Low Froude Number Flow Past Three-Dimensional Obstacles. Part II: Upwind Flow Reversal Zone

Abstract

The present paper contains a continuation of our study of the flow of a density-stratified fluid past three-dimensional obstacles for Froude number ?O(1). Linear theory (large Froude number) and potential-flow-type theory (small Froude number) are both invalid in this range, which is of particular relevance to natural, atmospheric flows past large mesoscale mountains. The present study was conceived to provide a systematic investigation of the basic aspects of this flow. Thus, we have excluded the effects of friction, rotation, nonuniform ambient flow, and the complexity of realistic terrain. In Part I of this study we focused on the pair of vertically oriented vortices forming on the lee side when the Froude number decreases below 0.5 (approximately), and argued that their formation may be understood in terms of nonlinear aspects of inviscid gravity waves, i.e., without invoking traditional arguments on the separation of the friction boundary layer. Herein we examine the zone of flow reversal on the windward side of the obstacle, which is also a characteristic feature of the low-Froude number flow. We find that flow stagnation and a tendency for flow reversal upwind of a symmetric bell-shaped obstacle is well predicted by linear inviscid gravity-wave theory. This finding stands in contrast with ?horseshoe-vortex? arguments (attributed to frictional boundary layer separation) often invoked in the literature. We also perform experiments on obstacles of varying aspect ratio, ? (across-stream length/along-stream length). Here the utility of the linear theory is less clear: considering cases with Fr = 0.33 and, ? ? ∞, we find that for ? ≤ 1 upstream-propagating columnar modes are essentially absent, however, for ? increasing beyond unity, they appear with increasing strength. It has been argued in the literature that having a sufficiently strong columnar mode is the means by which the upwind flow is brought to stagnation. The absence of this effect for ? ≤ 1 (even though there is upwind-flow stagnation) and its appearance for ? > 1 indicate the coexistence of two distinct gravity-wave effects that decelerate the flow upwind of an obstacle at low Froude number.