Entropy and Adjoint Methods

Abstract

Aerodynamic drag can be partially approximated by the entropy flux across fluid domain boundaries with a formula due to Oswatitsch. In this paper, we build the adjoint solution that corresponds to this representation of the drag and investigate its relation to the entropy variables, which are linked to the integrated residual of the entropy transport equation. For inviscid isentropic flows, the resulting adjoint variables are identical to the entropy variables, an observation originally due to Fidkowski and Roe, while for nonisentropic flows there is a significant difference that is explicitly demonstrated with analytic solutions in the shocked quasi-1D case. Both approaches are also investigated for viscous and inviscid flows in two and three dimensions, where the adjoint equations and boundary conditions are derived. The application of both approaches to mesh adaptation is investigated, with especial emphasis on inviscid flows with shocks.

Exact Inviscid Drag-Adjoint Solution for Subcritical Flows

“…This relation can be understood if we recall that entropy variables are also identical, for subcritical flows, to the Oswatitsch adjoint [10], which is dual to drag measured as integrated entropy flux in the domain outer boundaries. This relation of drag to entropy variables had been pointed out and exploited before in [17].…”

On The Mesh Divergence Of Inviscid Adjoint Solutions

“…Lift adjoint solution for a NACA0012 airfoil at 1measuring far-field entropy flux, shows the same behavior as the near-field drag(Figure 8). This is important as the output function is not based on near-field computations and, accordingly, the wall boundary condition is simply ( , ) 0x y S n   in this case[7].…”

Airfoil inverse design based on laminar compressible adjoint lattice Boltzmann method

“…The adjoint method is one of the best and most well-known approaches in aerodynamic shape optimization, which has been well developed based on the Euler and Navier-Stokes equations. [1][2][3][4][5] Derivation of the adjoint equation depends on the flow governing equation. Thus, accuracy and performance of the adjoint method is strongly affected by the flow solvers.…”

“…Researchers have made great studies in this area, providing more and more accurate methods for optimizing aerodynamics based on the CFD. The adjoint method is one of the best and most well‐known approaches in aerodynamic shape optimization, which has been well developed based on the Euler and Navier–Stokes equations 1–5 . Derivation of the adjoint equation depends on the flow governing equation.…”