Page 114 - Introduction to Computational Fluid Dynamics

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4.10 APPLICATIONS
For free-shear flows, again x 0 must be chosen to avoid the elliptic flow region 11:7 93
very close to where a jet or a wake originates. For advice on the choice of x 0 and
the u(y) profile, the reader is referred to Schlichting [65].
4.9.3 Choice of Step Size and Iterations
Iterative calculation is required to deal with nonlinearities arising out of implicit-
ness. In the present procedure, nonlinearities arise from four sources:

1. They can arise from dependence of coefficients and sources in Equation 4.43
on other scalar s. Thus, the source term R k in the equation for ω k may depend
on T , and ρ, and
may depend on ω k and T .
2. At a downstream station, y j are not a priori known and therefore the values y n ,
y s required in several evaluations are not known. These y j s can be evaluated
d
only after the profile is established.
j
3. In external boundary layers and free-shear flows, the flow width at a downstream
station is not known and we wish to select the largest width among all s. This
is done via Equation 4.74.
4. In internal flows, the pressure gradient is not known at a downstream station.

By choosing a small enough x, one can make the procedure completely non-
iterative. This can be achieved by evaluating AN, AS, and S in terms of upstream
values. We, however, prefer partial linearization. Thus, whereas the different s
required in the evaluation of AN, AS, and S are taken from the upstream station, y j
are established through an iterative solution of equations for all relevant s. With
this choice, experience shows that we may choose

u
x
0.25δ . (4.118)
2
This choice ensures both economy and accuracy. However, situations may arise
when larger step sizes are also permissible.

4.10 Applications

Flat Plate Boundary Layer
Figure 4.5 shows computed results of friction coefficient Cf x and Stanton number
St x for a flat plate boundary layer. Computations were begun with a laminar velocity
= 10 3
(with λ = 0 in Equation 4.114) and temperature profiles prescribed at Re x 0
with JN = 102. Such a large number of (nonuniform) grid points are necessary
to resolve the profiles in the vicinity of the wall and in the turbulent range. In the
mixing length model, transition is sensed by the Cebeci model (Equations 4.102
and 4.103). In the LRE model, the transition is sensed automatically. It is seen that
the mixing length model predicts transition at a higher Re x than the LRE model.