An extension of Osher's Riemann solver for chemical and vibrational non-equilibrium gas flows

In this paper we study an extension of Osher's Riemann solver to mixtures of perfect gases whose equation of state is of the form encountered in hypersonic applications. As classically, one needs to compute the Riemann invariants of the system to evaluate Osher's numerical flux. For the case of interest here it is impossible in general to derive simple enough expressions which can lead to an efficient calculation of fluxes. The key point here is the definition of approximate Riemann invariants to alleviate this difficulty. Some of the properties of this new numerical flux are discussed. We give 1D and 2D applications to illustrate the robustness and capability of this new solver. We show by numerical examples that the main properties of Osher's solver are preserved; in particular, no entropy fix is needed even for hypersonic applications.     

A contribution to the great Riemann solver debate

The aims of this paper are threefold: to increase the level of awareness within the shock-capturing community of the fact that many Godunov-type methods contain subtle flaws that can cause spurious solutions to be computed; to identify one mechanism that might thwart attempts to produce very-high-resolution simulations; and to proffer a simple strategy for overcoming the specific failings of individual Riemann solvers.     

A linearized Riemann solver for the steady supersonic Euler equations

A very simple linearization of the solution to the Riemann problem for the steady supersonic Euler equations is presented. When used locally in conjunction with the Godunov method, computing savings by a factor of about four relative to the use of exact Riemann solvers can be achieved. For severe flow regimes, however, the linearization loses accuracy and robustness. We then propose the use of a Riemann solver adaptation procedure. This retains the accuracy and robustness of the exact Riemann solver and the computational efficiency of the cheap linearized Riemann solver. Numerical results for two- and three-dimensional test problems are presented.     

Harten-Lax-van Leer-contact (HLLC) approximation Riemann solver with elastic waves for one-dimensional elastic-plastic problems

A Harten-Lax-van Leer-contact (HLLC) approximate Riemann solver is built with elastic waves (HLLCE) for one-dimensional elastic-plastic flows with a hypoelastic constitutive model and the von Mises’ yielding criterion. Based on the HLLCE, a third-order cell-centered Lagrangian scheme is built for one-dimensional elastic-plastic problems. A number of numerical experiments are carried out. The numerical results show that the proposed third-order scheme achieves the desired order of accuracy. The third-order scheme is used to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves. The numerical results are compared with a reference solution and the results obtained by other authors. The comparison shows that the presented high-order scheme is convergent, stable, and essentially non-oscillatory. Moreover, the HLLCE is more efficient than the two-rarefaction Riemann solver with elastic waves (TRRSE).     

Calculations of 3D compressible flows using an efficient low diffusion upwind scheme

A newly suggested E‐CUSP upwind scheme is employed for the first time to calculate 3D flows of propulsion systems. The E‐CUSP scheme contains the total energy in the convective vector and is fully consistent with the characteristic directions. The scheme is proved to have low diffusion and high CPU efficiency. The computed cases in this paper include a transonic nozzle with circular‐to‐rectangular cross‐section, a transonic duct with shock wave/turbulent boundary layer interaction, and a subsonic 3D compressor cascade. The computed results agree well with the experiments. The new scheme is proved to be accurate, efficient and robust for the 3D calculations of the flows in this paper. Copyright © 2005 John Wiley & Sons, Ltd.     

A finite volume scheme for two-phase compressible flows

In gas-particle two-phase flows, when the concentration of the disperesed phase is low, certain assumptions may be made which simplify considerably the equations one has to solve. The gas and particle flows are then linked only via the interaction terms. One may therefore uncouple the full system of equations into two subsystems: one for the gas phase, whose homogeneous part reduces to the Euler equations; and a second system for the particle motion, whose homogeneous part is a degenerate hyperbolic system. The equations governing the gas phase flow may be solved using a high-resolution scheme, while the equations describing the motion of the dispersed phase are treated by a donor-cell method using the solution of a particular Riemann problem. Coupling is then achieved via the right-hand-side terms. To illustrate the capabilities of the proposed method, results are presented for a case specially chosen from among the most difficult to handle, since it involves certain geometrical difficulties, the treatment of regions in which particles are absent and the capturing of particle fronts.     

An approximate‐state Riemann solver for the two‐dimensional shallow water equations with porosity

PorAS, a new approximate‐state Riemann solver, is proposed for hyperbolic systems of conservation laws with source terms and porosity. The use of porosity enables a simple representation of urban floodplains by taking into account the global reduction in the exchange sections and storage. The introduction of the porosity coefficient induces modified expressions for the fluxes and source terms in the continuity and momentum equations. The solution is considered to be made of rarefaction waves and is determined using the Riemann invariants. To allow a direct computation of the flux through the computational cells interfaces, the Riemann invariants are expressed as functions of the flux vector. The application of the PorAS solver to the shallow water equations is presented and several computational examples are given for a comparison with the HLLC solver. Copyright © 2009 John Wiley & Sons, Ltd.     

The Osher scheme for non-equilibrium reacting flows

An extension of the Osher upwind scheme to non-equilibrium reacting flows is presented, Owing to the presence of source terms, the Riemann problem is no longer self-similar and therefore its approximate solution becomes tedious. With simplicity in mind, a linearized approach which avoids an iterative solution is used to define the intermediate states and sonic points. The source terms are treated explicitly. Numerical computations are presented to demonstrate the feasibility, efficiency and accuracy of the proposed method. The test problems include a ZND (Zeldovich-Neumann-Doring) detonation problem for which spurious numerical solutions which propagate at mesh speed have been observed on coarse grids. With the present method, a change of limiter causes the solution to change from the physically correct CJ detonation solution to the spurious weak detonation solution.     

用鲁棒Riemann求解器和运动重叠网格计算旋翼粘性绕流

发展了一种基于鲁棒Riemann求解器和运动重叠网格技术计算直升机悬停旋翼流场的方法。基于惯性坐标系,悬停旋翼流场是非定常流场,控制方程为可压缩Reynolds平均Navier-Stoke方程,其对流项采用Roe近似Reimann求解器离散,使用改进的五阶加权基本无振荡格式进行高阶重构,非定常时间推进采用含牛顿型LUSGS子迭代的全隐式双时间步方法。为实施旋转运动和便于捕捉尾迹,计算采用运动重叠网格技术。计算得到的桨叶表面压力分布及桨尖涡涡核位置都与实验结果吻合较好。数值结果表明:所发展方法对桨尖涡具有较高的分辨率,对激波具有较好的捕捉能力,该方法可进一步推广到前飞旋翼粘性绕流的计算。     

An approximate solution of the Riemann problem for a realisable second-moment turbulent closure

Abstract. An approximate solution of the Riemann problem associated with a realisable and objective turbulent second-moment closure, which is valid for compressible flows, is examined. The main features of the continuous model are first recalled. An entropy inequality is exhibited, and the structure of waves associated with the non-conservative hyperbolic convective system is briefly described. Using a linear path to connect states through shocks, approximate jump conditions are derived, and the existence and uniqueness of the one-dimensional Riemann problem solution is then proven. This result enables to construct exact or approximate Riemann-type solvers. An approximate Riemann solver, which is based on Gallou?t's recent proposal is eventually presented. Some computations of shock tube problems are then discussed. Received 2 March 1999 / Accepted 24 August 2000     

A problem for diffusion with relaxation in polymers

A nonlinear problem for penetrant diffusion with relaxation in polymers is considered. A numerical approach to solving this type of problems is developed. The proposed numerical scheme based on a finite element domain approximation and a time difference method can be used for numerical simulation of the considered penetrant diffusion in 2-D and 3-D domains. A numerical procedure and a corresponding computer code are created and tested for some examples in 1-D and 2-D domains.     

A Riemann solver for the two-dimensional MHD equations

This paper presents how the equations of magnetohydrodynamics (MHD) in primitive form should be written in conservative form with the inclusion of a divergence source along with a divergence wave and how a physically correct sonic fix can be embedded directly in the fluxes. The numerical scheme was applied to a blast wave problem in which a circular energetic plasma is released in a free and magnetized medium with a reflected wall. The results show that the method with the new sonic fix can handle the divergence condition on the magnetic field and produces an almost uniform shock compression in all directions, resolving the shocks and discontinuities rather sharply. © 1997 John Wiley & Sons, Ltd.     

管系内一维可压缩流动数值建模与特性分析

针对复杂管系内可压缩流体,基于有限体积法,采用HLLC(Harten-Lax-van Leer Contact)格式和黎曼求解器构建了有限控制体数值离散方法,引入虚拟节点用于连接有限控制体,借助虚拟节点给出控制体之间数值通量的计算格式,发展了一种管道内一维流动数值建模方法.针对含有分支管路的管系,在管道连接部位构建了分支管路拟一维流动数值计算模型.基于所发展的一维流动数值方法,建立了变径管道和含60°分支管道内流动计算模型,验证了该方法的收敛性和有效性;基于虚拟节点的数值格式处理变径管激波问题具有一定精度优势.研究了变径管和分支管模型中可压缩流体激波、稀疏波等的传播机理,分析了管径对相邻支管压力的影响,为工程管路设计提供了参考.     

Variable density bore interaction with block obstacles

A horizontally variable density flow model is used to simulate hydraulic bore interactions with idealised urban obstacles. The 2D non-linear shallow water equations are solved using a second-order Monotonic Upstream-centered Schemes for Conservation Laws-Hancock Godunov-type HLLC approximate Riemann scheme. Validation test results are reported for wave propagation over a hump, a constant-density circular dam break and two 1D dam breaks involving different spatial distributions of solute concentration. Detailed parameter studies are then considered for hydraulic bore interactions with single and multiple-square obstacles under subcritical, critical and supercritical flow conditions. In all cases, reflected and diffracted wave patterns are generated immediately after the bore impacts the obstacle(s). Later, the incident bore reconstitutes itself downstream of the obstacle(s). Variable density flows are also considered, with the upstream volumetric concentrations set to values corresponding to water–sediment mixture densities of 1165 and 1495 kg/m³. It is found that the upstream Froude number, gap spacing between obstacles and upstream to downstream density difference influence the strength of the bore–structure interaction, run-up at the front face of the obstacle(s), and subsequent wave–wave interactions.     

A Riemann solver and upwind methods for a two‐phase flow model in non‐conservative form

We present a theoretical solution for the Riemann problem for the five‐equation two‐phase non‐conservative model of Saurel and Abgrall. This solution is then utilized in the construction of upwind non‐conservative methods to solve the general initial‐boundary value problem for the two‐phase flow model in non‐conservative form. The basic upwind scheme constructed is the non‐conservative analogue of the Godunov first‐order upwind method. Second‐order methods in space and time are then constructed via the MUSCL and ADER approaches. The methods are systematically assessed via a series of test problems with theoretical solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Flux difference splitting for the Euler equations in one spatial co-ordinate with area variation

An approximate (linearized) Riemann solver is presented for the solution of the Euler equations of gas dynamics in one spatial co-ordinate. This includes cylindrically and spherically symmetric geometries and also applies to narrow ducts with area variation. The method is Roe's flux difference splitting with a technique for dealing with source terms. The results of two problems, a spherically divergent infinite shock and a converging cylindrical shock, are presented. The numerical results compare favourably with those of Noh's recent survey and also with those of Ben-Artzi and Falcovitz using a more complicated Riemann solver.     

Simple treatment of non‐aligned boundaries in a Cartesian grid shallow flow model

A simple method is proposed for treating curved or irregular boundaries in Cartesian grid shallow flow models. It directly evaluates fictional values in ‘ghost’ cells adjacent to boundary cells and requires no interpolation or generation of cut cells. The boundary treatment is implemented in a dynamically adaptive quadtree grid‐based solver of the hyperbolic shallow water equations and validated against several test cases with analytical or alternative numerical solutions. The method is easy to code, accurate, and demonstrably effective in dealing with irregular computational domains in shallow flow simulations. Results are presented for still water in a basin of complicated geometry, steady hydraulic jump in an open channel with a converging sidewall, wind‐induced circulation in a circular shallow lake, and shock wave diffraction in a channel containing a contraction and expansion. Copyright © 2007 John Wiley & Sons, Ltd.