Kinematic waves (or continuity waves) occur when the one-dimensional flow rate of some quantity depends on the "density" of that quantity [Lighthill and Whitham (1995), Wallis (1969)]. For example, if the flow rate, (cars per hour), on a motorway is a function of the number of cars per mile, n, the speed of a kinematic wave is:
The partial derivative indicates that other parameters, such as the width of the roadway or the radius of bends, which might influence , are kept constant. In another example, the volumetric flow rate per unit width of a viscous fluid, of thickness δ and viscosity η in the laminar regime down an inclined plane at an angle θ to the horizontal, in an environment of stationary gas of low density, is:
and the kinematic wave speed is:
which is three times the average velocity, .
Similarly, the flow per unit width in a broad river of constant slope and depth, y, is:
if the shear stress on the bottom of the river is related to the mean velocity by
with Cf being the coefficient of friction.
For this case, if Cf is constant, the kinematic wave speed is:
or 3/2 times the average velocity.
Waves of finite amplitude are called kinematic (or continuity) shocks. For the motorway example, their speed is:
where subscripts 1 and 2 refer to conditions on either side of the shock and the algebraic sign gives the direction of propagation. A convenient graphical representation of Uw and Us is as slopes of tangents or chords on a plot of versus n. For a shock wave to be maintained as a sharp discontinuity, kinematic waves must run towards it on both sides. For example, if Uw1 << Us << Uw2, state 2 will be on the negative side of the shock.
Similarly, for a falling liquid film
and the shock will form when a thicker film is above a thinner film and rides over it, as does a surge wave in a river.
In two-phase flow, similar relationships apply, but they are subject to additional constraints. For incompressible flow of two components in a duct of constant area, for example, the constraint is that the overall volumetric flow rate, superficial velocity or volumetric flux, be unchanged across the wave. For gas-liquid flow, the wave speed is then
where the superscript S denotes "superficial" velocity.
In simple cases where relative motion can be described by a drift flux, jgf, that is determined by a balance of forces — as in a bubble column or foam drainage system — the wave speed is:
where jgf is the volumetric drift flux of bubbles in a coordinate system moving at the volumetric average velocity, and α is the void fraction. (See Drift Flux Models.) The corresponding speed of a kinematic shock wave is:
When UGS + ULS = 0, as in a batch phase separation, jgf becomes the volumetric flux of bubbles, UGS, and the equations resemble those for single-phase flow. This is the basis of Kynch’s (1952) theory of batch sedimentation.
Sometimes it is convenient to express wave speed in terms of forces on the components. For example, in stratified gas-liquid flow in an inclined pipe, the balance between buoyancy, interfacial and wall shear stresses may be expressed as:
where ÃL=AL/D2, ÃG=AG/D2, , and are dimensionless cross-sectional areas and wetted perimeters of the liquid stream, gas stream and interface. The symbols fWL, fWG and fi represent the friction factors for the liquid on the wall, the gas on the wall and at the interface, respectively. The kinematic wave speed is then [Wu et al. (1987), Crowley et al. (1992)]:
where F is treated as a function of the three variables ÃL, ULS and UGS.
Kinematic waves describe transients in highly-damped fluid systems, for which the contribution of inertia terms to the momentum balance is small (e.g., sedimentation or fluidization at low particle Reynolds Number). Otherwise, they usually correspond to the limit of "long" waves while smaller scale effects are influenced by other factors such as inertia and surface tension. A criterion for stability in the latter ease is that kinematic waved do not override "dynamic waves" which predominate at the other extreme when frictional forces are neglected. For example, for turbulent flow of a liquid film down an inclined plane, the dynamic wave speed is . Instability, leading to roll-wave formation, occurs when kinematic wave speed exceeds this value, i.e., when
For instance, if Cf = 0.005, the critical slope is tan−1 (.04) or 2.3° and roll-waves will form on steeper slopes.
Kinematic waves may also be the mechanism whereby end-conditions can influence a flow, determining when a solution propagates up- or downstream, or causing a critical condition when the wave speed is zero, as in two-phase countercurrent flow "flooding" [Wallis (1969)]. (See also Flooding and Flow Reversal.)
Crowley, C. J. et al. (1992) Validation of a one-dimensional wave model for the stratified-to-slug flow regime transition, with consequences for wave growth and slug frequency. Int. J. Multiphase Flow. 18: 249-271. DOI: 10.1016/0301-9322(92)90087-W
Kynch, G. H. (1952) A Theory of Sedimentation. Trans. Farad. Soc. 48:166-176.
Lighthill, M. J. and Whitham, G. B. (1955) On kinematic waves, I. Flood movements in long rivers, Roy. Soc. Proc. 229: 281-316.
Wallis, G. B. (1969) One-dimensional Two-phase Flow. McGraw-Hill. New York.
Wu, H. L. et al. (1987) Row pattern transitions in a two-phase gas/condensate flow at high pressure in an 8" horizontal pipe. In Proc. 3rd Int. Conf. on Multiphase Flow. The Hague. The Netherlands. 13-21.