On the physical mechanism of centrifugal-gravity wave resonant instability in swirling free surface rotating Polygons

TL;DR

This paper analyzes the wave-resonant instability in swirling flows within rotating cylinders, revealing how centrifugal and gravity waves interact and under what conditions they lead to instability or phase-locking.

Contribution

It provides an explicit linear analysis of wave interactions in swirling flows, identifying the conditions for resonant instability and phase-locking between centrifugal and gravity waves.

Findings

Resonant instability occurs only between counter-propagating centrifugal and gravity waves.

Far field velocity influences wave propagation and amplitude growth.

Near-resonant phase-locking does not produce instability.

Abstract

We present an explicit analysis of wave-resonant instability of swirling flows inside fast rotating cylindrical containers. The linear dynamics are decomposed into the interaction between the horizontal inner centrifugal edge waves, the outer vertical gravity waves with the aim of understanding the dynamics of the centrifugal waves. We show how the far field velocity induced respectively by the centrifugal and the gravity waves affect each other's propagation rates and amplitude growth. We follow this with an analysis of the instability in terms of a four wave interaction, two centrifugal and two gravity ones, and explain why the resonant instability can be obtained only between a pair of two counter-propagating waves, one centrifugal and one gravity. Furthermore, a near resonant regime which does not yield instability is shown to result from a phase-locking configuration between a pair of a counter-propagating centrifugal wave and a pro-propagating gravity one, where the interaction affects the waves' propagation rates but not the amplitude growth.