A novel extension to classical 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 energy Casimir functional $\mathcal{A}_{rd}$ of Wang (2009) [75]. A weighted kinetic energy integral of the perturbation, which has a form of $\Delta\mathcal{A}_{rd}$, the difference between the reduced Arnol'd functional and its base flow value, is used as a Lyapunov functional. 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 time evolution of $\Delta\mathcal{A}_{rd}$ depends only on the viscous effects of the perturbation's dynamics inside the flow domain. The requirement for the temporal decay of $\Delta\mathcal{A}_{rd}$ leads to two novel sufficient conditions for the nonlinear stability of the circular Couette flow in response to axisymmetric viscous perturbations. The linearized version of these conditions for infinitesimally small perturbations recovers the recent linear stability results by Kloosterziel (2010) [39]. 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 (1933) [76], and extend the classical nonlinear stability results of Serrin (1959) [62] and Joseph & Hung (1971) [35]. 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 setup 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 co-axial rotating cylinders and for the circular Couette flow. We demonstrate via numerical simulations that the respective analytical upper bounds are relevant to the dynamics of various axisymmetric perturbations tested, where they are strictly obeyed in both examples. This present study provides specific parameter regions for which the circular Couette flow is nonlinearly stable for all Reynolds numbers.