一种基于AUSM分裂的真正多维HLL格式

文章给出了一种真正多维的HLL Riemann解算器.采用AUSM分裂将通量分解成为对流通量和压力通量, 其中对流通量的计算采用迎风格式, 压力通量的计算采用HLL格式, 且将HLL格式的耗散项中的密度差用压力差代替, 从而使得格式能够分辨接触间断.为了实现数值格式真正多维的特性, 分别计算了网格界面中点和角点上的数值通量, 并且采用Simpson公式加权组合中点和角点上的数值通量得到网格界面的数值通量.为了减少重构角点处状态时的模板宽度, 计算中采用基于SDWLS梯度的线性重构获得2阶空间精度, 而时间离散采用2阶保强稳Runge-Kutta方法.数值实验表明, 相比于传统的一维HLL格式, 文章的真正多维HLL格式具有能够分辨接触间断, 以及更大的时间步长等优点.与其他能够分辨接触间断的格式(例如HLLC格式)不同, 真正多维的HLL格式在计算二维问题时不会出现激波不稳定现象.      

求解多维欧拉方程的二阶非结构网格混合旋转Riemann求解器

将基于旋转近似Riemann求解器的二阶精度迎风型有限体积方法推广到非结构网格,采用基于网格中心的有限体积法,梯度的计算采用基于节点的方法引入更多的控制体模板,限制器的构造采用与非结构化网格相适应的形式.在求解Riemann问题时,沿具有一定物理意义的两个迎风方向,即控制体界面两侧速度差矢量方向及与之正交的方向.能够完全消除基于Riemann求解器的通量差分裂格式存在的激波不稳定或"红斑"现象.为减小计算量,采用HLL和Roe FDS混合旋转格式.     

运动网格上的水下爆炸数值模拟方法

 在对水下爆炸过程的数值模拟中,把水-气介质界面当作一种特殊的内部边界,并且在该边界上的网格节点都有双重定义,分别对应边界两侧的不同介质状态,通过利用HLLC（Harten,Lax,van Leer,Contact）方法求解任意ALE（Arbitrary Largrangian Eulerian Method,拉格朗日-欧拉方程）方程组,以及运用界面跟踪和动网格方法,对一维球对称水下爆炸模型进行了数值模拟。结果表明,所得到的数值计算结果与实验结果吻合得较好,并且在扩展到二维柱对称水下爆炸模型时,同样得到了满意的结果。     

新型上风格式(AUSM+)的性能分析及其应用

研究了新型上风格式AUSM+的分辨率、效率等性能,并用它与Roe、vanLeer上风格式数值模拟了前向台阶激波反射流动,通过对激波、膨胀波、接触间断及其间相互干扰的复杂波系的模拟对比,分析探讨了AUSM+格式的低数值耗散、间断高分辨率等特性。     

A Switch Function-Based Gas-Kinetic Scheme for Simulation of Inviscid and Viscous Compressible Flows

In this paper, a switch function-based gas-kinetic scheme (SF-GKS) is presented for the simulation of inviscid and viscous compressible flows. With the finite volume discretization, Euler and Navier-Stokes equations are solved and the SF-GKS is applied to evaluate the inviscid flux at cell interface. The viscous flux is obtained by the conventional smooth function approximation. Unlike the traditional gas-kinetic scheme in the calculation of inviscid flux such as Kinetic Flux Vector Splitting (KFVS), the numerical dissipation is controlled with a switch function in the present scheme. That is, the numerical dissipation is only introduced in the region around strong shock waves. As a consequence, the present SF-GKS can well capture strong shock waves and thin boundary layers simultaneously. The present SF-GKS is firstly validated by its application to the inviscid flow problems, including 1-D Euler shock tube, regular shock reflection and double Mach reflection. Then, SF-GKS is extended to solve viscous transonic and hypersonic flow problems. Good agreement between the present results and those in the literature verifies the accuracy and robustness of SF-GKS.     

A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows

In this paper we present a genuinely two-dimensional HLLC Riemann solver. On logically rectangular meshes, it accepts four input states that come together at an edge and outputs the multi-dimensionally upwinded fluxes in both directions. This work builds on, and improves, our prior work on two-dimensional HLL Riemann solvers. The HLL Riemann solver presented here achieves its stabilization by introducing a constant state in the region of strong interaction, where four one-dimensional Riemann problems interact vigorously with one another. A robust version of the HLL Riemann solver is presented here along with a strategy for introducing sub-structure in the strongly-interacting state. Introducing sub-structure turns the two-dimensional HLL Riemann solver into a two-dimensional HLLC Riemann solver. The sub-structure that we introduce represents a contact discontinuity which can be oriented in any direction relative to the mesh.Media player

基于近似Riemann解的有限体积ALE方法

研究二维平面坐标系和二维轴对称坐标系中四边形网格上可压缩流体力学的有限体积ALE(Arbitrary Lagrangian Eulerian)方法.数值方法采用节点中心有限体积法,数值通量采用适用于任意状态方程的HLLC(Harten-Lax-Van Leer-Collela)通量.空间二阶精度通过用WENO(weighted essentially non-oscillatory)方法对原始变量进行重构获得,时间离散采用两步显式Runge-Kutta格式.数值例子显示,方法具有良好的激波分辨能力和高精度的数值逼近能力.     

A Composite Scheme for Gas Dynamics in Lagrangian Coordinates

One cycle of a composite finite difference scheme is defined as several time steps of an oscillatory scheme such as Lax–Wendroff followed by one step of a diffusive scheme such as Lax–Friedrichs. We apply this idea to gas dynamics in Lagrangian coordinates. We show numerical results in two dimensions for Noh's infinite strength shock problem and the Sedov blast wave problem, and for several one-dimensional problems including a Riemann problem with a contact discontinuity. For Noh's problem the composite scheme produces a better result than that obtained with a more conventional Lagrangian code.     

一种适用于磁流体力学切向间断的HLLC黎曼算子

磁场的存在使得磁流体力学特征波模不同于流体力学, 因此直接由流体力学HLLC黎曼算子导出的HLLC双中间态在交界的间断面会出现不守恒的问题。通常降级采用HLL单磁场中间态代替HLLC双磁场中间态以实现守恒和计算稳定, 代价是切向间断的模拟精度不足。本文对此进行改进, 在模拟切向间断时仍然保留原有的HLLC双磁场中间态, 同时各守恒量仍然能够满足Toro相容条件; 改进型HLLC算子在间断两侧的磁场分量存在差异, 因此能够更精确还原切向间断面。基于数值测试, 包括一维激波管和切向间断的时变模拟, 以及地球磁层三维数值模拟, 将模拟结果进行对比, 结果表明: 相比于已发展的HLLC算子, 改进型HLLC算子对切向间断具有更好的捕捉精度, 能够达到或接近耗时更多的HLLD算子的模拟精度。     

Integrating 1D and 2D hydrodynamic,sediment transport model for dam-break flow using finite volume method

The purpose of this study is to set up a dynamically linked 1D and 2D hydrodynamic and sediment transport models for dam break flow.The 1D-2D coupling model solves the generalized shallow water equations,the non-equilibrium sediment transport and bed change equations in a coupled fashion using an explicit finite volume method.It considers interactions among transient flow,strong sediment transport and rapid bed change by including bed change and variable flow density in the flow continuity and momentum equations.An unstructured Quadtree rectangular grid with local refinement is used in the 2D model.The intercell flux is computed by the HLL approximate Riemann solver with shock captured capability for computing the dry-to-wet interface for all models.The effects of pressure and gravity are included in source term in this coupling model which can simplify the computation and eliminate numerical imbalance between source and flux terms.The developed model has been tested against experimental and real-life case of dam-break flow over fix bed and movable bed.The results are compared with analytical solution and measured data with good agreement.The simulation results demonstrate that the coupling model is capable of calculating the flow,erosion and deposition for dam break flows in complicated natural domains.     

CFL Conditions for Runge–Kutta discontinuous Galerkin methods on triangular grids

We study time step restrictions due to linear stability constraints of Runge–Kutta Discontinuous Galerkin methods on triangular grids. The scalar advection equation is discretized in space by the Discontinuous Galerkin method with either the Lax–Friedrichs flux or the upwind flux, and integrated in time with various Runge–Kutta schemes designed for linear wave propagation problems or non-linear applications. Von–Neumann-like analyses are performed on structured periodic grids made up of congruent elements, to investigate the influence of element shape on the stability restrictions. We assess CFL conditions based on different element size measures, among which only the radius of the inscribed circle and the shortest height prove appropriate, although they are not totally independent of the triangle shape. We explain their general behaviour with respect to element quality, and report the corresponding Courant numbers with both types of flux and polynomial order p ranging from 1 to 10, for use as guidelines in practical simulations. We also compare the performance of the Lax–Friedrichs flux and the upwind flux, and we draw general conclusions about the relative computational efficiency of RK schemes. The application of CFL conditions to two examples involving respectively an unstructured and a hybrid grid confirms our results, although it shows that local stability criteria tend to yield too restrictive conditions.     

A high-order accurate hybrid scheme using a central flux scheme and a WENO scheme for compressible flowfield analysis

A high-order accurate hybrid central-WENO scheme is proposed. The fifth order WENO scheme [G.S. Jiang, C.W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202–228] is divided into two parts, a central flux part and a numerical dissipation part, and is coupled with a central flux scheme. Two sub-schemes, the WENO scheme and the central flux scheme, are hybridized by means of a weighting function that indicates the local smoothness of the flowfields. The derived hybrid central-WENO scheme is written as a combination of the central flux scheme and the numerical dissipation of the fifth order WENO scheme, which is controlled adaptively by a weighting function. The structure of the proposed hybrid central-WENO scheme is similar to that of the YSD-type filter scheme [H.C. Yee, N.D. Sandham, M.J. Djomehri, Low-dissipative high-order shock-capturing methods using characteristic-based filters, J. Comput. Phys. 150 (1999) 199–238]. Therefore, the proposed hybrid scheme has also certain merits that the YSD-type filter scheme has. The accuracy and efficiency of the developed hybrid central-WENO scheme are investigated through numerical experiments on inviscid and viscous problems. Numerical results show that the proposed hybrid central-WENO scheme can resolve flow features extremely well.     

一种基于HLLC算法的Mie-Grüneisen多介质混合模型

建立一种修正的HLLC （Harten-Lax-Van Leer-Contact）格式下稳定的Mie-Grüneisen多介质混合计算模型.Mie-Grüneisen混合模型中的通量包括守恒和非守恒两个部分，原始的HLLC格式对守恒部分适用，但是原始的HLLC格式直接用于非守恒部分，很难控制住数值振荡的产生.在原始格式中，间断面的移动速度为间断网格的左侧或右侧速度，修正后替换为网格中的平均速度，经过修正后，对HLLC格式重新进行推导，并随之扩展到二维问题.数值实验表明，利用修改后的HLLC格式，Mie-Grüneisen混合模型可以取得较好的稳定性和准确性.     

A multislope MUSCL method on unstructured meshes applied to compressible Euler equations for axisymmetric swirling flows

A finite volume method for the numerical solution of axisymmetric inviscid swirling flows is presented. The governing equations of the flow are the axisymmetric compressible Euler equations including swirl (or tangential) velocity. A first-order scheme is introduced where the convective fluxes at cell interfaces are evaluated by the Rusanov or the HLLC numerical flux while the geometric source terms are discretizated to provide a well-balanced scheme i.e. the steady-state solutions with null velocity are preserved. Extension to the second-order space approximation using a multislope MUSCL method is then derived. To test the numerical scheme, a stationary solution of the fluid flow following the radial direction has been established with a zero and nonzero tangential velocity. Numerical and exact solutions are compared for classical Riemann problems where we employ different limiters and effectiveness of the multislope MUSCL scheme is demonstrated for strongly shocked axially symmetric flows like in spherical bubble compression problem. Two other tests with axisymmetric geometries are performed: the supersonic flow in a tube with a cone and the axisymmetric blunt body with a free stream.     

High-order kinetic flux vector splitting schemes in general coordinates for ideal quantum gas dynamics

A class of high-order kinetic flux vector splitting schemes are presented for solving ideal quantum gas dynamics based on quantum statistical mechanics. The collisionless quantum Boltzmann equation approach is adopted and both Bose–Einstein and Fermi–Dirac gases are considered. The formulas for the split flux vectors are derived based on the general three-dimensional distribution function in velocity space and formulas for lower dimensions can be directly deduced. General curvilinear coordinates are introduced to treat practical problems with general geometry. High-order accurate schemes using weighted essentially non-oscillatory methods are implemented. The resulting high resolution kinetic flux splitting schemes are tested for 1D shock tube flows and shock wave diffraction by a 2D wedge and by a circular cylinder in ideal quantum gases. Excellent results have been obtained for all examples computed.     

High order conservative Lagrangian schemes with Lax–Wendroff type time discretization for the compressible Euler equations

In this paper, we explore the Lax–Wendroff (LW) type time discretization as an alternative procedure to the high order Runge–Kutta time discretization adopted for the high order essentially non-oscillatory (ENO) Lagrangian schemes developed in 3 and 5. The LW time discretization is based on a Taylor expansion in time, coupled with a local Cauchy–Kowalewski procedure to utilize the partial differential equation (PDE) repeatedly to convert all time derivatives to spatial derivatives, and then to discretize these spatial derivatives based on high order ENO reconstruction. Extensive numerical examples are presented, for both the second-order spatial discretization using quadrilateral meshes [3] and third-order spatial discretization using curvilinear meshes [5]. Comparing with the Runge–Kutta time discretization procedure, an advantage of the LW time discretization is the apparent saving in computational cost and memory requirement, at least for the two-dimensional Euler equations that we have used in the numerical tests.     

High-order finite-volume methods for the shallow-water equations on the sphere

This paper presents a third-order and fourth-order finite-volume method for solving the shallow-water equations on a non-orthogonal equiangular cubed-sphere grid. Such a grid is built upon an inflated cube placed inside a sphere and provides an almost uniform grid point distribution. The numerical schemes are based on a high-order variant of the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) pioneered by van Leer. In each cell the reconstructed left and right states are either obtained via a dimension-split piecewise-parabolic method or a piecewise-cubic reconstruction. The reconstructed states then serve as input to an approximate Riemann solver that determines the numerical fluxes at two Gaussian quadrature points along the cell boundary. The use of multiple quadrature points renders the resulting flux high-order. Three types of approximate Riemann solvers are compared, including the widely used solver of Rusanov, the solver of Roe and the new AUSM⁺-up solver of Liou that has been designed for low-Mach number flows. Spatial discretizations are paired with either a third-order or fourth-order total-variation-diminishing Runge–Kutta timestepping scheme to match the order of the spatial discretization. The numerical schemes are evaluated with several standard shallow-water test cases that emphasize accuracy and conservation properties. These tests show that the AUSM⁺-up flux provides the best overall accuracy, followed closely by the Roe solver. The Rusanov flux, with its simplicity, provides significantly larger errors by comparison. A brief discussion on extending the method to arbitrary order-of-accuracy is included.     

A high-order gas-kinetic Navier–Stokes flow solver

The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge–Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge–Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. Many recently developed high-order methods require a Navier–Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier–Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to construct a dissipative wave structure starting from an initially discontinuous flow condition on a time scale being larger than the particle collision time. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier–Stokes equations (BGK-NS). The novelty for the easy extension from a second-order to a higher order is due to the simple particle transport and collision mechanism on the microscopic level. This paper will present a hierarchy to construct such a high-order method. The necessity to couple spatial and temporal evolution nonlinearly in the flux evaluation can be clearly observed through the numerical performance of the scheme for the viscous flow computations.     

A Positive Conservative Method for Magnetohydrodynamics Based on HLL and Roe Methods

The exact Riemann problem solutions of the usual equations of ideal magnetohydrodynamics (MHD) can have negative pressures, if the initial data has ·B0. This creates a problem for numerical solving because in a first-order finite-volume conservative Godunov-type method one cannot avoid jumps in the normal magnetic field component even if the magnetic field was divergenceless in the three-dimensional sense. We show that by allowing magnetic monopoles in MHD equations and properly taking into account the magnetostatic contribution to the Lorentz force, an additional source term appears in Faraday's law only. Using the Harten–Lax–vanLeer (HLL) Riemann solver and discretizing the source term in a specific manner, we obtain a method which is positive and conservative. We show positivity by extensive numerical experimentation. This MHD-HLL method is positive and conservative but rather diffusive; thus we show how to hybridize this method with the Roe method to obtain a much higher accuracy while still retaining positivity. The result is a fully robust positive conservative scheme for ideal MHD, whose accuracy and efficiency properties are similar to the first order Roe method and which keeps ·B small in the same sense as Powell's method. As a special case, a method with similar characteristics for accuracy and robustness is obtained for the Euler equations as well.     

On phonon focussing in silicon

Phonon flux density in silicon as a function of direction is calculated. It is observed that the flux density corresponding to (quasi-)longitudinal polarization branch exhibits moderate dependence on direction. But in the case of the two (quasi-)transverse branches, the density vs direction curves are characterized with sharp peaks indicating strong concentration of phonons in certain directions, and correspondingly there is marked deconcentration in other directions.