Acoustic modes of rapidly rotating ellipsoids subject to centrifugal gravity

TL;DR

This paper investigates how centrifugal gravity and rotation influence acoustic modes in fluid-filled ellipsoids, providing a refined wave equation and numerical solutions relevant for experimental rotation rate measurements.

Contribution

It derives an exact wave equation for acoustic velocity in rotating fluids under centrifugal gravity and applies polynomial spectral methods for solutions, advancing understanding of acoustic modes in rotating ellipsoids.

Findings

Centrifugal acceleration affects acoustic frequencies at high rotational Mach numbers.

The polynomial spectral method accurately solves the wave problem in ellipsoids.

Experimental regimes with high compressibility gases can reach measurable effects.

Abstract

The acoustic modes of a rotating fluid-filled cavity can be used to determine the effective rotation rate of a fluid (since the resonant frequencies are modified by the flows). To be accurate, this method requires a prior knowledge of the acoustic modes in rotating fluids. Contrary to the Coriolis force, centrifugal gravity has received much less attention in the experimental context. Motivated by on-going experiments in rotating ellipsoids, we study how global rotation and buoyancy modify the acoustic modes of fluid-filled ellipsoids in isothermal (or isentropic) hydrostatic equilibrium. We go beyond the standard acoustic equation, which neglects solid-body rotation and gravity, by deriving an exact wave equation for the acoustic velocity. We then solve the wave problem using a polynomial spectral method in ellipsoids, which is compared with finite-element solutions of the primitive fluid-dynamic equations. We show that the centrifugal acceleration has measurable effects on the acoustic frequencies when $M_\Omega \gtrsim 0.3$, where $M_\Omega$ is the rotational Mach number defined as the ratio of the sonic and rotational time scales. Such a regime can be reached with experiments rotating at a few tens of Hz, by replacing air with a highly compressible gas (e.g. SF$_6$ or C$_4$F$_8$).