Extension to nonlinear stability theory of the circular Couette flow

Abstract

A nonlinear stability analysis of the viscous circular Couette flow to axisymmetric finite-amplitude perturbations under axial periodic boundary conditions is developed. The analysis is based on investigating the properties of a reduced Arnol'd energyCasimir function A rd of Wang (Phys. Fluids, vol. 2, 2009, 084104). A weighted kinetic energy of the perturbation, which has a form of A rd , the difference between the reduced Arnol'd function and its base flow value, is used as a Lyapunov function. We show that all the inviscid flow effects as well as all the viscous-dependent terms that are related to the flow boundaries vanish. The evolution of A rd depends only on the viscous effects of the perturbation's dynamics inside the flow domain. The requirement for the temporal decay of A rd leads to two novel sufficient conditions for the nonlinear stability of the circular Couette flow in response to axisymmetric perturbations. The linearized version of these conditions for infinitesimally small perturbations recovers the recent linear stability results by Kloosterziel (J. Fluid Mech., vol. 652, 2010, pp. 171-193). By examining the nonlinear stability conditions, we establish a definite operational region of the viscous circular Couette flow that is independent of the fluid viscosity. In this region of operation, the flow is nonlinearly stable in response to perturbations of any size, provided that the initial total circulation function is above a minimum level determined by the operational conditions of the base flow. Comparisons with historical studies show that our results shed light on the experimental measurements of Wendt (Ing.). When the flow is nonlinearly stable and evolves axisymmetrically for all time, then it always decays asymptotically in time to the circular Couette flow determined uniquely by the set-up of the rotating cylinders. Finally, we derive upper-bound estimates on the decay rate of finite-amplitude perturbations for the solid-body rotation flow between two coaxial rotating cylinders and for the circular Couette flow. We demonstrate via numerical simulations that the theoretical upper bound is relevant to the dynamics of various axisymmetric perturbations tested, where it is strictly obeyed. This present study provides new physical insights into a classical flow problem that was studied for many decades.

Extension of classical stability theory to viscous planar wall-bounded shear flows

“…It is thus of great interest to generalize the theory for inclusion of viscous flows, those with corresponding dynamical systems that are sufficiently close to Hamiltonian systems in their inviscid limits. Pursuing this endeavour, a first extension of Arnold's inviscid theory to viscous flows was carried out by Yau, Wang & Rusak (2016) for viscous circular Taylor-Couette flow confined between two concentric rotating cylinders. Through regarding the original inviscid Arnold's function as a Lyapunov function of the associated viscous dissipative system, the viscous extension had led to discovery of a definite flow operation domain in which the Taylor-Couette flow is nonlinearly stable to arbitrary finite-amplitude axisymmetric perturbations.…”