Self-sustaining critical layer/shear layer interaction in annular Poiseuille–Couette flow at high Reynolds number

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 numberRbased 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 ratioδin the range 0 < δ < 1 and a positive sliding velocityV, a numerical solution of the Rayleigh equation exists for sliding velocities in the range 0 < V < 1 − δ2 + 2δ2lnδ, whereas ifV < 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.