Optimization and Computational Fluid Dynamics - Department of ...

50 Gábor Janiga
The modified k–ω turbulence model proposed by Wilcox [79] in 1998 is
employed in this study. The transport equations for k and ω can be written
as, respectively:
∂k
∂t
∂ω
∂t
∂k
+ Uj
∂xj
∂ω
+ Uj = α
∂xj
ω
∂Ui
= τij − β
∂xj
∗ kω + ∂
�
(ν + σ
∂xj
∗ νT) ∂k
�
∂xj
k τij
∂Ui
− βω
∂xj
2 + ∂
∂xj
�
(ν + σνT) ∂ω
∂xj
�
, (2.18)
. (2.19)
The model includes several auxiliary relations and closure coefficients 1 .In
this study, five parameters of the model are selected for the optimization,
listed here with their standard values:
α =13/25 ,β0 =9/125 ,β ∗ 0 =9/100 ,σ=1/2 ,σ∗ =1/2 . (2.20)
A more detailed description of this turbulence model as well as the extension
of this model to compressible flows is extensively discussed in Wilcox [79].
2.5.2 Numerical Results
After calculating one set of values for the four selected Reynolds numbers, the
time-averaged turbulent velocity profiles are compared with the DNS results.
The differences between the four corresponding profiles are measured using
the area enclosing the curves. Due to the fine grid, this area can be approximated
quite well using a simple numerical integration based on the rectangle
rule. The four corresponding numerical parameters to optimize (minimize)
are these area values for the four different Reynolds numbers.
The main drawback associated to EAs in general remains their cost in
terms of computing time because they require a large number of evaluations
on different configurations. Here, a fast computational procedure is chosen in
this investigation to speed-up the optimization. The numerical simulations
for a fully developed channel flow are performed using the simplified computational
code from Wilcox [79]. This program is based on a simple numerical
integration and one single simulation takes only a few seconds on a standard
PC.
Five values have been selected as the input parameters of the optimization
procedure. The model parameters α, β0, β ∗ 0 , σ and σ∗ mayfreelyvary
between 0 and 1. A graphical representation in five dimensions is difficult.
Therefore, parallel coordinates are used to show the connection between the
1 In this work, these closure coefficients are referred to as model parameters. Calling them
“model constant” is confusing, since these values are probably not constant and possibly
problem-dependent.