Optimization and Computational Fluid Dynamics - Department of ...

198 Laurent Dumas
Fig. 7.5 Numerical flow field around a realistic car (Peugeot 206)
∂ūj ∂ūiūj
+ = −
∂t ∂xi
1 ∂¯p
+
ρ ∂xj
∂
� �
∂ūi
(ν + νt) +
∂xi ∂xj
∂ūj
�
−
∂xi
2
3 δi,jk
�
on Ω
(7.5)
where νt is called the eddy viscosity and is related to the turbulent kinetic
energy k and its rate of dissipation ε by
k
νt = Cµ
2
ε
. (7.6)
In this expression, Cµ is a constant and the new variables k and ε are obtained
from a set of two equations [17].
In the present configuration, it has been observed that the second-order
closure model called Reynolds Stress Model (RSM) with an adequate wall
function gives better results than the k–ε method [16]. This model will be
preferred in the forthcoming simulations despite the small computational
overcost on the order of 40% A first example of such flow computations
around a realistic car is given in Fig. 7.5.
7.2.2.2 The Cost Function to Minimize
The cost function that will have to be minimized is the drag coefficient already
introduced in (7.1). It can be rewritten in the following way after separating
the pressure part and the viscous part:
�� � � ��
p − p∞ ndσ τ · νdσ
Cd =
Sc
1
2 ρV 2 ∞ S
+
Sc
1
2 ρV 2 ∞ S
(7.7)