Energy curves for multi-layer flow through obstructions

Abstract

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...

Classification of unidirectional three-layer flow over a hump

“…a. Vue en plan du canal b. Ecoulement d'une couche unique c. Ecoulement en trois couches uniform flow in a sloping channel [10]. The present choice of a downstream contraction in flow width provides an opportunity to present a solution for three-layer flow through a contraction and to demonstrate the general features of a special type of internal hydraulic control that occurs in regions of decreasing flow width [12]. This type of control is called a virtual control [17].…”

“…The intensity and vertical extent of the mixing and its effect on water quality depends on the type and location of the hydraulic controls. In a companion paper, Denton [12] presented a three-layer model of gradually-varying flow through an obstruction. At a given flow cross-section, there can be as many as seven different maxima and minima in the variation of layer energy with layer thickness.…”

“…Depending upon the particular flow problem being analysed, the solution will involve one of these energy extrema at each hydraulic control cross-section. The variation of these energy extrema can also be used to determine the location of the hydraulic controls [10,12]. In this paper Demon's [12] model will be used to investigate the complete range of hydraulic control configurations, or flow regimes, for three-layer flow over a bottom hump in a channel of uniform width.…”

“…The variation of these energy extrema can also be used to determine the location of the hydraulic controls [10,12]. In this paper Demon's [12] model will be used to investigate the complete range of hydraulic control configurations, or flow regimes, for three-layer flow over a bottom hump in a channel of uniform width. The response of the flow to changes in the hump height and other flow param eters also depends on the choice of upstream and downstream boundary conditions.…”

“…Exchange flows, such as the flow through the Strait of Gibraltar, are more complicated because there can be two controls where the flow is critical with respect to the same internal wave mode [2,3,10,11,13]. Three-layer flow supports two different types of internal wave: a second-mode wave which deflects the two interfaces in opposite directions and a faster first-mode internal wave which causes interface deflections of the same sign [6,12]. Both internal waves travel much slower than JOURNAL OF HYDRAULIC RESEARCH, VOL.…”