On the stability of plane Couette flow to infinitesimal disturbances

Abstract

It has been conjectured for many years that plane Couette flow is stable to infinitesimal disturbances although this has never been proved. In this paper we use a, combination of asymptotic analysis and numerical computation to examine the associated Orr-Sommerfeld differential problem in a systematic manner. We obtain new evidence that the conjecture is, in all probability, correct. In particular we find that, at a fixed large value of the Reynolds number R, as in an experiment, if a disturbance of wavenumber α has a damping rate - αci then – ci has a minimum value of order R−½ when α is of order R½. We believe that this result may be an essential prerequisite to an understanding of the stability of plane Couette flow to finite-amplitude disturbances.

Subcritical Transition to Turbulence in Plane Channel Flows

“…Evidently, the three-dimensional disturbances quickly achieve a form that grows exponentially in time for R < 1000. The growth rate of the threedimensional disturbances is rapid with their amplitude increasing by a factor of about 10 in a Here E 2D is the total energy (relative to the laminar flow) in wave numbers of the form (na 9 0) 9 while E 3D is the total energy in wave numbers (na f (3). For R £ 1000 we obtain growth and for R = 500 decay.…”

“…A similar problem was studied in Ref. 3, where profile steepening because of cavity formation could be predicted. However, the calculations were based on the driven nonlinear Schrodinger equation ignoring ion inertia.…”

“…While experiments 1 show that incompressible plane Poiseuille and plane couette flow may undergo transition to turbulence at Reynolds numbers R of order 1000, linear stability analysis of these plane parallel flows gives critical Reynolds numbers of 5772 for plane Poiseuille flow 2 and «>for plane couette flow. 3 This discrepancy between theory and experiment suggests that the +1 mechanism of transition is not properly represented by parallel-flow linear stability analysis. In this Letter, we present a new linear threedimensional mechanism that predicts transition at Reynolds numbers in good agreement with experiment for both plane Poiseuille and plane couette flows.…”

On the linear stability of compressible plane Couette flow

“…Numerical studies of the linear stability problem have been carried out by Grohne (1954), Gallagher andMercer (1962, 1964), Deardorff (1963), Davey (1973), and Gallagher (1974); however, all these studies found no evidence of instability. A number of analytic studies have also been carried out on this problem.…”

Dynamics of vorticity defects in shear

“…However, at large Reynolds number these modes fall into two classes. There are relatively slowly decaying modes concentrated in boundary layers next to the walls (Davey 1973) and there are rapidly decaying modes with oscillatory structure in the centre of the channel (Schmid et al 1992). The boundary modes are exponentially small in the main body of the flow, whereas the central modes are not localized within the defect region.…”