Dual-consistency study for Green-Gauss gradient schemes in an unstructured Navier-Stokes method

A multi-pass reconstruction method for the discrete-adjoint residual is presented that computes the adjoint of the viscous fluxes based on Green-Gauss gradients in an unstructured finite-volume RANS method. The intermediate discrete-adjoint multipliers of the multi-pass reconstruction are the dual viscous stresses containing the dual Green-Gauss gradients. Since the latter are explicitly evaluated on the fly, meaningful discrete-adjoint operators can be identified and compared against their primal counterparts. Numerical experiments are carried out for a 1D diffusion problem, 2D and 3D RANS cases on a sequence of grids to verify the consistency of the dual Green-Gauss gradients. They are compared against a rediscretisation of the adjoint Green-Gauss gradients known from the continuous-adjoint approach. In that sense, the multi-pass residual reconstruction method provides a deeper insight into the effective dual discretisation, an important part of which is the dual Green-Gauss gradient.