Stationary perturbations of Couette–Poiseuille flow: the flow development in long cavities and channels

Abstract

We consider stationary perturbations to Couette–Poiseuille flows. These may be considered to be related to far downstream/upstream entry/end effects in flow inside long cavities and channels. Three distinct classes of basic flow are considered, all of which are exact solutions of the Navier–Stokes equations. We first study the problem in the case of Poiseuille flow, and are able to explain a previous discrepancy between fully numerical results, and asymptotic theory valid for large Reynolds numbers, R. The second case, which may be derived from a combination of an imposed streamwise pressure gradient and sliding of the upper channel wall, is for the particular situation where the flow on the lower surface is on the verge of reversing direction. The third case is relevant to the flow inside a long driven cavity (with closed ends, no imposed streamwise pressure gradient and no net mass flux). The flow is driven exclusively by a sliding top wall and mass conservation demands that the flow is no longer unidirectional.For low Reynolds numbers, the stationary eigenvalues in all cases considered are complex (and hence are not monotonic in the streamwise direction). Indeed as R → 0 the eigenvalues become completely independent of the base profile. As the Reynolds number is increased, the eigenvalues generally undergo a number of branching processes switching between being complex and real (and vice versa) in nature, and at large Reynolds numbers fall broadly into three distinct categories, namely O(1), O(R−1/7) and O(1/R). In this limit the eigenvalues may be either complex or real (tending to monotonic eigensolutions in the streamwise direction).Of particular interest are certain of the O(1) eigensolutions for the ‘driven-cavity’ problem, in the high-Reynolds-number limit; these turn out to be highly oscillatory (WKB-type) over much of the cavity section.In all three cases, we use a combination of numerical and asymptotic techniques, and a thorough comparison between results thus obtained is made.