Acoustic modes of rapidly rotating ellipsoids subject to centrifugal gravity

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 greater than or similar to 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., SF6 or C4F8).

Automated summary

Using the acoustic modes of a rotating fluid-filled cavity can determine the effective rotation rate of a fluid, requiring a prior knowledge of the acoustic modes in rotating fluids for accuracy. Unlike the Coriolis force, centrifugal gravity has received less attention in experiments. This study investigates how global rotation and buoyancy modify the acoustic modes of fluid-filled ellipsoids in hydrostatic equilibrium.