Stratified separated flow around a mountain with an inversion layer below the mountain top

Abstract

This paper presents analytical and numerical results for separated stratified inviscid flow over and around an isolated mountain in the limit of small Froude number. The vertical density profile consists of a lower strongly stratified layer whose depth is just less than that of the mountain. It is separated from a semi-infinite upper stably stratified layer by a thin, highly stable, inversion layer. The paper aims to provide, for this particular profile, a thorough analysis of the three-dimensional separated flow over a mountain top with strong stratification. The Froude numbers F and F I of the lower layer and the interface are small with F I F 1, but the upper-layer Froude number is arbitrary. The flow at each height in the lower layer is governed by the twodimensional Euler equations and moves horizontally around the mountain. It is given by a modification of a previous model using Kirchhoff free-streamline theory for the separated flow region downstream of the mountain. The pressure variations associated with the lower-layer flow are of the same order as the dynamic head and induce significant displacements of the inversion layer. When the inversion is near the top of the mountain these deflections are of the same order as the height of the projecting part of the mountain top and combine with the flow over the mountain top to excite vertically propagating internal waves in the upper layer. The resultant pressure field, vertical stream surface displacements, and surface streamlines in the upper layer are described consistently in the hydrostatic limit. Many of the features of the upper flow, including the perturbations of the critical dividing streamlines, are similar to those in flows with uniform stable stratification at low Froude number. Comparisons are made with experiments and approximate models for these summit flows based on the assumption that the dividing streamlines have small vertical displacement.

Wake of inertial waves of a horizontal cylinder in horizontal translation

“…Since vertical motion is the primary cause of internal wave motion, only the portion of the topography protruding above the dividing streamsurface contributes to the wave radiation, acting as a thin cut-off obstacle to which the weak-streamwise-perturbation approximation may be applied. This approach has first been evoked by Newley, Pearson and Hunt [36], Hunt et al [37] and Greenslade [38], then implemented by Hunt, Vilenski and Johnson [39], Voisin [17] and Dalziel et al [40], the latter also presenting experiments for hemispherical topography. As a result, the weak-streamwise-perturbation approximation turns out to also be applicable to three-dimensional bluff topography in a stratified flow, so long as the stratification is strong.…”

“…The mathematical origin of the nonuniformity and the derivation of the uniform far-field expansion are briefly discussed in Appendix C. The uniform expansion gives to the wave field some extension upstream, however still smaller than with the full model. Another noticeable feature of expansion (37)(38)(39), visible in Fig. 6, is an unphysical vertical shift of the streamlines between their original position upstream and their final position downstream.…”

“…In (c) and (d), these angles of zero amplitude do not exist since the roots of the spectrum q1(kx) correspond to evanescent waves. In (a-d), we also show with dashed lines a few lines of constant phase of the far-field wake (37)(38)(39). We actually plot the parametric curve (41)(42) for ϕs + π/2 = π, 2π, 3π, 4π, .…”

“…39) for the spectrum (51) (the spectrum (51) being imaginary introduces the +π/2 phase shift). locity field predicted in the far-field approximation (37)(38)(39), still using the weak-streamwiseperturbation spectrum of the cylinder (51), for the same values of Ro and Re as in Fig. 5.…”

The structure of low-Froude-number lee waves over an isolated obstacle

“…While the difference in elevation of upstream and downstream stagnation points has been discussed previously (e.g. Hunt & Snyder 1980;Hunt, Vilenski & Johnson 2006), part of its importance has remained unexplored and only becomes clear if we return to the principle of stationary phase to consider what waves can actually be sustained in such a situation.…”