Energy dissipation near the outflow boundary in the vanishing viscosity limit

TL;DR

This paper investigates how energy dissipates near the outflow boundary in the zero-viscosity limit of incompressible flows, revealing proportional relationships involving boundary conditions and enstrophy behavior.

Contribution

It provides a detailed analysis of energy dissipation and enstrophy near the boundary in the vanishing viscosity limit, connecting boundary conditions to dissipation rates.

Findings

Energy dissipation rate is proportional to boundary suction and tangential velocity difference.

Enstrophy within a boundary layer scales with total enstrophy and is inversely proportional to viscosity.

Enstrophy production rate near the boundary increases as viscosity decreases.

Abstract

We consider the incompressible Navier-Stokes and Euler equations in a bounded domain with non-characteristic boundary condition, and study the energy dissipation near the outflow boundary in the zero-viscosity limit. We show that in a general setting, the energy dissipation rate is proportional to $\bar U \bar V ^2$, where $\bar U$ is the strength of the suction and $\bar V$ is the tangential component of the difference between the Euler and the Navier-Stokes solutions on the outflow boundary. Moreover, we show that the enstrophy within a layer of order $\nu / \bar U$ is comparable with the total enstrophy. The rate of enstrophy production near the boundary is inversely proportional to $\nu$.