A non-perturbative approach to spatial instability of weakly non-parallel shear flows

Abstract

Boundary-layer instability is influenced by non-parallel-flow effects, that is, the streamwise variation of the mean flow not only modifies directly the growth rate but also distorts the shape (i.e., the transverse distribution) of the disturbance, thereby affecting the growth rate indirectly. In this paper, we present a simple but effective local approach to spatial instability of three-dimensional boundary layers, which takes into account both direct and indirect effects of non-parallelism. Unlike existing WKB/multi-scale type of methods, the present approach is non-perturbative in which non-parallelism does not need to be a higher-order correction to the leading-order prediction by the parallel-flow approximation. The non-parallel-flow effects are accounted for by expanding the local mean flow and the shape function as Taylor series, and this leads to a sequence of extended eigenvalue problems, depending on the order of truncation of the Taylor series. These eigenvalue problems can be solved effectively by standard numerical methods. In the case of the Blasius boundary layer, the predictions are verified and confirmed by direct numerical simulation results as well as by the non-parallel theory of Gaster. The non-parallel-flow effects on the eigenvalues, eigenfunctions, and neutral curves for planar and oblique Tollmien-Schlichting (T-S) waves are discussed. The distortion of the eigenfunction is found to have a significant effect, which may be stabilizing or destabilizing depending on the ranges of the Reynolds number and frequency.

A local scattering approach for the effect of abrupt changes on boundary-layer instability and transition: a numerical method

“…The reason is attributed to the fact that the inlet disturbance is, with non-parallelism being ignored, incompatible with the solution of LSP, in which non-parallelism is taken into account. In order to overcome this difficulty, Huang & Wu 16 proposed a non-perturbative approach to spatial instability of weakly non-parallel shear flows. This approach takes into account both the direct and indirect effects of non-parallelism, associated with the streamwise variations of the mean flow and the shape function respectively.…”

“…In principle, it is possible to extend the formulation to an arbitrary order so that the prediction can be made as accurate as one likes. 16 The eigenvalue problem EEVn is solved for a given ω at the inlet and outlet. At the inlet, the perturbation is represented by an eigen mode with a given amplitude (which may be set to unity without losing generality).…”

A local scattering approach for the effects of abrupt changes on boundary-layer instability and transition: a finite-Reynolds-number formulation for isolated distortions

“…As was mentioned in the formulation ( §2), a transfer matrix T, which relates the eigenfunctions of the incident and transmitted instability modes (see (2.30)) needs to Local scattering approach: isolated distortions 29 be known in advance. We now show that the existence of this transfer matrix can be established by using the so-called EEV approach for linear instability of weakly nonparallel flows proposed by Huang & Wu (2015). The approach yields the transfer matrix rather naturally.…”

“…Nonparallel-flow effects on linear stability have often been accounted for by a perturbative approach (Gaster 1974), which is applicable only when non-parallelism causes a small correction to the growth rate. Recently, a non-perturbative approach, free from this restriction, was proposed by Huang & Wu (2015).…”

“…In the EEV approach (Huang & Wu 2015), the local eigenfunction in the vicinity of an arbitrary streamwise location x a is expanded as a Taylor series…”