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
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|                                       | The autocorrelation can be expanded about the origin in a MacClaurin series; i.e.,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                                                                                                               | The autocorrelation can be expanded about the origin in a MacClaurin series; i.e.,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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|                                       | Consider for example the one-dimensional spatial correlation which is obtained by measuring the correlation between the temperature at two points along a line in the x-direction, say,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                                                                                                               | Consider for example the one-dimensional spatial correlation which is obtained by measuring the correlation between the temperature at two points along a line in the x-direction, say,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | B^{(1)}_{\theta,\theta} \left( r \right) \equiv \left\langle \theta \left( x_{1} + r , x_{2} , x_{3} , t  \right) \theta \left( x_{1} , x_{2} , x_{3} , t  \right) \right\rangle                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(60)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | The superscript "(1)" denotes "the coordinate direction in which the separation occurs". This distinguishes it from the vector separation of <math> B_{\theta,\theta} </math> above. Also, note that the correlation at zero separationis just the variance; i.e.,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | B^{(1)}_{\theta,\theta} \left( 0 \right) = \left\langle \theta^{2} \right\rangle                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(61)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | The integral scale in the <math> x </math>-direction can be defined as:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | L^{(1)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x + r, y,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(62)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | It is clear that there are at least two more integral scales which could be defined by considering separations in the y and z directions. Thus                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | L^{(2)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y + r,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(63)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | L^{(3)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y,z + r,t \right) \theta \left( x,y,z,t \right) \right\rangle dr                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(64)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | In fact, an integral scale could be defined for ''any'' direction simply by choosing the components of the separation vector <math> \stackrel{\rightarrow}{r} </math>. This situation is even more complicated when correlations of vector quantities are considered. For example, consider the correlation of the velocity vectors at two points, <math> B_{i,j} \left( \stackrel{\rightarrow}{r} \right) </math>. Clearly  <math> B_{i,j} \left( \stackrel{\rightarrow}{r} \right) </math> is not a single correlation, but rather nine separate correlations: <math> B_{1,1} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{1,2} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{1,3} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{2,1} \left( \stackrel{\rightarrow}{r} \right) </math> , <math> B_{2,2} \left( \stackrel{\rightarrow}{r} \right) </math> , etc. For each of these an integral scale can be defined once a direction for the separation vector is chosen. For example, the integral scales associated with <math> B_{1,1} </math> for the principal directions are                                                                                                                                          |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | L^{(1)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( r,0,0 \right) dr                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(65)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(66)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | L^{(3)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,0,r \right) dr                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(67)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | Similar integral scales can be defined for the other componentsof the correlation tensor. Two of particular importance in the development of the turbulence theory are:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(68)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | L^{(1)}_{2,2} \equiv \frac{1}{\left\langle u^{2}_{2} \right\rangle} \int^{\infty}_{0} B_{2,2} \left( r,0,0 \right) dr                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(69)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | In general, each of these integral scales will be different, unless restrictions beyond simple homogeneity are placed on the process (e.g., like ''isotropy'' discussed below). Thus, it is important to specify precisely which integral scale is being referred to; i.e., which components of the vector quantities are being used and in which direction the integration is being performed.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | Similar considerations apply to the Taylor microscales, regardless of whether they are being determined from the correlations at small separations, or from the mean square fluctuating gradients. The two most commonly used Taylor microscales are often referred to as <math> \lambda_{f} </math> and <math> \lambda_{g} </math> and are defined by                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | \lambda^{2}_{f} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{1}  \right]^{2} \right\rangle }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(70)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | \lambda^{2}_{g} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{2}  \right]^{2} \right\rangle }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(71)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | The subscripts f and g refer to the autocorrelation coefficients defined by:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | f \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1} + r,x_{2},x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( r,0,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(72)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | g \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1},x_{2}+r,x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( 0,r,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(73)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | It is straightforward to show from the definitions that <math> \lambda_{f} </math> and <math> \lambda_{g} </math> are related to the curvature of the <math> f </math> and <math> g </math> correlation functions at <math> r=0 </math>. Specifically,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | \lambda^{2}_{f}= \frac{2}{d^{2} f / dr^{2} |_{r=0}  }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(74)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | \lambda^{2}_{g}= \frac{2}{d^{2} g / dr^{2} |_{r=0}  }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(75)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | Since both <math> f </math> and <math> g </math> are symmetrical functions of <math> r </math>,  <math> df/dr </math> and <math> dg/dr </math> must be zero at <math> r=0 </math>. It follows immediately that the leading <math> r </math>-dependent term in the expansions about the origin of both autocorrelations are of parabolic form; i.e.,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | f \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{f}} + \cdots                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | </td><td width="5%">(76)</td></tr></table>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | <table width="70%"><tr><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | :<math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                                                                                                             | g \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{g}} + \cdots                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
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## Latest revision as of 09:19, 25 February 2008

Nature of turbulence
Statistical analysis

Reynolds averaged equation
Turbulence kinetic energy
Stationarity and homogeneity
Homogeneous turbulence
Wall bounded turbulent flows
Study questions ... template not finished yet!

## Processes statistically stationary in time

Many random processes have the characteristic that their statistical properties do not appear to depend directly on time, even though the random variables themselves are time-dependent. For example, consider the signals shown in Figures 2.2 and 2.5

When the statistical properties of a random process are independent of time, the random process is said to be stationary. For such a process all the moments are time-independent, e.g., $\left\langle \tilde{ u \left( t \right)} \right\rangle = U$, etc. In fact, the probability density itself is time-independent, as should be obvious from the fact that the moments are time independent.

An alternative way of looking at stationarity is to note that the statistics of the process are independent of the origin in time. It is obvious from the above, for example, that if the statistics of a process are time independent, then $\left\langle u^{n} \left( t \right) \right\rangle = \left\langle u^{n} \left( t + T \right) \right\rangle$ , etc., where $T$ is some arbitrary translation of the origin in time. Less obvious, but equally true, is that the product $\left\langle u \left( t \right) u \left( t' \right) \right\rangle$ depends only on time difference $t'-t$ and not on $t$ (or $t'$ ) directly. This consequence of stationarity can be extended to any product moment. For example $\left\langle u \left( t \right) v \left( t' \right) \right\rangle$ can depend only on the time difference $t'-t$. And $\left\langle u \left( t \right) v \left( t' \right) w \left( t'' \right)\right\rangle$ can depend only on the two time differences $t'- t$ and $t'' - t$ (or $t'' - t'$ ) and not $t$ , $t'$ or $t''$ directly.

## Autocorrelation

One of the most useful statistical moments in the study of stationary random processes (and turbulence, in particular) is the autocorrelation defined as the average of the product of the random variable evaluated at two times, i.e. $\left\langle u \left( t \right) u \left( t' \right)\right\rangle$. Since the process is assumed stationary, this product can depend only on the time difference $\tau = t' - t$. Therefore the autocorrelation can be written as:

$C \left( \tau \right) \equiv \left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle$  (1)

The importance of the autocorrelation lies in the fact that it indicates the "memory" of the process; that is, the time over which is correlated with itself. Contrast the two autocorrelation of deterministic sine wave is simply a cosine as can be easily proven. Note that there is no time beyond which it can be guaranteed to be arbitrarily small since it always "remembers" when it began, and thus always remains correlated with itself. By contrast, a stationary random process like the one illustrated in the figure will eventually lose all correlation and go to zero. In other words it has a "finite memory" and "forgets" how it was. Note that one must be careful to make sure that a correlation really both goes to zero and stays down before drawing conclusions, since even the sine wave was zero at some points. Stationary random process always have two-time correlation functions which eventually go to zero and stay there.

Example 1.

Consider the motion of an automobile responding to the movement of the wheels over a rough surface. In the usual case where the road roughness is randomly distributed, the motion of the car will be a weighted history of the road's roughness with the most recent bumps having the most influence and with distant bumps eventually forgotten. On the other hand, if the car is travelling down a railroad track, the periodic crossing of the railroad ties represents a determenistic input an the motion will remain correlated with itself indefinitely, a very bad thing if the tie crossing rate corresponds to a natural resonance of the suspension system of the vehicle.

Since a random process can never be more than perfectly correlated, it can never achieve a correlation greater than is value at the origin. Thus

$\left| C \left( \tau \right) \right| \leq C\left( 0 \right)$  (2)

An important consequence of stationarity is that the autocorrelation is symmetric in the time difference $\tau = t' - t$. To see this simply shift the origin in time backwards by an amount $\tau$ and note that independence of origin implies:

$\left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle = \left\langle u \left( t - \tau \right) u \left( t \right) \right\rangle$  (3)

Since the right hand side is simply $C \left( - \tau \right)$, it follows immediately that:

$C \left( \tau \right) = C \left( - \tau \right)$  (4)

## Autocorrelation coefficient

It is convenient to define the autocorrelation coefficient as:

$\rho \left( \tau \right) \equiv \frac{ C \left( \tau \right)}{ C \left( 0 \right)} = \frac{\left\langle u \left( t \right) u \left( t + \tau \right) \right\rangle}{ \left\langle u'^{2} \right\rangle }$  (5)

where

$\left\langle u^{2} \right\rangle = \left\langle u \left( t \right) u \left( t \right) \right\rangle = C \left( 0 \right) = var \left[ u \right]$  (6)

Since the autocorrelation is symmetric, so is its coefficient, i.e.,

$\rho \left( \tau \right) = \rho \left( - \tau \right)$  (7)

It is also obvious from the fact that the autocorrelation is maximal at the origin that the autocorrelation coefficient must also be maximal there. In fact from the definition it follows that

$\rho \left( 0 \right) = 1$  (8)

and

$\rho \left( \tau \right) \leq 1$  (9)

for all values of $\tau$ .

## Integral scale

One of the most useful measures of the length of a time a process is correlated with itself is the integral scale defined by

$T_{int} \equiv \int^{\infty}_{0} \rho \left( \tau \right) d \tau$  (10)

It is easy to see why this works by looking at Figure 5.2. In effect we have replaced the area under the correlation coefficient by a rectangle of height unity and width $T_{int}$ .

## Temporal Taylor microscale

The autocorrelation can be expanded about the origin in a MacClaurin series; i.e.,

$C \left( \tau \right) = C \left( 0 \right) + \tau \frac{ d C }{ d t }|_{\tau = 0} + \frac{1}{2} \tau^{2} \frac{d^{2} C}{d t^{2} }|_{\tau = 0} + \frac{1}{3!} \tau^{3} \frac{d^{3} C}{d t^{3} }|_{\tau = 0}$  (11)

But we know the aoutocorrelation is symmetric in $\tau$ , hence the odd terms in $\tau$ must be identically to zero (i.e., $dC / dt |_{\tau = 0} = 0$ , $d^{3}C / dt^{3} |_{\tau = 0} = 0$, etc.). Therefore the expansion of the autocorrelation near the origin reduces to:

$C \left( \tau \right) = C \left( 0 \right) + \frac{1}{2} \tau^{2} \frac{d^{2} C}{d t^{2} }|_{\tau = 0} + \cdots$  (12)

Similary, the autocorrelation coefficient near the origin can be expanded as:

$\rho \left( \tau \right) = 1 + \frac{1}{2}\frac{d^{2}\rho}{d t^{2}}|_{\tau = 0} \tau^{2}+ \cdots$  (13)

where we have used the fact that $\rho \left( 0 \right) = 1$ . If we define $' = d / dt$ we can write this compactly as:

$\rho \left( \tau \right) = 1 + \frac{1}{2} \rho '' \left( 0 \right) \tau^{2} + \cdots$  (14)

Since $\rho \left( \tau \right)$ has its maximum at the origin, obviously $\rho'' \left( 0 \right)$ must be negative.

We can use the correlation and its second derivative at the origin to define a special time scale, $\lambda_{\tau}$ (called the Taylor microscale) by:

$\lambda^{2}_{\tau} \equiv - \frac{2}{\rho'' \left( 0 \right)}$  (15)

Using this in equation 14 yields the expansion for the correlation coefficient near the origin as:

$\rho \left( \tau \right) = 1 - \frac{\tau^{2}}{\lambda^{2}_{\tau}} + \cdots$  (16)

Thus very near the origin the correlation coefficient (and the autocorrelation as well) simply rolls off parabolically; i.e.,

$\rho \left( \tau \right) \approx 1 - \frac{\tau^{2}}{\lambda^{2}_{\tau}}$  (17)

This parabolic curve is shown in Figure 3 as the osculating (or 'kissing') parabola which approaches zero exactly as the autocorrelation coefficient does. The intercept of this osculating parabola with the $\tau$ -axis is the Taylor microscale, $\lambda_{\tau}$.

The Taylor microscale is significant for a number of reasons. First, for many random processes (e.g., Gaussian), the Taylor microscale can be proven to be the average distance between zero-crossing of a random variable in time. This is approximately true for turbulence as well. Thus one can quickly estimate the Taylor microscale by simply observing the zero-crossings using an oscilloscope trace.

The Taylor microscale also has a special relationship to the mean square time derivative of the signal, $\left\langle \left[ d u / d t \right]^{2} \right\rangle$. This is easiest to derive if we consider two stationary random signals at two different times say $u = u \left( t \right)$ and $u' = u' \left( t' \right)$. The derivative of the first signal is $d u / d t$ and the second $d u' / d t'$. Now lets multiply these together and rewrite them as:

$\frac{du'}{dt'} \frac{du}{dt} = \frac{d^{2}}{dtdt'} u \left( t \right) u' \left( t' \right)$  (18)

where the right-hand side follows from our assumption that $u$ is not a function of $t'$ nor $u'$ a function of $t$.

Now if we average and interchenge the operations of differentiation and averaging we obtain:

$\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{dtdt'} \left\langle u \left( t \right) u' \left( t' \right) \right\rangle$  (19)

Here comes the first trick: we simply take $u'$ to be exactly $u$ but evaluated at time $t'$. So $u \left( t \right) u' \left( t' \right)$ simply becomes $u \left( t \right) u \left( t' \right)$ and its average is just the autocorrelation, $C \left( \tau \right)$. Thus we are left with:

$\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{dtdt'} C \left( t' - t \right)$  (20)

Now we simply need to use the chain-rule. We have already defined $\tau = t' - t$. Let's also define $\xi = t' + t$ and transform the derivatives involving $t$ and $t'$ to derivatives involving $\tau$ and $\xi$. The result is:

$\frac{d^{2}}{dtdt'} = \frac{d^{2}}{d \xi^{2}} - \frac{d^{2}}{d \tau^{2}}$  (21)

So equation 20 becomes

$\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = \frac{d^{2}}{d \xi^{2}}C \left( \tau \right) - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)$  (22)

But since $C$ is a function only of $\tau$, the derivative of it with respect to $\xi$ is identically zero. Thus we are left with:

$\left\langle \frac{du'}{dt'} \frac{du}{dt} \right\rangle = - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)$  (23)

And finally we need the second trick. Let's evaluate both sides at $t = t'$ (or $\tau = 0$ ) to obtain the mean square derivative as:

$\left\langle \left( \frac{du}{dt} \right)^{2} \right\rangle = - \frac{d^{2}}{d \tau^{2}} C \left( \tau \right)|_{ \tau = 0}$  (24)

But from our definition of the Taylor microscale and the facts that $C \left( 0 \right) = \left\langle u^{2} \right\rangle$ and $C \left( \tau \right) = \left\langle u^{2} \right\rangle \rho \left( \tau \right)$, this is exactly the same as:

$\left\langle \left( \frac{du}{dt} \right)^{2} \right\rangle = 2 \frac{ \left\langle u^{2} \right\rangle}{\lambda^{2}_{\tau}}$  (25)

This amasingly simple result is very important in the study of turbulence, especially after we extend it to spatial derivatives.

## Time averages of stationary processes

It is common practice in many scientific disciplines to define a time average by integrating the random variable over a fixed time interval, i.e. ,

$U_{T} \equiv \frac{1}{T} \int^{T_{2}}_{T_{1}} u \left( t \right) dt$  (26)

For the stationary random processes we are considering here, we can define $T_{1}$ to be the origin in time and simply write:

$U_{T} \equiv \frac{1}{T} \int^{T}_{0} u \left( t \right) dt$  (27)

where $T = T_{2} - T_{1}$ is the integration time.

Figure 5.4. shows a portion of a stationary random signal over which such an integration might be performed. The ime integral of $u \left( t \right)$ over the integral $\left( O, T \right)$ corresponds to the shaded area under the curve. Now since $u \left( t \right)$ is random and since it formsthe upper boundary of the shadd area, it is clear that the time average, $U_{T}$ is a lot like the estimator for the mean based on a finite number of independent realization, $X_{N}$ we encountered earlier in section Estimation from a finite number of realizations (see Elements of statistical analysis)

It will be shown in the analysis presented below that if the signal is stationary, the time average defined by equation 27 is an unbiased estimator of the true average $U$. Moreover, the estimator converges to $U$ as the time becomes infinite; i.e., for stationary random processes

$U = \lim_{T \rightarrow \infty} \frac{1}{T} \int^{T}_{0} u \left( t \right) dt$  (28)

Thus the time and ensemble averages are equivalent in the limit as $T \rightarrow \infty$, but only for a stationary random process.

## Bias and variability of time estimators

It is easy to show that the estimator, $U_{T}$, is unbiased by taking its ensemble average; i.e.,

$\left\langle U_{T} \right\rangle = \left\langle \frac{1}{T} \int^{T}_{0} u \left( t \right) dt \right\rangle = \frac{1}{T} \int^{T}_{0} \left\langle u \left( t \right) \right\rangle dt$  (29)

Since the process has been assumed stationary, $\left\langle u \left( t \right) \right\rangle$ is independent of time. It follows that:

$\left\langle U_{T} \right\rangle = \frac{1}{T} \left\langle u \left( t \right) \right\rangle T = U$  (30)

To see whether the etimate improves as $T$ increases, the variability of $U_{T}$ must be examined, exactly as we did for $X_{N}$ earlier in section Bias and convergence of estimators (see chapter The elements of statistical analysis). To do this we need the variance of $U_{T}$ given by:

$\begin{matrix} var \left[ U_{T} \right] & = & \left\langle \left[ U_{T} - \left\langle U_{T} \right\rangle \right]^{2} \right\rangle = \left\langle \left[ U_{T} - U \right]^{2} \right\rangle \\ & = & \frac{1}{T^{2}} \left\langle \left\{ \int^{T}_{0} \left[ u \left( t \right) - U \right] \right\}^{2} \right\rangle \\ & = & \frac{1}{T^{2}} \left\langle \int^{T}_{0} \int^{T}_{0} \left[ u \left( t \right) - U \right] \left[ u \left( t' \right) - U \right] dtdt' \right\rangle \\ & = & \frac{1}{T^{2}} \int^{T}_{0} \int^{T}_{0} \left\langle u'\left( t \right) u'\left( t' \right) \right\rangle dtdt' \\ \end{matrix}$  (31)

But since the process is assumed stationary $\left\langle u' \left( t \right) u' \left( t' \right) \right\rangle = C \left( t' - t \right)$ where $C \left( t' - t \right) = \left\langle u^{2} \right\rangle \rho \left( t'-t \right)$ is the correlation coefficient. Therefore the integral can be rewritten as:

$\begin{matrix} var \left[ U_{T} \right] & = & \frac{1}{T^{2}} \int^{T}_{0} \int^{T}_{0} C \left( t' - t \right) dtdt' \\ & = & \frac{ \left\langle u^{2} \right\rangle }{ T^{2} } \int^{T}_{0} \int^{T}_{0} \rho \left( t' - t \right) dtdt' \\ \end{matrix}$  (33)

Now we need to apply some fancy calculus. If new variables $\tau= t'-t$ and $\xi= t'+t$ are defined, the double integral can be transformed to (see Figure 5.5):

$var \left[ U_{T} \right] = \frac{var \left[ u \right]}{2 T^{2}} \left[ \int^{T}_{0} d \tau \int^{T-\tau}_{\tau} d \xi \rho \left( \tau \right) + \int^{0}_{-T} d \tau \int^{T+\tau}_{-\tau} d \xi \rho \left( \tau \right) \right]$  (35)

where the factor of $1/2$ arises from the Jacobian of the transformation. The integrals over $d \xi$ can be evaluated directly to yield:

$var \left[ U_{T} \right] = \frac{var \left[ u \right]}{2 T^{2}} \left\{ \int^{T}_{0} \rho \left( \tau \right) \left[ T - \tau \right] d \tau + \int^{0}_{-T} \rho \left( \tau \right) \left[ T + \tau \right] \right\}$  (36)

By noting that the autocorrelation is symmetric, the second integral can be transformed and added to the first to yield at last the result we seek as:

$var \left[ U_{T} \right] = \frac{var \left[ u \right]}{T} \int^{T}_{-T} \rho \left( \tau \right) \left[ 1 - \frac{ \left| \tau \right| }{T} \right] d \tau$  (37)

Now if our averaging time, $T$, is chosen so large that $\left| \tau \right| / T << 1$ over the range for which $\rho \left( \tau \right)$ is non-zero, the integral reduces:

$\begin{matrix} var \left[ U_{T} \right] & \approx & \frac{2 var \left[ u \right]}{T} \int^{T}_{0} \rho \left( \tau \right) d \tau \\ & = & \frac{2 T_{int}}{T} var \left[ u \right] \\ \end{matrix}$  (38)

where $T_{int}$ is the integral scale defined by equation 10. Thus the variability of our estimator is given by:

$\epsilon^{2}_{U_{T}} = \frac{2T_{int}}{T}$  (39)

Therefore the estimator does, in fact, converge (in mean square) to the correct result as the averaging time, $T$ increases relative to the integral scale, $T_{int}$.

There is a direct relationship between equation 39 and equation 52 in chapter The elements of statistical analysis ( section Bias and convergence of estimators) which gave the mean square variability for the ensemble estimate from a finite number of statistically independent realizations, $X_{N}$. Obviously the effective number of independent realizations for the finite time estimator is:

$N_{eff} = \frac{2T_{int}}{T}$  (40)

so that the two expressions are equivalent. Thus, in effect, portions of the record separated by two integral scales behave as though they were statistically independent, at least as far as convergence of finite time estimators is concerned.

Thus what is required for convergence is again, many independent pieces of information. This is illustrated in Figure 5.6. That the length of the recordn should be measured in terms of the integral scale should really be no surprise since it is a measure of the rate at which a process forgets its past.

Example

It is desired to mesure the mean velocity in a turbulent flow to within an rms error of 1% (i.e. $\epsilon = 0.01$ ). The expected fluctuation level of the signal is 25% and integral scale is estimated as 100 ms. What is the required averaging time?

From equation 39

$\begin{matrix} T & = & \frac{2T_{int}}{\epsilon^{2}} \frac{var \left[ u \right]}{U^{2}} \\ & = & 2 \times 0.1 \times (0.25)^{2} / (0.01)^{2} = 125 sec \\ \end{matrix}$  (41)

Similar considerations apply to any other finite time estimator and equation 55 from chapter Statistical analysis can be applied directly as long as equation 40 is used for the number of independent samples.

It is common common experimental practice to not actually carry out an analog integration. Rather the signal is sampled at fixed intervals in time by digital means and the averages are computed as for an esemble with a finite number of realizations. Regardless of the manner in which the signal is processed, only a finite portion of a stationary time series can be analyzed and the preceding considerations always apply.

It is important to note that data sampled more rapidly than once every two integral scales do not contribute to the convergence of the estimator since they can not be considered independent. If $N$ is the actual number of samples acquired and $\Delta t$ is the time between samples, then the effective number of independent realizations is

$N_{eff} = \left\{ \begin{array}{lll} N \Delta t /T_{int} & if & \Delta t < 2T_{int} \\ N & if & \Delta t \geq 2T_{int} \\ \end{array} \right.$  (42)

It should be clear that if you sample faster than $\Delta t = 2T_{int}$ you are processing unnecessary data which does not help your statistics converge.

You may wonder why one would ever take data faster than absolutely necessary, since it simply it simply fills up your computer memory with lots of statistically redundant data. When we talk about measuring spectra you will learn that for spectral measurements it is necessary to sample much faster to avoid spactral aliasing. Many wrongly infer that they must sample at these higher rates even when measuring just moments. Obviously this is not the case if you are not measuring spectra.

## Random fields of space and time

To this point only temporally varying random fields have been discussed. For turbulence however, random fields can be functions of both space and time. For example, the temperature $\theta$ could be a random scalar function of time $t$ and position $\stackrel{\rightarrow}{x}$, i.e.,

$\theta = \theta \left( \stackrel{\rightarrow}{x} , t \right)$  (43)

The velocity is another example of a random vector function of position and time, i.e.,

$\stackrel{\rightarrow}{u} = \stackrel{\rightarrow}{u} \left( \stackrel{\rightarrow}{x},t \right)$  (44)

or in tensor notation,

$u_{i} = u_{i} \left( \stackrel{\rightarrow}{x},t \right)$  (45)

In the general case, the ensemble averages of these quantities are functions of both positon and time; i.e.,

$\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv \Theta \left( \stackrel{\rightarrow}{x},t \right)$  (46)

$\left\langle u_{i} \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv U_{i} \left( \stackrel{\rightarrow}{x},t \right)$  (47)

If only stationary random processes are considered, then the averages do not depend on time and are functions of $\stackrel{\rightarrow}{x}$ only; i.e.,

$\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv \Theta \left( \stackrel{\rightarrow}{x} \right)$  (48)

$\left\langle u_{i} \left( \stackrel{\rightarrow}{x},t \right) \right\rangle \equiv U_{i} \left( \stackrel{\rightarrow}{x}\right)$  (49)

Now the averages may not be position dependent either. For example, if the averages are independent of the origin in position, then the field is said to be homogeneous. Homogenity (the noun corresponding to the adjective homogeneous) is exactly analogous to stationarity except that position is now the variable, and not time.

It is, of course, possible (at least in concept) to have homogeneous fields which are either stationary or non stationary. Since position, unlike time, is a vector quantity it is also possible to have only partial homogeneity. For example, a field can be homogeneous in the $x_{1}-$ and $x_{3}-$ directions, but not in the $x_{2}-$ direction so that $U_{i}=U_{i}(X_{2})$ only. In fact, it appears to be dynamically impossible to have flows which are honogeneous in all variables and stationary as well, but the concept is useful, nonetheless.

Homogeneity will be seen to have powerful consequences for the equations govering the averaged motion, since the spatial derivative of any averaged quantity must be identically zero. Thus even homogeneity in only one direction can considerably simplify the problem. For example, in the Reynolds stress transport equation, the entire turbulence transport is exactly zero if the field is homogeneous.

## Multi-point statistics in homogeneous field

The concept of homogeneity can also be extended to multi-point statistics. Consider for example, the correlation between the velocity at one point and that at another as illustrated in Figure 5.7. If the time dependence is suppressed and the field is assumed statistically homogeneous, this correlation is a function only of the separation of the two points, i.e.,

$\left\langle u_{i} \left( \stackrel{\rightarrow}{x} , t \right) u_{j} \left( \stackrel{\rightarrow}{x'} , t \right) \right\rangle \equiv B_{i,j} \left( \stackrel{\rightarrow}{r} \right)$  (50)

where $\stackrel{\rightarrow}{r}$ is the separation vector defined by

$\stackrel{\rightarrow}{r} = \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{x}$  (51)

or

$r_{i} = x'_{i} - x_{i}$  (52)

Note that the convention we shall follow for vector quantities is that the first subscript on $B_{i,j}$ is the component of velocity at the first position, $\stackrel{\rightarrow}{x}$ , and the second subscript is the component of velocity at the second, $\stackrel{\rightarrow}{x'}$. For scalar quantities we shall simply put a simbol for the quantity to hold the place. For example, we would write the two-point temperature correlation in a homogeneous field by:

$\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) \theta \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{\theta , \theta} \left( \stackrel{\rightarrow}{r} \right)$  (53)

A mixed vector/scalar correlation like the two-point temperature velocity correlation would be written as:

$\left\langle u_{i} \left( \stackrel{\rightarrow}{x} , t \right) \theta \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{i,\theta } \left( \stackrel{\rightarrow}{r} \right)$  (54)

On the other hand, if we meant for the temperature to be evaluated at $\stackrel{\rightarrow}{x}$ and the velocity at $\stackrel{\rightarrow}{x'}$ we would have to write:

$\left\langle \theta \left( \stackrel{\rightarrow}{x},t \right) u_{i} \left( \stackrel{\rightarrow}{x'},t \right) \right\rangle \equiv B_{ \theta, i } \left( \stackrel{\rightarrow}{r} \right)$  (55)

Now most books don't bother with the subscript notation, and simply give each new correlation a new symbol. At first this seems much simpler; and it is as long as you are only dealing with one or two different correlations. But introduce a few more, then read about a half-dozen pages, and you will find you completely forget what they are or how they were put together. It is usually very important to know exactly what you are talking about, so we will use this comma system to help us remember.

It is easy to see that the consideration of vector quantities raises special considerations. For example, the correlation between a scalar function of position at two points is symmetrical in $\stackrel{\rightarrow}{r}$ , i.e.,

$B_{\theta,\theta} \left( \stackrel{\rightarrow}{r} \right) = B_{\theta,\theta} \left( - \stackrel{\rightarrow}{r} \right)$  (56)

This is easy to show from the definition of $B_{\theta,\theta}$ and the fact that the field is homogeneous. Simply shift each of the position vectors by the same amount $- \stackrel{\rightarrow}{r}$ as shown in Figure 5.8 to obtain:

$\begin{matrix} B_{\theta,\theta}\left( \stackrel{\rightarrow}{r},t \right) & \equiv & \left\langle \theta\left( \stackrel{\rightarrow}{x}, t \right) \theta\left( \stackrel{\rightarrow}{x'}, t \right) \right\rangle \\ & = & \left\langle \theta \left( \stackrel{\rightarrow}{x} - \stackrel{\rightarrow}{r} , t \right) \theta \left( \stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{r} , t \right) \right\rangle \\ & = & B_{\theta,\theta}\left( - \stackrel{\rightarrow}{r},t \right) \\ \end{matrix}$  (57)

since $\stackrel{\rightarrow}{x'} - \stackrel{\rightarrow}{r} = \stackrel{\rightarrow}{x}$ ; i.e., the points are reversed and the separation vector is pointing the opposite way.

Such is not the case, in general, for vector functions of position. For example, see if you can prove to yourself the following:

$B_{\theta,i} \left( \stackrel{\rightarrow}{r} \right) = B_{i,\theta} \left( - \stackrel{\rightarrow}{r} \right)$  (58)

and

$B_{i,j} \left( \stackrel{\rightarrow}{r} \right) = B_{j,i} \left( - \stackrel{\rightarrow}{r} \right)$  (59)

Clearly the latter is symmetrical in the variable $\stackrel{\rightarrow}{r}$ only when $i = j$ .

These properties of the two-point correlation function will be seen to play an important role in determining the interrelations among the different two-point statistical quantities. They will be especially important when we talk about spectral quantities.

## Spatial integral and Taylor microscales

Just as for a stationary random process, correlations between spatially varying, but statistically homogeneous, random quantities ultimately go to zero;, i.e., they become uncorrelated as their locations become widely separated. Because position (o relative position) is a vector quantity, however, the correlation the carrelation may die off at different rates in different directions. Thus direction must be an important part of the definitions of the integral scales and microscales.

Consider for example the one-dimensional spatial correlation which is obtained by measuring the correlation between the temperature at two points along a line in the x-direction, say,

$B^{(1)}_{\theta,\theta} \left( r \right) \equiv \left\langle \theta \left( x_{1} + r , x_{2} , x_{3} , t \right) \theta \left( x_{1} , x_{2} , x_{3} , t \right) \right\rangle$  (60)

The superscript "(1)" denotes "the coordinate direction in which the separation occurs". This distinguishes it from the vector separation of $B_{\theta,\theta}$ above. Also, note that the correlation at zero separationis just the variance; i.e.,

$B^{(1)}_{\theta,\theta} \left( 0 \right) = \left\langle \theta^{2} \right\rangle$  (61)

The integral scale in the $x$-direction can be defined as:

$L^{(1)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x + r, y,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr$  (62)

It is clear that there are at least two more integral scales which could be defined by considering separations in the y and z directions. Thus

$L^{(2)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y + r,z,t \right) \theta \left( x,y,z,t \right) \right\rangle dr$  (63)

and

$L^{(3)}_{\theta} \equiv \frac{1}{ \left\langle \theta^{2} \right\rangle} \int^{\infty}_{0} \left\langle \theta \left( x,y,z + r,t \right) \theta \left( x,y,z,t \right) \right\rangle dr$  (64)

In fact, an integral scale could be defined for any direction simply by choosing the components of the separation vector $\stackrel{\rightarrow}{r}$. This situation is even more complicated when correlations of vector quantities are considered. For example, consider the correlation of the velocity vectors at two points, $B_{i,j} \left( \stackrel{\rightarrow}{r} \right)$. Clearly $B_{i,j} \left( \stackrel{\rightarrow}{r} \right)$ is not a single correlation, but rather nine separate correlations: $B_{1,1} \left( \stackrel{\rightarrow}{r} \right)$ , $B_{1,2} \left( \stackrel{\rightarrow}{r} \right)$ , $B_{1,3} \left( \stackrel{\rightarrow}{r} \right)$ , $B_{2,1} \left( \stackrel{\rightarrow}{r} \right)$ , $B_{2,2} \left( \stackrel{\rightarrow}{r} \right)$ , etc. For each of these an integral scale can be defined once a direction for the separation vector is chosen. For example, the integral scales associated with $B_{1,1}$ for the principal directions are

$L^{(1)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( r,0,0 \right) dr$  (65)

$L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr$  (66)

$L^{(3)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,0,r \right) dr$  (67)

Similar integral scales can be defined for the other componentsof the correlation tensor. Two of particular importance in the development of the turbulence theory are:

$L^{(2)}_{1,1} \equiv \frac{1}{\left\langle u^{2}_{1} \right\rangle} \int^{\infty}_{0} B_{1,1} \left( 0,r,0 \right) dr$  (68)

$L^{(1)}_{2,2} \equiv \frac{1}{\left\langle u^{2}_{2} \right\rangle} \int^{\infty}_{0} B_{2,2} \left( r,0,0 \right) dr$  (69)

In general, each of these integral scales will be different, unless restrictions beyond simple homogeneity are placed on the process (e.g., like isotropy discussed below). Thus, it is important to specify precisely which integral scale is being referred to; i.e., which components of the vector quantities are being used and in which direction the integration is being performed.

Similar considerations apply to the Taylor microscales, regardless of whether they are being determined from the correlations at small separations, or from the mean square fluctuating gradients. The two most commonly used Taylor microscales are often referred to as $\lambda_{f}$ and $\lambda_{g}$ and are defined by

$\lambda^{2}_{f} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{1} \right]^{2} \right\rangle }$  (70)

and

$\lambda^{2}_{g} \equiv 2 \frac{ \left\langle u^{2}_{1} \right\rangle }{ \left\langle \left[ \partial u_{1} / \partial x_{2} \right]^{2} \right\rangle }$  (71)

The subscripts f and g refer to the autocorrelation coefficients defined by:

$f \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1} + r,x_{2},x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( r,0,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }$  (72)

and

$g \left( r \right) \equiv \frac{\left\langle u_{1} \left( x_{1},x_{2}+r,x_{3} \right) u_{1} \left( x_{1},x_{2},x_{3} \right) \right\rangle}{ \left\langle u^{2}_{1} \right\rangle } = \frac{B_{1,1} \left( 0,r,0 \right)}{ B_{1,1} \left( 0,0,0 \right) }$  (73)

It is straightforward to show from the definitions that $\lambda_{f}$ and $\lambda_{g}$ are related to the curvature of the $f$ and $g$ correlation functions at $r=0$. Specifically,

$\lambda^{2}_{f}= \frac{2}{d^{2} f / dr^{2} |_{r=0} }$  (74)

and

$\lambda^{2}_{g}= \frac{2}{d^{2} g / dr^{2} |_{r=0} }$  (75)

Since both $f$ and $g$ are symmetrical functions of $r$, $df/dr$ and $dg/dr$ must be zero at $r=0$. It follows immediately that the leading $r$-dependent term in the expansions about the origin of both autocorrelations are of parabolic form; i.e.,

$f \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{f}} + \cdots$  (76)

and

$g \left( r \right) = 1 - \frac{r^{2}}{\lambda^{2}_{g}} + \cdots$  (77)

This is illustrated in Figure 5.9 which shows that the Taylor microscales are the intersection with the $r$-axis of a parabola fitted to the appropriate correlation function at the origin. Fitting a parabola is a common way to determine the Taylor microscale, but to do so you must make sure you resolve accurately to scales much smaller than it (typically an order of magnitude smaller is required). Otherwise you are simply determining the spatial filtering of your probe or numerical algorithm.

## Credits

This text was based on "Lectures in Turbulence for the 21st Century" by Professor William K. George, Professor of Turbulence, Chalmers University of Technology, Gothenburg, Sweden.
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