Optimization and Computational Fluid Dynamics - Department of ...

118 Nicolas R. Gauger
CL := 1
�
Cp(ny cosα − nx sinα)dl ,
Cref C
Cm :=
(5.6)
1
C2 �
Cp(ny(x − xm) − nx(y − ym)) dl .
ref C
(5.7)
If the geometry is now perturbed from C(X) toC(X + δX), then via the
solution of
where
∂(w + δw)
∂t
⇔ ∂(δw)
∂t
+ ∂(f + δf)
∂x
+ ∂(δf)
∂x
+ ∂(g + δg)
+ ∂(δg)
∂y
∂y
=0
=0 inD (5.8)
n ⊤ v =0 on C = C(X + δX) , (5.9)
the associated variation of pressure is as follows
Finally via
and
δCp = 2δp
γM2 ∞p∞ 2(p(X + δX) − p(X))
≈
γM2 ∞p∞ . (5.10)
δnx ≈ nx(X + δX) − nx(X) (5.11)
δny ≈ ny(X + δX) − ny(X) (5.12)
the variations of CD, CL and Cm are obtained as
2
δCD =
γM2 �
δp(nx cosα + ny sinα) dl
∞p∞Cref C
+ 1
�
Cp(δnx cosα + δny sinα) dl , (5.13)
Cref
δCL =
2
C
γM 2 ∞ p∞Cref
+ 1
Cref
�
C
�
C
δp(ny cosα − nx sinα) dl
Cp(δny cosα − δnx sinα) dl , (5.14)
2
δCm =
γM2 ∞p∞C2 �
δp(ny(x − xm) − nx(y − ym)) dl
ref C
�
Cpδ(ny(x − xm) − nx(y − ym)) dl . (5.15)
+ 1
C 2 ref
C
Proceeding as described above for the n perturbations δiX in each of the n
components of the design vector X, the gradient of the cost function I (e.g.,