Two-layer critical flow over a semi-circular obstruction

Forbes, Lawrence K.

doi:10.1007/bf00128906

Cited by 23 publications

(20 citation statements)

References 16 publications

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“…Instead of using the two-layer model with (10)- (11) we tried the alternative formulation (12) and (13) (mass equations (8)- (9) remain the same). This model produced the single layer solution for very long times without any sign of instability.…”

Section: Comparison With One-layer Shallow Water With Constant Densitymentioning

confidence: 99%

“…Only the thicknesses and the total momentum are conserved. There is an alternative formulation in conservation form, namely (12) and…”

Section: Passive Layer Solutionmentioning

confidence: 99%

“…In fact, the two-layer Green-Naghdi equations are unconditionally ill-posed for both the case of a free and of a rigid top surface, while the two-layer shallow water equations are of mixed type, that is, there are regions of state space for which the equations are hyperbolic and regions for which they are ill-posed. Nevertheless, there is interest in multilayer flows, for example in the theoretical development of [10], in the experiments reported in [11], in the steady state results of [12], and in the time-dependent work of [13].…”

Section: Introductionmentioning

confidence: 98%

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Analysis and Computation with Stratified Fluid Models

Liska¹,

Wendroff²

1997

Journal of Computational Physics

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Section: Comparison With One-layer Shallow Water With Constant Densitymentioning

confidence: 99%

“…Only the thicknesses and the total momentum are conserved. There is an alternative formulation in conservation form, namely (12) and…”

Section: Passive Layer Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Analysis and Computation with Stratified Fluid Models

Liska¹,

Wendroff²

1997

Journal of Computational Physics

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“…In particular, the problem of internal waves in a two-layer liquid is classed among these problems.There are numerous methods for solving problems of two-dimensional flow with different Bernoulli constants both in linear [1,2] and nonlinear [3][4][5][6][7][8][9] formulations. A considerable number of publications in this line are devoted to problems of internal waves.…”

mentioning

confidence: 99%

Two-layer gravity flow of a heavy fluid above a curvilinear bottom

Guzevskii¹

1996

J Appl Mech Tech Phys

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The class of hydrodynamic and aerodynamic free-surface problems with internal velocity-discontinuity surfaces is of great theoretical and practical interest.In problems of jet collisions, pneumonics, interaction of flows with different dynamic pressures, one deals with flows with internal discontinuity surfaces. In particular, the problem of internal waves in a two-layer liquid is classed among these problems.There are numerous methods for solving problems of two-dimensional flow with different Bernoulli constants both in linear [1,2] and nonlinear [3][4][5][6][7][8][9] formulations. A considerable number of publications in this line are devoted to problems of internal waves.An effective method has been proposed [10][11][12] for precise nonlinear calculations of plane and axisymmetric flows of an ideal fluid with free surfaces. The present paper extends this method to the case of flow with discontinuity surfaces of the Bernoulli constant.The method is described and tested by calculation of the problem of potential two-layer flow above a curvilinear bottom in the class of soliton-type solutions. The particular case of this problem with a rectilinear. bottom has been studied by a number of authors.The proposed method allows one to obtain a numerical solution of the problem for a curvilinear bottom of an arbitrary shape, while the most effective method of calculating such flows [9] is applicable only for the particular case of a bottom irregularity in the shape of a circular half-cylinder.The accuracy of the method is tested by comparison of numerical and exact solutions of a certain model problem.Let us consider a two-dimensional problem of steady two-layer potential flow of a heavy fluid over an obstruction located on a horizontal straight bottom. The parameters for the lower layer are denoted by subscript 1, and those for the upper layer, by subscript 2. The flow dia3ram is shown in Fig. Soliton-type solutions symmetric about the vertical axis x = xo/2 (xo is the length of the obstruction) are sought for the free-surface and interface shapes.Let V + and V 1-denote the flow velocity along the inner interface with approach to this interface from above and from below, respectively.The condition of pressure continuity for passage through the interface, according to the Bernoulli equation, leads to the following relation for velocities on the sides of the interface:Here Pl and P2 are the fluid densities, Frl = Vool/vf~ is Froude's number for the lower layer, g is the acceleration of gravity, and H1 is the interface ordinate at infinity. The constant-pressure condition along the free surface is equivalent to the following law of velocity distribution along it:V~ 2 =l-Fr--~2 H2-1 ,

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“…1 and 2. We note that numerical studies of steady flow of a two-layer fluid over a bump or a step bounded by a free or rigid upper boundary were carried out by Forbes, 3 Belward and Forbes, 4 Sha and Vanden-Broeck, 5 and Moni and King, 6 among others, and an asymptotic approach for the case of a rigid upper boundary was developed without surface tension by Shen 7 on the basis of the FKdV theory, and with surface tension by Choi, Sun, and Shen,8 where a forced modified KdV equation ͑FMKdV͒ was obtained. The FKdV theory fails when the coefficient of the nonlinear term or that of the third derivative in the FKdV vanishes.…”

Section: Introductionmentioning

confidence: 99%

Internal capillary-gravity waves of a two-layer fluid with free surface over an obstruction - forced extended KdV equation

1997

International Journal of Multiphase Flow

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A Galerkin method to strongly nonlinear KdV equations and Schrödinger equations AIP Conf. Proc. 88, 319 (1982); 10.1063/1.33642 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation. In this paper we study steady capillary-gravity waves in a two-layer fluid bounded above by a free surface and below by a horizontal rigid boundary with a small obstruction. Two critical speeds for the waves are obtained. Near the smaller critical speed, the derivation of the usual forced KdV equation ͑FKdV͒ fails when the coefficient of the nonlinear term in the FKdV vanishes. To overcome this difficulty, a new equation, called a forced extended KdV equation ͑FEKdV͒ governing interfacial wave forms, is obtained by a refined asymptotic method. Various solutions and numerical results of this equation are presented.

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Two-layer critical flow over a semi-circular obstruction

Cited by 23 publications

References 16 publications

Analysis and Computation with Stratified Fluid Models

Analysis and Computation with Stratified Fluid Models

Two-layer gravity flow of a heavy fluid above a curvilinear bottom

Internal capillary-gravity waves of a two-layer fluid with free surface over an obstruction - forced extended KdV equation

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