Steady rotating flows over a ridge

Esler, J. G.; Rump, O. J.; Johnson, E. R.

doi:10.1063/1.2130740

Cited by 7 publications

(9 citation statements)

References 28 publications

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“…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. The regime diagram of ERJ07b ͑Ref.…”

Section: Discussionmentioning

confidence: 98%

“…The focus is on the supercritical regime in which the Froude number F, defined as the ratio of the oncoming flow speed to the characteristic gravity wave phase speed, 7 is greater than unity. Following previous studies, [8][9][10][11][12] the rotating shallow water equations ͑rSWEs͒ are chosen as a model paradigm that can be regarded as representative of more general atmospheric and oceanic flows. For the atmosphere the physically relevant scenario described by the rSWE is that of a Boussinesq oneand-a-half layer fluid, in which a layer of finite depth underlies an infinite layer of slightly greater buoyancy.…”

Section: Introductionmentioning

confidence: 99%

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Supercritical rotating flow over topography

2009

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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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Section: Discussionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Supercritical rotating flow over topography

2009

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“…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.…”

Section: And 5 For An Overview)mentioning

confidence: 99%

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

2013

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It is well-known that in certain parameter regimes, the steady flow of a density stratified fluid over topography can yield large amplitude internal waves. We discuss an embedded boundary method to solve the Dubreil-Jacotin-Long (DJL) equation for steady-state, supercritical flows over topography in an inviscid, stratified fluid. The DJL equation is equivalent to the full set of stratified steady Euler equations and thus the waves we compute are exact nonlinear solutions. The numerical method presented yields far better scaling with increase in grid size than other iterative methods that have been used to solve this equation, and this in turn allows for a more thorough exploration of parameter space. For waves under the Boussinesq approximation, we contrast the properties of trapped waves over hill-like and valley-like topography, finding that the symmetry of freely propagating solitary waves when the stratification is reflected across the middepth is not present for trapped waves. We extend the derivation of the DJL equation to the non-Boussinesq case and discuss the effect of the new, non-Boussinesq terms on the structure of the trapped waves, finding that the sharp transition between large and small amplitude waves observed under the Boussinesq approximation is much more gradual when the Boussinesq approximation is relaxed. Finally, we demonstrate the existence of asymmetric steady states over hill-like topography where the flow is subcritical upstream of the topography but transitions to supercritical somewhere over the hill. Waves in this new class of exact solutions are related to so-called downstream recovery jumps predicted on the basis of hydrostatic (shallow water) theories, but when breaking does not occur the recovery jump does not stop propagating downstream and an asymmetric state across the topography maximum is reached for long times.

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“…Baines (1995). The rotating two-dimensional obstacle curves are particular to a parabolic obstacle, and are discussed in Esler et al (2005). Note that there are two curves on the supercritical side, as in the two-dimensional obstacle case a region of hysteresis exists in both non-rotating and rotating flow (e.g.…”

Section: Transcritical Flows In Rotating Shallow Watermentioning

confidence: 99%

“…Shrira 1986;Grimshaw et al 1998;Zeitlin, Medvedev & Plougonven 2003), which are known to be modified by dispersion (Ostrovsky 1978). The generation of these inertia-gravity waves in the relatively simple context of flow over a two-dimensional obstacle in the presence of rotation has been studied previously (Baines & Leonard 1989;Esler, Rump & Johnson 2005). Some basic qualitative effects of rotation in the transcritical regime are apparent from these studies: upstream-propagating hydraulic jumps are arrested a finite distance ahead of the obstacle, the amplitude of these jumps decreases, and a hydraulic jump appears downstream of the obstacle when the amplitude of the wavetrain of inertia-gravity waves excited downstream exceeds a limiting value.…”

Section: Introductionmentioning

confidence: 99%

Transcritical rotating flow over topography

Esler¹,

Rump²,

Johnson³

2007

J. Fluid Mech.

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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.

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Steady rotating flows over a ridge

Cited by 7 publications

References 28 publications

Supercritical rotating flow over topography

Supercritical rotating flow over topography

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

Transcritical rotating flow over topography

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