Page 106 - Introduction to Computational Fluid Dynamics

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4.7 SOURCE TERMS
selecting that for which the thickness is largest. Usually, the largest thickness 11:7 85
will correspond to the largest
, and for this selected , we evaluate
| JN − JN−1 | −3
∗
R = , = 10 (say), (4.73)
∗
where is a sufficiently small reference quantity. Since Equation 4.43 is it-
∗
eratively solved, Patankar [52] has suggested the following formula for exact
evaluation:
n
˙ m E (exact) = ˙ m E,std × R , (4.74)
where, from computational experience, n
0.1 is found to be a convenient value
in most cases. Thus, when Equation 4.43 has converged, ψ E , as evaluated from

Equation 4.72, will provide a correct estimate of total mass flow rate ψ EI = ψ E − ψ I
through the boundary layer at the given x. Once this mass flow rate is known, the
y dimension and hence the largest boundary layer thickness among all s can be
estimated.

4.7 Source Terms

4.7.1 Pressure Gradient

In external boundary layers, the pressure gradient is specified or indirectly evaluated
from
dp dU ∞
, (4.75)
=−ρ U ∞
dx dx
where U ∞ (x) is specified. In internal flows, however, a special procedure must be
adopted to specify the pressure gradient. The procedure relies on satisfying the
overall mass flow rate balance at every streamwise location x. Thus, in a general
duct, let A d (x) represent the duct area between the axis of symmetry (I boundary)
and the wall (E boundary). Then

dω
E 1
A d = rdy = ψ EI . (4.76)
I 0 ρ u
Therefore,

A d ω j
= C (constant) = . (4.77)
ψ EI ρ j u j
The task now is to replace u j in terms of the pressure gradient. To do this,
Patankar [52] writes the discretised version of the momentum equation as
AP j u j = AN j u j+1 + AS j u j−1 + D j − V j p x , (4.78)