# Energy dissipation caused by boundary layer instability at vanishing viscosity

## TL;DR

This paper explains how boundary layer instabilities at high Reynolds numbers lead to energy dissipation, supported by asymptotic analysis and numerical simulations of Navier-Stokes equations.

## Contribution

It introduces a new asymptotic analysis linking boundary layer instabilities to energy dissipation scaling at high Reynolds numbers.

## Key findings

- Unstable Rayleigh-Tollmien-Schlichting modes scale with Re as $Re^{3/8}$ to $Re^{1/2}$.
- Boundary vorticity flux scales as Re^1, influencing drag.
- Numerical simulations confirm the theoretical predictions.

## Abstract

A qualitative explanation for the scaling of energy dissipation by high Reynolds number fluid flows in contact with solid obstacles is proposed in the light of recent mathematical and numerical results. Asymptotic analysis suggests that it is governed by a fast, small scale Rayleigh-Tollmien-Schlichting instability with an unstable range whose lower and upper bounds scale as $Re^{3/8}$ and $Re^{1/2}$, respectively. By linear superposition the unstable modes induce a boundary vorticity flux of order $Re^1$, a key ingredient in detachment and drag generation according to a theorem of Kato. These predictions are confirmed by numerically solving the Navier-Stokes equations in a two-dimensional periodic channel discretized using compact finite differences in the wall-normal direction, and a spectral scheme in the wall-parallel direction.