Spectral portraits of the Orr--Sommerfeld operator with large Reynolds numbers

TL;DR

This paper analyzes the spectral behavior of the Orr--Sommerfeld operator in fluid dynamics as viscosity approaches zero, revealing that the spectrum's accumulation points and eigenvalue distributions align with the model problem's spectral graphs.

Contribution

It establishes the correspondence between the spectra of the Orr--Sommerfeld operator and a simplified model problem in the limit of small viscosity, including spectral graph structures.

Findings

Spectral accumulation points coincide between the model and the Orr--Sommerfeld operator.

Eigenvalue counting functions match along the spectral graphs.

Main spectral features are preserved in the zero-viscosity limit.

Abstract

A model problem of the form -i\epsilon y''+q(x)y=\lambda y, y(-1)=y(1)=0, is associated with well-known in hydrodynamics Orr--Sommerfeld operator. Here (\lambda) is the spectral parameter, (\epsilon) is the small parameter which is proportional to the viscocity of the liquid and to the reciprocal of the Reynolds number, and (q(x)) is the velocity of the stationary flow of the liquid in the channel (|x|\leqslant 1). We study the behaviour of the spectrum of the corresponding model operator as (\epsilon\to 0) with linear, quadratic and monotonous analytic functions. We show that the sets of the accumulation points of the spectra (the limit spectral graphs) of the model and the corresponding Orr--Sommerfeld operators coincide as well as the main terms of the counting eigenvalue functions along the curves of the graphs.