Optimization and Computational Fluid Dynamics - Department of ...

2 A Few Illustrative Examples of CFD-based Optimization 51
0.9
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0
Wilcox (1998)
β0 β ∗ 0 σ σ ∗ α
Fig. 2.20 The five input parameters of the optimization represented using parallel coordinates
corresponding values in Fig. 2.20. The original values proposed by Wilcox
[79] are also represented in this figure with symbols.
Figure 2.20 also shows that considerable modifications of the model parameters
have been tested by the optimization procedure, compared to the
original reference values proposed by Wilcox [79].
A total of 5,050 evaluations have been performed during the optimization
procedure. Plotting two arbitrary objectives (i.e., the errors for two different
Reynolds numbers) against each other, a strong linear dependence can be
observed (Fig. 2.21). This is satisfactory, since it demonstrates that, globally,
a good parameter set will be valid for different Reynolds numbers, which is
essential for practical purposes. However, zooming on the best results (lower
left corner) in order to see the details, a Pareto front corresponding to concurrent
objectives is detected (Fig. 2.22). This proves that one single, optimal
parameter set cannot be found that would be equally valid for all Reynolds
numbers.
The computed velocity profiles using the model constants in one optimum
configuration are shown in Fig. 2.23. These profiles are compared to DNS data
and to the results computed with the original parameter values. The proposed
modifications lead to a considerably better agreement with the DNS.
Figure 2.24 finally shows the four objectives of the optimization compared
with results obtained by the original model ([79], shown with symbols). Parallel
coordinates are used to represent the connection between the corresponding
values. It can be clearly seen that, in all four cases, better results are
obtained with an optimal parameter set compared with the values proposed
by Wilcox in 1998 for channel flows.