Generalized Rankine-Hugoniot Relations Research Articles

Delta-shock for the Chaplygin gas Euler equations with source terms

Abstract This article discusses the Riemann problem for the Chaplygin gas Euler equations that include the presence of two source terms. By means of variable substitution, two kinds of non-self-similar Riemann solutions involving delta-shock are constructed explicitly. For the delta-shock, the generalized Rankine–Hugoniot relations and the over-compressive entropy condition are clarified. Moreover, the position, propagation speed and strength of the delta-shock are given explicitly. It is discovered that the position of the delta-shock is a combination of an exponential function and a linear function, and the weight of the delta-shock is an exponential function of the time. Interestingly, even when the delta-shock is a straight line, the weight of the delta-shock is no longer a linear function of the time t. In addition, it is proved that the Riemann solutions converge to the corresponding ones of Chaplygin gas Euler equations with friction as k drops to zero, and the Riemann solutions converge to the corresponding ones of Chaplygin gas Euler equations as k and β tend to zero simultaneously. Furthermore, it is also shown that the limits of Riemann solutions are just the Riemann solutions to the transport equations with same source terms as the Chaplygin gas pressure falls to zero.

HLLEPJ and HLLCEPJ Riemann solvers for the Wilkins model of elastoplasticity

This paper presents three Riemann solvers for the Wilkins model of hypoelasticity with the Jaumann derivative correction. The solvers are developed as extensions of the HLL and HLLC Riemann solvers for the Euler hydrodynamic model and differ in the number of waves considered (two-wave HLLEPJ Riemann solver, three-wave HLLCEPJ Riemann solver and five-wave HLLCEPJ Riemann solver). The discontinuous solution of the Wilkins system of equations is obtained using generalized Rankine-Hugoniot relations developed on the basis of the path-conservative DLM approach. A special choice of the path in the phase space allows to avoid any iterative processes in the solver construction. The proposed Riemann solvers are implicated in the first order Finite Volume numerical scheme for testing on the set of 1D, 2D and 3D problems of elastoplastic flows.

The Riemann problem for a traffic flow model

A traffic flow model describing the formation and dynamics of traffic jams was introduced by Berthelin et al. [“A model for the formation and evolution of traffic jams,” Arch. Ration. Mech. Anal. 187, 185–220 (2008)], which consists of a pressureless gas dynamics system under a maximal constraint on the density and can be derived from the Aw–Rascle model under the constraint condition ρ≤ρ* by letting the traffic pressure vanish. In this paper, we give up this constraint condition and consider the following form: {ρt+(ρu)x=0,(ρu+εp(ρ))t+(ρu2+εup(ρ))x=0,in which p(ρ)=−1ρ. The Riemann problem for the above traffic flow model is constructively solved. The delta shock wave arises in the Riemann solutions, although the system is strictly hyperbolic, its first eigenvalue is genuinely nonlinear, and the second eigenvalue is linearly degenerate. Furthermore, we clarify the generalized Rankine–Hugoniot relations and δ-entropy condition. The position, strength, and propagation speed of the delta shock wave are obtained from the generalized Rankine–Hugoniot conditions. The delta shock may be useful for the description of the serious traffic jam. More importantly, it is proved that the limits of the Riemann solutions of the above traffic flow model are exactly those of the pressureless gas dynamics system with the same Riemann initial data as the traffic pressure vanishes.

Riemann problems for the 2-D pressureless gas dynamics with external forces

Abstract This paper is concerned with the pressureless gas dynamics with a space-dependent gravity or a time-dependent friction in two dimensions. The basic Riemann problems with two pieces of constant initial data are considered. By the characteristic analysis, the Riemann problems are constructively solved by two kinds of solutions: vacuum solution and delta-shock solution, which describe the phenomena of cavitation and concentration, respectively. Especially, the generalized Rankine–Hugoniot relations for delta-shock waves are clarified and applied to the Riemann problems.

Delta-shock for a class of strictly hyperbolic systems of conservation laws

Abstract In this paper, a class of strictly hyperbolic systems of conservation laws which arises in connection with enhanced oil recovery is studied. The Riemann problem is solved analytically. The Riemann solutions with two kinds of different structures involving the delta-shock are obtained. For delta-shock, the generalized Rankine–Hugoniot relations and over-compressive delta-entropy condition are clarified. Further, the existence and uniqueness of delta-shock are established. The theoretical analysis is tested accurately by the numerical results.

The Interaction of Waves in the Zero-Pressure Euler Equations with a Coulomb-Like Friction Term

In this study, the interaction of waves in the zero-pressure Euler equations with a Coulomb-like friction term is considered, which is equivalent to the Riemann problem with three constant initial states for the zero-pressure Euler equations. By solving generalized Rankine–Hugoniot relations under suitable entropy conditions, four different structures of explicit solutions are obtained uniquely, in which the interactions among contact discontinuity, vacuum, and delta shock are presented.

The Riemann problem for the Suliciu relaxation system with the double-coefficient Coulomb-like friction terms

We solve the Riemann problem for the Suliciu relaxation system with the Coulomb-like friction terms containing two constant friction coefficients. The exact Riemann solutions involving delta shock waves and contact discontinuities are obtained constructively. Via deriving the generalized Rankine–Hugoniot relations of delta shock waves in detail, all of the position, propagation speed and strength of the delta shock wave are given concretely under a suitable entropy condition. Based on this, we clearly analyze the influence of the Coulomb-like friction terms on the Riemann solutions of the Suliciu relaxation system.

Eulerian droplet model: Delta-shock waves and solution of the Riemann problem

We study an Eulerian droplet model which can be seen as the pressureless gas system with a source term, a subsystem of this model and the inviscid Burgers equation with source term. The condition for loss of regularity of a solution to Burgers equation with source term is established. The same condition applies to the Eulerian droplet model and its subsystem. The Riemann problem for the Eulerian droplet model is constructively solved by going through the solution of the Riemann problems for the inviscid Burgers equation with a source term and the subsystem, respectively. Under suitable generalized Rankine–Hugoniot relations and entropy condition, the existence of delta-shock solution is established. The existence of a solution to the generalized Rankine–Hugoniot conditions is proven. Some numerical illustrations are presented.

Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term

The Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term are studied. The Riemann solutions exactly include two kinds: delta-shock solutions and vacuum solutions. In order to see more clearly the influence of the source term on Riemann solutions, the generalized Rankine-Hugoniot relations of delta shock waves are derived in detail, and the position, propagation speed and strength of delta shock wave are given. It is also shown that, as the source term vanishes, the Riemann solutions converge to the corresponding ones of the homogeneous system, which is just the generalized zero-pressure flow model and contains the one-dimensional zero-pressure flow as a prototypical example. Furthermore, the generalized balance relations associated with the generalized mass and momentum transportation are established for the delta-shock solution. Finally, two typical examples are presented to illustrate the application of our results.

GENERALIZED MULTI-POLYTROPIC RANKINE–HUGONIOT RELATIONS AND THE ENTROPY CONDITION

ABSTRACT The study aims at a derivation of generalized Rankine–Hugoniot relations, especially that for the entropy, for the case of different upstream/downstream polytropic indices and their implications. We discuss the solar/stellar wind interaction with the interstellar medium for different polytropic indices. Moreover, we concentrate on the situation when the polytropic index changes across hydrodynamical shocks. First, we use a numerical monofluid approach with a constant polytropic index in the entire integration region to show the influence of the polytropic index on the thickness of the helio-/astrosheath and on the compression ratio. Second, the Rankine–Hugoniot relations for a polytropic index changing across a shock are derived analytically, particularly including a new form of the entropy condition. In application to the/an helio-/astrosphere, we find that the size of the helio-/astrosheath as a function of the polytropic index decreases in a monofluid model for indices less than and increases for higher ones and vice versa for the compression ratio. Furthermore, we demonstrate that changing polytropic indices across a shock are physically allowed only for sufficiently high Mach numbers and that in the hypersonic limit the compression ratio depends only on the downstream polytropic index, while the ratios of the temperature and pressure as well as the entropy difference depend on both the upstream and downstream polytropic indices.

Generalized δ-entropy condition to Riemann solution for Chaplygin gas in traffic flow

The Riemann problem for the Aw-Rascle model in the traffic flow with the Delta initial data for the Chaplygin gas is studied. The solutions are constructed globally under the generalized Rankine-Hugoniot relations, the δ-entropy conditions, and the generalized δ-entropy conditions. A new Delta wave, which is called a primary Delta wave, is defined in some solutions. The primary Delta wave satisfies the generalized Rankine-Hugoniot relations and the generalized δ-entropy conditions. It generates initially from the Delta initial data, which either evaluates a Delta wave, whose weight becomes stronger and stronger, or disappears at a finite time.

Riemann problem for a geometrical optics system

The paper solves analytically the Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in geometrical optics, in which the flux contains the nonconvex function possessing an infinite number of inflection points. Firstly, the generalized Rankine-Hugoniot relations and entropy condition of delta shock waves and left(right)-contact delta shock waves are proposed and clarified. Secondly, with the help of the convex hull, seven kinds of structures of Riemann solutions are obtained. The solutions fall into three broad categories with a series of geometric structures involving simultaneously contact discontinuities, vacuums and delta shock waves. Finally, numerical experiments confirm the theoretical analysis.

A phase decomposition approach and the Riemann problem for a model of two-phase flows

We present a phase decomposition approach to deal with the generalized Rankine–Hugoniot relations and then the Riemann problem for a model of two-phase flows. By investigating separately the jump relations for equations in conservative form in the solid phase, we show that the volume fractions can change only across contact discontinuities. Then, we prove that the generalized Rankine–Hugoniot relations are reduced to the usual form. It turns out that shock waves and rarefaction waves remain on one phase only, and the contact waves serve as a bridge between the two phases. By decomposing Riemann solutions into each phase, we show that Riemann solutions can be constructed for large initial data. Furthermore, the Riemann problem admits a unique solution for an appropriate choice of initial data.

The Riemann Problem for the one-dimensional, free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations

In this paper, the solution of the Riemann Problem for the one-dimensional, free-surface Shallow Water Equations over a bed step is analyzed both from a theoretical and a numerical point of view. Particular attention has been paid to the wave that is generated at the location of the bed discontinuity. Starting from the classical Shallow Water Equations, considering the bed level as an additional variable, and adding to the system an equation imposing its time invariance, we show that this wave is a contact wave, across which one of the Riemann invariants, namely the energy, is not constant. This is due to the fact that the relevant problem is nonconservative. We demonstrate that, in this type of system, Riemann Invariants do not generally hold in contact waves. Furthermore, we show that in this case the equations that link the flow variables across the contact wave are the Generalized Rankine–Hugoniot relations and we obtain these for the specific problem. From the numerical point of view, we present an accurate and efficient solver for the step Riemann Problem to be used in a finite-volume Godunov-type framework. Through a two-step predictor–corrector procedure, the solver is able to provide solutions with any desired accuracy. The predictor step uses a well-balanced Generalized Roe solver while the corrector step solves the exact nonlinear system of equations that consitutes the problem by means of an iterative procedure that starts from the predictor solution. In order to show the effectiveness and the accuracy of the proposed approach, we consider several step Riemann Problems and compare the exact solutions with the numerical results obtained by using a standard Roe approach far from the step and the novel two-step algorithm for the fluxes over the step, achieving good results.

Shock propagation along a two-layer interface in confined geometry

The paper presents direct numerical simulations of the propagation of a strong planar shock wave along a two-layer interface confined between two walls. Transient stage and steady state of the propagation are described. In the steady state regime, a stationary irregular refraction of the shock and a vorticity sheet are characterized. Physics of the stationary irregular refraction state is explained. Shock velocity is studied. Quantities are shown to verify a set of generalized Rankine–Hugoniot relations. The unknown coupling mechanism between shear layer and pressure step is predicted by a neural network which leads to a predictive, gray-box type model.

One-Dimensional Multiple-Temperature Gas-Kinetic Bhatnagar-Gross-Krook Scheme for Shock Wave Computation

Accurately computing the inner structure of normal shock waves or oblique shock waves is crucial for many hypersonic applications. As such, it will improve the prediction accuracy of aerodynamics properties and aerothermal effects on hypersonic vehicles and spacecraft during atmospheric entries. Because a shock wave usually has a thickness of a few mean free paths, it is very difficult to accurately compute the detailed nonequilibrium inner structure across a shock wave with a continuum method. This paper reports a gas-kinetic Bhatnagar-Gross-Krook scheme for computations of one-dimensional vibrationally nonequilibrium nitrogen flows through a planar shock wave. The present gas-kinetic Bhatnagar-Gross-Krook scheme solves for the shock structure with multiple temperatures, including two translational temperatures, one rotational temperature, and one vibrational temperature. The salient features of the present gas-kinetic Bhatnagar-Gross-Krook method are multifold. Its applicability covers a wide simulation regime, extending that of continuum flows to the transition flows; it is more computationally efficient in time than the traditional direct simulation Monte Carlo method for shock wave simulation. To provide proper downstream subsonic boundary conditions for very strong shock waves, it is required to determine a proper postshock equilibrium state in which all temperatures have accomplished relaxation processes to a common equilibrium temperature. Analytical expressions of a complete set of generalized Rankine-Hugoniot relations across a planar shock wave are obtained to account for the variant specific heat ratio due to inner energy excitations. Numerical simulation results by the present gas-kinetic Bhatnagar-Gross-Krook scheme and the direct simulation Monte Carlo method are found to be in good agreement.

Asymptotic solution of the thin viscous shock layer equations at low Reynolds numbers for a cold surface

Hypersonic three-dimensional viscous rarefied gas flow past blunt bodies in the neighborhood of the stagnation line is considered. The question of the applicability of the gasdynamic thin viscous shock layer model [1] is investigated for the transition flow regime from continuum to free-molecular flow. It is shown that for a power-law temperature dependence of the viscosity coefficient μ∼Tω the quantity (Ree)1/(1+ω), where e = (γ − 1)/2γ and γ is the specific heat ratio, is an important determining parameter of the hypersonic flow at low Reynolds numbers. In the case of a cold surface approximate asymptotic solutions of the thin viscous shock layer equations are obtained for noslip conditions on the surface and generalized Rankine-Hugoniot relations on the shock wave at low Reynolds numbers. These solutions give simple analytic expressions for the thermal conductivity and friction coefficients as functions of the determining flow parameters. As the Reynolds number tends to zero, the values of the thermal conductivity and friction coefficients determined by this solution tend to their values in free-molecular flow for an accommodation coefficient equal to unity. This tending of the thermal conductivity and friction coefficients to the free-molecular limit takes place for both two-and three-dimensional flows. The asymptotic solutions are compared with numerical calculations and experimental data.

Delta-shocks as limits of vanishing viscosity for multidimensional zero-pressure gas dynamics

In this paper we study zero-pressure gas dynamics, which is a nonstrict hyperbolic system of nonlinear conservation laws with delta-shock waves in solutions. By using the generalized Rankine-Hugoniot relations to solve the Riemann problem with two pieces of constant initial data, multidimensional planar delta-shock waves dependent upon a family of one parameter are obtained. Furthermore, we choose a unique entropy solution through the process of a viscosity vanishing, and obtain a stability for delta-shocks in multidimensions.

Laser-propelled ram accelerator

The concept of laser-propelled ram accelerator (L-RAMAC)' is proposed. Theoretically it is capable of achieving a higher launch speed than that by a chemical ram accelerator because a higher specific energy can be input to the propellant gas. The laser beam is supplied through the muzzle, focused as an annulus behind the base of the projectile. The performance of L-RAMAC is analized based on generalized Rankine-Hugoniot relations, suggesting that a superorbital muzzle speed is achievable out of this device.

Jump conditions across normal shock waves in pure vapour–droplet flows

Closed-form analytical jump conditions across normal shock waves in pure vapour–droplet flows have been derived for different boundary conditions. They are equally applicable to partly and fully dispersed shock waves. Collectively they may be called the generalized Rankine–Hugoniot relations for wet vapour. A phase diagram is constructed which specifies the type of shock structure obtained in vapour–droplet flow given some overall parameters. It is shown that in addition to the partly and fully dispersed shock waves that are possible in any relaxing medium, there also exists a class of shock waves in wet vapour in which the two-phase relaxing medium reverts to a single-phase non-relaxing one. An analytical expression for the limiting upstream wetness fraction below which complete evaporation will take place inside a shock of specified strength has been deduced. A new theory has been formulated which shows that, depending on the upstream wetness fraction, a continuous transition exists for the shock velocity between its frozen and fully equilibrium values. The mechanisms of entropy production inside a shock are also discussed.