Nonlinear evolution of the centrifugal instability using a semi-linear model

TL;DR

This paper investigates the nonlinear development of centrifugal instability in a vortex using a semi-linear model, which accurately predicts the evolution of flow and matches DNS results at high Reynolds numbers.

Contribution

The study introduces a semi-linear model with spatial averaging for centrifugal instability, extending previous models and including second harmonic effects for better accuracy.

Findings

Model predictions agree with DNS for various Reynolds numbers.

Reynolds stresses homogenize angular momentum towards a stable profile.

Final velocity profiles match theoretical predictions at high Reynolds numbers.

Abstract

We study the nonlinear evolution of the centrifugal instability developing on a columnar anticyclone with a Gaussian angular velocity using a semi-linear approach. The model consists in two coupled equations: one for the linear evolution of the most unstable perturbation on the axially averaged mean flow and another for the evolution of the mean flow under the effect of the axially averaged Reynolds stresses due to the perturbation. Such model is similar to the self-consistent model of \cite{Vlado14} except that the time averaging is replaced by a spatial averaging. The non-linear evolutions of the mean flow and the perturbations predicted by this semi-linear model are in very good agreement with DNS for the Rossby number $Ro=-4$ and both values of the Reynolds numbers investigated: $Re=800$ and $2000$ (based on the initial maximum angular velocity and radius of the vortex). An improved model taking into account the second harmonic perturbations is also considered. The results show that the angular momentum of the mean flow is homogenized towards a centrifugally stable profile via the action of the Reynolds stresses of the fluctuations. The final velocity profile predicted by \cite{Kloosterziel07} in the inviscid limit is extended to finite high Reynolds numbers. It is in good agreement with the numerical simulations.