Two-stage fourth-order gas-kinetic scheme for three-dimensional Euler and Navier-Stokes solutions

ABSTRACT

For the one-stage third-order gas-kinetic scheme (GKS), successful applications have been achieved for the three-dimensional compressible flows [Pan, L., K. Xu, Q. Li, and J. Li. 2016. “An Efficient and Accurate Two-stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations.” Journal of Computational Physics 326: 197–221]. The high-order accuracy of the scheme is obtained by integrating a multidimensional time-accurate gas distribution function over the cell interface within a time step without using Gaussian quadrature points and Runge-Kutta time-stepping technique. However, to the further increase of the order of the scheme, such as the fourth-order one, the one step formulation becomes very complicated for the multidimensional flow. Recently, a two-stage fourth-order GKS with high efficiency has been constructed for two-dimensional inviscid and viscous flow computations ([Li, J., and Z. Du. 2016. “A Two-stage Fourth Order Time-accurate Discretization for Lax-Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws.” SIAM Journal on Scientific Computing 38: 3046–3069]; Pan et al. 2016), and the scheme uses the time accurate flux function and its time derivatives. In this paper, a fourth-order GKS is developed for the three-dimensional flows under the two-stage framework. Based on the three-dimensional WENO reconstruction and flux evaluation at Gaussian quadrature points on a cell interface, the high-order accuracy in space is achieved first. Then, the two-stage time stepping method provides the high accuracy in time. In comparison with the formal third-order GKS [Pan, L., and K. Xu. 2015. “A Third-order Gas-kinetic Scheme for Three-dimensional Inviscid and Viscous Flow Computations.” Computers & Fluids 119: 250–260], the current fourth-order method not only improves the accuracy of the scheme, but also reduces the complexity of the gas-kinetic flux solver greatly. More importantly, the fourth-order GKS has the same robustness as the second-order shock capturing scheme [Xu, K. 2001. “A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and its Connection with Artificial Dissipation and Godunov Method.” Journal of Computational Physics 171: 289–335]. Numerical results validate the outstanding reliability and applicability of the scheme for three-dimensional flows, such as the cases related to turbulent simulations.     

THREE HIGH-ORDER SPLITTING SCHEMES FOR 3D TRANSPORT EQUATION

IntroductionWith the development of modern industry, various pollutants discharge into the air,rivers, lakes and oceans, which makes the environmental qualities worse and has bad effectson the mankind’s health and the sustained development of industry an…     

GENERALIZED FINITE SPECTRAL METHOD FOR 1D BURGERS AND KDV EQUATIONS

A generalized finite spectral method is proposed. The method is of high-order accuracy. To attain high accuracy in time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Her-mite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection- diffusion problem) and KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.     

Simulation of three-dimensional incompressible flows in generalized curvilinear coordinates using a high-order compact finite-difference lattice Boltzmann method

In the present study, a high-order compact finite-difference lattice Boltzmann method is applied for accurately computing 3-D incompressible flows in the generalized curvilinear coordinates to handle practical and realistic geometries with curved boundaries and nonuniform grids. The incompressible form of the 3-D nineteen discrete velocity lattice Boltzmann method is transformed into the generalized curvilinear coordinates. Herein, a fourth-order compact finite-difference scheme and a fourth-order Runge-Kutta scheme are used for the discretization of the spatial derivatives and the temporal term, respectively, in the resulting 3-D nineteen discrete velocity lattice Boltzmann equation to provide an accurate 3-D incompressible flow solver. A high-order spectral-type low-pass compact filtering technique is applied to have a stable solution. All boundary conditions are implemented based on the solution of the governing equations in the 3-D generalized curvilinear coordinates. Numerical solutions of different 3-D benchmark and practical incompressible flow problems are performed to demonstrate the accuracy and performance of the solution methodology presented. Herein, the 2-D cylindrical Couette flow, the decay of a 3-D double shear wave, the cubic lid-driven cavity flow with nonuniform grids, the flow through a square duct with 90° bend and the flow past a sphere at different flow conditions are considered for validating the present computations. Numerical results obtained show the accuracy and robustness of the present solution methodology based on the implementation of the high-order compact finite-difference lattice Boltzman method in the generalized curvilinear coordinates for solving 3-D incompressible flows over practical and realistic geometries.     

Verification of higher-order discontinuous Galerkin method for hexahedral elements

A high-order implementation of the Discontinuous Galerkin (dg) method is presented for solving the three-dimensional Linearized Euler Equations on an unstructured hexahedral grid. The method is based on a quadrature free implementation and the high-order accuracy is obtained by employing higher-degree polynomials as basis functions. The present implementation is up to fourth-order accurate in space. For the time discretization a four-stage Runge–Kutta scheme is used which is fourth-order accurate. Non-reflecting boundary conditions are implemented at the boundaries of the computational domain.The method is verified for the case of the convection of a 1D compact acoustic disturbance. The numerical results show that the rate of convergence of the method is of order $p+1$ in the mesh size, with p the order of the basis functions. This observation is in agreement with analysis presented in the literature. To cite this article: H. Özdemir et al., C. R. Mecanique 333 (2005).     

A Runge-Kutta–based WENO reconstruction on moving mesh for compressible fluids

In this paper, we construct a high-order moving mesh method based on total variation diminishing Runge-Kutta and weighted essential nonoscillatory reconstruction for compressible fluid system. Beginning with the integral form of fluid system, we get the semidiscrete system with an arbitrary mesh velocity. We use weighted essential nonoscillatory reconstruction to get the space accuracy on moving meshes, and the time accuracy is obtained by modified Runge-Kutta method; the mesh velocity is determined by moving mesh method. One- and two-dimensional numerical examples are presented to demonstrate the efficient and accurate performance of the scheme.     

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries (KdV) equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation (nonlinear convectiondiffusion problems) and the KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.     

对流扩散方程的摄动有限体积(PFV)方法及讨论

在有限体积(FV)方法的重构近似中,引入数值摄动处理,即把界面数值通量摄动展开成网格间距的幂级数,并利用积分方程自身的性质求出幂级数的系数,同时获得高精度迎风和中心型摄动有限体积(PFV)格式.对标量输运方程给出积分近似为二阶、重构近似为二、三和四阶迎风和中心型PFV格式,这些PFV格式的结构形式及使用基点数与一阶迎风格式完全一致,迎风PFV格式满足对流有界准则;二阶和四阶中心PFV格式对网格Peclet数的任意值均为正型格式,比常用的二阶中心格式优越.用一维标量输运和方腔流动算例说明PFV格式的优良性能,并把PFV方法与性质相近的摄动有限差分(PFD)方法及相关的高精度方法作了对比分析.     

气体动理学格式研究进展

介绍了近年来气体动理学格式(gas-kinetic scheme, GKS, 亦简称BGK 格式) 的主要研究进展, 重点是高阶精度动理学格式及适合从连续流到稀薄流全流域的统一动理学格式. 通过对速度分布函数的高阶展开和对初值的高阶重构, 构造了时间和空间均为三阶精度的气体动理学格式. 研究表明, 相比于传统的基于Riemann 解的高阶格式, 新格式不仅考虑了网格单元界面上物理量的高阶重构, 而且在初始场的演化阶段耦合了流体的对流和黏性扩散, 也能够保证解的高阶精度. 该研究为高精度计算流体力学(computatial uiddymamics, CFD) 格式的建立提供了一条新的途径. 通过分子离散速度空间直接求解Boltzmann 模型方程,在每个时间步长内将宏观量的更新和微观气体分布函数的更新紧密地耦合在一起, 建立了适合任意Knudsen(kn) 数的统一格式, 相比于已有的直接离散格式具有更高的求解效率. 最后, 本文还讨论了合理的物理模型对数值方法的重要性. 气体动理学方法的良好性能来自于Boltzmann 模型方程对计算网格单元界面上初始间断的时间演化的准确描述. 气体自由运动与碰撞过程的耦合是十分必要的. 通过分析数值激波层内的耗散机制,我们认识到采用Euler 方程的精确Riemann 解作为现代可压缩CFD 方法的基础具有根本的缺陷, 高马赫数下的激波失稳现象不可避免. 气体动理学格式为构造数值激波结构提供了一个重要的可供参考的物理机制.      

AUFS 格式在无网格方法中的应用简

将计算量小,激波分辨率高的AUFS (artificially upstream flux vector splitting) 格式应用于无网格方法. 所发展算法基于多项式基函数最小二乘无网格方法,采用线性基函数曲面拟合及AUFS 格式计算各离散点的空间导数,应用四阶Runge-Kutta 法进行时间显式推进. 为验证算法健壮性、精度以及计算效率,对Riemann 问题、超音速平面流动,以及不同攻角NACA0012 翼型跨音速流场进行了数值模拟,其结果同采用HLLC (Harten-Lax-van Leer-contact) 格式的无网格方法以及文献报道结果吻合较好,并且计算量较形式简单HLLC 格式减少约15%.     

Use of characteristics method with cubic interpolation for unsteady-flow computation

The specified-time-interval (STI) scheme has been used commonly in applying the method of characteristics (MOC) to unsteady open-channel flow problems. However, with the use of STI scheme, the numerical error for the simulation results can always be induced due to the interpolation used to approximate the characteristics trajectory. Hence, in order to remedy the numerical errors caused by the interpolation, one needs to seek some kind of interpolation technique with higher-order accuracy. Instead of the linear interpolation technique, which has been used very commonly and can induce serious numerical diffusion, the Holly--Preissmann two-point, method, which is a cubic interpolation technique with fourth-order of accuracy, is proposed here to integrate with the method of characteristics for the computation of one-dimensional unsteady flow in open channel. The concept of reachback and reachout in space and time directions for the characteristics is also introduced to assure the model stability. The computed results from this new model are compared with those computed by using the Preissmann four-point scheme and the multimode method of characteristics with linear interpolation.     

一类高精度TVD差分格式及其应用

构造了一维非线性双曲型守恒律的一个新的高精度、高分辨率的守恒型TvD差分格式。其构造思想是：首先，将计算区间划分为若干个互不相交的小区间，再根据精度要求等分小区间，通过各细小区间上的单元平均状态变量，重构各细小区间交界面上的状态变量，并加以校正；其次，利用近似Riemann解计算细小区间交界面上的数值通量，并结合高阶Runge—Kutta TVD方法进行时间离散，得到了高精度的全离散方法。证明了该格式的TVD特性。该格式适合于使用分量形式计算而无须进行局部特征分解。通过计算几个典型的问题，验证了格式具有高精度、高分辨率且计算简单的优点。     

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.     

Cavitating/non-cavitating flows simulation by third-order finite volume scheme and power-law preconditioning method

Equations of steady inviscid and laminar flows are solved by means of a third-order finite volume (FV) scheme. For this purpose, a cell-centered discretization technique is employed. In this technique, the flow parameters at the cell faces are computed using a third-order weighted averages procedure. A fourth-order artificial dissipation is used for stability of the solution. In order to achieve the steady-state situation, four-step Runge-Kutta explicit time integration method is applied. An advanced progressive preconditioning method, named the power-law preconditioning method, is used for faster convergence. In this method, the preconditioning matrix is adjusted automatically from the velocity and/or pressure flow-field by a power-law relation. Attention is directed towards accuracy and convergence of the schemes. The results presented in the paper focus on steady inviscid and laminar flows around sheet-cavitating and fully-wetted bodies including hydrofoils and circular/elliptical cylinder. Excellent agreements are obtained when numerical predictions are compared with other available experimental and numerical results. In addition, it is found that using the power-law preconditioner significantly increases the numerical convergence speed.     

非线性双曲型守恒律的高精度MmB差分格式

构造了一维非线性双曲型守恒律方程的一个高精度、高分辨率的广义G odunov型差分格式。其构造思想是:首先将计算区间划分为若干个互不相交的小区间,再根据精度要求等分小区间,通过各细小区间上的单元平均状态变量,重构各等分小区间交界面上的状态变量,并加以校正;其次,利用近似R iem ann解算子求解细小区间交界面上的数值通量,并结合高阶R unge-K u tta TVD方法进行时间离散,得到了高精度的全离散方法。证明了该格式的Mm B特性。然后,将格式推广到一、二维双曲型守恒方程组情形。最后给出了一、二维Eu ler方程组的几个典型的数值算例,验证了格式的高效性。     

Simulations of the unsteady separated flow past a normal flat plate

Well-resolved two-dimensional numerical simulations of the unsteady separated flow past a normal flat plate at low Reynolds numbers have been performed using a fractional step procedure with high-order spatial discretization. A fifth-order upwind-biased scheme is used for the convective terms and the diffusive terms are represented by a fourth-order central difference scheme. The pressure Poisson equation is solved using a direct method based on eigenvalue decomposition of the coefficient matrix. A systematic study of the flow has been conducted with high temporal and spatial resolutions for a series of Reynolds numbers. The interactions of the vortices shed form the shear layers in the near-and far-wake regions are studied. For Reynolds numbers less than 250 the vortices are observed to convect parallel to the freestream. However, at higher Reynolds numbers (500 and 1000), complex interactions including vortex pairing, tearing and deformations are seen to occur in the far-wake region. Values of the drag coefficient and the wake closure length are presented and compared with previous experimental and numerical studies.     

Application of a fourth-order relaxation scheme to hyperbolic systems of conservation laws

A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourth-order central weighted essentially nonoscillatory (CWENO) reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments. The project supported by the National Natural Science Foundation of China (60134010) The English text was polished by Yunming Chen.     

High-order divergence-free velocity reconstruction for free surface flows on unstructured Voronoi meshes

In this paper, we present an efficient semi-implicit scheme for the solution of the Reynolds-averaged Navier-Stokes equations for the simulation of hydrostatic and nonhydrostatic free surface flow problems. A staggered unstructured mesh composed by Voronoi polygons is used to pave the horizontal domain, whereas parallel layers are adopted along the vertical direction. Pressure, velocity, and vertical viscosity terms are taken implicitly, whereas the nonlinear convective terms as well as the horizontal viscous terms are discretized explicitly by using a semi-Lagrangian approach, which requires an interpolation of the three-dimensional velocity field to integrate the flow trajectories backward in time. To this purpose, a high-order reconstruction technique is proposed, which is based on a constrained least squares operator that guarantees a globally and pointwise divergence-free velocity field. A comparison with an analogous reconstruction, which is not divergence-free preserving, is also presented to give evidence of the new strategy. This allows the continuity equation to be satisfied up to machine precision even for high-order spatial discretizations. The reconstructed velocity field is then used for evaluating high-order terms of a Taylor method that is here adopted as ODE integrator for the flow trajectories. The proposed semi-implicit scheme is validated against a set of academic test problems, and proof of convergence up to fourth-order of accuracy in space is shown.     

Fhree-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method, a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper. Numerical characteristics of the scheme are studied by the Fourier analysisl Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convectiondiffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.     

A hybrid scheme for the numerical simulation of shock/discontinuity problems

To address accuracy issues for direct numerical simulation, a hybrid scheme based on the weighted compact scheme (WCS) and weighted essentially non-oscillatory (WENO) scheme is developed. The new hybrid method incorporates the advantages of both schemes. Time integration is performed using the fourth-order total variation diminishing Runge–Kutta method with a characteristic filter. The accuracy of the scheme is assessed using several benchmark problems. Results show that the proposed scheme produces a more accurate solution for problems involving shocks and discontinuities in comparison with the traditional shock-capturing methods.