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.     

On a regularization of the compressible Euler equations for an isothermal gas

We consider a Leray-type regularization of the compressible Euler equations for an isothermal gas. The regularized system depends on a small parameter α>0. Using Riemann invariants, we prove the existence of smooth solutions for the regularized system for every α>0. The regularization mechanism is a non-linear bending of characteristics that prevents their finite-time crossing. We prove that, in the α→0 limit, the regularized solutions converge strongly. However, based on our analysis and numerical simulations, the limit is not the unique entropy solution of the Euler equations. The numerical method used to support this claim is derived from the Riemann invariants for the regularized system. This method is guaranteed to preserve the monotonicity of characteristics.     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

A General Cavitation Model for the Highly Nonlinear Mie-Grüneisen Equation of State

A general one-fluid cavitation model is proposed for a family of Mie-Grüneisen equations of state (EOS), which can provide a wide application of cavitation flows, such as liquid-vapour transformation and underwater explosion. An approximate Riemann problem and its approximate solver for the general cavitation model are developed. The approximate solver, which provides the interface pressure and normal velocity by an iterative method, is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian grids. Several numerical examples, including Riemann problems and underwater explosion applications, are presented to validate the cavitation model and the corresponding approximate solver.     

Numerical solutions of transonic two-dimensional flows at a ninety-degree wedge

We present numerical results on self-similar two-dimensional Riemann problems governed by the compressible Euler system and the nonlinear wave system, which give rise to a transonic shock. We consider a configuration for a vertical incident shock moving to the right above a rectangular object. The incident shock then interacts with a sonic circle soon after it moves beyond the object, and creates a transonic region. We implement Lax–Liu positive schemes and Strang splitting, and obtain linear correlations of the incident shock strength and the shock strength at the vertical wall. We further implement Roe average methods and finite volume methods on quadrilateral grids to capture a contact discontinuity of the Euler system near the corner of the object. The contact discontinuity creates a new supersonic state and a transonic shock inside the transonic region.     

A well-balanced approximate Riemann solver for compressible flows in variable cross-section ducts

A well-balanced approximate Riemann solver is introduced in this paper in order to compute approximations of one-dimensional Euler equations in variable cross-section ducts. The interface Riemann solver is grounded on the VFRoe-ncv scheme, and it enforces the preservation of Riemann invariants of the steady wave. The main properties of the scheme are detailed. We provide numerical results to assess the validity of the scheme, even when the cross-section is discontinuous. A first series is devoted to analytical test cases, and the last results correspond to the simulation of a bubble collapse.     

Interaction of three and four rarefaction waves of the pressure-gradient system

We study the two-dimensional pressure-gradient system, a subsystem of the two-dimensional compressible Euler system. We consider the problem of interaction of four rarefaction waves which is one case of two-dimensional Riemann problems. It is known that, when two planar waves interact, there exists a smooth solution in the interaction region. In this paper, we establish the existence of a smooth solution in the hyperbolic domain of determinacy, in which we encounter the interaction of simple and planar waves and shock prevention in simple waves.     

TWO-DIMENSIONAL RIEMANN PROBLEMS： FROM SCALAR CONSERVATION LAWS TO COMPRESSIBLE EULER EQUATIONS

In this paper we survey the authors＇ and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections： 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.     

Riemann boundary value problems and reflection of shock for the Chaplygin gas

In this paper,we study two-dimensional Riemann boundary value problems of Euler system for the isentropic and irrotational Chaplygin gas with initial data being two constant states given in two sectors respectively,where one sector is a quadrant and the other one has an acute vertex angle.We prove that the Riemann boundary value problem admits a global self-similar solution,if either the initial states are close,or the smaller sector is also near a quadrant.Our result can be applied to solving the problem of shock reflection by a ramp.     

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.     

A relaxation Riemann solver for compressible two-phase flow with phase transition and surface tension

The dynamics of two-phase flows depend crucially on interfacial effects like surface tension and phase transition. A numerical method for compressible inviscid flows is proposed that accounts in particular for these two effects. The approach relies on the solution of Riemann-like problems across the interface that separates the liquid and the vapour phase. Since the analytical solutions of the Riemann problems are only known in particular cases an approximative Riemann solver for arbitrary settings is constructed. The approximative solutions rely on the relaxation technique.The local well-posedness of the approximative solver is proven. Finally we present numerical experiments for radially symmetric configurations that underline the reliability and efficiency of the numerical scheme.     

Asymptotic Expansions and Numerical Methods for Compressible and Low Mach Number Fluid Flow

The results of a formal asymptotic low Mach number analysis [5, 6] of the Euler equations of gas dynamics are used to extend the validity of a numerical method from the simulation of compressible inviscid flow fields to the low Mach number regime. Although, different strategies are applicable [7, 8, 5, 9] in this context we focus our view to a preconditioning technique recently proposed by Guillard and Viozat [16]. We present a finite volume approximation of the governing equations using a Lax‐Friedrichs scheme whereby a preconditioning of the incorporated numerical dissipation is employed. A discrete asymptotic analysis proves the validity of the scheme in the low Mach number regime.     

The limits of Riemann solutions for the isentropic Euler system with extended Chaplygin gas

The Riemann problem for the isentropic Euler system with the state equation for the extended Chaplygin gas is considered, and the Riemann solutions are constructed completely for all the cases. The limiting relations of Riemann solutions for the isentropic Euler system with the state equation from the extended Chaplygin gas to the Chaplygin gas are derived in detail when the corrected term tends to zero. The formation of delta shock wave solution and two-contact-discontinuity solution is investigated during the process of taking the limit.     

二相流计算的一种差分算法

考虑两相流的力学行为,忽略相间的耗散作用,建立了Euler型的基本控制方程．状态方程采用刚性状态方程．基于Abgrall提出的准则,在流动区域内,对可压两相流提出了一个精度较高的Euler型数值方法,数值格式是Godunov型格式,对守恒型和非守恒型方程采用HLLC型和Lax-Friedrichs型近似Riemann解算器,引入了速度驰豫和压强驰豫过程来代替两相间的相互作用．在一维情形下给出数值算例,并且和Saurel的算例进行了比较,结果表明该算法不但精确而且稳定,且在间断处没有数值振荡．     

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.     

The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

The formation of vacuum state and delta shock wave in the solutions to the Riemann problem for the simplified pressureless Euler system is considered under the linear approximations of flux functions. The method is to perturb the non‐strictly hyperbolic system into a nearby strictly hyperbolic system by introducing appropriately the linear approximations of flux functions. The solutions to the Riemann problem for the approximated system can be constructed explicitly and then the formation of vacuum state and delta shock wave can be observed by taking the perturbation parameter tend to zero in the solutions.     

An overview of some recent results on the Euler system of isentropic gas dynamics

This overview is concerned with the well-posedness problem for the isentropic compressible Euler equations of gas dynamics. The results we present are in line with the programof investigatingthe efficiency of different selection criteria proposed in the literature in order to weed out non-physical solutions to more-dimensional systems of conservation laws and they build upon the method of convex integration developed by De Lellis and Székelyhidi for the incompressible Euler equations. Mainly following [5], we investigate the role of the maximal dissipation criterion proposed by Dafermos in [6]: we prove how, for specific pressure laws, some non-standard (i.e. constructed via convex integration methods) solutions to the Riemann problem for the isentropic Euler system in two space dimensions have greater energy dissipation rate than the classical self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos does not favour in general the self-similar solutions.     

一维多介质可压缩流动的守恒型界面追踪方法

本文针对一维问题的ProntTracking方法,提出了一种较易实现的守恒型界面追踪方法.利用双波近似求解Riemann问题来确定界面处的数值通量,在固定的网格上采用统一的有限体积格式进行内点和交界面点的计算,通过守恒插值以及守恒量的重新分配,保证数值解在全场实现一致守恒,将该方法应用于一维多介质可压缩流动的模拟,给出了满意的数值模拟结果.     

Real-time simulation of temperature in hot rolling rolls

Model-based process control is widely used in metal working. Often, simplified real-time capable numerical models replace or support measurement. In this work, we investigate the simulation of temperature in hot rolling rolls based on a 3-D model of a roll which is fast enough to act as soft sensor during operation. The involved discretization of the heat equation in cylindrical space is done via simple finite volumes and propagation in time is done via a forward Euler method. This allows a parallel and efficient implementation for GPGPUs. We validate our code against analytic solutions and measurements, provide a detailed performance analysis, and show simulation results for realistic rolls.     

Van der Waals气体Euler方程的Riemann问题及基本波的相互作用

研究了修正的等熵Van der Waals气体动力学Euler方程Riemann问题及其基本波的相互作用.利用Maxwell提出的等面积法则,将Van der Waals气体状态方程修正为与实际相符,从而守恒律方程组从混合型转化为双曲型.利用广义特征线分析法,构造性地得到了Riemann问题的解是存在的.进一步,得到了基本波相互作用.