Interaction of a Singular Surface with a Characteristic Shock in a Relaxing Gas with Dust Particles

A system of hyperbolic differential equations outlining one-dimensional planar, cylindrical symmetric and spherical symmetric flow of a relaxing gas with dust particles is considered. Singular surface theory used to study different aspects of wave propagation and its culmination to the steepened form. The evolutionary behavior of the characteristic shock is studied. A particular solution of the governing system of equations is used to discuss the steepened wave form, characteristic shock and their interaction. The results of the interaction between the steepened wave front and the characteristic shock using the general theory of wave interaction are discussed. Also, the influence of relaxation and dust parameters on the steepened wave front, the formation of a characteristic shock, reflected and transmitted waves after interaction and a jump in shock acceleration are investigated.

Riemann solution for a class of morphodynamic shallow water dam-break problems

This paper investigates a family of dam-break problems over an erodible bed. The hydrodynamics is described by the shallow water equations, and the bed change by a sediment-conservation equation, coupled to the hydrodynamics by a sediment transport (bed-load) law. When the initial states $\boldsymbol{U}_{l}$ and $\boldsymbol{U}_{r}$ are sufficiently close to each other the resulting solutions are consistent with the theory proposed by Lax (Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, 1973, SIAM), that for a Riemann problem of $n$ equations there are $n$ waves associated with the $n$ characteristic families. However, for wet–dry dam-break problems over a mobile bed, there are three governing equations, but only two waves. One wave vanishes because of the presence of the dry bed. When initial left and right bed levels ($B_{l}$ and $B_{r}$) are far apart, it is shown that a semi-characteristic shock may occur, which happens because, unlike in shallow water flow on a fixed bed, the flux function is non-convex. In these circumstances it is shown that it is necessary to reconsider the usual shock conditions. Instead, we propose an implied internal shock structure the concept of which originates from the fact that the stationary shock over a fixed-bed discontinuity can be regarded as a limiting case of flow over a sloping fixed bed. The Needham & Hey (Phil. Trans. R. Soc. Lond. A, vol. 334, 1991, pp. 25–53) approximation for the ambiguous integral term $\int \!h\,\text{d}B$ in the shock condition is improved based on this internal shock structure, such that mathematically valid solutions that incorporate a morphodynamic semi-characteristic shock are arrived at.