Transcritical rotating flow over topography

Abstract

The flow of a one-and-a-half layer fluid over a three-dimensional obstacle of non-dimensional 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 = B1/2M−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.