Non-modal stability in sliding Couette flow

Abstract

AbstractThe problem of an incompressible flow between two coaxial cylinders with radii$a$and$b$subjected to a sliding motion of the inner cylinder in the axial direction is considered. The energy stability and the non-modal stability have been investigated for both axisymmetric and non-axisymmetric disturbances. For the non-modal stability, we focus on two problems: response to external excitations and response to initial conditions. The former is studied by examining the$\epsilon $-pseudospectrum, and the latter by examining the energy growth function$G(t)$. Unlike the results of the modal analysis in which the stability of sliding Couette flow is determined by axisymmetric disturbances, the energy analysis shows that a non-axisymmetric disturbance has a critical energy Reynolds number for all radius ratios$\eta = a/ b$. The results for non-modal stability show that rather large transient growth occurs over a wide range of azimuthal wavenumber$n$and streamwise wavenumber$\ensuremath{\alpha} $, even though the Reynolds number is far below its critical value. For the problem of response to external excitations, the response is sensitive to low-frequency external excitations. For all values of the radius ratio, the maximum response is achieved by non-axisymmetric and streamwise-independent disturbances when the frequency of external forcing$\omega = 0$. For the problem of response to initial conditions, the optimal disturbance is in the form of helical streaks at low Reynolds numbers. With the increase of$\mathit{Re}$, the optimal disturbance becomes very close to straight streaks. For each$\eta $, the maximum energy growth of streamwise-independent disturbances is of the order of${\mathit{Re}}^{2} $, and the optimal time is of the order of$\mathit{Re}$. This relation is qualitatively similar to that for plane Couette flow, plane Poiseuille flow and pipe Poiseuille flow. Direct numerical simulations are applied to investigate the transition of the streamwise vortex (SV) scenario at$\mathit{Re}= 1000$and 1500 for various$\eta $. The initial disturbances are the optimal streamwise vortices predicted by the non-modal analysis. We studied the streak breakdown phase of the SV scenarios by examining the instability of streaks. Moreover, we have investigated the sustainment of the energy of disturbances in the SV scenario.