Abstract We studied the shallow water equations of nonlinear conservation laws. First we studied the parametrisation of nonlinear elementary waves and hence we present the solution to the Riemann problem. We also prove the uniqueness of the Riemann solution. The Riemann invariants are formulated. Moreover we give an interesting application of the Riemann invariants. We present the shallow water system in a diagonal form, which admits the existence of a global smooth solution for these equations. The other application is to introduce new conservation laws for the shallow water equations.

On the Shallow Water Equations

Propagation of a strong incident shock through a bed of particles results in complex wave dynamics such as a reflected shock, a transmitted shock, and highly unsteady flow inside the particle bed. In this paper we present three-dimensional numerical simulations of shock propagation in air over a random bed of particles. We assume the flow is inviscid and governed by the Euler equations of gas dynamics. Simulations are carried out by varying the volume fraction of the particle bed at a fixed shock Mach number. We compute the unsteady inviscid streamwise and transverse drag coefficients as a function of time for each particle in the random bed for different volume fractions. We show that (i) there are significant variations in the peak drag for the particles in the bed, (ii) the mean peak drag as a function of streamwise distance through the bed decreases with a slope that increases as the volume fraction increases, and (iii) the deviation from the mean peak drag does not correlate with local volume fraction. We also present the local Mach number and pressure contours for the different volume fractions to explain the various observed complex physical mechanisms occurring during the shock–particle interactions. Since the shock interaction with the random bed of particles leads to transmitted and reflected waves, we compute the average flow properties to characterize the strength of the transmitted and reflected shock waves and quantify the energy dissipation inside the particle bed. Finally, to better understand the complex wave dynamics in a random bed, we consider a simpler approximation of a planar shock propagating in a duct with a sudden area change. We obtain Riemann solutions to this problem, which are used to compare with fully resolved numerical simulations.

Propagation of a Strong Shock Over a Random Bed of Spherical Particles

ECOGEN: An open-source tool for multiphase, compressible, multiphysics flows

On the wave interactions in the drift-flux equations of two-phase flows

 We determine completely the exact Riemann solutions for the shallow water equations with a bottom step including the dry bed problem The nonstrict hyperbolicity of this first order system of partial differential equations leads to resonant waves and non unique solutions To address these difficulties we construct the L–M and R–M curves in the state space For the bottom step elevated from left to right, we classify the L–M curve into five different cases and the R–M curve into two different cases based on the subcritical and supercritical Froude number of the Riemann initial data as well as the jump of the bottom step The behaviors of all basic cases of the L–M and R–M curves are fully analyzed We observe that the non–uniqueness of the Riemann solutions is due to bifurcations on the L–M or R–M curves The possible Riemann solutions include classical waves and resonant waves as well as dry bed solutions that are solved in a uniform framework for any given initial data

/pdf/exact-riemann-solutions-to-shallow-water-equations-qakdx7316f.pdf

Exact Riemann solutions to shallow water equations

We determine completely the exact Riemann solutions for the system of Euler equations in a duct with discontinuous varying cross-section. The crucial point in solving the Riemann problem for hyperbolic system is the construction of the wave curves. To address the difficulty in the construction due to the nonstrict hyperbolicity of the underlying system, we introduce the L-M and R-M curves in the velocity-pressure phase plane. The behaviors of the L-M and R-M curves for six basic cases are fully analyzed. Furthermore, we observe that in certain cases the L-M and R-M curves contain the bifurcation which leads to the nonuniqueness of the Riemann solutions. Nevertheless, all possible Riemann solutions including classical as well as resonant solutions are solved in a uniform framework for any given initial data.

/pdf/exact-riemann-solutions-to-compressible-euler-equations-in-12knzzt25q.pdf

Exact riemann solutions to compressible euler equations in ducts with discontinuous cross-section

Efficient and robust relaxation procedures for multi-component mixtures including phase transition

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Exact Riemann solutions to shallow water equations

Exact riemann solutions to compressible euler equations in ducts with discontinuous cross-section

Efficient and robust relaxation procedures for multi-component mixtures including phase transition