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# Approximation Schemes for convective term - structured grids - definitions

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|                                       |                                                                         | +                              | == Goals of this section ==                                                                                                                      |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       | Here we shall develop a commone definitions and regulations because of  |                                | Here we shall develop a commone definitions and regulations because of                                                                           |
|                                       |                                                                         |                                |                                                                                                                                                  |
| -                                     | * in different articles was used defferent definitions and notations    | +                              | * in different issues was used different definitions and notations                                                                               |
|                                       | * we are searching for common approach and generalisation               |                                | * we are searching for common approach and generalisation                                                                                        |
|                                       |                                                                         |                                |                                                                                                                                                  |
| -                                     | == Usual using definition for convected variable ==                     | +                              | '''Please note:''' ''as we still developing this section, you can find the rest of non-unificated definitions''                                  |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | == Usual used definition for convected variable ==                                                                                               |
|                                       |                                                                         |                                |                                                                                                                                                  |
|                                       | <math>                                                                  |                                | <math>                                                                                                                                           |
| Line 12:                              |                                                                         | Line 16:                       |                                                                                                                                                  |
|                                       | <math>                                                                  |                                | <math>                                                                                                                                           |
|                                       | \boldsymbol{\phi}                                                       |                                | \boldsymbol{\phi}                                                                                                                                |
| -                                     | </math>                                                                 | +                              | </math>                                                                                                                                          |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | we shall use here <math>\boldsymbol{\phi}</math>                                                                                                 |
|                                       |                                                                         |                                |                                                                                                                                                  |
|                                       | == definition of considered face, upon wich approximation is applied == |                                | == definition of considered face, upon wich approximation is applied ==                                                                          |
|                                       |                                                                         |                                |                                                                                                                                                  |
| -                                     | usually (in the most articles) west face of the control volume <math>   | +                              | usually (in the most articles) west face <math>                                                                                                  |
| -                                     | \boldsymbol{w} </math> is considered (''without loss of generality'')   | +                              | \boldsymbol{w} </math> of the control volume  is considered ''without loss of generality''                                                       |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | for which '''flux is directed from the left to the right''' i.e. <math> \boldsymbol{U_{f} \triangleright 0 } </math>                             |
|                                       |                                                                         |                                |                                                                                                                                                  |
| -                                     | for which '''flux is directed from the left to the right'''             |                                |                                                                                                                                                  |
|                                       |                                                                         |                                |                                                                                                                                                  |
|                                       | we shall define it as <math> \boldsymbol{f} </math>                     |                                | we shall define it as <math> \boldsymbol{f} </math>                                                                                              |
|                                       |                                                                         |                                |                                                                                                                                                  |
| -                                     | and convected variable at face as <math> \boldsymbol{\phi_{f}} </math>  | +                              | and convected variable at face of CV as <math> \boldsymbol{\phi_{f}} </math>                                                                     |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | also you can find in literature such definition as <math> \boldsymbol{i+1/2} </math> , but we suggested it non suitable, because of complication |
|                                       |                                                                         |                                |                                                                                                                                                  |
|                                       | == indicators of the local velocity direction ==                        |                                | == indicators of the local velocity direction ==                                                                                                 |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | approximation scheme can be written in the next form                                                                                             |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <table width="100%"><tr><td>                                                                                                                     |
|                                       |                                                                         | +                              | :<math>                                                                                                                                          |
|                                       |                                                                         | +                              | \phi_{w}=\sigma^{+}_{w}\phi_{W}	+ \sigma^{-}_{w}\phi_{P}                                                                                         |
|                                       |                                                                         | +                              | </math>                                                                                                                                          |
|                                       |                                                                         | +                              | </td><td width="5%">(1)</td></tr></table>                                                                                                        |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | where <math>\sigma^{+}_{w}</math> and <math>\sigma^{-}_{w}</math> are the indicators of the local velocity direction such that                   |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <table width="100%"><tr><td>                                                                                                                     |
|                                       |                                                                         | +                              | :<math>                                                                                                                                          |
|                                       |                                                                         | +                              | \sigma^{+}_{w} = 0.5 \left( 1 + \frac{\left|U_{w} \right|}{U_{w}} \right)                                                                        |
|                                       |                                                                         | +                              | </math>                                                                                                                                          |
|                                       |                                                                         | +                              | </td><td width="5%">(1)</td></tr></table>                                                                                                        |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <table width="100%"><tr><td>                                                                                                                     |
|                                       |                                                                         | +                              | :<math>                                                                                                                                          |
|                                       |                                                                         | +                              | \sigma^{-}_{w} = 1 - \sigma^{+}_{w}                                                                                                              |
|                                       |                                                                         | +                              | </math>                                                                                                                                          |
|                                       |                                                                         | +                              | </td><td width="5%">(1)</td></tr></table>                                                                                                        |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | and of course                                                                                                                                    |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <table width="100%"><tr><td>                                                                                                                     |
|                                       |                                                                         | +                              | <math>                                                                                                                                           |
|                                       |                                                                         | +                              | \left( U_{w} \neq 0  \right)                                                                                                                     |
|                                       |                                                                         | +                              | </math>                                                                                                                                          |
|                                       |                                                                         | +                              | </td><td width="5%">(1)</td></tr></table>                                                                                                        |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | also used such definitions as <math>U^{+}_{w}</math> and <math>U^{-}_{w}</math>                                                                  |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | we offer to use                                                                                                                                  |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <math>U^{+}_{f}</math> and <math>U^{-}_{f}</math>                                                                                                |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | therefore unnormalised form of approximation scheme can be written                                                                               |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <table width="100%"><tr><td>                                                                                                                     |
|                                       |                                                                         | +                              | <math>                                                                                                                                           |
|                                       |                                                                         | +                              | \phi_{f}=U^{+}_{f}\phi_{W} + U^{-}_{f}\phi_{P}                                                                                                   |
|                                       |                                                                         | +                              | </math>                                                                                                                                          |
|                                       |                                                                         | +                              | </td><td width="5%">(1)</td></tr></table>                                                                                                        |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | or in more general form                                                                                                                          |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <table width="100%"><tr><td>                                                                                                                     |
|                                       |                                                                         | +                              | <math>                                                                                                                                           |
|                                       |                                                                         | +                              | \phi_{f}=U^{+}_{f}\phi_{C} + U^{-}_{f}\phi_{D}                                                                                                   |
|                                       |                                                                         | +                              | </math>                                                                                                                                          |
|                                       |                                                                         | +                              | </td><td width="5%">(1)</td></tr></table>                                                                                                        |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | == definitions for NV diagram ==                                                                                                                 |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | we discovered such definitions as                                                                                                                |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <math>\boldsymbol{ \hat{\phi}_{i+1/2} }</math> is a function of <math>\boldsymbol{ \hat{\phi}_{i}} </math>                                       |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <math>\boldsymbol{ \hat{\phi_{w}} }</math> is a function of <math>\boldsymbol{ \hat{\phi}_{W}} </math>                                           |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | we shall use here                                                                                                                                |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <math>\boldsymbol{ \hat{\phi_{f}} }</math> is a function of <math>\boldsymbol{ \hat{\phi}_{C}} </math>                                           |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | == node stencil ==                                                                                                                               |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | Bear in mind this stencil                                                                                                                        |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | ----                                                                                                                                             |
|                                       |                                                                         | +                              | <i> Return to [[Numerical methods | Numerical Methods]] </i>                                                                                     |
|                                       |                                                                         | +                              |                                                                                                                                                  |
|                                       |                                                                         | +                              | <i> Return to [[Approximation Schemes for convective term - structured grids]] </i>                                                              |

## Goals of this section

Here we shall develop a commone definitions and regulations because of

• in different issues was used different definitions and notations
• we are searching for common approach and generalisation

Please note: as we still developing this section, you can find the rest of non-unificated definitions

## Usual used definition for convected variable

$\boldsymbol{f}$

$\boldsymbol{\phi}$

we shall use here $\boldsymbol{\phi}$

## definition of considered face, upon wich approximation is applied

usually (in the most articles) west face $\boldsymbol{w}$ of the control volume is considered without loss of generality

for which flux is directed from the left to the right i.e. $\boldsymbol{U_{f} \triangleright 0 }$

we shall define it as $\boldsymbol{f}$

and convected variable at face of CV as $\boldsymbol{\phi_{f}}$

also you can find in literature such definition as $\boldsymbol{i+1/2}$ , but we suggested it non suitable, because of complication

## indicators of the local velocity direction

approximation scheme can be written in the next form

$\phi_{w}=\sigma^{+}_{w}\phi_{W} + \sigma^{-}_{w}\phi_{P}$  (1)

where $\sigma^{+}_{w}$ and $\sigma^{-}_{w}$ are the indicators of the local velocity direction such that

$\sigma^{+}_{w} = 0.5 \left( 1 + \frac{\left|U_{w} \right|}{U_{w}} \right)$  (1)

$\sigma^{-}_{w} = 1 - \sigma^{+}_{w}$  (1)

and of course

$\left( U_{w} \neq 0 \right)$  (1)

also used such definitions as $U^{+}_{w}$ and $U^{-}_{w}$

we offer to use

$U^{+}_{f}$ and $U^{-}_{f}$

therefore unnormalised form of approximation scheme can be written

$\phi_{f}=U^{+}_{f}\phi_{W} + U^{-}_{f}\phi_{P}$  (1)

or in more general form

$\phi_{f}=U^{+}_{f}\phi_{C} + U^{-}_{f}\phi_{D}$  (1)

## definitions for NV diagram

we discovered such definitions as

$\boldsymbol{ \hat{\phi}_{i+1/2} }$ is a function of $\boldsymbol{ \hat{\phi}_{i}}$

$\boldsymbol{ \hat{\phi_{w}} }$ is a function of $\boldsymbol{ \hat{\phi}_{W}}$

we shall use here

$\boldsymbol{ \hat{\phi_{f}} }$ is a function of $\boldsymbol{ \hat{\phi}_{C}}$

## node stencil

Bear in mind this stencil

Return to Numerical Methods

Return to Approximation Schemes for convective term - structured grids
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