Nonmodal Tollmien-Schlichting waves

TL;DR

This paper investigates the nonmodal instability mechanisms of Tollmien-Schlichting waves in shear flows, revealing new eigenvalue formulations and energy dispersion effects that influence flow stability beyond traditional modal analysis.

Contribution

It introduces a novel nonmodal framework based on eigenvalue problems for energy bounds and phase speed, providing deeper insight into flow stability and perturbation evolution.

Findings

Eigenvalue problems for energy bounds and phase speed are derived.

Flow stability depends on energy dispersion among base perturbations.

Analysis applied to three shear flows demonstrating the theoretical results.

Abstract

The instability of flows via two-dimensional perturbations is analyzed theoretically and numerically in a nonmodal framework. The analysis is based on results obtained in [Verschaeve et al. (2018)] showing the inviscid character of the growth mechanism of these waves. In particular, it is shown that the formulation of this growth mechanism naturally reduces to the eigenvalue problem for the energy bound formulated by [Davis and von Kerczek (1973)]. This eigenvalue equation thus allows for a broader interpretation. It provides the discrete growth rates for the base flow in question. In addition to this eigenvalue problem, a corresponding eigenvalue problem for the phase speed of the perturbations can be extracted from the equations found in [Verschaeve et al. (2018)]. These two eigenvalue equations relate to the Hermitian and skew-Hermitian part, respectively, of the nonmodal equations, cf. [Schmid (2007)]. In contrast to traditional Orr-Sommerfeld modal analysis, the above eigenvalue equations define an orthogonal set of eigenfunctions allowing to decompose the perturbation into base perturbations. As a result of this decomposition, it can be shown that the evolution of two-dimensional perturbations is governed by two mechanisms: A first one, responsible for extracting and returning energy from and to the base flow, in addition to viscous dissipation and, a second one, responsible for dispersing energy among the different base perturbations constituting the perturbation. As a general result, we show that the stability of a flow is not only determined by the growth rates of the base perturbations, but it is also closely related to its ability to disperse energy away from the base perturbations with positive growth. We illustrate the above results by means of three shear flows.