Triangular instability of a strained Batchelor vortex

TL;DR

This paper studies the triangular instability of a Batchelor vortex under strain and axial flow, combining theoretical analysis with numerical simulations to identify unstable modes and how axial flow influences their growth.

Contribution

It provides a detailed analysis of how axial flow affects the triangular instability modes of a Batchelor vortex, including the emergence of new unstable modes and the evolution of dominant instability patterns.

Findings

Unstable mode pairs depend on axial flow strength.

Additional mode combinations become unstable with increasing axial flow.

The dominant unstable mode shifts as axial flow varies.

Abstract

We investigate the triangular instability of a Batchelor vortex subjected to a stationary triangular strain field generated by three satellite vortices, in the presence of weak axial flow. The analysis combines theoretical predictions with numerical simulations. Theoretically, the instability arises from resonant coupling between two quasi-neutral Kelvin modes with azimuthal wavenumbers $m$ and $m+3$ with the background strain. Numerically, we solve the linearized Navier-Stokes equations around a quasi-steady base flow to identify the most unstable modes, and compare their growth rates and frequencies with theoretical predictions for a Reynolds number $Re = 10^4$ and a straining strength $\epsilon = 0.008$. In the absence of axial flow, only the mode pair $(m_A, m_B) = (-1,2)$ (and its symmetric counterpart) is unstable. However, we show that additional combinations such as $(0,3)$, $(1,4)$, and $(2,5)$, which are otherwise strongly damped by the critical layer in the absence of axial flow, also become unstable once axial flow exceeds a certain threshold, as the critical layer damping is significantly reduced. Furthermore, we show that the most unstable mode in the no-axial-flow case, originating from the second branch of $m = -1$ and the first branch of $m = 2$, becomes less unstable as axial flow increases. It is eventually overtaken by a mode from the first branches of both wavenumbers, which then remains the dominant unstable mode across a wide range of axial flow strengths and Reynolds numbers. A comprehensive instability diagram as a function of the axial flow parameter is presented.