A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes

The aim of this paper is to present a kinetic formulation of a model for the coupling of transient free surface and pressurised flows. Firstly, we revisit the system of Saint-Venant equations for free surface flow: we state some properties of Saint-Venant equations, we propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises (in some general case) an energy and preserves the still water steady state. Secondly, we propose a model for pressurised flows in a Saint-Venant-like conservative formulation. We then propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises in any case an energy and preserves the still water steady state. Finally, we propose a dual model that couples these two types of flow.     

Regularized shallow water equations and an efficient method for numerical simulation of shallow water flows

Regularized shallow water equations are derived as based on a regularization of the Navier-Stokes equations in the form of quasi-gasdynamic and quasi-hydrodynamic equations. Efficient finite-difference algorithms based on the regularized shallow water equations are proposed for the numerical simulation of shallow water flows. The capabilities of the model are examined by computing a test Riemann problem, the flow over an obstacle, and asymmetric dam break.     

正变坐标系下完全深度平均湍流方程组

本文针对宽浅型水域，对三维湍流时均方程组逐项进行深度平均，推导出包含自由水面和地形影响的深度平均流动控制方程组．本文还同时获得了深度平均形式的ｋ－ε湍流模型方程组．因计入了水流的三维效应，该模型称为完全深度平均模型．考虑到天然水域几何边界复杂，本文运用较简便的方法，将上述模型方程组交换至正交坐标系下．所得控制方程组可以直接运用于对实际问题的数值模拟．     

A robust central scheme for the shallow water flows with an abrupt topography based on modified hydrostatic reconstructions

We study a second-order central scheme for the shallow water flows with a discontinuous bottom topography based on modified hydrostatic reconstructions (HRs). The first HR scheme was proposed in Audusse et al, which may be missing the effect of the large discontinuous bottom topography. We introduce a modified HR method to cope with this numerical difficulty. The new scheme is well-balanced for still water solutions and can guarantee the positivity of the water depth. Finally, several numerical results of classical problems of the shallow water equations confirmed these properties of the new scheme. Especially, the new scheme yields superior results for the shallow water downhill flow over a step.     

A new symmetric interior penalty discontinuous Galerkin formulation for the Serre–Green–Naghdi equations

In this work, we investigate the construction of a new discontinuous Galerkin discrete formulation to approximate the solution of Serre–Green–Naghdi (SGN) equations in the one-dimensional horizontal framework. Such equations describe the time evolution of shallow water free surface flows in the fully nonlinear and weakly dispersive asymptotic approximation regime. A new non-conforming discrete formulation belonging to the family of symmetric interior penalty discontinuous Galerkin methods is introduced to accurately approximate the solutions of the second order elliptic operator occurring in the SGN equations. We show that the corresponding discrete bilinear form enjoys some consistency and coercivity properties, thus ensuring that the corresponding discrete problem is well-posed. The resulting global discrete formulation is then validated through an extended set of benchmarks, including convergence studies and comparisons with data taken from experiments.     

New Exact Solutions of the Navier–Stokes Equations for Axisymmetric Automodel Flows of Fluids

We investigate self-similar solutions of the Navier–Stokes equations for the axisymmetric flow of a viscous incompressible fluid. The original equations are transformed by the Slezkin method. On the basis of analysis of physical properties of the flow and the Slezkin general equation, we show that, in parallel with the known solutions of this equation, there exist several other solutions with physical meaning. We consider the simplest case of irrotational flows for which current lines may be circles, ellipses, parabolas, and hyperbolas. Unlike the Landau and Squire solutions, these flows are interpreted as nonjet flows of fluid flowing into and out of a homogeneous porous axially symmetric body.     

A mathematical model for unsteady mixed flows in closed water pipes

We present the formal derivation of a new unidirectional model for unsteady mixed flows in nonuniform closed water pipes.In the case of free surface incompressible flows,the FS-model is formally obtained,using formal asymptotic analysis,which is an extension to more classical shallow water models.In the same way,when the pipe is full,we propose the P-model,which describes the evolution of a compressible inviscid flow,close to gas dynamics equations in a nozzle.In order to cope with the transition between a free surface state and a pressured(i.e.,compressible) state,we propose a mixed model,the PFS-model,taking into account changes of section and slope variation.     

A new composite scheme for two-layer shallow water flows with shocks

This paper is devoted to solve the system of partial differential equations governing the flow of two superposed immiscible layers of shallow water flows. The system contains source terms due to bottom topography, wind stresses, and nonconservative products describing momentum exchange between the layers. The presence of these terms in the flow model forms a nonconservative system which is only conditionally hyperbolic. In addition, two-layer shallow water flows are often accompanied with moving discontinuities and shocks. Developing stable numerical methods for this class of problems presents a challenge in the field of computational hydraulics. To overcome these difficulties, a new composite scheme is proposed. The scheme consists of a time-splitting operator where in the first step the homogeneous system of the governing equations is solved using an approximate Riemann solver. In the second step a finite volume method is used to update the solution. To remove the non-physical oscillations in the vicinity of shocks a nonlinear filter is applied. The method is well-balanced, non-oscillatory and it is suitable for both low and high values of the density ratio between the two layers. Several standard test examples for two-layer shallow water flows are used to verify high accuracy and good resolution properties for smooth and discontinuous solutions.     

Two-dimensional numerical modelling of shallow water flows over multilayer movable beds

The two-dimensional modelling of shallow water flows over multi-sediment erodible beds is presented. A novel approach is developed for the treatment of multiple sediment types in morphodynamics. The governing equations include the two-dimensional shallow water equations for hydrodynamics, an Exner-type equation for morphodynamics, a two-dimensional transport equation for the suspended sediments, and a set of empirical equations for entrainment and deposition. Multilayer sedimentary beds are formed of different erodible soils with sediment properties and new exchange conditions between the bed layers are developed for the model. The coupled equations yield a hyperbolic system of balance laws with source terms. As a numerical solver for the system, we implement a fast finite volume characteristics method. The numerical fluxes are reconstructed using the method of characteristics which employs projection techniques. The proposed finite volume solver is simple to implement, satisfies the conservation property and can be used for two-dimensional sediment transport problems in non-homogeneous isotropic beds without need of complicated three-dimensional equations. To assess the performance of the proposed models, we present numerical results for a wide variety of shallow water flows over sedimentary layers. Comparisons to experimental data for dam-break problems over movable beds are also included in this study.     

A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows

We introduce a new model for shallow water flows with non-flat bottom. A prototype is the Saint Venant equation for rivers and coastal areas, which is valid for small slopes. An improved model, due to Savage–Hutter, is valid for small slope variations. We introduce a new model which relaxes all restrictions on the topography. Moreover it satisfies the properties (i) to provide an energy dissipation inequality, (ii) to be an exact hydrostatic solution of Euler equations. The difficulty we overcome here is the normal dependence of the velocity field, that we are able to establish exactly. Applications we have in mind concern, in particular, computational aspects of flows of granular material (for example in debris avalanches) where such models are especially relevant. To cite this article: F. Bouchut et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Hydraulic Theory and Particle-Driven Gravity Currents

Hydraulic theory, as it has been applied to compositionally driven gravity flows, involves the single simplifying assumption that the pressure in the fluid is hydrostatic [1]. This assumption provides, as a consequence, a depth independent horizontal velocity field. This approach has led to a greatly increased understanding of many of the phenomena associated with these complex flows, including issues surrounding internal hydraulic jumps and energy loss [2]. Recently, investigations into flow and deposition of particles from particle-driven gravity currents have been carried out using an approach that employs the hydraulic theory that had proved so successful in the case of homogeneous flows [3]–[7]. Unfortunately, as we show here, there is a fundamental contradiction in adopting this simplifying assumption when particles drive the flow. This contradiction is essentially that one cannot have a hydrostatic pressure that arises from the presence of particles while at the same time maintaining a depth-independent horizontal velocity field, as was assumed in references [3]–[7].     

A gas-kinetic scheme for shallow-water equations with source terms

In this paper, the Kinetic Flux Vector Splitting (KFVS) scheme is extended to solving the shallow water equations with source terms. To develop a well-balanced scheme between the source term and the flow convection, the source term effect is accounted in the flux evaluation across cell interfaces. This leads to a modified gas-kinetic scheme with particular application to the shallow water equations with bottom topography. Numerical experiments show better resolution of the unsteady solution than conventional finite difference method and KFVS method with little additional cost. Moreover, some positivity properties of the gas-kinetic scheme is established.     

Three-Dimensional Nonlinear Dispersive Waves on Shear Flows

The Green–Naghdi equations describing three-dimensional water waves are considered. Assuming that transverse variations of the flow occur at a much shorter lengthscale than variations along the wave propagation direction, we derive simplified asymptotic equations from the Green–Naghdi model. For steady flows, we show that the approximate model reduces to a one-dimensional Hamiltonian system along each stream line. Exact solutions describing a wide class of free-boundary flows depending on several arbitrary functions of one argument are found. The numerical results showing different patterns of steady three-dimensional waves are presented.     

A discontinuous Galerkin method for a new class of Green–Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies.     

Three‐layer flows in the shallow water limit

We formulate and discuss the shallow water limit dynamics of the layered flow with three layers of immiscible fluids of different densities bounded above and below by horizontal walls. We obtain a resulting system of four equations, which may be nonlocal in the non‐Boussinesq case. We provide a systematic way to pass to the Boussinesq limit, and then study those equations, which are first‐order PDEs of mixed type, more carefully. We show that in a symmetric case the solutions remain on an invariant surface and using simple waves we illustrate that this is not the case for nonsymmetric cases. Reduced models consisting of systems of two equations are also proposed and compared to the full system.     

On free surfaces in hydrostatic flows

In the following note we investigate the occurrence of non-trivial free-surfaces for hydrostatic flows. We show that, for travelling-wave solutions of the full Euler equations where we invoke the hydrostatic approximation, no non-flat travelling wave flows occur unless the resulting flow is outside the domain of current local or global existence results for travelling water waves.     

Relaxation approximation to bed-load sediment transport

In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit–explicit Runge–Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems.     

Three‐Dimensional Surface Water Waves Governed by the Forced Benney–Luke Equation

This paper studies the propagation of three‐dimensional surface waves in water with an ambient current over a varying bathymetry. When the ambient flow is near the critical speed, under the shallow water assumptions, a forced Benney–Luke (fBL) equation is derived from the Euler equations. An asymptotic approximation of the water's reaction force over the varying bathymetry is derived in terms of topographic stress. Numerical simulations of the fBL equation over a trough are compared to those using a forced Kadomtsev–Petviashvilli equation. For larger variations in the bathymetry that upstream‐radiating three‐dimensional solitons are observed, which are different from the upstream‐radiating solitons simulated by the forced Kadomtsev–Petviashvilli equation. In this case, we show the fBL equation is a singular perturbation of the forced Kadomtsev–Petviashvilli equation which explains the significant differences between the two flows.     

Finite-difference method for the system of equations of tide dynamics

For the system of shallow water equations describing the tidal wave propagation, we construct a finite-difference scheme on an unstructured grid. We analyze the properties of the resulting system of equations; in particular, we show that, after the elimination of part of the variables, the matrix of the system becomes an M-matrix.     

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.