Notes on Computational Fluid Dynamics: General Principles

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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. $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , 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 $\text{[math]}$ , ensures that $\text{[math]}$ is inﬂuenced instantaneously at all points in the domain by a perturbation in $\text{[math]}$ at any point.

The resulting instantaneous change in $\text{[math]}$ then causes $\text{[math]}$ 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 $\text{[math]}$ is inﬂuenced instantaneously everywhere in the domain by a perturbation at any point. In other words, the speed of sound $\text{[math]}$ , 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 $\text{[math]}$ , with a region of inﬂuence $\text{[math]}$ in the direction of ﬂow. The relation can be obtained from scale similarity when $\text{[math]}$ and $\text{[math]}$ are similar in magnitude, i.e. $\text{[math]}$ in Eq. (2.69 ).

By the same argument, the region of inﬂuence for diﬀusion coincides with $\text{[math]}$ and $\text{[math]}$ being similar in magnitude, i.e. $\text{[math]}$ . A diﬀusion “front” travels a distance according to $\text{[math]}$ ( $\text{[math]}$ means “of the order of magnitude”).

The diﬀusion equation, $\text{[math]}$ in one dimension ( $\text{[math]}$ ) has a solution²⁴ $\text{[math]}$ where $\text{[math]}$ , $\text{[math]}$ are constants and $\text{[math]}$ 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 $\text{[math]}$ , consistent with similarity arguments above.

The coeﬃcient $\text{[math]}$ depends on where on the $\text{[math]}$ curve the front is located. One option is $\text{[math]}$ , 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