Propagation of discontinuouswaves along a dry bed

The possibility of simulating the process of propagation of discontinuous waves along a dry bed on the basis of the equations of the first approximation of shallow water theory is studied. It is shown that the consistent losses of the free-stream total momentum and energy can be found from the mass conservation law within the framework of the shallow water equations. As an example, solutions of the problem of dam break with a dry bed in the lower pool and the problem of impingement of a discontinuous wave on a coastal shelf are constructed. These exact solutions are compared with the results of laboratory experiments.     

Discontinuous solutions of the shallow water equations on a rotatable attracting sphere

The shallow water equations on a rotatable attracting sphere represent a system of hyperbolic equations on a compact manifold. These equations are derived in a spherical coordinate system from the integral laws of mass and total momentum conservation with account for the Coriolis and centrifugal forces. An analysis of the stability of discontinuous solutions with discontinuous waves and contact discontinuities is made using the closing law of total energy conservation, which represents a convex extension of the basic conservation-law system. The classes of stationary, one-dimensional (latitude-dependent only) exact solutions with contact discontinuities and discontinuous waves are constructed. Within the framework of the one-dimensional equations the test problem of wave flows resulting from the simultaneous break of two dams confining a fluid at rest in the vicinities of the poles is numerically modeled.     

Open-channel waves generated by propagation of a discontinuous wave over a bottom step

This paper presents the results of theoretical and experimental studies of open-channel waves generated by the propagation of a discontinuous dam-break wave over a bottom step. The cases where the initial tailwater level is higher than the step height (the step is under water) and where this value is smaller than the step height (at the initial time, water is absent on the step) are considered. Exact solutions are constructed using modified first-approximation equations of shallow-water theory, which admit the propagation of discontinuous waves in a dry channel. On the stationary hydraulic jump formed above the bottom step, the total free-stream energy is assumed to be conserved. These solutions agree with experimental data on various parameters (types of waves, wave propagation velocity, asymptotic depths behind the wave fronts). __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 31–44, January–February, 2008.     

Another Proof of Stability for Global Weak Solutions of 2D Degenerated Shallow Water Models

We consider non-linear viscous shallow water models with varying topography, extra friction terms and capillary effects, in a two-dimensional framework. Water-depth dependent laminar and turbulent friction coefficients issued from an asymptotic analysis of the three-dimensional free-surface Navier–Stokes equations are considered here. A new proof of stability for global weak solutions is given in periodic domain Ω = T², adapting the method introduced by J. Simon in [15] for the non-homogeneous Navier–Stokes equations. Existence results for such solutions can be obtained from this stability analysis.     

Numerical simulation of wave flows caused by a shoreside landslide

A mathematical model is developed for formation and propagation of discontinuous waves caused by sliding of a shoreside landslide into water. The model is based on the equations of a two-layer “shallow liquid” with specially introduced “dry friction” in the low layer, which allows one to describe the joint motion of the landslide and water. An explicit difference scheme approximating these equations is constructed, and it is used to develop a numerical algorithm for simulating the motion of the free boundaries of both the landslide and water (in particular, the propagation of a water wave along a dry channel, incidence of the wave on the lakeside, and flow over obstacles). Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 109–117, July–August, 1999.     

Asymptotic models of internal stationary waves

Equations of stationary long waves on the interface between a homogeneous fluid and an exponentially stratified fluid are considered. An equation of the second-order approximation of the shallow water theory inheriting the dispersion properties of the full Euler equations is used as the basic model. A family of asymptotic submodels is constructed, which describe three different types of bifurcation of solitary waves at the boundary points of the continuous spectrum of the linearized problem. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 151–161, July–August, 2008.     

Convergence acceleration of segregated algorithms using dynamic tuning additive correction multigrid strategy

A convergence acceleration method based on an additive correction multigrid–SIMPLEC (ACM‐S) algorithm with dynamic tuning of the relaxation factors is presented. In the ACM‐S method, the coarse grid velocity correction components obtained from the mass conservation (velocity potential) correction equation are included into the fine grid momentum equations before the coarse grid momentum correction equations are formed using the additive correction methodology. Therefore, the coupling between the momentum and mass conservation equations is obtained on the coarse grid, while maintaining the segregated structure of the single grid algorithm. This allows the use of the same solver (smoother) on the coarse grid. For turbulent flows with heat transfer, additional scalar equations are solved outside of the momentum–mass conservation equations loop. The convergence of the additional scalar equations is accelerated using a dynamic tuning of the relaxation factors. Both a relative error (RE) scheme and a local Reynolds/Peclet (ER/P) relaxation scheme methods are used. These methodologies are tested for laminar isothermal flows and turbulent flows with heat transfer over geometrically complex two‐ and three‐dimensional configurations. Savings up to 57% in CPU time are obtained for complex geometric domains representative of practical engineering problems. Copyright © 1999 John Wiley & Sons, Ltd.     

A coupled lattice Boltzmann model for the shallow water‐contamination system

A coupled numerical method for the direct simulation of shallow water dynamics and pollutant transport is formulated and implemented. The conservation equations of shallow water dynamics equations and the convection–diffusion equations are solved using the lattice Boltzmann (LB) method. The local equilibrium distribution of the pollutant has no terms of second order in flow velocity. And the relaxation time of the pollutant deviates from a constant for the flows with variable free surface water depth. The numerical tests show that this scheme strictly obeys the conservation law of mass and momentum. Excellent agreement is obtained between numerical predictions and analytical solutions in the pure diffusion problem and convection–diffusion problem. Furthermore, the influences on the accuracy of the lattice size and the diffusivity are also studied. The results indicate that the variation in the free surface water depth cannot affect the conservation of the model, and the model has the ability to simulate the complex topography problem. The comparison shows that the LB scheme has the capacity to solve the complex convection–diffusion problem in shallow water. Copyright © 2008 John Wiley & Sons, Ltd.     

Waves ahead of a vertical plate in a channel

The behavior of disturbances propagating with supercritical speed ahead of a plate in a channel is analyzed on the basis of the experimental results obtained by the authors and data taken from the literature. In particular, the transition from smooth to breaking waves has been found to occur at higher propagation speeds than follows from the first approximation of shallow water theory. It has also been found that for waves widely encountered in practice the value of the propagation speed agrees well with the limiting propagation speed of solitary waves obtained on the basis of the complete equations of potential fluid flow. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 82–90, January–February, 1999. The study was carried out with the support of the Russian Foundation for Basic Research (project No. 95-01-01164) and by the Integration Program of the Siberian Branch of the Russian Academy of Sciences under grant No. 97-43.     

Semi-analytic solution of the two-dimensional turbulent energy equation in round tubes using an analytic velocity profile and its experimental validation

 A semi-analytic solution of the temperature development of single-phase, turbulent viscous flows inside smooth round tubes is performed. The special feature of the theoretical analysis revolves around two single universal functions of analytic form for the accurate characterization of the turbulent diffusivity of momentum and the turbulent velocity profile in the entire cross-section of a round tube. Using this valuable information that emanates from the analytic solution of the one-dimensional momentum balance equation, the two-dimensional energy balance equation was reformulated into an adjoint system of ordinary differential equations of first–order with constant coefficients. Each equation in the system of differential equations governs the axial variation of the average temperature of a finite volume of fluid of annular shape. Exploiting the linearity of the system of differential equations, an analytic solution of it was obtained via the matrix eigenvalue method with LAPACK, a library of Fortran 77 subroutines for numerical linear algebra. Reliable series have been determined for the axial variation of the two thermal quantities of importance: (a) the time-mean bulk temperature and (b) the local Nusselt number. The semi-analytic nature of the local Nusselt number distribution is advantageous because it may be viewed as an analytic-based correlation equation. Prediction of the local Nusselt numbers for turbulent air flows compare satisfactorily with the comprehensive correlation equations and the abundant experimental data that are accessible from the literature. The air flows are regulated by a wide spectrum of turbulent Reynolds numbers. Received on 4 June 2001 RID="★" ID="★" Current address Mechanical Engineering Dept. The University of Vermont Burlington, VT 05405, USA     

Stability of roll waves in open channel flowsStabilité de trains d'ondes dans un canal découvert

The stability of finite amplitude roll waves that may develop at a liquid free surface in inclined open channels of arbitrary cross-section is studied. In the framework of shallow water theory with turbulent friction the modulation equations for wave series are derived and a nonlinear stability criterion is obtained. To cite this article: A. Boudlal, V.Yu. Liapidevskii, C. R. Mecanique 330 (2002) 291–295.     

Influence of a strong discontinuity on the propagation of second order discontinuities

We study the propagation of second order weak discontinuities in quasi-linear hyperbolic systems of equations with discontinuous coefficients. The general theory is applied to shallow water waves.     

Nonlinear excitations and ＂peakons＂ of a （2＋1）-dimensional generalized Broer-Kaup system

Shallow water waves and a host of long wave phenomena are commonly investigated by various models of nonlinear evolution equations. Examples include the Korteweg–de Vries, the Camassa–Holm, and the Whitham–Broer–Kaup (WBK) equations. Here a generalized WBK system is studied via the multi-linear variable separation approach. A special class of wave profiles with discontinuous derivatives (“peakons”) is developed. Peakons of various features, e.g. periodic, pulsating or fractal, are investigated and interactions of such entities are studied. The project supported by the National Natural Science Foundation of China (10475055, 10547124 and 90503006), and the Hong Kong Research Grant Council Contract HKU 7123/05E.     

Evolution of the turbulent wakes of spherical bodies in stably stratified media

A turbulent wake model, based on the Reynolds, energy and turbulence dissipation equations together with the closing relations for the turbulent transport coefficients, is proposed. A comparative investigation of swirled momentumless wakes with zero and nonzero angular momentum is carried out. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 13–22, January–February, 1994.     

The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin–Bona–Mahoney and Korteweg–de Vries equations. In particular, they accommodate wave breaking phenomena.     

An approximate‐state Riemann solver for the two‐dimensional shallow water equations with porosity

PorAS, a new approximate‐state Riemann solver, is proposed for hyperbolic systems of conservation laws with source terms and porosity. The use of porosity enables a simple representation of urban floodplains by taking into account the global reduction in the exchange sections and storage. The introduction of the porosity coefficient induces modified expressions for the fluxes and source terms in the continuity and momentum equations. The solution is considered to be made of rarefaction waves and is determined using the Riemann invariants. To allow a direct computation of the flux through the computational cells interfaces, the Riemann invariants are expressed as functions of the flux vector. The application of the PorAS solver to the shallow water equations is presented and several computational examples are given for a comparison with the HLLC solver. Copyright © 2009 John Wiley & Sons, Ltd.     

An adaptive ALE method for underwater explosion simulations including cavitation

The aim of this paper is to develop a numerical procedure for simulating a simplified mathematical model of underwater explosion phenomena. The Euler set of equations is selected as the governing equations and the ideal gas and Tammann equations of state (EOS) are used to obtain pressure in the gas bubble and the surrounding water zone, respectively. The modified Schmidt EOS is used to simulate the cavitation regions. An arbitrary Lagrangian–Eulerian method is used to integrate the governing equations over an unstructured moving grid. A mesh adapting technique is applied to increase the accuracy as well as for better capturing of flow physics. Moreover, a least-square smoother is employed to moderate the undesirable effects of gas–water interface irregularities. The numerical results verify that the proposed method is capable of predicting complex physics involved in a spherical underwater explosion. The method also shows a very good performance in smoothing the interface while minimizing the loss of mass and momentum in two-dimensional problems.     

Large eddy simulation of vertical turbulent jets under JONSWAP waves

The effect of random waves on vertical plane turbulent jets is studied numerically and the mechanism behind the interaction of the jet and waves is analyzed. The large eddy simulation method is used and the σ-coordinate system is adopted. Turbulence is modeled by a dynamic coherent eddy model. The σ-coordinate transformation is introduced to map the irregular physical domain with a wavy free surface and an uneven bottom onto a regular computational domain. The fractional step method is used to solve the fil...     

Unstructured finite volume discretization of two‐dimensional depth‐averaged shallow water equations with porosity

This paper deals with the numerical discretization of two‐dimensional depth‐averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small‐scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small‐scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two‐dimensional shallow water equations with porosity, both of them are high‐order schemes. The numerical schemes proposed are well‐balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high‐order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed. Copyright © 2009 John Wiley & Sons, Ltd.