Ekman boundary layers in a fluid filled precessing cylinder

F Pizzi,A Giesecke,F Stefani

doi:10.1063/5.0037922

Abstract

The fluid flow in a precessing cylinder is investigated numerically with focus on the Ekman boundary layers in the strongly forced regime. Not surprisingly, in that regime, we find deviations from the linear theory due to significant modifications of the base flow in terms of an axisymmetric geostrophic mode whose rotation is opposite to that of the container. The transition of the bulk flow from a three-dimensional non-axisymmetric base flow to a geostrophic axisymmetric pattern is reflected in the scaling of both the sidewall boundary layers and the Ekman boundary layers on top and bottom of the cylinder. In our simulations, the Ekman layers surpass the threshold of the first instability (class A) and show an increase in the thickness together with a marked vertical flow advection inside the boundary layer in a limited range of the forcing magnitude. However, due to numerical restrictions in our simulations, which limit the range of achievable Ekman numbers, no developed boundary layer turbulence is found. An estimation by extrapolation shows that, for this purpose, Ekman numbers smaller by a factor of two have to be achieved.

Highlights

Precession driven flows are frequent phenomena that occur, e.g., in fuel tanks of rotating rockets, large scale vortices in the atmosphere, or the liquid outer part of the Earth’s core
Precession gives rise to complex three-dimensional flow structures originating from the interactions of free inertial modes, boundary layers, and the base flow
While allowing for a convenient discussion of solutions in terms of inertial modes umnk with corresponding ωmnk, the solution of the Navier–Stokes equation in the container frame, Eq (1a), comes along with some disadvantages when executing numerical simulations

Summary

Introduction

Precession driven flows are frequent phenomena that occur, e.g., in fuel tanks of rotating rockets, large scale vortices in the atmosphere, or the liquid outer part of the Earth’s core. Corresponding studies started more than 100 years ago with the pioneering work of Poincaré (1910), in which a precessing inviscid flow in a spheroidal cavity was mathematically described in terms of a uniform vorticity solution. An extension of the Poincaré solution was developed by Busse (1968) who added viscous effects in boundary layers and non-linear effects. Precession gives rise to complex three-dimensional flow structures originating from the interactions of free inertial modes, boundary layers, and the base flow. Inertial modes are intrinsic features of rotating fluid systems caused by the restoring effect of the Coriolis force

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