非结构网格二阶有限体积法中黏性通量离散格式精度分析与改进

非结构网格二阶有限体积离散方法广泛应用于计算流体力学工程实践中,研究非结构网格二阶精度有限体积离散方法的计算精度具有现实意义.计算精度主要受到网格和计算方法的影响,本文从单元梯度重构方法、黏性通量中的界面梯度计算方法两个方面考察黏性流动模拟精度的影响因素.首先从理论上分析了黏性通量离散中的"奇偶失联"问题,并通过基于标量扩散方程的制造解方法验证了"奇偶失联"导致的精度下降现象,进一步通过引入差分修正项消除了"奇偶失联"并提高了扩散方程计算精度;其次,在不同类型、不同质量的网格上进行基于扩散方程的制造解精度测试,考察单元梯度重构方法、界面梯度计算方法对扩散方程计算精度的影响,结果显示,单元梯度重构精度和界面梯度计算方法均对扩散方程计算精度起重要作用;最后对三个黏性流动算例(二维层流平板、二维湍流平板和二维翼型近尾迹流动)进行网格收敛性研究,初步验证了本文的结论,得到了计算精度和网格收敛性均较好的黏性通量计算格式.     

基于制造解的非结构二阶有限体积离散格式的精度测试与验证

随着计算机技术的飞速进步,计算流体力学得到迅猛发展,数值计算虽能够快速得到离散结果,但是数值结果的正确性与精度则需要通过严谨的方法来进行验证和确认.制造解方法和网格收敛性研究作为验证与确认的重要手段已经广泛应用于计算流体力学代码验证、精度分析、边界条件验证等方面.本文在实现标量制造解和分量制造解方法的基础上,通过将制造解方法精度测试结果与经典精确解(二维无黏等熵涡)精度测试结果进行对比,进一步证实了制造解精度测试方法的有效性,并将两种制造解方法应用于非结构网格二阶精度有限体积离散格式的精度测试与验证,对各种常用的梯度重构方法、对流通量格式、扩散通量格式进行了网格收敛性精度测试.结果显示,基于Green-Gauss公式的梯度重构方法在不规则网格上会出现精度降阶的情况,导致流动模拟精度严重下降,而基于最小二乘(least squares)的梯度重构方法对网格是否规则并不敏感.对流通量格式的精度测试显示,所测试的各种对流通量格式均能达到二阶精度,且各方法精度几乎相同;而扩散通量离散中界面梯度求解方法的选择对流动模拟精度有显著影响.     

非结构网格质量对梯度重构及无粘流动模拟精度的影响

综合利用理论分析和数值测试手段,研究了非结构格心型有限体积离散中梯度重构算法的计算精度,分别给出了非结构算法中常用的基于Green-Gauss公式(GG方法)和基于Least squares方法(LSQ方法)的梯度重构方法达到至少一阶精度的条件。其中,GG方法在面积分低阶项不能互相抵消的情况下,要求面心插值精度达到至少二阶;而LSQ方法对于任意网格均能实现梯度重构一阶精度。在各向同性网格上的梯度重构精度数值测试验证了数学推导结论;进一步通过制造解方法量化无粘流动数值离散误差,结合网格收敛性测试研究了网格质量(网格点随机扰动、网格弯曲度和网格倾斜度等因素)以及网格类型(三角形和四边形)对无粘流动模拟精度的影响,验证了理论分析结论。     

基于非结构化同位网格的SIMPLE算法

通过基于非结构化网格的有限体积法对二维稳态Navier—Stokes方程进行了数值求解。其中对流项采用延迟修正的二阶格式进行离散；扩散项的离散采用二阶中心差分格式；对于压力-速度耦合利用SIMPLE算法进行处理；计算节点的布置采用同位网格技术，界面流速通过动量插值确定。本文对方腔驱动流、倾斜腔驱动流和圆柱外部绕流问题进行了计算，讨论了非结构化同位网格有限体积法在实现SIMPLE算法时，迭代次数与欠松弛系数的关系、不同网格情况的收敛性、同结构化网格的对比以及流场尾迹结构。通过和以往结果比较可知，本文的方法是准确和可信的。     

HyperFLOW软件非结构网格亚跨声速湍流模拟的验证与确认

计算流体力学(computational fluid dynamics,CFD)数值模拟在航空航天等领域发挥越来越重要的作用,然而CFD数值模拟结果的可信度仍然需要通过不断地验证与确认来提高.本文给出了从制造解精度测试、简单到复杂外形湍流模拟网格收敛性研究等三个方面开展CFD软件验证与确认的方法,并对自主研发的CFD软件平台HyperFLOW在非结构网格上模拟亚跨声速湍流问题的能力进行了验证与确认.首先通过基于Euler方程和标量扩散方程的制造解精度测试,分别验证了HyperFLOW在非结构网格上对Euler方程和黏性项的求解精度,结果表明其能够在任意非结构网格上达到设计的二阶精度. 其次,通过NASATurbulence Modeling Resource中的湍流平板、二维翼型近尾迹流动、二维Bump等几个典型的亚声速湍流算例的网格收敛性研究,量化考察了数值结果的观测精度阶和网格收敛性指数,并与国外知名CFD解算器CFL3D,FUN3D的计算结果进行了对比,验证了HyperFLOW对简单湍流问题的模拟能力,且具有良好的网格收敛性和计算精度(阶). 最后,通过NASA CommonResearchModel标模定升力系数的网格收敛性研究和升阻极曲线预测,验证了软件在复杂外形亚跨声速湍流流动数值模拟中也具有良好的可信度.      

适用于任意网格拓扑和质量的格心有限体积法

针对格心有限体积法的离散精度易受网格类型影响的问题,基于最小二乘原理,提出了一种适用于任意网格拓扑和网格质量的有限体积方法.在解的光滑区能保证二阶精度,可光滑、陡峭地捕捉激波等强间断面.精度与网格无关,且算法统一的特性使其非常适用于网格自适应和多重网格等计算应用,同时也降低了网格生成和流场求解的复杂性.跨音速算例的网格含有多种网格拓扑,计算结果表明发展的线性重构方法(linearreconstruction method,LRM)适用于不同的网格拓扑,计算得到的激波位置准确、陡峭,未产生数值振荡.运用Ringleb流动考查了该方法对低质量网格的收敛性,与传统方法相比,线性重构方法(LRM)不仅平均误差较小,而且误差随网格尺度的收敛性也更好,其精度接近二阶.三段翼型的黏性绕流计算进一步表明网格质量对其精度的影响较小.     

基于非结构/混合网格模拟黏性流的高阶精度DDG/FV混合方法

间断Galerkin有限元方法(discontinuous Galerkin method, DGM) 因具有计算精度高、模板紧致、易于并行等优点, 近年来已成为非结构/混合网格上广泛研究的高阶精度数值方法. 但其计算量和内存需求量巨大, 特别是对于网格规模达到百万甚至数千万的大型三维实际复杂外形问题, 其计算量和存储量对计算资源的消耗是难以承受的. 基于“混合重构”的DG/FV 格式可以有效降低DGM 的计算量和存储量. 本文将DDG 黏性项离散方法推广应用于DG/FV 混合算法, 得到新的DDG/FV混合格式, 以进一步提高DG/FV混合算法对于黏性流动模拟的计算效率. 通过Couette流动、层流平板边界层、定常圆柱绕流, 非定常圆柱绕流和NACA0012 翼型绕流等二维黏性流算例, 优化了DDG 通量公式中的参数选择, 验证了DDG/FV 混合格式对定常和非定常黏性流模拟的精度和计算效率, 并与广泛使用的BR2-DG 格式的计算结果和效率进行对比研究. 一系列数值实验结果表明, 本文构造的DDG/FV混合格式在二维非结构/混合网格的Navier-Stokes 方程求解中, 在达到相同的数值精度阶的前提下, 相比BR2-DG格式, 对于隐式时间离散的定常问题计算效率提高了2 倍以上, 对于显式时间离散的非定常问题计算效率提高1.6 倍, 并且在一些算例中, 混合格式具有更优良的计算稳定性. DDG/FV 混合格式提升了计算效率和稳定性, 具有良好的应用前景.      

HyperFLOW软件非结构网格亚跨声速湍流模拟的验证与确认

计算流体力学(computational fluid dynamics, CFD)数值模拟在航空航天等领域发挥越来越重要的作用,然而CFD数值模拟结果的可信度仍然需要通过不断地验证与确认来提高.本文给出了从制造解精度测试、简单到复杂外形湍流模拟网格收敛性研究等三个方面开展CFD软件验证与确认的方法,并对自主研发的CFD软件平台HyperFLOW在非结构网格上模拟亚跨声速湍流问题的能力进行了验证与确认.首先通过基于Euler方程和标量扩散方程的制造解精度测试,分别验证了HyperFLOW在非结构网格上对Euler方程和黏性项的求解精度,结果表明其能够在任意非结构网格上达到设计的二阶精度.其次,通过NASA Turbulence Modeling Resource中的湍流平板、二维翼型近尾迹流动、二维Bump等几个典型的亚声速湍流算例的网格收敛性研究,量化考察了数值结果的观测精度阶和网格收敛性指数,并与国外知名CFD解算器CFL3D,FUN3D的计算结果进行了对比,验证了HyperFLOW对简单湍流问题的模拟能力,且具有良好的网格收敛性和计算精度(阶).最后,通过NASA Common Research Model标模定升力系数的网格收敛性研究和升阻极曲线预测,验证了软件在复杂外形亚跨声速湍流流动数值模拟中也具有良好的可信度.     

基于非结构网格求解二维浅水方程的高精度有限体积方法

采用HLL格式,在三角形非结构网格下采用有限体积离散,建立了求解二维浅水方程的高精度的数值模型.本文采用多维重构和多维限制器的方法来获得高精度的空间格式以及防止非物理振荡的产生,时间离散采用三阶Runge-Kutta法以获得高阶的时间精度.基于三角形网格,底坡源项采用简单的斜底模型离散,为保证计算格式的和谐性,对经典的HLL格式计算的数值通量中的静水压力项进行了修正.算例证明本文提出的方法的和谐性并具有高精度的间断捕捉能力和稳定性.     

NND格式在非结构网格中的推广

在张涵信提出的无波动、无自由参数的差分格式（ＮＮＤ格式）的基础上，构造了适用于非结构网格的二阶精度ＮＮＤ有限体积格式，解决了现有非结构网格方法中为抑制激波附近的波动而必须引入含自由参数的人工粘性项的困难，并采用网格自适应技术以提高效率．通过对二维平板激波反射和前台阶在管道内的流动问题的计算，表明本方法可有效地用于Ｅｕｌｅｒ方程的求解．     

An approach for choosing discretization schemes and grid size based on the convection-diffusion equation

A new approach for selecting proper discretization schemes and grid size is presented. This method is based on the convection-diffusion equation and can provide insight for the Navier-Stokes equation. The approach mainly addresses two aspects, i.e., the practical accuracy of diffusion term discretization and the behavior of high wavenumber disturbances. Two criteria are included in this approach. First, numerical diffusion should not affect the theoretical diffusion accuracy near the length scales of interest. This is achieved by requiring numerical diffusion to be smaller than the diffusion discretization error. Second, high wavenumber modes that are much smaller than the length scales of interest should not be amplified. These two criteria provide a range of suitable scheme combinations for convective flux and diffusive flux and an ideal interval for grid spacing. The effects of time discretization on these criteria are briefly discussed.     

A Fully—Implicit 2‐D Navier-Stokes Algorithm for Unstructured Grids

A fully-implicit algorithm is developed for the two-dimensional, compressible, Favre-averaged Navier-Stokes equations. It incorporates the standard k-? turbulence model of Launder and Spalding and the low Reynolds number correction of Chien. The equations are solved using an unstructured grid of triangles with the flow variables stored at the centroids of the cells. A generalization of wall functions including pressure gradient effects is implemented to solve the near-wall region for turbulent flows using a separate algorithm and a hybrid grid. The inviscid fluxes are obtained from Roe's flux difference split method. Linear reconstruction of the flow variables to the cell faces provides second-order spatial accuracy. Turbulent and viscous stresses as well as heat transfer are obtained from a discrete representation of Gauss's theorem. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, Trapezoidal or 3-Point Backward differencing. An incomplete LU factorization of the Jacobian matrix is implemented as a preconditioning method. The accuracy of the code and the efficiency of the solution strategy are presented for three test cases: a supersonic turbulent mixing layer, a supersonic laminar compression corner and a supersonic turbulent compression corner.     

A collocated grid,projection method for time‐accurate calculation of low‐Mach number variable density flows in general curvilinear coordinates

A time‐accurate algorithm is proposed for low‐Mach number, variable density flows on curvilinear grids. Spatial discretization is performed on collocated grid that offers computational simplicity in curvilinear coordinates. The flux interpolation technique is used to avoid the pressure odd–even decoupling of the collocated grid arrangement. To increase the stability of the method, a two‐step predictor–corrector time integration scheme is employed. At each step, the projection method is used to calculate the hydrodynamic pressure and to satisfy the continuity equation. The robustness and accuracy of the method is illustrated with a series of numerical experiments including thermally driven cavity, polar cavity, three‐dimensional cavity, and direct numerical simulation of non‐isothermal turbulent channel flow. Copyright © 2012 John Wiley & Sons, Ltd.     

A technique of flux reconstruction at the interfaces of nonconforming grids for aeroacoustic simulations

A flux reconstruction technique is presented to perform aeroacoustic computations using implicit high-order spatial schemes on multiblock structured grids with nonconforming interfaces. The use of such grids, with mesh spacing discontinuities across the block interfaces, eases local mesh refinements, simplifies the mesh generation process, and thus facilitates the computation of turbulent flows. In this work, the spatial discretization consists of sixth-order finite-volume implicit schemes with low-dispersion and low-dissipation properties. The flux reconstruction is based on the combination of noncentered schemes with local interpolations to define ghost cells and compute flux values at the grid interfaces. The flow variables in the ghost cells are calculated from the flow field in the grid cells using a meshless interpolation with radial basis functions. In this study, the flux reconstruction is applied to both plane and curved nonconforming interfaces. The performance of the method is first evaluated by performing two-dimensional simulations of the propagation of an acoustic pulse and of the convection of a vortex on Cartesian and wavy grids. No significant spurious noise is produced at the grid interfaces. The applicability of the flux reconstruction to a three-dimensional computation is then demonstrated by simulating a jet at a Mach number of 0.9 and a diameter-based Reynolds number of 4×10⁵ on a Cartesian grid. The nonconforming grid interface located downstream of the jet potential core does not appreciably affect the flow development and the jet sound field, while reducing the number of mesh points by a factor of approximately two.     

An investigation of artificial dissipation for the cell-vertex finite volume method

This paper is concerned with an investigation of artificial dissipation models that are used with the cell-vertex finite volume approximation of the compressible Euler and Navier-Stokes equations. Based on the observation that first and second-order upwind schemes can be written as a central discretization plus an appropriately scaled dissipative flux, a matrix scaling of second and fourth-differences is implemented in an artificial dissipation model that also uses a procedure to limit the applicability of shock-capturing dissipation. Analysis of the model and the associated boundary conditions is given to determine the effect on accuracy. Numerical results are given for transonic Euler flow past a NACA0012 aerofoil profile which demonstrate the improved shock-capturing capability of the model. Results for laminar subsonic viscous flow over a flat plate show that the matrix-dissipation model reduces the amount of spurious artificial dissipation within boundary layers.     

An implicit high-order preconditioned flux reconstruction method for low-Mach-number flow simulation with dynamic meshes

A fully implicit high-order preconditioned flux reconstruction/correction procedure via reconstruction (FR/CPR) method is developed to solve the compressible Navier-Stokes equations at low Mach numbers. A dual-time stepping approach with the second-order backward differentiation formula (BDF2) is employed to ensure temporal accuracy for unsteady flow simulation. When dynamic meshes are used to handle moving/deforming domains, the geometric conservation law is implicitly enforced to eliminate errors due to the resolution discrepancy between BDF2 and the spatial FR/CPR discretization. The large linear system resulted from the spatial and temporal discretizations is tackled with the restarted generalized minimal residual solver in the PETSc (portable, extensible toolkit for scientific computation) library. Through several benchmark steady and unsteady numerical tests, the preconditioned FR/CPR methods have demonstrated good convergence and accuracy for simulating flows at low Mach numbers. The new flow solver is then used to study the effects of Mach number on unsteady force generation over a plunging airfoil when operating in low-Mach-number flows. It is observed that weak compressibility has a significant impact on thrust generation but has a negligible effect on lift generation of an oscillating airfoil.     

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.