Analysis of finite-volume discrete adjoint fields for two-dimensional compressible Euler flows

TL;DR

This paper investigates the properties and consistency of discrete and continuous adjoint fields for 2D compressible Euler flows, providing new conditions for finite-volume schemes and analyzing divergence issues near walls.

Contribution

It establishes adjoint consistency conditions for the 2D Jameson-Schmidt-Turkel scheme and offers a new heuristic approach to study adjoint discretization effects.

Findings

Derived adjoint consistency conditions for a specific finite-volume scheme.

Identified the physical source term responsible for numerical divergence near walls.

Linked adjoint divergence to flow response to stagnation pressure and entropy changes.

Abstract

This work deals with a number of questions relative to the discrete and continuous adjoint fields associated with the compressible Euler equations and classical aerodynamic functions. The consistency of the discrete adjoint equations with the corresponding continuous adjoint partial differential equation is one of them. It is has been established or at least discussed only for a handful of numerical schemes and a contribution of this article is to give the adjoint consistency conditions for the 2D Jameson-Schmidt-Turkel scheme in cell-centred finite-volume formulation. The consistency issue is also studied here from a new heuristic point of view by discretizing the continuous adjoint equation for the discrete flow and adjoint fields. Both points of view prove to provide useful information. Besides, it has been often noted that discrete or continuous inviscid lift and drag adjoint exhibit numerical divergence close to the wall and stagnation streamline for a wide range of subsonic and transonic flow conditions. This is analyzed here using the physical source term perturbation method introduced in reference [Giles and Pierce, AIAA Paper 97-1850, 1997]. With this point of view, the fourth physical source term of appears to be the only one responsible for this behavior. It is also demonstrated that the numerical divergence of the adjoint variables corresponds to the response of the flow to the convected increment of stagnation pressure and diminution of entropy created at the source and the resulting change in lift and drag.