Self-sustaining critical layer/shear layer interaction in annular Poiseuille–Couette flow at high Reynolds number
File(s)
Accepted version
Author(s)
Kumar, Rishi
Walton, Andrew
Type
Journal Article
Abstract
The nonlinear stability of annular Poiseuille–Couette flow through a cylindrical annulus subjected to axisymmetric and helical disturbances is analysed theoretically at asymptotically large Reynolds number R based on the radius of the outer cylinder and the constant axial pressure gradient applied. The inner cylinder moves with a prescribed positive or negative velocity in the axial direction. A distinguished scaling for the disturbance size Δ = O(R−4/9) is identified at which the jump in vorticity across the fully nonlinear critical layer is in tune with that induced across a near-wall shear layer. The disturbance propagates at close to the velocity of the inner cylinder and possesses a wavelength comparable to the radius of the outer cylinder. The dynamics of the critical layer, shear layer and the Stokes layer adjacent to the stationary wall are discussed in detail. In the majority of the pipe, the disturbance is governed predominantly by inviscid dynamics with the pressure perturbation satisfying a form of Rayleigh’s equation. For a radius ratio δ in the range 0 < δ < 1 and a positive sliding velocity V, a numerical solution of the Rayleigh equation exists for sliding velocities in the range 0 < V < 1 − δ2 + 2δ2lnδ, whereas if V < 0, solutions exist for 1 − δ2 + 2lnδ < V < 0. The amplitude equations for both these situations are derived analytically, and we further find that the corresponding asymptotic structures break down when the maximum value of the basic flow becomes located at the inner and outer walls, respectively.
Date Issued
2020-03-01
Date Acceptance
2020-02-13
Citation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2020, 476 (2235)
ISSN
1364-5021
Publisher
Royal Society, The
Journal / Book Title
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume
476
Issue
2235
Copyright Statement
© 2020 The Author(s). Published by the Royal Society. All rights reserved.
Subjects
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Publication Status
Published
Article Number
20190600
Date Publish Online
2020-03-18