Optimization and Computational Fluid Dynamics - Department of ...

90 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
The gradient of Faug with respect to the design variables is finally given by
δFaug
δbi
= 1
�
�
2 δ(dS)
(p − ptar) + (p − ptar)
2 Sw δbi Sw
δp
dS
δbi
� �� �
BCW
� �
δU
−
Ω
T
−
δbi
∂U
�
T �
δxm
A
∂xm δbi
T �
∂Ψ
k dΩ
∂xk
� �� �
�
−
�
+
Sw
Sw
∂U
∂xk
T
FAE
An T Ψ δxk
dS +
δbi
�
δp
dS
Ψk+1nk
Sw δbi
� �� �
BCW
(Ψk+1p −Ψ T f k) δ(nkdS)
�
δU
+
δbi Si,o
T
(A
δbi
T nΨ)dS . (4.26)
� �� �
BCIO
In Eq. (4.26), the field integral marked with FAE, which depends on the flow
variable sensitivities, is eliminated giving rise to the field adjoint equations,
Eq. (4.19). Terms marked with BCW and BCIO are also eliminated by
satisfying the adjoint boundary conditions (Eqs. (4.20) and (4.21)). The remaining
terms determine the sensitivity derivatives of the objective function
with respect to the design variables, Eq. (4.23).
4.4 Inverse Design Using the Navier-Stokes Equations
In real flows, the inverse design of optimal aerodynamic shapes is based on
the Navier-Stokes equations
∂U
∂t
∂finv
k +
∂xk
− ∂fvis
k
∂xk
= 0 (4.27)
where fvis k = � 0 , τ T k ,umτkm
�T + qk is the vector of viscous fluxes, τ k are
are the thermal heat flux components.
Using the same objective function, the extra work to be done to develop the
direct approach for the computation of δF is to account for the sensitivities
δbi
of the viscous stresses, too. Using the continuous adjoint approach, based on
the partial derivatives of the Navier-Stokes equations with respect to bi, the
sensitivity derivatives of F are finally given by [53]
the viscous stresses and qk = k ∂T
∂xk