The Derivation of a Time-Harmonic Adjoint Method

Recent years have seen a remarkable progress in aerodynamic design optimisation based on the adjoint methods to efficiently compute the sensitivities of steady-state objective functions to changes in design variables. Following this success, this book establishes an unsteady adjoint analysis whose purpose is the calculation of the linear worksum typically used in turbomachinery aeroelastic applications. The emphasis falls on the algorithmic development of the adjoint time-harmonic method, which is derived from a nonlinear flow analysis using the discrete approach. In this approach, the discrete nonlinear flow equations are first linearised and the discrete linear harmonic operator is used to form the corresponding adjoint equations. The derivation of the adjoint harmonic equations and boundary conditions involves linear algebraic manipulations of the discrete flow problems, in which the strong boundary condition at the solid wall plays an important role and directly determines the structure of both the adjoint analysis and the iterative solution procedure.