The dispersion relation of Tollmien-Schlichting waves

TL;DR

This work analyzes the dispersion relation for Tollmien–Schlichting waves arising from small-viscosity instabilities of shear flows in half-space and strip geometries. By formulating the linearized Navier–Stokes problem into Orr–Sommerfeld and Rayleigh equations and decomposing solutions into fast and slow modes, the authors derive a boundary-mounded dispersion relation and study its asymptotic properties as $ν\to 0$, $α\to0$, and $c\to0$ with bounded $α/c$. They identify two marginal stability curves (upper and lower) with distinct scaling laws for $α$ and show how the critical layer behavior changes, including a detached layer on the upper branch and different scaling regimes in the strip. The results provide a detailed, pedagogical account of the dispersion relation and its asymptotics, clarifying the onset and structure of Tollmien–Schlichting instabilities in these canonical geometries.

Abstract

It is well-known that shear flows in a strip or in the half plane are unstable for the Navier-Stokes equations if the viscosity $ν$ is small enough, provided the horizontal wave number $α$ lies in a small interval, between the so called lower and upper marginal stability curves. The corresponding instabilities are called Tollmien-Schlichting waves. In this letter, we give a simple presentation of the dispersion relation of these waves and study its mathematical properties.

Theorems & Definitions (8)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 4.1
• proof