On the Properties of Solutions of the Adjoint Euler Equations

The behavior of analytic and numerical adjoint solutions is examined for the quasi-1D Euler equations. For shocked ow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock and that an internal adjoint boundary condition is required at the shock. A Green's function approach is used to derive the analytic adjoint solutions corresponding to isentropic and shocked transonic ow, revealing a logarithmic singularity at the sonic throat and connrming the expected properties at the shock. Numerical solutions obtained using both discrete and continuous adjoint formulations reveal that there is no need to explicitly enforce the adjoint shock boundary condition. Adjoint methods are demonstrated to play an important role in the error estimation of integrated quantities such as lift and drag.