STABILITY OF VISCOUS CONTACT WAVE FOR COMPRESSIBLE NAVIER-STOKES SYSTEM OF GENERAL GAS WITH FREE BOUNDARY

In this paper, we study the large time behavior of solutions to the nonisentropic Navier-Stokes equations of general gas, where polytropic gas is included as a special case, with a free boundary. First we construct a viscous contact wave which approximates to the contact discontinuity, which is a basic wave pattern of compressible Euler equation, in finite time as the heat conductivity tends to zero. Then we prove the viscous contact wave is asymptotic stable if the initial perturbations and the strength of the contact wave are small. This generalizes our previous result [6] which is only for polytropic gas.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TOWARD THE SUPERPOSITION OF CONTACT DISCONTINUITY AND SHOCK WAVE FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is established under some smallness conditions. To do this, we first construct a new viscous contact wave such that the momentum equation is satisfied exactly and then determine the shift of the viscous shock wave. By using them together with an inequality concerning the heat kernel in the half space, we obtain the desired a priori estimates. The proof is based on the elementary energy method by the anti-derivative argument.

ZERO DISSIPATION LIMIT TO CONTACT DISCONTINUITY FOR THE COMPRESSIBLE NAVIER-STOKES SYSTEM OF GENERAL GAS

The zero dissipation limit to the contact discontinuities for one-dimensional com- pressible Navier-Stokes equations was recently proved for ideal polytropic gas （see Huang et al. [15, 22] and Ma [31]）, but there is few result for general gases including ideal polytropic gas. We prove that if the solution to the corresponding Euler system of general gas satisfying （1.4） is piecewise constant with a contact discontinuity, then there exist smooth solutions to Navier-Stokes equations which converge to the inviscid solutions at a rate of k1/4 as the heat-conductivity coefficient k tends to zero. The key is to construct a viscous contact wave of general gas suitable to our proof （see Section 2）. Notice that we have no need to restrict the strength of the contact discontinuity to be small.

CONVERGENCE RATE OF SOLUTIONS TO STRONG CONTACT DISCONTINUITY FOR THE ONE-DIMENSIONAL COMPRESSIBLE RADIATION HYDRODYNAMICS MODEL

This paper is concerned with a singular limit for the one-dimensional compress- ible radiation hydrodynamics model. The singular limit we consider corresponds to the physical problem of letting the Bouguer number infinite while keeping the Boltzmann number constant. In the case when the corresponding Euler system admits a contact discontinuity wave, Wang and Xie （2011） [12] recently verified this singular limit and proved that the solution of the compressible radiation hydrodynamics model converges to the strong contact 1 discontinuity wave in the L∞-norm away from the discontinuity line at a rate of ε1/4, as the reciprocal of the Bouguer number tends to zero. In this paper, Wang and Xie＇s convergence rate is improved to ε7/8 by introducing a new a priori assumption and some refined energy estimates. Moreover, it is shown that the radiation flux q tends to zero in the L∞-norm away from the discontinuity line, at a convergence rate as the reciprocal of the Bouguer number tends to zero.

On the Vortex Sheets of Compressible Flows

This paper provides a review of the recent results on the stability of vortex sheets in compressible flows.Vortex sheets are contact discontinuities of the underlying flows.The vortex sheet problem is a free boundary problem with a characteristic boundary and is challenging in analysis.The formulation of the vortex sheet problem will be introduced.The linear stability and nonlinear stability for both the two-dimensional two-phase compressible flows and the two-dimensional elastic flows are summarized.The linear stability of vortex sheets for the three-dimensional elastic flows is also presented.The difficulties of the vortex sheet problems and the ideas of proofs are discussed.

ANALYSIS OF THE STRUCTURAL PROPERTIES OF THE SOLUTIONS TO SPEED GRADIENT TRAFFIC FLOW MODEL

In this paper,we carry out an analysis of the structural properties of the solutions to the speed gradient(SG)traffic flow model.Under the condition that the relaxation effect can be neglected,it is shown that a 1-shock or a 1-rarefaction is associated with the first characteristic,but on the other hand,a contact discontinuity rather than a 2-shock or a 2-rarefaction is associated with the second characteristic.Since the existence of a 2-shock or 2-rarefaction violates the physical mechanism of the traffic flow,the SG model is more reasonable.If the relaxation effect cannot be neglected,it is somewhat difficult to carry out the analytical analysis and the numerical simulation results should be obtained.

ZERO DISSIPATION LIMIT TO A RIEMANN SOLUTION FOR THE COMPRESSIBLE NAVIER-STOKES SYSTEM OF GENERAL GAS

For the general gas including ideal polytropic gas, we study the zero dissipation limit problem of the full 1-D compressible Navier-Stokes equations toward the superposition of contact discontinuity and two rarefaction waves. In the case of both smooth and Riemann initial data, we show that if the solutions to the corresponding Euler system consist of the composite wave of two rarefaction wave and contact discontinuity, then there exist solutions to Navier-Stokes equations which converge to the Riemman solutions away from the initial layer with a decay rate in any fixed time interval as the viscosity and the heat-conductivity coefficients tend to zero. The proof is based on scaling arguments, the construction of the approximate profiles and delicate energy estimates. Notice that we have no need to restrict the strengths of the contact discontinuity and rarefaction waves to be small.

Stability of Viscous Contact Wave for Compressible Navier-Stokes Equations with a Large Initial Perturbation

The viscous contact wave for the compressible Navier-Stokes equations has recently been shown to be asymptotically stable provided that all the L2 norms of initial perturbations, their derivatives and/or anti-derivatives axe small. The main purpose of this paper is to study the asymptotic stability and convergence rate of the viscous contact wave with a large initial perturbation. For this purpose, we introduce a positive number l in the construction of a smooth approximation of the contact discontinuity for the compressible Euler equations and then we make the quantity l to be sufficiently large in order to control the growth induced by the nonlinearity of the system and the interaction of waves from different families. This makes for us to estimate the L2 norms of the solution and its derivative for perturbation system without assuming that L2 norms of the anti-derivatives and the derivatives of initial perturbations are small.

ANTI-DIFFUSIVE FINITE DIFFERENCE WENO METHODS FOR SHALLOW WATER WITH TRANSPORT OF POLLUTANT

In this paper we further explore and apply our recent anti-diffusive flux corrected high order finite difference WENO schemes for conservation laws [18] to compute the SaintVenant system of shallow water equations with pollutant propagation, which is described by a transport equation. The motivation is that the high order anti-diffusive WENO scheme for conservation laws produces sharp resolution of contact discontinuities while keeping high order accuracy for the approximation in the smooth region of the solution. The application of the anti-diffusive high order WENO scheme to the Saint-Venant system of shallow water equations with transport of pollutant achieves high resolution

The Unified Coordinate System in Computational Fluid Dynamics

A fundamental issue in CFD is the role of coordinates and,in particular,the search for“optimal”coordinates.This paper reviews and generalizes the recently developed unified coordinate system(UC).For one-dimensional flow,UC uses a material coordinate and thus coincides with Lagrangian system.For two-dimensional flow it uses a material coordinate,with the other coordinate determined so as to preserve mesh othorgonality(or the Jacobian),whereas for three-dimensional flow it uses two material coordinates,with the third one determined so as to preserve mesh skewness(or the Jacobian).The unified coordinate system combines the advantages of both Eulerian and the Lagrangian system and beyond.Specifically,the followings are shown in this paper.(a)For 1-D flow,Lagrangian system plus shock-adaptive Godunov scheme is superior to Eulerian system.(b)The governing equations in any moving multi-dimensional coordinates can be written as a system of closed conservation partial differential equations(PDE)by appending the time evolution equations–called geometric conservation laws–of the coefficients of the transformation(from Cartesian to the moving coordinates)to the physical conservation laws;consequently,effects of coordinate movement on the flow are fully accounted for.(c)The system of Lagrangian gas dynamics equations is written in conservation PDE form,thus providing a foundation for developing Lagrangian schemes as moving mesh schemes.(d)The Lagrangian system of gas dynamics equations in two-and three-dimension are shown to be only weakly hyperbolic,in direct contrast to the Eulerian system which is fully hyperbolic;hence the two systems are not equivalent to each other.(e)The unified coordinate system possesses the advantages of the Lagrangian system in that contact discontinuities(including material interfaces and free surfaces)are resolved sharply.(f)In using the UC,there is no need to generate a body-fitted mesh prior to computing flow past a body;the mesh is automatically generated by the flow.Numerical examples are given to confirm these properties.Relations of the UC approach with the Arbitrary-Lagrangian-Eulerian(ALE)approach and with various moving coordinates approaches are also clarified.

Generalized Riemann problem for Euler system Dedicated to Professor LI TaTsien on the Occasion of His 80th Birthday

This article is a survey on the progress in the study of the generalized Riemann problems for MD Euler system. A new result on generalized Riemann problems for Euler systems containing all three main nonlinear waves(shock, rarefaction wave and contact discontinuity) is also introduced.

Global Stability of Multi-wave Configurations for the Compressible Non-isentropic Euler System

This paper is contributed to the structural stability of multi-wave configurations to Cauchy problem for the compressible non-isentropic Euler system with adiabatic exponentγ∈(1,3].Given some small BV perturbations of the initial state,the author employs a modified wave front tracking method,constructs a new Glimm functional,and proves its monotone decreasing based on the possible local wave interaction estimates,then establishes the global stability of the multi-wave configurations,consisting of a strong 1-shock wave,a strong 2-contact discontinuity,and a strong 3-shock wave,without restrictions on their strengths.

Numerical Study of Hydrogen Peroxide Thermal Decomposition in a Shock Tube

Hydrogen peroxide(H_2O_2) has its significance during the combustion of heavy hydrocarbons in the internal combustion(IC) engines. Owing to its importance the measurements of H_2O_2 dissociation rate have been reported mostly using the shock tube apparatus. These types of experimental measurements are although quite reliable but require high cost. On the other hand, numerical simulations provide low cost and reliable solutions especially using computation fluid dynamics(CFD) software. In the current study an experimental shock tube flow is modeled using open access platform OpenFOAM to investigate the thermal decomposition of H_2O_2. Using two different convective schemes, limited Linear and upwind, the propagation of shock wave and resultant dissociation reaction are simulated. The results of the simulations are compared with the experimental data. It is observed that the rate constant measured using the simulation data deviates from the experimental results in the low temperature range and approaches the experimental values as the temperature is raised.