Adjoint Equations in Cfd: Duality, Boundary Conditions and Solution Behaviour

The rst half of this paper derives the adjoint equations for inviscid and viscous compressible ow, with the emphasis being on the correct formulation of the adjoint boundary conditions and restrictions on the permissible choice of operators in the linearised functional. It is also shown that the boundary conditions for the adjoint problem can be simpliied through the use of a linearised perturbation to generalised coordinates. The second half of the paper constructs the Green's functions for the quasi-1D and 2D Euler equations. These are used to show that the adjoint variables have a logarithmic singularity at the sonic line in the quasi-1D case, and a weak inverse square-root singularity at the upstream stagnation streamline in the 2D case, but are continuous at shocks in both cases.