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.
Highlights
Understanding bulk mean flow generation mechanisms in rotating flows in weakly nonlinear and weakly or intermittently unstable regimes is important, since they appear in a variety of geophysical and technical bulk flows, and contribute to our understanding of turbulence.056603-2 Ghasemi V. et al.Phys
Considering the retrograde azimuthal mean flows induced by bulk compared with the closed annulus case (BC I), in libration (III), IV, and V with ε = 0.6 (Figure 3(b)), one can infer that the mean flow induced by the Görtler vortices affects the bulk flow likewise the flow driven by the nonlinearities in the oscillatory Ekman boundary layer
In order to validate proportionality (17) we performed a series of coarse numerical simulations (Table III) and explored dependency of the azimuthal mean flow on the dimensionless parameters E, ω, and ε
Summary
Understanding bulk mean flow generation mechanisms in rotating flows in weakly nonlinear and weakly or intermittently unstable regimes is important, since they appear in a variety of geophysical and technical bulk flows, and contribute to our understanding of turbulence. Kloosterziel et al (2007) investigated development of inertial instability in initially barotropic vortices in a uniformly rotating and stratified fluid They stated that centrifugally unstable vortices propagating beyond the unstable region may mix angular momentum and produce a basin-scale vortex with a stable velocity profile spreading along the axis of rotation, i.e., with homogenized angular momentum. We distinguish between the azimuthal mean flows generated by the nonlinearities in the oscillatory Ekman layer and by the Görtler vortices, using the librational boundary condition for the lids and/or cylinder side walls, and periodic and closed boundary conditions in axial direction.
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