This paper reports experimental observations of finite amplitude interfacial waves forced by a surface-mounted obstacle towed through a two-layer fluid both when the fluid is otherwise at rest and when the fluid is otherwise rotating as a solid body. The experimental apparatus is sufficiently wide so that sidewall effects are negligible even in near-critical flow when the towing speed is close to the interfacial long-wave speed and the transverse extent of the forced wavefield is large. The observations are modelled by a simple forced Benjamin–Davis–Acrivos equation and comparison between integrations of both linear and nonlinear problems shows the fundamental nonlinearity of the near-critical flow patterns. In both the experiments and integrations rotation strongly confines the wavefield to extend laterally over distances only of order of the Rossby radius and also introduces finite-amplitude sharply pointed lee waves in supercritical flow.
Non-dispersive and weakly dispersive single-layer flows over axisymmetric obstacles, of non-dimensional height M measured relative to the layer depth, are investigated. The case of transcritical flow, for which the Froude number F of the oncoming flow is close to unity, and that of supercritical flow, for which F > 1, are considered. For transcritical flow, a similarity theory is developed for small obstacle height and, for non-dispersive flow, the problem is shown to be isomorphic to that of the transonic flow of a compressible gas over a thin aerofoil. The non-dimensional drag exerted by the obstacle on the flow takes the form D(Γ) M5/3, where Γ = (F-1)M−2/3 is a transcritical similarity parameter and D is a function which depends on the shape of the ‘equivalent aerofoil’ specific to the obstacle. The theory is verified numerically by comparing results from a shock-capturing shallow-water model with corresponding solutions of the transonic small-disturbance equation, and is found to be generally accurate for M≲0.4 and |Γ| ≲ 1. In weakly dispersive flow the equivalent aerofoil becomes the boundary condition for the Kadomtsev–Petviashvili equation and (multiple) solitary waves replace hydraulic jumps in the resulting flow patterns.For Γ ≳ 1.5 the transcritical similarity theory is found to be inaccurate and, for small M, flow patterns are well described by a supercritical theory, in which the flow is determined by the linear solution near the obstacle. In this regime the drag is shown to be $c_d M^2/(F\sqrt{F^2-1})$, where cd is a constant dependent on the obstacle shape. Away from the obstacle, in non-dispersive flow the far-field behaviour is known to be described by the N-wave theory of Whitham and in dispersive flow by the Korteweg–de Vries equation. In the latter case the number of emergent solitary waves in the wake is shown to be a function of ${\cal A}= 3M/(2\delta^2 \sqrt{F^2-1})$, where δ is the ratio of the undisturbed layer depth to the radial scale of the obstacle.
A model describing rotating single-layer flows over a parabolic ridge is investigated. A method of constructing steady solutions is introduced, and is used to extend previous results and determine exact regime diagrams describing the qualitative nature of the solution. Analytic expressions for the boundaries between transcritical flow and supercritical and subcritical flows are given as a function of obstacle height, Froude number of the upstream flow, and the flow inverse Burger number ͑a nondimensional number proportional to the square of the rotation rate͒. For fixed obstacle height, the nature of the supercritical transition is found to change as the rotation rate increases, with a hysteresis region like that in nonrotating flow being present only at lower rotation rates. At higher rotation rates, solutions with stationary jumps over the obstacle become stable, and abrupt transitions between supercritical and transcritical flow no longer occur. An exact analytic expression is also found for transcritical flow over the obstacle, which is closely related to the solutions for nonlinear inertia-gravity waves of limiting amplitude found by Shrira. For sufficiently high ridges in initially supercritical flow, a wave train of nonlinear inertia-gravity waves of limiting amplitude appears behind a downstream hydraulic jump.
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.
The flow of a one-and-a-half layer Boussinesq fluid over an 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 nondimensional parameter B ͑inverse Burger number͒. The supercritical regime in which the Froude number F, the ratio of the flow speed to the interfacial gravity wave speed, is significantly greater than one is considered in the shallow water ͑small aspect ratio͒ limit. The linear drag exerted by the obstacle on the flow is shown to be M 2 / ͑F ͱ F 2 −1͒ ϫ f͑B / F 2 −1͒, where f is a function specific to each obstacle. Explicit expressions are given for several common obstacle shapes, and the results are checked against nonlinear flows simulated by a shock-capturing finite volume numerical scheme. For flows within the supercritical regime ͑F −1ӷ M 2/3 ͒ the linear drag result is found to remain accurate up to ͑at least͒ M Ϸ 0.7. The success of the linear drag theory can be explained because, in the supercritical regime, strong nonlinearities are displaced to the wake regions at the flanks of the obstacle. In the presence of weak rotation and for small obstacle height the development of the nonlinear wakes is governed by the Ostrovsky-Hunter ͑OH͒ equation. Across a wide range of parameter space the wake pattern is determined by a single parameter  =3M / B ͱ F 2 − 1. Numerical solutions of the OH equation illustrate the dependence of flow patterns and wave breaking regions on . Results are again verified by comparison with numerical solutions of the full nonlinear rotating shallow water equations.
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