一种隐式预处理方法及其在定常和非定常流动数值模拟中的应用

将Choi-Merkle矩阵预处理方法与LU-SGS隐式方法、双时间法以及多重网格方法结合,发展适用于绕飞行器定常和非定常粘性流动的高效隐式预处理计算方法和程序.介绍一种针对定常和非定常流动的LU-SGS隐式预处理方法的统一表述方法.在不改变流动解的前提下,对Navier-Stokes方程的伪时间导数项实施Choi-Merkle矩阵预处理,从而改善可压缩控制方程在低速情况下的系统刚性,使基于LU-SGS时间推进格式的数值模拟方法同时适用于从极低马赫数到可压缩范围内的数值模拟.对Jameson中心格式的人工粘性进行相应的修改,以提高低速流动的计算精度.翼型、机翼以及翼身组合体绕流的数值模拟研究表明,隐式预处理方法获得了很高的计算效率,可使马赫数0.1左右的低速流动计算时间减少50%以上;通过对现有可压缩计算程序进行小量改动,便可使其均匀覆盖整个低速流动范围,提高CFD程序在飞行器绕流数值模拟中的实用性.     

求解非定常不可压N-S方程的预处理方法

应用预处理技术,对不可压非定常N-S方程使用双时间推进法求解.当沿物理时间层推进时,连续性方程和动量方程沿伪时间方向使用隐式线Gauss-Seidel迭代法求解,对流项采用三阶迎风差分法离散.通过对不同Reynolds数、不同深宽比下非定常驱动腔内流动的模拟,数值研究了预处理法计算非定常不可压粘性流动的收敛特性,分析了沿伪时间层的迭代收敛速度对流场Reynolds数的依赖特征.     

隐式格式求解拟压缩性非定常不可压Navier-Stokes方程

采用Rogers发展的双时间步拟压缩方法,数值求解不可压非定常问题.数值通量分别采用三阶精度Roe格式和二阶精度Harten-Yee的TVD格式离散.为了加快收敛,提高求解效率,试验了几种隐式格式(ADI-LU,LGS,LU-SGS).针对经典的低雷诺数(Re=200)圆柱绕流问题,比较了不同隐式方法的计算结果和求解效率,以及两种数值离散格式计算结果的异同.最后采用Roe格式数值求解了两种典型的低速非定常流动问题:绕转动圆柱(ω=1)低雷诺数流动;NACA0015翼型等速拉起数值模拟.     

基于多步高阶差分格式求解时域Maxwell方程

提出基于无穷维哈密尔顿系统及分裂算子理论的多步高阶差分格式,求解时域Maxwell方程.在时间方向上,针对Maxwell方程采用不同阶数的辛算法进行差分离散;在空间方向上,采用四阶差分格式进行差分离散.探讨多步高阶差分格式的稳定性及数值色散性,最后给出数值计算结果.结果表明,五级四阶格式为最有效的多步高阶差分格式,具有高精度、占用较少的计算机资源等优点,适用于长时间的数值模拟.     

分区拼接网格算法数值模拟超声速复杂流场

以分区拼接网格数值模拟方法为研究对象,对二阶Godunov方法在拼接网格中流场的数值模拟进行了研究,发展了适用于Godunov格式的通量守恒型算法,结合二阶Godunov有限体积法离散非定常Euler方程,数值模拟了捆绑火箭及航天飞机的超声速流场,计算结果正确描述了流场中的激波相交、反射等干扰特性.     

基于非结构网格的高精度投影法

本文提出了一种基于非结构同位网格的求解非定常不可压缩流动的高精度投影算法。采用单元中心非结构网格,利用动量插值方法实现同位网格上的压力速度耦合,对流项和扩散项的时间离散均采用C-N格式,空间离散则分别采用QUICK格式和中心差分。运用二维衰减涡流动、圆柱绕流和顶盖振荡驱动流等经典算例对算法进行了考核,结果表明本文算法与实验结果或经典数值解良好吻合,时间和空间均达到了二阶以上的收敛精度。     

三维定常问题的高阶紧致差分格式向非定常问题的推广

将已经建立的求解三维定常对流扩散方程的高阶紧致差分格式直接推广到三维非定常对流扩散方程的数值求解,时间导数项利用二阶向后欧拉差分公式,所得到的高阶隐式紧致差分格式时间为二阶精度,空间为四阶精度,并且是无条件稳定的.数值实验结果验证了本文方法的精确性和稳健性.     

应用GAO-YONG可压缩湍流模式数值模拟RAE2822翼型绕流

应用Gao-Yong可压缩湍流模式,数值模拟RAE2822二维翼型在两种不同来流情况下的跨音速粘性绕流问题.湍流模式的对流项用ROE格式离散,扩散项用中心差分格式离散,空间离散后的控制方程用多步Runge-Kutta显式时间推进格式求解.计算结果预测了翼型表面的压力系数的分布、平均速度剖面、激波的位置、马赫数等值线等情况.同时,对翼型表面激波与边界层相互干扰以及转捩问题进行分析计算,结果表明,Gao-Yong可压缩湍流模式结合适当的数值方法能够成功地模拟翼型跨音速粘性流动.最后,基于Gao-Yong可压缩湍流模式各项异性湍流粘性的机理,初步提出一种预测转捩起始位置的方法.     

带副翼的翼身组合体绕流的Euler和N-S方程解

将对接分区网格与分区求解算法结合,有效地求解了带副翼偏转的翼身组合体绕流的N S方程.数值方法中选用VanLeer分裂格式离散无粘通量项,采用中心差分格式来离散粘性通量项.分区交界面采用了一种满足通量守恒的内边界耦合条件.数值算例表明该方法是求解带操纵面偏转的翼身组合体绕流的有效方法.     

基于气体动理论的二维Karman涡街数值模拟

基于从稀薄流到连续流的跨流域气体动理论统一算法（gas-kinetic unified algorithm,GKUA）,通过数值求解考虑转动自由度激发的Boltzmann-Rykov模型方程,得到了一种跨流域非定常流动数值模拟的方法.该求解方法以Boltzmann模型方程为控制方程,在常温状态下如果考虑转动能激发的情况则选用Rykov模型.文中数值求解Rykov模型时,首先基于转动能模对速度分布函数积分以消去分子转动能量这一自变量,在速度空间应用自适应离散速度坐标法与数值积分演化更新计算技术,在位置空间应用3阶WENO空间离散格式和3阶显式Runge-Kutta时间推进.针对经典的二维Karman涡街流动现象进行数值模拟,说明该跨流域非定常流动模拟算法对于连续流区低速流动的适应性.      

High-resolution viscous flow simulations at arbitrary Mach number

A characteristic-based unsteady viscous flow solver is developed with preconditioning that is uniformly applicable for Mach numbers ranging from essentially incompressible to supersonic. A preconditioned flux-difference formulation for nondimensional primitive variables is a key element of the present approach. The simple primitive-variable numerical flux is related to Roe’s flux-difference scheme and preserves contact discontinuities using primitive variables, with or without preconditioning. Preconditioning by a single-parameter diagonal matrix conditions the system eigenvalues in terms of nondimensional local velocity and local temperature. An iterative implicit solution algorithm is given for the preconditioned formulation and is used for several simple test and validation cases. These include an inviscid shock-tube case, flat-plate boundary layer flow at low Mach number, viscous flow past a circular cylinder at low Reynolds number and with different thermal boundary conditions, and validation cases for incompressible and transonic flows.     

A hybrid particle approach for continuum and rarefied flow simulation

A hybrid particle scheme is presented for the simulation of compressible gas flows involving both continuum regions and rarefied regions with strong translational nonequilibrium. The direct simulation Monte Carlo (DSMC) method is applied in rarefied regions, while remaining portions of the flowfield are simulated using a DSMC-based low diffusion particle method for inviscid flow simulation. The hybrid scheme is suitable for either steady state or unsteady flow problems, and can simulate gas mixtures comprising an arbitrary number of species. Numerical procedures are described for strongly coupled two-way information transfer between continuum and rarefied regions, and additional procedures are outlined for the determination of continuum breakdown. The hybrid scheme is evaluated through a comparison with DSMC simulation results for a Mach 6 flow of N₂ over a cylinder, and good overall agreement is observed. Large potential efficiency gains (over three orders of magnitude) are estimated for the hybrid algorithm relative to DSMC in a simple example involving a rarefied expansion flow through a small nozzle into a vacuum chamber.     

A discontinuous Galerkin method for inviscid low Mach number flows

In this work we extend the high-order discontinuous Galerkin (DG) finite element method to inviscid low Mach number flows. The method here presented is designed to improve the accuracy and efficiency of the solution at low Mach numbers using both explicit and implicit schemes for the temporal discretization of the compressible Euler equations. The algorithm is based on a classical preconditioning technique that in general entails modifying both the instationary term of the governing equations and the dissipative term of the numerical flux function (full preconditioning approach). In the paper we show that full preconditioning is beneficial for explicit time integration while the implicit scheme turns out to be efficient and accurate using just the modified numerical flux function. Thus the implicit scheme could also be used for time accurate computations. The performance of the method is demonstrated by solving an inviscid flow past a NACA0012 airfoil at different low Mach numbers using various degrees of polynomial approximations. Computations with and without preconditioning are performed on different grid topologies to analyze the influence of the spatial discretization on the accuracy of the DG solutions at low Mach numbers.     

The momentum interpolation method based on the time-marching algorithm for All-Speed flows

The time-marching approach has clear physical meaning and strict mathematical nature and has been applied in computation of compressible flows widely and extended to many uniform algorithms for All-Speed flows. Remedy for its weakness in the problem of checkerboard decoupling of pressure field for incompressible flows is envisaged with the time-marching momentum interpolation method (MIM) taken into account in this paper. Existing preconditioning methods for suppressing decoupling and time-marching MIM are analyzed for this purpose, and algorithms of time-marching MIM are proposed for steady and unsteady flows and for All-Speed flows. Asymptotic analysis shows that the supposed time-marching MIM has at least a third-order accuracy, better than the existing time-marching coupling methods, which only have an accuracy of the same order as the adopted scheme has. Effects of the time step sizes on the ability of the time-marching MIM to suppress the checkerboard pressure decoupling are particularly discussed in terms of the dual-time stepping approach, and it is revealed how the decreased sizes of either the pseudo- or physical-time step increases the possibility of decoupling and how Choi’s modification, in which the history of the interface velocity is decided by itself instead of the arithmetic average of the velocities on its adjacent nodes, removes the unphysical pressure oscillation with small size of the physical time step but leads to divergence with the pseudo-time step as well. As a remedy for the pseudo-time step, such methods are recommended as implicit methods and the local-time step method with a proposed modification of the time-marching MIM preventing accuracy loss due to very large time step size. Numerical experiments support the theoretical analyses and show the validity of the time-marching MIM proposed.     

用格子Boltzmann模型模拟可压缩完全气体流动

采用一种新的格子Boltzmann模型模拟超音速流动。在这种模型中,粒子的速度不受限制,可以取得很广。而平衡分布函数的支集却相对集中,使模型得以简化。粒子速度的这种自适应特性允许流体以较高的马赫数流动。通过引入粒子的势能使得该模型适用于具有任意比热比的完全气体。利用Chapman-Enskog方法,从BGK型Boltzmann方程推导出Navier-Stokes方程。在六边形网格上模拟了马赫数为3的前台阶绕流,得到了合理的结果。     

Implicit preconditioned WENO scheme for steady viscous flow computation

A class of lower–upper symmetric Gauss–Seidel implicit weighted essentially nonoscillatory (WENO) schemes is developed for solving the preconditioned Navier–Stokes equations of primitive variables with Spalart–Allmaras one-equation turbulence model. The numerical flux of the present preconditioned WENO schemes consists of a first-order part and high-order part. For first-order part, we adopt the preconditioned Roe scheme and for the high-order part, we employ preconditioned WENO methods. For comparison purpose, a preconditioned TVD scheme is also given and tested. A time-derivative preconditioning algorithm is devised and a discriminant is devised for adjusting the preconditioning parameters at low Mach numbers and turning off the preconditioning at intermediate or high Mach numbers. The computations are performed for the two-dimensional lid driven cavity flow, low subsonic viscous flow over S809 airfoil, three-dimensional low speed viscous flow over 6:1 prolate spheroid, transonic flow over ONERA-M6 wing and hypersonic flow over HB-2 model. The solutions of the present algorithms are in good agreement with the experimental data. The application of the preconditioned WENO schemes to viscous flows at all speeds not only enhances the accuracy and robustness of resolving shock and discontinuities for supersonic flows, but also improves the accuracy of low Mach number flow with complicated smooth solution structures.     

基于预处理HLLEW格式的全速域数值算法

基于HLLEW(Harten-Lax-Van Leer-Einfeldt-Wada)格式引入预处理技术发展适合求解全速域流场的三维Navier-Stokes求解器.引入低速预处理技术,重新构造HLLEW格式的耗散项,给出预处理后的HLLEW格式,并根据预处理后的雅克比矩阵构造相应的隐式时间推进方程.利用预处理方法求解NACA 4412低速不可压流动与RAE 2822跨声速可压缩流动,并与实验结果及原有方法的计算结果对比.结果表明:预处理HLLEW格式不仅提高低速不可压缩流动的计算效率和精度,也保持了对可压缩流动的处理能力,是一种适用于全速域流场数值模拟的有效方法.     

An Asymptotic-Preserving all-speed scheme for the Euler and Navier–Stokes equations

We present an Asymptotic-Preserving ‘all-speed’ scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretization, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime. The divergence-free condition on the velocity in the incompressible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and two-dimensional numerical results provide a validation of the Asymptotic-Preserving ‘all-speed’ properties.