Fluid flows in a librating cylinder

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-Görtler 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.

Mean flow generation by Görtler vortices in a rotating annulus with librating side walls

“…In contrast, in our setup inner (as a convex wall) and outer (as a concave wall) cylinder side walls librate longitudinally, causing the Stokes boundary layer (Sauret et al, 2012 andNoir et al, 2010). For librational frequencies up to a limit value, the Stokes boundary layer becomes centrifugally unstable-once the librational amplitude becomes supercritical-in the prograde (retrograde) phase of a librational cycle over the inner (outer) cylinder side wall, thus generating Görtler vortices.…”

“…Nonlinear effects in the oscillatory Ekman layer induce an azimuthal mean flow in the fluid bulk (Busse, 2010;Sauret et al, 2012;. Busse (2010) using an analytical solution, Sauret et al (2012) performing 2D-numerical simulation using commercial software COMSOL Multiphysics, and Noir et al (2010) conducting a laboratory experiments using direct flow visualization with kalliroscope particles, all illustrated the bulk mean flow driven due to the nonlinearities in the a) ghaseabo@tu-cottbus.de oscillatory Ekman layer. Noir et al (2009) found that for moderate libration amplitude in a weakly nonlinear regime, the centrifugally unstable Stokes boundary layer becomes susceptible to Görtler vortices.…”

Wave field and zonal flow of a librating disk

“…This situation, where no axial flux is permitted in the bulk, has already been encountered in the literature for a bounded configuration without stratification. Wang (1970) and Sauret et al (2012) showed that the zonal flow created by librating a cylinder satisfies this property. As a consequence, the value of the mean angular velocity Ω 2 is equal to the value obtained for a librating cylinder.…”

“…When there is no internal shear layer, they are dominantly generated by the nonlinear interactions occurring in the boundary layer (Busse 1968). They can be calculated exactly for a cylinder (Wang 1970;Sauret et al 2012), a sphere and a spherical shell (Busse 2010;Sauret & Le Dizès 2013). In an infinite domain, we shall see that the mean flow corrections generated in the boundary are no longer purely azimuthal.…”

“…As a consequence, the value of the mean angular velocity Ω 2 is equal to the value obtained for a librating cylinder. It corresponds to the function Ω 2 (ω) given in the appendix of Sauret et al (2012). Expression (4.8) tells us that the outer mean flow is a solid-body rotation close to the disk boundary (r < 1), while it is expected to vanish at this order on the boundary outside the disk (r > 1).…”