Fluid flows in a librating cylinder

TL;DR

This paper investigates fluid flow in a librating cylinder, analyzing boundary layer instabilities, inertial wave generation, and mean flow corrections through theoretical and numerical methods, revealing new mechanisms and scaling laws.

Contribution

It introduces a viscous stability criterion for centrifugal instability and explores the nonlinear regime and inertial wave generation in librating flows.

Findings

Centrifugal instability leads to Taylor-Görler vortices near the boundary.

Mean flow correction is a uniform rotation below instability threshold.

Resonant excitation of inertial modes results in complex mean flow structures.

Abstract

The flow in a cylinder driven by time harmonic oscillations of the rotation rate, called longitudinal librations, is investigated. Using a theoretical approach and axisymmetric numerical simulations, we study two distinct phenomena appearing in this librating flow. First, we investigate the occurrence of a centrifugal instability near the oscillating boundary, leading to the so-called Taylor-G\"ortler vortices. A viscous stability criterion is derived and compared to numerical results obtained for various libration frequencies and Ekman numbers. The strongly nonlinear regime well above the instability threshold is also documented. We show that a new mechanism of spontaneous generation of inertial waves in the bulk could exist when the sidewall boundary layer becomes turbulent. Then, we analyse the librating flow below the instability threshold and characterize the mean zonal flow correction induced by the nonlinear interaction of the boundary layer flow with itself. In the frequency regime where inertial modes are not excited, we show that the mean flow correction in the bulk is a uniform rotation, independent of the Ekman number and cylinder aspect ratio, in perfect agreement with the analytical results of Wang [J. Fluid. Mech., 41, pp. 581 - 592, 1970]. When inertial modes are resonantly excited, the mean flow correction is found to have a more complex structure. Its amplitude still scales as the square of the libration amplitude but now depends on the Ekman number.