Libby–Fox perturbations and the semi-analytic adjoint solution for laminar viscous flow along a flat plate

The properties of the solution to the adjoint two-dimensional boundary layer equations on a flat plate are investigated from the viewpoint of Libby–Fox theory, which describes the algebraic perturbations to the Blasius boundary layer. The adjoint solution is obtained from the Green’s function of the perturbation equation as a sum over the infinite perturbation modes of the Blasius solution. The explicit representation of the adjoint solution allows us to derive constraints on the eigenvalues and eigenfunctions, explicitly compute the adjoint transpose convection term, and evaluate flow sensitivities for shape design, initial-value perturbations and active flow control. The extension of the analysis to the case with non-zero pressure gradient, corresponding to the Falkner–Skan solution, is also briefly discussed.