Exact and Asymptotic Conditions on Traveling Wave Solutions of the Navier-Stokes Equations

TL;DR

This paper establishes necessary exact and asymptotic conditions for traveling wave solutions of the Navier-Stokes equations across various flow geometries, aiding the understanding of transition and vortex structures at high Reynolds numbers.

Contribution

It introduces new exact and asymptotic conditions for traveling wave solutions in Navier-Stokes flows, applicable across different geometries and Reynolds number regimes.

Findings

Derived exact conditions valid for all Reynolds numbers.

Connected asymptotic conditions to vortex structures at high Re.

Performed computations up to Re=100000 in pipe flow.

Abstract

We derive necessary conditions that traveling wave solutions of the Navier-Stokes equations must satisfy in the pipe, Couette, and channel flow geometries. Some conditions are exact and must hold for any traveling wave solution irrespective of the Reynolds number ($Re$). Other conditions are asymptotic in the limit $Re\to\infty$. The exact conditions are likely to be useful tools in the study of transitional structures. For the pipe flow geometry, we give computations up to $Re=100000$ showing the connection of our asymptotic conditions to critical layers that accompany vortex structures at high $Re$.