The Occurrence of Critical Flow and Hydraulic Jumps in a Multi-Layered Fluid System

Benton, George S.

doi:10.1175/1520-0469(1954)011<0139:toocfa>2.0.co;2

Cited by 26 publications

(43 citation statements)

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“…An analysis for Cj and Vj, based on the solution of the equations of motion for the long wave characteristics [2,7] is presented in Appendix A. Numerical solution of these equations yields the corresponding wave speed Froude numbers Fy The variations in F 0 , F u and F 2 with upper-layer thickness and the corresponding variation in G 2 at the channel cross-section (z s = 0, /?…”

Section: Energy Variation With Layer Thicknessmentioning

confidence: 92%

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Energy curves for multi-layer flow through obstructions

Denton

1989

Journal of Hydraulic Research

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One method for obtaining solutions for single-and multi-layer flow over bottom humps and through sidewall contractions is to consider the variation in mechanical energy with layer thickness at key cross-sections. However, previous models of gradually-varying multi-layer flow have been restricted to flows which remain subcritical with respect to all but the slowest internal wave speed. This paper shows how the energy extrema method can be used over the full range of flow conditions; from subcritical with respect to all internal wave speeds to supercritical with respect to the free surface wave. The complete energy curves for three-layer flow are found to have up to seven different energy extrema at a given cross-section. The energy extrema are classified according to whether the flow is critical with respect to the first-or second-mode internal wave or the free-surface wave, whether the energy is a local maximum or minimum, and which layers are active. Froude numbers based on the actual wave speeds are used to determine whether the flow at a given cross-section is subcritical, critical or supercritical with respect to each of the three possible wave modes. All seven energy extrema represent possible energy conditions at a hydraulic control. Which energy extrema are applicable for a particular three-layer flow depends on the obstruction geometry and the upstream and downstream boundary conditions. RESUME Une methode pour calculer l'écoulement a une ou plusieurs couches, par dessus un seuil de fond et au passage de contractions latérales, consiste a prendre en compte la variation d'énergie mécanique en fonction de Pépaisseur de la couche a certaines sections remarquables. Cependant, les modèles existants de multicouches en regime graduellement varié sont limités aux écoulements sous-critiques partout sauf en ce qui concerne la célérité d'onde interne la plus faible. Cet article montre comment la methode des extremas d'énergie peut être appliquée a toute la gamme des conditions d'écoulement: depuis les écoulements souscritiques en ce qui conceme toutes les célérités d'ondes internes jusqu'aux écoulements super-critiques en ce qui concerne les ondes a la surface libre. Les courbes completes d'énergie pour un écoulement a trois couches peuvent présenter jusqu'a sept extremas différents d'énergie en une section donnée. Les extremas d'énergie sont classes soit selon que l'écoulement est critique en ce qui concerne les ondes internes de premier et du second mode ou les ondes de surface, soit selon que l'énergie est un maximum ou un minimum local et en fonction des couches qui sont actives. Des nombres de Froude construits avec les célérités d'ondes réelles sont utilises pour determiner en une section donnée si l'écoulement est sous-critique, critique ou super-critique en ce qui concerne chacun des trois modes d'ondes possibles. Chacun des sept extremas d'énergie représente une condition d'énergie possible a une section de controle; la distinction entre les différentrs extremas applicables a un écoulement particulier a tro...

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Section: Energy Variation With Layer Thicknessmentioning

confidence: 92%

“…Similarly it can be shown that the critical flow condition for three-layer flow is (Benton [7]), d_EJ_ ~dhf…”

Section: Critical Flow Conditionmentioning

confidence: 99%

Energy curves for multi-layer flow through obstructions

Denton

1989

Journal of Hydraulic Research

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“…Therefore we have here a baroclinic shock, the thickness variations being of opposite sign layerwise (as well as the velocity ones), and the signature in the barotropic field being much weaker than in the baroclinic field (by a factor of 20: see figure 16b). The baroclinic (or internal) hydraulic jump in two-layer models has been the subject of numerous studies beginning with Benton (1954) and Yih & Guha (1955). Criteria for the shock formation derived in these works are based on layerwise 'local' Froude numbers, defined as |v i |/ √ gh i (see also Armi 1986).…”

Section: Adjustment For Rossby Number Ro = 04 321 Weakly Stratifimentioning

confidence: 99%

Existence and properties of ageostrophic modons and coherent tripoles in the two-layer rotating shallow water model on the -plane

Lahaye¹,

Zeitlin²

2012

J. Fluid Mech.

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We study formation and properties of new coherent structures: ageostrophic modons in the two-layer rotating shallow water model. The ageostrophic modons are obtained by 'ageostrophic adjustment' of the exact modon solutions of the twolayer quasi-geostrophic equations with the free surface, which are used to initialize the full two-layer shallow water model. Numerical simulations are performed using a well-balanced high-resolution finite volume numerical scheme. For large enough Rossby numbers, the initial configurations undergo ageostrophic adjustment towards asymmetric ageostrophic quasi-stationary coherent dipoles. This process is accompanied by substantial emission of inertia-gravity waves. The resulting dipole is shown to be robust and survives frontal collisions. It contains captured inertia-gravity waves and, for higher Rossby numbers and weak stratification, carries a (baroclinic) hydraulic jump at its axis. For stronger stratifications and high enough Rossby numbers, 'rider' coherent structures appear as a result of adjustment, with a monopole in one layer and a dipole in another. Other ageostrophic coherent structures, such as two-layer tripoles and two-layer modons with nonlinear scatter plot, result from the collisions of ageostrophic modons. They are shown to be long-living and robust, and to capture waves.

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“…This has been done by Armi [2], Benton [4] and Wood and Lai [18], for example, and for ease of reference we summarize the results here. In particular, it will be seen that shallow-water theory leads to an approximate definition of critical flow for a two-layer system (equation (3.7)), which is useful in discussing solutions to the fully non-linear problem.…”

Section: Shallow-water Approximationmentioning

confidence: 99%

“…The occurrence of critical flow in a two-layer or even a multi-layer fluid system is a problem of importance in meteorology and oceanography (see Melville and Helfrich [12]), and has been considered in the context of one-dimensional hydraulic theory by Benton [4] and Wood and Lai [18], for example. Armi [2] has investigated the flow of a system of two fluids over bottom topography in a channel in which width variations may also occur, and the hydraulics of exchange flows (which involve opposite flow directions in each fluid layer) in a channel having both bottom and width variations is discussed by Armi and Farmer [3] and Farmer and Armi [6].…”

Section: Introductionmentioning

confidence: 99%

Two-layer critical flow over a semi-circular obstruction

Forbes

1989

J Eng Math

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Steady, trans-critical flow of a two-fluid system over a semi-circular cylinder on the bottom of a channel is considered. Each fluid is assumed to be inviscid and incompressible and to flow irrotationally, but the fluids have different densities, so that one flows on top of the other. Consequently, a sharp interface exists between the fluids, in addition to a free surface at the top of the upper fluid. Trans-critical flow is investigated, in which waves are absent from the system, but the upstream and downstream fluid depths differ in each fluid layer. The problem is formulated using conformal mapping and a system of three integrodifferential equations, and solved numerically with the aid of Newton's method. The free-surface shape and that of the interface are obtained along with the Froude numbers in each fluid layer. Results of computation are presented and discussed.

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The Occurrence of Critical Flow and Hydraulic Jumps in a Multi-Layered Fluid System

Cited by 26 publications

References 0 publications

Energy curves for multi-layer flow through obstructions

Energy curves for multi-layer flow through obstructions

Existence and properties of ageostrophic modons and coherent tripoles in the two-layer rotating shallow water model on the -plane

Two-layer critical flow over a semi-circular obstruction

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