Solution of Riemann problem for dusty gas flow

In the present paper, the exact solution of quasilinear hyperbolic system of equations governing the propagation of weak shock waves in a one-dimensional dusty adiabatic gas flow with generalized geometries is derived. Here, the density behind the shock front is assumed to vary according to a power law of the distance. An analytical expression for the total energy carried by weak shock wave in dusty gas is also derived.

The main aim of this paper is, to obtain the analytical solution of the Riemann problem for a quasi-linear system of equations, which describe the one-dimensional unsteady flow of an ideal polytropic dusty gas in magnetogasdynamics without any restriction on the initial data. By using the Rankine-Hugoniot (R-H) and Lax conditions, the explicit expressions of elementary wave solutions (i. e., shock waves, simple waves and contact discontinuities) are derived. In the flow field, the velocity and density distributions for the compressive and rarefaction waves are discussed and shown graphically. It is also shown how the presence of small solid particles and magnetic field affect the velocity and density across the elementary waves. It is an interesting fact about this study that the results obtained for the Riemann problem are in closed form.

The solution for the flow of an incompressible viscous dusty gas, induced by two infinitely extended parallel plates when the lower plate is at rest and the upper plate begins to oscillate harmonically in its own plane, is obtained, It is found that (i) with the increase in the mass concentration both the velocities of the dusty gas and the particle decreases, (ii) the velocity of the dusty gas increases and that of the particle decreases with the increase in neta.

The Riemann problem for the Umami Chaplygin Aw-Rascle model of traffic flow with Coulomb-like friction is solved by introducing the reasonable flux-function. Three kinds of non-self-similar Riemann solutions involving delta-shock are given exactly by the means of variable substitution method. The serious traffic congestion may be explained by the delta-shock, for which, the generalized Rankine–Hugoniot relation and entropy condition are clarified. Since the friction term is involved, all waves containing the rarefaction wave, contact discontinuity, shock wave, and delta-shock are bent into parabolic curves. By analyzing the limiting behavior of the solutions, it is discovered that as the Umami Chaplygin gas pressure and flux-function vanish simultaneously, the Riemann solution containing the rarefaction wave and the contact discontinuity tends to the vacuum solution of the zero-pressure Euler equations with friction, and the solution including the shock wave and the contact discontinuity tends to the delta-shock solution of the zero-pressure Euler equations with friction.

The Riemann problem for two-dimensional gas dynamics with isentropic or polytropic gas is considered. The initial data is constant in each quadrant and chosen so that only a rarefaction wave, shock wave, or slip line connects two neighboring constant initial states. With this restriction sixteen (respectively, fifteen) genuinely different wave combinations for isentropic (respectively, polytropic) gas exist. For each configuration the numerical solution is analyzed and illustrated by contour plots. Additionally, the required relations for the initial data and the symmetry properties of the solutions are given. The chosen calculations correspond closely to the cases studied by T. Zhang and Y. Zheng [SIAM J. Math. Anal., 21 (1990), pp. 593–630], so that the analytical theory can be directly compared to our numerical study.

High order WENO finite difference scheme with adaptive dual order ideal weights for hyperbolic conservation laws

The viscous effects on unsteady shock wave propagation are investigated by numerical solution of the Boltzmann model kinetic equations. The kinetic equations are solved for two unsteady non-equilibrium flow problems, namely, the one-dimensional Riemann problem and a two-dimensional viscous shock-tube. The numerical method comprises the discrete velocity method in the velocity space and the finite volume discretization in physical space using various flux schemes. The discrete version of H-theorem is applied for analysis of accuracy of the numerical solution as well as of the onset of non-equilibrium. Simulations show that the maximum entropy generation rate in viscous shock tube occurs in the boundary layer / shock wave interaction region. The entropy generation rate is used to determine the time-variation of the speed of propagation of shock, contact discontinuity and rarefaction waves.

Two-Dimensional Riemann Problem for a Hyperbolic System of Nonlinear Conservation Laws: II. Initial Data Involving Some Rarefaction Waves

We investigate the solution of the nonlinear junction Riemann problem for the one-dimensional shallow water equations (SWEs) in a simple star network made of three rectangular channels. We consider possible bottom discontinuities between the channels and possible differences in the channels width. In the literature, the solution of the Riemann problem at the junction is investigated for the symmetric case without bottom steps and channels width variations. Here, the solution is extended to a more general situation such that neither the equality of the channels width nor the symmetry of the flow are assumed in the downstream channels. The analysis is performed under sub-criticality conditions and the results are summarized in a main theorem, while a series of numerical examples are presented and support our conclusions.

Further development of the dynamic adaptation method for gas dynamics problems that describe multiple interactions of shock waves, rarefaction waves, and contact discontinuities is consid- ered. Using the Woodward-Colella problem and a nonuniformly accelerating piston as examples, the efficiency of the proposed method is demonstrated for the gas dynamics problems with shock wave and contact discontinuity tracking. The grid points are distributed under the control of the diffusion approx- imation. The choice of the diffusion coefficient for obtaining both quasi-uniform and strongly nonuni- form grids for each subdomain of the solution is validated. The interaction between discontinuities is resolved using the Riemann problem for an arbitrary discontinuity. Application of the dynamic adapta- tion method to the Woodward-Colella problem made it possible to obtain a solution on a grid consisting of 420 cells that is almost identical to the solution obtained using the WENO5m method on a grid con- sisting of 12 800 cells. In the problem for a nonuniformly accelerating piston, a proper choice of the dif- fusion coefficient in the transformation functions makes it possible to generate strongly nonuniform grids, which are used to simulate the interaction of a series of shock waves using shock wave and contact discontinuity tracking. The dynamic adaptation methods (13-21) are less complicated and make it possible to generate grids with a constant (19) or variable (18) number of grid points. In these methods, the control of the distribution of the grid points is achieved by using information about the dynamic behavior of the solution to be found. This makes it possible to concentrate a large number of grid points in the regions where the solution under- goes sharp changes (13). A close relationship between the dynamic behavior of the solution and the location of the grid points requires that the coordinates of the grid points be recalculated at each time layer. This fact places high requirements upon the consistency of the solution dynamics with the grid points motion. For this reason, the algorithms that do not use any fitting parameters have certain advantages (15). These fea- tures are especially pronounced in the generation of adaptive grids for unsteady gas dynamics problems (see (16-22)), which describe rapidly changing processes. The solution of hyperbolic differential equations,

In the field of computational fluid dynamics, many numerical algorithms have been developed to simulate inviscid, compressible flows problems. Among those most famous and relevant are based on flux vector splitting and Godunov-type schemes. Previously, this system was developed through computational studies by Mawlood [1]. However the new test cases for compressible flows, the shock tube problems namely the receding flow and shock waves were not investigated before by Mawlood [1]. Thus, the objective of this study is to develop a high-order compact (HOC) finite difference solver for onedimensional Euler equation. Before developing the solver, a detailed investigation was conducted to assess the performance of the basic third-order compact central discretization schemes. Spatial discretization of the Euler equation is based on flux-vector splitting. From this observation, discretization of the convective flux terms of the Euler equation is based on a hybrid flux-vector splitting, known as the advection upstream splitting method (AUSM) scheme which combines the accuracy of flux-difference splitting and the robustness of flux-vector splitting. The AUSM scheme is based on the third-order compact scheme to the approximate finite difference equation was completely analyzed consequently. In one-dimensional problem for the first order schemes, an explicit method is adopted by using time integration method. In addition to that, development and modification of source code for the one-dimensional flow is validated with four test cases namely, unsteady shock tube, quasi-one-dimensional supersonic-subsonic nozzle flow, receding flow and shock waves in shock tubes. From these results, it was also carried out to ensure that the definition of Riemann problem can be identified. Further analysis had also been done in comparing the characteristic of AUSM scheme against experimental results, obtained from previous works and also comparative analysis with computational results generated by van Leer, KFVS and AUSMPW schemes. Furthermore, there is a remarkable improvement with the extension of the AUSM scheme from first-order to third-order accuracy in terms of shocks, contact discontinuities and rarefaction waves.

In MHD, as in ordinary gas dynamics, there exist surfaces across which the physical fields change discontinuously and on either side of which the field is constant (e.g., see Landau and Lifshitz, 1960; Burlaga, 1971a, Richter et al., 1985a). These surfaces include both nonpropagating structures (tangential discontinuities and contact discontinuities) and propagating structures (rotational discontinuities and shock waves). In reality, each type of surface has an internal structure determined by plasma processes, but this structure is ignored in the MHD fluid approximation.

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

The equations of planar motion of an elastic string form a $4 \times 4$ system of first order conservation laws. Two of the characteristic fields correspond to genuinely nonlinear longitudinal shocks and rarefaction waves, involving changes in the tension in the string, but not the slope. The other two fields correspond to contact discontinuities, across which the slope of the string jumps, reflecting the absence of any resistance to bending. Here, the tension T is related to the local elongation $\xi > 1$ in such a way as to ensure strict hyperbolicity: $T''(\xi ) > T{{(\xi )} / {\xi \geq 0}}$. The other main assumption is chosen to reflect properties of typical materials such as nylon and rubber. That is, $T'(\xi) $ is negative for $\xi > \xi _I $ and positive for $\xi > \xi _I $ for some $\xi _I > 1 $. The principal result of the paper is that the Riemann problem has a unique solution among combinations of centered waves, with a natural entropy condition placed on shocks. That is, any initial jump discontinuity in the tension, slope and velocity of the string can be resolved into combinations of longitudinal waves and contact discontinuities. This is illustrated for the plucked string problem, whose solution (valid until a wave first hits an end of the string) necessarily involves longitudinal waves.

The expansion of a dense plasma into an ambient plasma with a step-like distribution is studied with a series of one-dimensional particle-in-cell (PIC) simulations. Several electron-to-proton temperature ratios Te/Tp in the ambient plasma and density ratios nr between the dense and ambient plasmas are examined. The collisionless plasma is unmagnetized. If nr≥10, hybrid structures form, which are composed of an electrostatic shock for the ambient plasma and a double layer for the expanding dense plasma. If nr=5, the electrostatic shocks become separated from the rarefaction waves by a contact discontinuity. If Te/Tp=5, the steepening of the electrostatic shock is halted by dissipation of wave energy by Landau damping. Its transition layer is wide, it propagates at the ion acoustic speed in the downstream frame of reference, and it hardly compresses the ambient plasma that crosses its transition layer. The contact discontinuity is wide, and the total density varies across its layer, creating a thermoelectric field. With a temperature ratio of Te/Tp=100, shock steepening is halted by wave dispersion. The shock transition layer is thin; it doubles the density of the ambient plasma that crosses it as expected, and the contact discontinuity is thin and has a uniform density across its layer. PIC simulations with nr=5 and several ratios Te/Tp between 5 and 100 show how the electrostatic shock changes from being dissipative to dispersive. Non-linear Landau damping allows the dispersive shock collapse, changing it into an ensemble of waves resembling solitary ion acoustic waves.