Optimal streaks in a Falkner–Skan boundary layer

Abstract

This paper deals with the optimal streaky perturbations (which maximize the perturbed energy growth) in a wedge flow boundary layer, which involves a favorable streamwise pressure gradient. These three-dimensional perturbations are governed by a system of linearized boundary layer equations around the Falkner–Skan base flow. Based on an asymptotic analysis of this system near the free-stream and the leading edge singularity, we show that for acute wedge semi-angle, all solutions converge after a streamwise transient to a single streamwise-growing solution of the linearized equations, whose initial condition near the leading edge is given by an eigenvalue problem first formulated in this context by Tumin [Phys. Fluids 13, 1521 (2001)]. Such a solution may be regarded as a streamwise evolving most unstable streaky mode, in analogy with the usual eigenmodes in strictly parallel flows, and shows an approximate self-similarity, which was partially known and is completed in this paper. An important consequence of this result is that the optimization procedure based on the adjoint equations heretofore used to define optimal streaks is not necessary. Instead, a simple low-dimensional optimization process is proposed and used to obtain optimal streaks. Comparison with previous results by Tumin and Ashpis [AIAA J. 41, 2297 (2003)] shows an excellent agreement. The unstable streaky mode exhibits transient growth if the wedge semi-angle is smaller than a critical value that is slightly larger than π/6, and decays otherwise. Thus, the cases of right and obtuse wedge semi-angles exhibit less practical interest, but they show a qualitatively different behavior, which is briefly described in the supplementary material.