Viscous Shallow Water Equations Research Articles

Steady solutions to viscous shallow water equations. The case of heavy water.

In this note, we show the existence of regular solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we are able to construct a solution, provided the total mass is sufficiently large. The main mathematical part is located in the construction of solutions. Uniqueness is impossible to obtain, since the gradient of the velocity is of magnitude of the force. The investigation is connected to the corresponding singular limit as Mach number goes to zero and methods for weak solutions to the compressible Navier-Stokes system.

Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications

This paper provides the full proof of the results announced by the authors in [C. R. Acad. Sciences (2016)]. We introduce an original relative entropy for compressible Navier-Stokes equations with density dependent viscosities and discuss some possible applications such as inviscid limit or low Mach number limit. We first consider the case µ() = µµ and λ() = 0 and a pressure law under the form p() = aa γ with γ > 1, which corresponds in particular to the formulation of the viscous shallow water equations. We present some mathematical results related to the weak-strong uniqueness, the convergence to a dissipative solution of compressible or incompressible Euler equations. Moreover, we show the convergence of the viscous shallow water equations to the inviscid shallow water equations in the vanishing viscosity limit and further prove convergence to the incompressible Euler system in the low Mach limit. This extends results with constant viscosities recently initiated by E. Feireisl, B.

Note on the stability of viscous roll waves

In this note, we announce a complete classification of the stability of periodic roll-wave solutions of the viscous shallow water equations, from their onset at Froude number F≈2 up to the infinite Froude limit. For intermediate Froude numbers, we obtain numerically a particularly simple power-law relation between F and the boundaries of the region of stable periods, which appears potentially useful in hydraulic engineering applications. In the asymptotic regime F→2 (onset), we provide an analytic expression of the stability boundaries, whereas in the limit F→∞, we show that roll waves are always unstable.

On the Stability of Weak Solution for Compressible Primitive Equations

In this paper, we study a compressible Primitive Equations (CPEs) of the ocean in the two dimensional space, with horizontal periodic and vertical mixed boundary conditions. Thanks to an effective change of variables, we obtain a new CPEs model, which is similar as viscous shallow water equation. Using a new entropy estimate, we prove the stability of weak solutions for this new two dimensional CPEs model.

A Galerkin method with spherical splines for the shallow water equations on a sphere: error analysis

We analyse a Galerkin method for the viscous shallow water equations on the unit sphere using spherical splines. A priori error estimates for semi-discrete and fully discrete schemes are derived. In particular, the proposed fully discrete scheme has second order accuracy in time. Numerical experiments support our theoretical results.

A tsunami simulation of Hakata Bay using the viscous shallow-water equations

The tsunami generated by the Great East Japan Earthquake caused serious damage to the coastal areas of the Tohoku district. Numerical simulations are used to predict damage caused by tsunamis. Shallow-water equations are generally used in numerical simulations of tsunami propagation from the open sea to the coast. This research focuses on viscous shallow-water equations and attempts to generate a computational method using finite element techniques based on the previous investigations of Kanayama and Ohtsuka (Coast Eng Jpn 21:157–171, 1978). First, the viscous shallow-water equation system is derived from the Navier–Stokes equations, based on the assumption of hydrostatic pressure in the direction of gravity. Next a numerical scheme is shown. Finally, tsunami simulations of Hakata Bay and Tohoku-Oki are shown using the approach.

A streamline diffusion finite element method for the viscous shallow water equations

We formulate and prove a priori error bounds for the streamline diffusion finite element method or Streamline Upwind Petrov–Galerkin (SUPG) method applied to the shallow water equations. We derive an error estimate for linear approximations in both velocity and water elevation and comment on higher-order approximations.

Optimal Control of a Viscous Shallow Water Equation

This paper is concerned with a viscous shallow water equation, which includes both the viscous Camassa-Holm equation and the viscous Degasperis-Procesi equation as its special cases. The optimal control under boundary conditions is given, and the existence of optimal solution to the equation is proved.

Phragmén–Lindelöf alternative results for the shallow water equations for transient compressible viscous flow

In this paper, the spatial behavior of the viscous shallow water equations in a two-dimensional channel domain is studied. A differential inequality for suitable energy associated with the solutions of the viscous shallow water equations in a semi-infinite channel is derived, from which we show the Phragmén–Lindelöf alternative results. The main tool used is the energy method. Our results can be viewed as a version of Saint-Venant’s principle to the transient compressible viscous flow.

Finite Element Analysis of Tsunami by Viscous Shallow-Water Equations

In this paper, viscous shallow-water equations are derived from three-dimensional Navier-Stokes equations under the hydrostatic assumption. The viscous shallow-water equations are approximated by the finite element method based on our numerical scheme developed in 1978. This approach is used to simulate a tsunami in Hakata Bay. Results show a reasonable estimate of the tsunami arrival time.

Special Issue on Modeling and Simulation

Special Issue on Modeling and Simulation

Shallow water equations: viscous solutions and inviscid limit

We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C2 test-functions, are confined in a compact set in H−1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.

NM-JP-1 Tsunami simulation of Hakata bay using the viscous shallow-water equations

NM-JP-1 Tsunami simulation of Hakata bay using the viscous shallow-water equations

Simulation of the spread of a viscous fluid using a bidimensional shallow water model

In this paper we propose a numerical method to solve the Cauchy problem based on the viscous shallow water equations in an horizontally moving domain. More precisely, we are interested in a flooding and drying model, used to modelize the overflow of a river or the intrusion of a tsunami on ground. We use a nonconservative form of the two-dimensional shallow water equations, in eight velocity formulation and we build a numerical approximation, based on the Arbitrary lagrangian eulerian formulation, in order to compute the solution in the moving domain.

K12 2層粘性浅水波方程式の有限要素法による数値解析(Kl計算力学)

K12 2層粘性浅水波方程式の有限要素法による数値解析(Kl計算力学)

Nonsimilar solutions of the viscous shallow water equations governing weak hydraulic jumps

The steady viscous shallow water equations are often used for the study of hydraulic jumps. We cast these as a single parametric ordinary differential equation with global continuity as a constraint. The solution provides both the local velocity profile and the downstream evolution of the film height. Moreover this is an exact approach, in contrast with existing approaches which encounter a closure problem and need modeling. There is only one solution which is supercritical initially. This shows a jumplike behavior at a Froude number close to unity, in consonance with predictions of inviscid theory. At low Froude number, it is shown that two solutions are possible, one with a separated profile and one without. Flow downstream of a real hydraulic jump must switch to the second solution, calling into question the validity of the shallow water approach in resolving the region of the switch. A series solution of the velocity profile shows that the first correction to a streamwise-varying parabolic profile is a quartic term. Circular and planar solutions are qualitatively similar.

Mathematical derivation of viscous shallow-water equations with zero surface tension

The purpose of this paper is to derive rigorously the so called viscous shallow water equations given for instance page 958-959 in [A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys, 69 (1997), 931–980]. Such a system of equations is similar to compressible Navier-Stokes equations for a barotropic fluid with a non-constant viscosity. To do that, we consider a layer of incompressible and Newtonian fluid which is relatively thin, assuming no surface tension at the free surface. The motion of the fluid is described by 3d Navier-Stokes equations with constant viscosity and free surface. We prove that for a set of suitable initial data (asymptotically close to “shallow water initial data”), the Cauchy problem for these equations is well-posed, and the solution converges to the solution of viscous shallow water equations. More precisely, we build the solution of the full problem as a perturbation of the strong solution to the viscous shallow water equations. The method of proof is based on a Lagrangian change of variable that fixes the fluid domain and we have to prove the well-posedness in thin domains: we have to pay a special attention to constants in classical Sobolev inequalities and regularity in Stokes problem.

Cauchy problem for viscous shallow water equations with surface tension

We are concerned with the Cauchy problem for a viscous shallow water system with a third-order surface-tension term. The global existence and uniqueness of the solution in the space of Besov type is shown for the initial data close to a constant equilibrium state away from the vacuum by using the Friedrich's regularization and compactness arguments.

二層粘性浅水波方程式と保存則(流体工学,流体機械)

In our previous papers, the two-layer viscous shallow-water equations were derived from the three-dimensional Navier-Stokes equations under the hydrostatic assumption. Also, it was noted that the combination of upper and lower equations in the two-layer model produces the classical one-layer equations if the density of each layer is the same. Then, the two-layer equations were approximated by a finite element method which followed our numerical scheme established for the one-layer model in 1978. Also, it was numerically demonstrated that the interfacial instability generated when the densities are the same can be eliminated by providing a sufficient density difference. In this paper, we newly show that conservation laws are still valid in the two-layer model. Also, we show results of a new physical experiment for the interfacial instability.

Schwarz Waveform Relaxation Algorithms for the Linear Viscous Equatorial Shallow Water Equations

We are interested in numerically solving the viscous shallow water equations with a small Coriolis force on a large domain. Specifically, we develop and analyze Schwarz waveform relaxation algorithms: we split the domain of computation into two subdomains and with appropriate transmission conditions an iterative procedure leads to the global solution. We first analyze the Dirichlet-type transmission conditions—computing the convergence rate in Fourier–Laplace variables and proving it is less than 1 yields the convergence of the algorithm. The algorithm requires an overlap between subdomains, and the convergence is slow. We propose a better algorithm for which the overlap is unnecessary: the transmission conditions are now an approximation of the absorbing boundary conditions. We also prove convergence; if the domains overlap, the arguments are similar to the Dirichlet-type problem, if not, we use variational arguments. A numerical scheme is then proposed and numerical results are shown which highlight the...