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

Abstract

Time periodic variation of the rotation rate of an annulus induces in supercritical regime an unstable Stokes boundary layer over the cylinder side walls, generating Görtler vortices in a portion of a libration cycle as a discrete event. Numerical results show that these vortices propagate into the fluid bulk and generate an azimuthal mean flow. Direct numerical simulations of the fluid flow in an annular container with librating outer (inner) cylinder side wall and Reynolds-averaged Navier–Stokes (RANS) equations as diagnostic equations are used to investigate generation mechanism of the retrograde (prograde) azimuthal mean flow in the bulk. First, we explain, phenomenologically, how absolute angular momentum of the bulk flow is mixed and changed due to the propagation of the Görtler vortices, causing a new vortex of basin size. Then we investigate the RANS equations for intermediate time scale of the development of the Görtler vortices and for long time scale of the order of several libration periods. The former exhibits sign selection of the azimuthal mean flow. Investigating the latter, we predict that the azimuthal mean flow is proportional to the libration amplitude squared and to the inverse square root of the Ekman number and libration frequency and then confirms this using the numerical data. Additionally, presence of an upscale cascade of energy is shown, using the kinetic energy budget of fluctuating flow.

Study of turbulence and interacting inertial modes in a differentially rotating spherical shell experiment

“…From Koch et al (2013) and Calkins et al (2010) it is known that Görtler vortices in spherical shells with librating inner boundaries get excited in the "blow-up" region of the equatorial boundary layer and spreading outward. Moreover, Ghasemi et al (2016) showed that Görtler vortices propagating from the boundary to the bulk of the fluid can drive a significant mean flow. With this knowledge, we propose the following scenario: For moderate Rossby numbers Ro > Ro c , the flow is sub-critical and dominated by large-scale features.…”

“…The flow becomes supercritical. Subsequently, the vortices propagate towards the vertical Stewartson layer and, according to Ghasemi et al (2016), amplify the mean flow around the tangent cylinder. This scenario would first explain the E 1/5 scaling of Ro c and second the massive enhancement of the zonal mean flow around the tangent cylinder at Ro c (figure 7).…”

Triadic resonances in the wide-gap spherical Couette system

“…The Rayleigh stability criterion in a rotating frame is given by (Rayleigh 1917;Kloosterziel & van Heijst 1991;Ghasemi et al 2016),…”

Inertial instability of the time-periodic Kolmogorov flow in a rotating fluid with the full account of the Coriolis force

“…The studies of the inertial stability of the time-periodic Kolmogorov flow carried out in this work have applications to laboratory experiments with rotating annular vessels in the presence of 'longitudinal' libration (a time-periodic change in the angular velocity of rotation of the cylindrical side wall of the annulus) (Ghasemi et al 2016). For a wide gap between cylinders, the problem considered in this article can serve as a simplified analog of the problem of the inertial stability of the flow in the so-called second Stokes problem (see Davis 1976, Frenkel 1991, when the exponential decay, with distance from the wall, of the amplitude of the spatially periodic oscillations in the second Stokes problem is neglected.…”

“…This is done in the second part of this work. Throughout the work we will consider that the Kolmogorov flow is not steady, unlike in Kurgansky (2021), but that its amplitude is modulated over time according to a periodic law, for example, due to the action of tidal forces or longitudinal libration, as the latter occurs in a laboratory experiment or on other planets (and their satellites) (Noir et al 2009, Ghasemi et al 2016. We note that the study of inertial (symmetric) instability of time-periodic flows having a linear velocity profile, with an exact account of the Coriolis force and the arbitrary orientation of the flow with respect to the zonal direction, was carried out in Kurgansky (2022).…”