Generalized adjoint consistent treatment of wall boundary conditions for compressible flows

• Using numerical fluxes on boundary faces like on interior faces increases stability.
• Extension of the adjoint consistency analysis to any consistent wall boundary flux.
• Adjoint consistent discretization of integral quantities like force coefficients.
• Associated treatment of local quantities like surface pressure and skin friction.

In this article, we revisit the adjoint consistency analysis of Discontinuous Galerkin discretizations of the compressible Euler and Navier–Stokes equations with application to the Reynolds-averaged Navier–Stokes and k–ωk–ω turbulence equations. Here, particular emphasis is laid on the discretization of wall boundary conditions. While previously only one specific combination of discretizations of wall boundary conditions and of aerodynamic force coefficients has been shown to give an adjoint consistent discretization, in this article we generalize this analysis and provide a discretization of the force coefficients for any   consistent discretization of wall boundary conditions. Furthermore, we demonstrate that a related evaluation of the cpcp- and cfcf-distributions is required. The freedom gained in choosing the discretization of boundary conditions without loosing adjoint consistency is used to devise a new adjoint consistent discretization including numerical fluxes on the wall boundary which is more robust than the adjoint consistent discretization known up to now.While this work is presented in the framework of Discontinuous Galerkin discretizations, the insight gained is also applicable to (and thus valuable for) other discretization schemes. In particular, the discretization of integral quantities, like the drag, lift and moment coefficients, as well as the discretization of local quantities at the wall like surface pressure and skin friction should follow as closely as possible the discretization of the flow equations and boundary conditions at the wall boundary.

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