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

TL;DR

This paper investigates the adjoint boundary layer equations on a flat plate using Libby-Fox perturbation theory, deriving solutions from Green's functions and analyzing eigenvalues, with extensions to non-zero pressure gradients.

Contribution

It introduces a novel analysis of the adjoint boundary layer equations using Libby-Fox perturbations and Green's functions, extending to Falkner-Skan flows.

Findings

Derived the adjoint solution from Green's functions and perturbation modes.

Established constraints on eigenvalues and eigenfunctions of the perturbation problem.

Extended the analysis to flows with non-zero pressure gradients.

Abstract

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 that 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 analysis of the solution allows us to obtain constraints on the eigenvalues and eigenfunctions. The extension of the analysis to the case with non-zero pressure gradient, corresponding to the Falkner-Skan solution, is also briefly discussed.