Optimization and Computational Fluid Dynamics - Department of ...

2 A Few Illustrative Examples of CFD-based Optimization 49
y [mm]
18
16
14
12
10
8
6
4
2
0
-2
CO
0.085
0.08
0.075
0.07
0.065
0.06
0.055
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
HCO
8.4E-05
7.9E-05
7.4E-05
6.9E-05
6.4E-05
5.9E-05
5.4E-05
4.9E-05
4.4E-05
3.9E-05
3.4E-05
2.9E-05
2.4E-05
1.9E-05
1.4E-05
9E-06
4E-06
-10 -5 0
x [mm]
5 10
Fig. 2.19 Mass fraction field of CO (left) and HCO (right) at the optimal point
The Reynolds-averaged turbulent flow variables can be separated as:
ui (x,t)=Ui (x,t)+u ′ i (x,t) (2.13)
where the vector Ui is the time-averaged velocity and u ′ i
is the fluctuation
term. The time-averaged mass conservation equation reads:
∂Ui
= 0 (2.14)
∂xi
and the momentum equations can be written as:
ρ ∂Ui ∂Ui
+ ρUi = −
∂t ∂xj
∂P
+
∂xi
∂
� �
∂Ui
μ +
∂xj ∂xj
∂Uj
��
+
∂xi
∂
(−ρu
∂xj
′ iu′ j ) . (2.15)
According to the Boussinesq hypothesis, the Reynolds stress-tensor (corresponding
to the last term in Eq .(2.15)) can be modeled as:
−u ′ iu′ �
∂Ui
j = νt +
∂xj
∂Uj
�
−
∂xi
2
3 kδij (2.16)
where νt is usually called turbulent eddy viscosity, expressed as:
νt = k
ω
. (2.17)