Spatial decay of the vorticity field of time-periodic viscous flow past a body

TL;DR

This paper investigates how the vorticity field in a time-periodic viscous flow past a body decays spatially, showing exponential decay outside the wake and algebraic decay inside, with implications for steady-state behavior.

Contribution

It provides a detailed analysis of the spatial decay rates of vorticity in time-periodic Navier-Stokes flows, including exponential decay outside the wake and algebraic decay of the mean vorticity inside.

Findings

Vorticity decays exponentially outside the wake region.

Inside the wake, the mean vorticity decays algebraically.

The purely periodic component of vorticity decays faster than the mean.

Abstract

We study the asymptotic spatial behavior of the vorticity field, $\omega(x,t)$, associated to a time-periodic Navier-Stokes flow past a body, $\mathscr B$, in the class of weak solutions satisfying a Serrin-like condition. We show that, outside the wake region, $\mathcal R$, $\omega$ decays pointwise at an exponential rate, uniformly in time. Moreover, denoting by $\bar{\omega}$ its time-average over a period and by $\omega_P:=\omega-\bar{\omega}$ its purely periodic component, we prove that inside $\mathcal R$, $\bar{\omega}$ has the same algebraic decay as that known for the associated steady-state problem, whereas $\omega_P$ decays even faster, uniformly in time. This implies, in particular, that "sufficiently far" from $\mathscr B$, $\omega(x,t)$ behaves like the vorticity field of the corresponding steady-state problem.