Transcritical rotating flow over topography

Abstract

The flow of a one-and-a-half layer fluid over a three-dimensional obstacle of nondimensional height M, relative to the lower layer depth, is investigated in the presence of rotation, the magnitude of which is measured by a non-dimensional parameter B (inverse Burger number). The transcritical regime in which the Froude number F , the ratio of the flow speed to the interfacial gravity wave speed, is close to unity is considered in the shallow-water (small-aspect-ratio) limit. For weakly rotating flow over a small isolated obstacle (M → 0) a similarity theory is developed in which the behaviour is shown to depend on the parameters Γ = (F −1)M −2/3 and ν = B 1/2 M −1/3 . The flow pattern in this regime is determined by a nonlinear equation in which Γ and ν appear explicitly, termed here the 'rotating transcritical small-disturbance equation' (rTSD equation, following the analogy with compressible gasdynamics). The rTSD equation is forced by 'equivalent aerofoil' boundary conditions specific to each obstacle. Several qualitatively new flow behaviours are exhibited, and the parameter reduction afforded by the theory allows a (Γ, ν) regime diagram describing these behaviours to be constructed numerically. One important result is that, in a supercritical oncoming flow in the presence of sufficient rotation (ν & 2), hydraulic jumps can appear downstream of the obstacle even in the absence of an upstream jump. Rotation is found to have the general effect of increasing the amplitude of any existing downstream hydraulic jumps and reducing the lateral extent and amplitude of upstream jumps. Numerical results are compared with results from a shock-capturing shallow-water model, and the (Γ, ν) regime diagram is found to give good qualitative and quantitative predictions of flow patterns at finite obstacle height (at least for M . 0.4). Results are compared and contrasted with those for a two-dimensional obstacle or ridge, for which rotation also causes hydraulic jumps to form downstream of the obstacle and acts to attenuate upstream jumps.

Supercritical rotating flow over topography

“…11,13,36 The rotating towing-tank experiments of Johnson et al 36 demonstrated that transcritical rotating flows feature a "sharp-crested" nonlinear wave to the rear of the obstacle that is not present in nonrotating flow. In transcritical twodimensional flow over a ridge, the analytical results of Esler et al 11 showed that increasing rotation leads to the appearance and growth of a hydraulic jump downstream of the obstacle that can exceed the amplitude of the upstream jump.…”

“…Following Esler et al, 13 the finite volume software package conservation laws software package 18 ͑CLAW-PACK͒ is used to obtain the solutions. Explicit details of how CLAWPACK can be adapted to solve the shallow water equations in two dimensions are given by LeVeque, 18 and here the rotation and topographic forcing terms are treated using the method of Strang splitting, following Kuo and Polvani.…”

“…1 ͑right͒, is used as the definition of the supercritical regime of rSWE flow over obstacles in ERJ07b. 13 In the supercritical regime the two hydraulic jumps can be considered to be "rotation modified" versions of the N-waves of Whitham, 27 that have been shown to characterize the wakes of nonrotating shallow water flows over isolated obstacles. 10,12 The fact that the nonlinearity is displaced away from the obstacle in this fashion suggests that linear theory will accurately capture the drag exerted by the obstacle on the flow, as will be demonstrated next.…”

“…Numerical simulations showed the transition between the two regimes to be abrupt. The effect of rotation on transcritical flows has been studied in detail elsewhere ͑Esler et al, 13 ERJ07b hereafter͒, and consequently the aim of the present work is to determine its effect in the supercritical regime, where F տ 1+⌫ + M 2/3 .…”

Trapped internal waves over topography: Non-Boussinesq effects, symmetry breaking and downstream recovery jumps

“…12 Steady states for subcritical and supercritical flows over topography (using the more traditional definition of criticality) have been previously studied in the context of shallow water theory. [31][32][33][34] One of these steady states includes a downstream recovery jump when the flow upstream of the topography is subcritical but transitions to supercritical as it passes over the topography (i.e., a finite amplitude topography effect). A form of the downstream recovery jump was also observed by Stastna et al 35 in rotating continuously stratified flows.…”