Görtler vortices in the Rayleigh layer on an impulsively started cylinder

Abstract

Centripetal instabilities in two flows involving time-dependent Rayleigh layers on a rotating circular cylinder are examined. In one case we consider the stability of the flow induced in an infinite expanse of quiescent fluid when the cylinder is impulsively given a constant angular velocity; in the other problem the angular velocity increases as the square root of time so that the undisturbed flow has a constant wall shear. For both situations linear neutral stability curves for vortex motions are calculated by quasi-steady ͑or frozen-time͒ methods, with these results justified, where possible, by Wentzel-Kramers-Brillouin techniques. The topology of the neutral curve for the ramped angular velocity configuration allows a rigorous description of small wavelength, weakly and fully nonlinear vortex structures to be obtained. Our results are compared with the equivalent cases that arise in the study of unsteady thermal Rayleigh layers induced by the sudden heating of a horizontal flat plate.

Unsteady flow in a rotating torus after a sudden change in rotation rate

“…In figure 17 we also show the results of a frozen-time eigenvalue analysis applied to (6.9) for which, rather than solving the initial-value problem, we fix the base flow and obtain an eigenvalue set parameterized by T and K. Clearly such an approach is not valid in general, as the time scale over which the base flow develops is the same as the time scale over which the instability grows. However, as is well known, the results of these two approaches agree for the upper-branch modes, for which K ∼ T −1/8 , as described by Mackerrell et al (2002).…”

“…These time/length scales have also been obtained for the onedimensional flow induced in the Rayleigh layer on an impulsively rotated infinite cylinder by Otto (1993) and Mackerrell, Blennerhassett & Bassom (2002). We seek a multi-scale solution in which the base flow varies around the torus on the slow scale θ but the perturbation depends on the fast scale Θ.…”

Relative energy stability analysis on the onset of Taylor-Görtler vortices in impulsively accelerating Couette flow

“…In the latter method, the instantaneous stability of the primary-velocity profile is analyzed and the state of motion at a particular instant examined. As discussed MacKerrell [7], this static approximation has two critical contradictions: First, that the growth rate of the disturbance is larger than that of base quantity under the quasi-steady state ansatz, and second that the growth rate of the disturbance is zero under the neutral stability condition. To overcome this contradiction, they propose the energy method as an alternative.…”

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

“…Even though such an approach was successfully applied to this problem recently in [18], and very recently in [202], in our opinion our construction is more flexible and intuitive. The main advantage of our proof, compared to those of the latter references, is that, in the process, we also establish existence and uniqueness of a solution of problem (157), which seems to be a new and useful result (recall Remark 17). Although a sizable literature has been devoted to the study of the Painlevé equation (see [99,129,149]), we understand that the solution of this problem was not previously known.…”

“…Since then, it has been (formally) used to describe layered structures in problems involving crystalline interphase boundaries [18,190,191], patterns of convection in rectangular platform containers [75], self-similar parabolic optical solitary waves [41], and the Navier-Stokes and continuity equations for axisymmetric flow [157].…”