Expansion of a wedge of non-ideal gas into vacuum

We study the problem of expansion of a wedge of non-ideal gas into vacuum in a two-dimensional bounded domain. The non-ideal gas is characterized by a van der Waals type equation of state. The problem is modeled by standard Euler equations of compressible flow, which are simplified by a transformation to similarity variables and then to hodograph transformation to arrive at a second order quasilinear partial differential equation in phase space; this, using Riemann variants, can be expressed as a non-homogeneous linearly degenerate system provided that the flow is supersonic. For the solution of the governing system, we study the interaction of two-dimensional planar rarefaction waves, which is a two-dimensional Riemann problem with piecewise constant data in the self-similar plane. The real gas effects, which significantly influence the flow regions and boundaries and which do not show-up in the ideal gas model, are elucidated; this aspect of the problem has not been considered until now.     

Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of δ-shocks and vacuum states are presented as the pressure decreases.     

Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation

In this paper,firstly,by solving the Riemann problem of the zero-pressure flow in gas dynamics with a flux approximation,we construct parameterized delta-shock and constant density solutions,then we show that,as the flux perturbation vanishes,they converge to the delta-shock and vacuum state solutions of the zero-pressure flow,respectively.Secondly,we solve the Riemann problem of the Euler equations of isentropic gas dynamics with a double parameter flux approximation including pressure.Furthermore,we rigorously prove that,as the two-parameter flux perturbation vanishes,any Riemann solution containing two shock waves tends to a delta-shock solution to the zero-pressure flow;any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state.Finally,numerical results are given to present the formation processes of delta shock waves and vacuum states.     

Two-dimensional Riemann problem of the Euler equations to the Van der Waals gas around a sharp corner

In this paper, we study the Riemann problem of the two-dimensional (2D) pseudo-steady supersonic flow with Van der Waals gas around a sharp corner expanding into vacuum. The essence of this problem is the interaction of the centered simple wave with the planar rarefaction wave, which can be solved by a Goursat problem or a mixed characteristic boundary value and slip boundary value problem for the 2D self-similar Euler equations. We establish the hyperbolicity and a priori C¹ estimates of the solution through the methods of characteristic decompositions and invariant regions. Moreover, we construct the pentagon invariant region in order to obtain the global solution. In addition, based on the generality of the Van der Waals gas, we construct the subinvariant regions and get the hyperbolicity of the solution according to the continuity of the subinvariant region. At last, the global existence of solution to the gas expansion problem is obtained constructively.     

Discharge of gas into vacuum with temperature at the boundary subject to exponential law

A two-dimensional self-similar problem of discharge of a heat conducting gas Into vacuum is analyzed. The temperature at the boundary of gas and vacuum is assumed to change as an exponential function of time. The coefficient of thermal conductivity depends exponentially on temperature and density. The initial gas density is assumed to be finite and constant. With definite values of exponents this problem is self-similar i.e. the system of partial differential equations can be reduced to the solution of a system of ordinary equations.

The self-modeling properties of solutions of this kind of problems has been noted earlier in [1 and 2]. The problem analyzed here is a particular case of the problem of piston motion considered in [3]. In this problem, however, there appears at the boundary of gas and vacuum a new singular point which does not occur in the piston problem.

A numerical solution of the boundary value problem defined by a system of ordinary equations is made difficult by the presence in the latter of singular points, and of discontinuities in the sought solution. These difficulties have been overcome by a qualitative analysis of the behavior of integral curves, and by the selection of a suitable method of numerical integration.

It is shown in this work that, depending on the initial parameters of the problem, there may exist two kinds of solutions. This had been noted earlier in [1, 3 and 4]. Examples of these are presented here. The degeneration of the solution into a trivial one, when the thermal conductivity coefficient is either invariant of density, or increases with increasing density, is pointed out.     

Non-reflecting boundary conditions for the compressible Euler and Navier-Stokes equations in the finite volume method

The paper is concerned with the numerical implementation of the inlet and outlet boundary conditions in the finite volume method for the solution of the 3D Euler and Navier-Stokes equations. The explicit time marching procedure is described. The classical Riemann problem is modified for physically relevant boundary conditions with the aim to keep conservation laws. This technique was used in [2]. The initial condition in the Riemann problem is replaced by the suitable one-sided boundary condition. This results in the acceleration of the numerical method itself. On the inlet the pressure and the density and the angle of attack or velocity vector and the entropy are prescribed. On the outlet the pressure or normal component of the velocity or temperature or mass flow are investigated in such a way to obtain the unique solution of the modified Riemann problem. Various combinations of inlet and outlet boundary conditions are investigated. This results in the sufficiently precise approximation of real flow boundary conditions. Numerical examples illustrating the usefulness of the proposed approach for cascade flow are presented. Another numerical example is shown in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解的极限

研究了带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解的极限.由于非齐次项的影响,带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解不再是自相似的.当压力和磁感强度同时消失时,它的解会收敛到零压流输运方程组的Riemann解,解中会出现δ-激波和真空现象.同时研究还得到了仅当磁感强度消失时,它的解会收敛到非齐次广义Chaplygin气体Euler方程组的Riemann解,并且解中只出现δ-激波.     

Existence of solutions to gas expansion problem through a sharp corner for 2-D Euler equations with general equation of state

In this article, we study the gas expansion problem by turning a sharp corner into a vacuum for the two-dimensional (2-D) pseudosteady compressible Euler equations with a convex equation of state. This problem can be considered as the interaction of a centered simple wave with a planar rarefaction wave. To obtain the global existence of a solution up to the vacuum boundary of the corresponding 2-D Riemann problem, we consider several Goursat-type boundary value problems for 2-D self-similar Euler equations and use the ideas of characteristic decomposition and bootstrap method. Further, we formulate 2-D-modified shallow water equations newly and solve a dam-break-type problem for them as an application of this work. Moreover, we also recover the results from the available literature for certain equations of states that provide a check that the results obtained in this article are actually correct.     

THE VANISHING PRESSURE LIMIT OF SOLUTIONS TO THE SIMPLIFIED EULER EQUATIONS FOR ISENTROPIC FLUIDS

In this paper,the Riemann problem of the 1-D reduced model for the 2-D Euler equations is considered and the Riemann solutions are obtained.It is proved that,as the pressure vanishes,they converge to two kinds of Riemann solutions to the 1D reduced model for the 2-D transport equations:one contains δ-shocks,the other contains vacuum.     

Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases

Semi-hyperbolic patches are the regions in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves. This type of region appears frequently in the two-dimensional Riemann problem for the Euler equations and its simplified models and a few other situations. We construct a semi-hyperbolic patch of solution to the two-dimensional nonlinear wave system with Chaplygin gas equation of state by approaching the problem as a Goursat-type boundary value problem which has a sonic curve as the degenerate boundary.     

Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases

The Riemann solutions for the Euler system of conservation laws of energy and momentum in special relativity for polytropic gases are considered. It is rigorously proved that, as pressure vanishes, they tend to the two kinds of Riemann solutions to the corresponding pressureless relativistic Euler equations: the one includes a delta shock, which is formed by a weighted δ-measure, and the other involves vacuum state.     

Global Solution of an Initial-Value Problem for Two-Dimensional Compressible Euler Equations

This paper is concerned with the existence of global continuous solutions of the expansion of a wedge of gas into a vacuum for compressible Euler equations. By hodograph transformation, we first prove that the flow is governed by a partial differential equation of second order, which is further reduced to a system of two nonhomogeneous linearly degenerate equations in the phase space under an irrotationality condition. Then this conclusion is applied to solving the problem that a wedge of gas expands into a vacuum, which is actually a Goursat-type problem for these two equations in the supersonic domain.     

On the modeling and simulation of boundary flow through partially open pipe ends

The determination of boundary conditions for the Euler equations of gas dynamics in a pipe with partially open pipe ends is considered. The boundary problem is formulated in terms of the exact solution of the Riemann problem and of the St. Venant equation for quasi-steady flow so that a pressure-driven calculation of boundary conditions is defined. The resulting set of equations is solved by a Newton scheme. The proposed algorithm is able to solve for all inflow and outflow situations including choked and supersonic flow.Received: August 7, 2002; revised: November 11, 2002     

Well‐posedness for compressible Euler equations with physical vacuum singularity

An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristic speeds u ± c coincide and have unbounded spatial derivative since c behaves like x^1/2 close to the boundary. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local‐in‐time well‐posedness of one‐dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity. © 2009 Wiley Periodicals, Inc.     

A new averaging scheme for the riemann problem in pure water

The numerical investigation of shock phenomena in gas or liquid media where enthalpy is the preferred thermodynamic variable poses special problems. When an expression for internal energy is available, the usual procedure is to employ a splitting scheme to remove source terms from the Euler equations, then upwind-biased shock capturing algorithms are built around the Riemann problem for the conservative system which remains. However, when the governing equations arc formulated in terms of total enthalpy, treatment of a pressure time derivative as a source term leads to a Riemann problem for a system where one equation is not a conservation law. The present research establishes that successful upwind-biased shock capturing schemes can be based upon the pseudo-conservative system. A new averaging scheme for solving the associated Riemann problem is developed. The method is applied to numerical simulations of shock wave propagation in pure water.     

Singular behavior of vacuum states for compressible fluids

In this survey paper, we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics, that is, Euler equations, Euler–Poisson equations and Navier–Stokes equations. The main concern is the well-posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas. Furthermore, the relation of the solutions for the gas dynamics with vacuum to those of the Boltzmann equation will also be discussed. In fact, the results obtained so far for vacuum states are far from being complete and satisfactory. Therefore, this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems. Moreover, the problems considered here are limited to the author's interest and knowledge in this area.     

Non-formation of a vacuum state in a standing piston problem

When boundary data is introduced, additional terms are introduced into the weak formulation of the Navier-Stokes conservation law. We examine the example of single standing piston problem. The single piston problem corresponds to a fixed boundary problem.It is intuitively clear when a single piston filled with gas is pulled apart, even though gas becomes sparse in density, a vacuum state is never formed, because of viscosity. To study this rigorously, the Navier-Stokes equations are used to describe the gas's density and velocity, subject to the presence of viscosity. We prove that, given reasonable assumptions on the boundary data, vacuum states cannot form, if they are not present initially.     

Coupling of compressible and incompressible flow regions using the multiple pressure variables approach

In many cases, multiphase flows are simulated on the basis of the incompressible Navier–Stokes equations. This assumption is valid as long as the density changes in the gas phase can be neglected. Yet, for certain technical applications such as fuel injection, this is no longer the case, and at least the gaseous phase has to be treated as a compressible fluid. In this paper, we consider the coupling of a compressible flow region to an incompressible one based on a splitting of the pressure into a thermodynamic and a hydrodynamic part. The compressible Euler equations are then connected to the Mach number zero limit equations in the other region. These limit equations can be solved analytically in one space dimension that allows to couple them to the solution of a half‐Riemann problem on the compressible side with the help of velocity and pressure jump conditions across the interface. At the interface location, the flux terms for the compressible flow solver are provided by the coupling algorithms. The coupling is demonstrated in a one‐dimensional framework by use of a discontinuous Galerkin scheme for compressible two‐phase flow with a sharp interface tracking via a ghost‐fluid type method. The coupling schemes are applied to two generic test cases. The computational results are compared with those obtained with the fully compressible two‐phase flow solver, where the Mach number zero limit is approached by a weakly compressible fluid. For all cases, we obtain a very good agreement between the coupling approaches and the fully compressible solver. Copyright © 2014 John Wiley & Sons, Ltd.     

Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations

As a ladder step to study transonic problems, we investigate two families of degenerate Goursat-type boundary value problems arising from the two-dimensional pseudo-steady isothermal Euler equations. The first family is about the genuinely two-dimensional full expansion of gas into a vacuum with a wedge; the other is a semi-hyperbolic patch that starts on sonic curves and ends at transonic shocks. Both the vacuum and the sonic sets cause parabolic degeneracy that results in substantial difficulties such as singularities of solutions and uniform a priori estimates. Main ingredients in this study are various characteristic decompositions for the pseudo-steady Euler equations in order to obtain necessary a priori estimates. Furthermore, we are able to verify the uniform H?lder continuity of solutions with exponent 1/2 for the gas expansion problem and up to 2/7 for the semi-hyperbolic problem.     

Well‐posedness of Compressible Euler Equations in a Physical Vacuum

An important problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of a vacuum. In particular, physical vacuum, in which the boundary moves with a nontrivial finite normal acceleration, naturally arises in the study of the motion of gaseous stars or shallow water. Despite its importance, there are only a few mathematical results available near a vacuum. The main difficulty lies in the fact that the physical systems become degenerate along the vacuum boundary. In this paper, we establish the local‐in‐time well‐posedness of three‐dimensional compressible Euler equations for polytropic gases with a physical vacuum by considering the problem as a free boundary problem. © 2015 Wiley Periodicals, Inc.