Flow Patterns and Drag in Near-Critical Flow over Isolated Orography

Abstract

Since early manned space flight orographically forced cloud patterns have been described in terms of the single isolated shock structure of shallow-water flow or, equivalently, compressible fluid flow. Some of these observations show, behind an initial “bow wave,” a series of almost parallel wave crests. This paper considers the simplest extension of shallow-water theory that retains not only nonlinear steepening of waves but includes departures from hydrostatic balance, and thus wave dispersion, showing that the single shocks of shallow-water theory are transformed into multiple parallel finite-amplitude wave crests. The context of the discussion is the forced Kadomtsev–Petviashvili equation from classical ship wave dynamics, which plays the same role in two-dimensional near-critical fluid flow as the more familiar Korteweg–de Vries equation in one-dimensional flow. The drag and flow regimes in near-critical flow over isolated orography are described in terms of the three governing parameters of the flow: the deviation of the flow speed from critical, the strength of nonhydrostatic effects, and the strength of orographic forcing.

Non-dispersive and weakly dispersive single-layer flow over an axisymmetric obstacle: the equivalent aerofoil formulation

“…In contrast to the previous work, here we use the boundary conditions derived from the equivalent aerofoil of the obstacles (2.18) to force the KP equation (2.15) in order to find solutions consistent with flow over an axisymmetric three-dimensional obstacle. The weakly dispersive results do not differ qualitatively from those of Johnson & Vilenski (2004) or from those for flows forced from a moving pressure distribution (Katsis & Akylas 1987;Lee & Grimshaw 1990) but are of particular interest in that they allow the results for weakly dispersive and non-dispersive flows to be compared directly, by examining the results from the TSD equation (2.19) under identical forcing. Also, unlike the results of Johnson & Vilenski (2004) the current results can be definitively associated with the flow over a specific physical obstacle (in Johnson & Vilenski (2004) the obstacle's cross-stream dimension varies as −1/2 ).…”

“…The weakly dispersive results do not differ qualitatively from those of Johnson & Vilenski (2004) or from those for flows forced from a moving pressure distribution (Katsis & Akylas 1987;Lee & Grimshaw 1990) but are of particular interest in that they allow the results for weakly dispersive and non-dispersive flows to be compared directly, by examining the results from the TSD equation (2.19) under identical forcing. Also, unlike the results of Johnson & Vilenski (2004) the current results can be definitively associated with the flow over a specific physical obstacle (in Johnson & Vilenski (2004) the obstacle's cross-stream dimension varies as −1/2 ). Figure 7 shows for comparison the steady height fields for non-dispersive and weakly dispersive transcritical flows past the equivalent aerofoil corresponding to the paraboloid obstacle.…”

“…A similar, but clockwise, rotation occurs for the coordinates in y < 0. In terms of these new coordinates, Johnson & Vilenski (2004) noted that provided the obstacle height decays sufficiently rapidly at large distances then as y → ± ∞ the solution (2.24) converges to…”

“…Johnson & Vilenski (2004) previously derived equation (2.31) for the case of supercritical flow over an elongated ridge in the weakly dispersive limit (the Kadomtsev-Petviashvili limit, see (2.15)). Here, it has been shown that the solution applies equally to the more general problem of supercritical flow over a small isolated obstacle (M 1) at all Froude numbers satisfying F > 1,…”

“…The domain sizes used are 40L × 40 −1/2 L to 60L × 60 −1/2 L, increasing as Γ approaches zero. The third numerical model, used to solve the KP equation (2.15), is a standard pseudo-spectral model similar to those described by Fornberg & Driscoll (1999) and Johnson & Vilenski (2004). Equation (2.15) is solved on a doubly periodic domain with size either…”