Adjoint Optimization of Volumetric Sources in Steady, Supersonic Flow: Energy Addition

Adjoint optimization methods have proven successful for the control of turbulence and boundary layers and in the design of airfoils and aircraft. In this paper, the adjoint equations are extended to the problem of controlling steady, inviscid, supersonic flow with volumetric source terms in the Euler equations, that is, mass, momentum, and energy addition. The adjoint solutions are shown to indicate both the optimal location for the source placement and cost gradient. In the particular case of drag reduction by energy deposition, the gradient is proportional to energy efficiency and becomes a key factor in optimization. The general form of the problem makes these results applicable to all forms of volumetric control with the goals of drag reduction, lift enhancement, and the generation of pitching moments.