Properties of Adjoint Solutions of the Full-potential Equations for Two-Dimensional Subcritical Flow

Abstract

The adjoint full-potential equations are studied for two-dimensional (2D) steady subcritical flows. In contrast with the incompressible case, explicit closed-form solutions are generally not available in the compressible setting, so the emphasis is placed here on the underlying structure. Using the Green's-function approach and the relation between the adjoint full-potential and compressible adjoint Euler equations, we identify the adjoint potential and stream function with linear combinations of the Euler adjoint variables associated with point mass and vorticity sources. For lift-based cost functions, the corresponding adjoint solutions contain two unknown functions that encode the effect of perturbations to the Kutta condition. We show that these functions obey the linearized full-potential equations, are linked by generalized Cauchy-Riemann equations, and reduce in the incompressible limit to the Poisson kernel of the Laplacian on the exterior of the circle and its harmonic conjugate. Their properties are examined analytically and through numerical adjoint solutions. Finally, a continuous formulation of the Kutta condition for the adjoint full-potential equations is discussed and interpreted in terms of singular boundary forcing, Green-function kernels, and an equivalent Lagrange-multiplier formulation.