Asymmetric Vortex Sheet

TL;DR

This paper derives an analytical steady solution for asymmetric vortex sheets in incompressible flows across arbitrary dimensions, revealing how vorticity profiles and layer widths depend on viscosity and dimension.

Contribution

It introduces a novel analytical solution for vortex sheets with asymmetric vorticity profiles in any dimension, connecting Hermite polynomials to the vorticity structure.

Findings

Vorticity confined to a thin layer with Gaussian and power decay in higher dimensions.

Layer width shrinks as viscosity to the power of 3/5 in dimensions greater than 3.

Solution reduces to constant vorticity flow in 2D.

Abstract

We present a steady analytical solution of the incompressible Navier-Stokes equation for arbitrary viscosity in an arbitrary dimension $d$ of space. It represents a $d-1$ dimensional vortex "sheet" with an asymmetric profile of vorticity as a function of the normal coordinate $z$. This profile is related to the Hermite polynomials $H_\mu(z)$ which are analytically continued to the negative fractional index $\mu = -\frac{d}{d-1}$. In $d=2$ dimensions, the solution degenerates to a constant vorticity flow. In $ d \ge 3$ dimensions, the vorticity is confined to the thin layer around the hyperplane with Gaussian decay on one side of the hyperplane and the power decay on another side. One can adjust the common scale of velocity so that the dissipation will stay finite at vanishing viscosity. In this limit, the width $w$ of the viscous lawyer will shrink to zero as $\nu^{\frac{3}{5}}$ for arbitrary dimension $d>3$. In $d=3$ dimensions, this power law is also accompanied by powers of the logarithm.