Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 91
δF
=
δbi
1
�
�
2 δ(dS)
(p − ptar) + (Ψk+1p − Ψ
2 Sw δbi Sw
T fk) δ(nkdS)
δbi
�
− Ψ
Sw
T
� inv ∂f k −
∂xm
∂fvis
�
k δxm
nkdS
∂xm δbi
�
�
Ψk+1
δ(nkdS)
+ τkmδ(nknm)dS + ΨΛqk
Sw
nk
Sw δbi
� � �
∂uk ∂Ψm+1
− μ +
∂xl ∂xk
∂Ψk+1
� �
∂Ψn+1 δxl
+ λδkm nmdS
∂xm ∂xn δbi
(4.28)
Sw
(Λ = 4 in 2D and Λ = 5 in 3D). In viscous flows, Ψ should satisfy the field
adjoint equations
∂Ψ
∂t − AT ∂Ψ
k − M
∂xk
−T K = 0 (4.29)
where M is the transformation matrix from the non-conservative to the conservative
flow variables and K corresponds to the adjoint viscous stresses,
where
K1=− T
�
∂
k
ρ ∂xk
∂ΨΛ
�
KΛ=
∂xk
T
�
∂
k
p ∂xk
∂ΨΛ
�
∂xk
� �
∂Ψm+1
μ +
∂xm ∂xk
∂Ψk+1
� �
∂Ψl+1
+λδkm
∂xm ∂xl
+ ∂
� �
� �
∂ΨΛ ∂ΨΛ ∂ΨΛ
μ um +uk +λδkmul
∂xm ∂xk ∂xm ∂xl
Kk+1= ∂
−τkm
∂ΨΛ
∂xm
. (4.30)
Along the solid walls, the boundary conditions for the adjoint variables which
correspond to the velocity components read
Ψk+1 = −(p − ptar)nk , k=1, ..., Λ . (4.31)
The boundary condition for ΨΛ, which corresponds to the energy equation,
is either homogeneous Dirichlet (constant wall temperature) or Neumann
(adiabatic walls). The inlet/outlet adjoint boundary conditions are still given
by Eq. (4.21), neglecting locally the spatial derivatives of flow variables.
4.5 Viscous Losses Minimization in Internal Flows
In this section, the continuous adjoint approach is adapted to shape optimization
problems in internal flows where the minimization of viscous losses
is targeted. Viscous losses can be expressed as either total pressure drop or
entropy generation. Both are discussed in this section. Concerning the two