2.22 Region of influence

A CFD calculation is performed by solving partial differential equations, such e.g. for conservation of mass, momentum and energy, over a solution domain. It requires suitable boundary conditions, discussed in Chapter 4 .

The solution is influenced by any change in a field value, e.g. eqn, eqn, eqn, at some point in the domain, e.g. at an inlet boundary. The form of the equation determines the way in which these changes, or perturbations, propagate across the domain over time.

PICT\relax \special {t4ht=

Momentum and mass conservation for an incompressible fluid can be represented by Eq. (2.67 ) and Eq. (2.48 ). The form of Eq. (2.48 ), including only a Laplacian derivative eqn, ensures that eqn is influenced instantaneously at all points in the domain by a perturbation in eqn at any point.

The resulting instantaneous change in eqn then causes eqn to be redistributed everywhere by Eq. (2.67 ), with further short-range changes due to advection and diffusion (discussed next).

The outcome for an incompressible fluid is that eqn is influenced instantaneously everywhere in the domain by a perturbation at any point. In other words, the speed of sound eqn, corresponding to propagation of disturbances, is infinite.

Advection-diffusion equations

Energy conservation can be represented by Eq. (2.65 ), in which perturbations propagate at a characteristic speed due to the advection and diffusion.

Advection propagates at speed eqn, with a region of influence eqn in the direction of flow. The relation can be obtained from scale similarity when eqn and eqn are similar in magnitude, i.e. eqn in Eq. (2.69 ).

By the same argument, the region of influence for diffusion coincides with eqn and eqn being similar in magnitude, i.e. eqn. A diffusion “front” travels a distance according to eqn (eqn means “of the order of magnitude”).

The diffusion equation, eqn in one dimension (eqn) has a solution24 eqn where eqn, eqn are constants and eqn is the error function below.

PICT\relax \special {t4ht=

This means that for heat conduction problems, e.g. in solids, the distance travelled by the thermal front is eqn, consistent with similarity arguments above.

The coefficient eqn depends on where on the eqn curve the front is located. One option is eqn, where the solution is within 0.5% of the asymptote of 1.

24Horatio Carslaw and John Jaeger, Conduction of Heat in Solids.

Notes on CFD: General Principles - 2.22 Region of influence