界面不稳定性的自适应欧拉数值计算

采用欧拉网格自适应算法数值模拟Richtmyer Meshkov和Rayleigh Taylor不稳定多介质流界面,获得了高精度界面特征。对不同流体引入不同位标函数跟踪界面运动,将位标函数方程与流体动力学方程耦合求解,在笛卡儿坐标系中运用二阶精度有限体积算法,保持流场守恒条件下,通过采用多层网格级对笛卡儿网格嵌套细化,从而实现多介质流体界面的高精度自适应跟踪。给出的方法逻辑简单,可以大大节省CPU时间。     

The residual formulation of the method of manufactured solutions for computationally efficient solution verification

The method of manufactured solutions (MMS) is a solution verification methodology for determining whether the implementation of a discretization method is achieving its theoretical order of accuracy. This methodology combines the advantages of grid refinement studies and comparison with exact solution, by modifying the governing equations solved within a code by adding a source term to drive the solution towards a predetermined analytic function. By solving the modified equations on a sequence of grids and comparing the differences between the converged solution and manufactured solution, the order of accuracy of the implementation can be investigated. However, in its current form, converged solutions on a sequence of grids are required which can be quite costly and difficult to obtain. In this paper, by comparing the MMS to the method for determining the theoretical order of accuracy of a discretization method, the residual formulation of the MMS is developed. This new formulation only requires that the residual of the discretized governing equations to be calculated and not the solution to the discretized equations, thus avoiding the computational cost and difficulties inherent in obtaining converged solutions. Furthermore, since only the residuals are interrogated, individual components of the flow solver can be tested, in isolation, allowing the MMS to be used more effectively in locating errors within the code. This new approach is demonstrated to yield the same order of accuracy as the original MMS using three different cases—one-dimensional porous media equation, one-dimensional St Venant equations and two-dimensional unstructured Navier–Stokes simulations.     

Galerkin finite element simulation of convection driven by rotation and gravitation

The performance of the Galerkin finite element method when applied to time-dependent convection involving rotation, self-gravitation and the normal gravity field in a horizontal cylinder is discussed in this paper. The governing equations, the parameters of the problem and our implementation of the numerical schemes are presented. The accuracy, spatial scale of resolution, flexibility and robustness of the resulting code show the element method as a valuable tool for research in this area or in related problems in astrophysical fluid dynamics. The numerical difficulties in large-amplitude flows are associated with an error-control scheme for time integration and the ‘short-time’ wiggles in transient Dirichlet problems. Coarse grids give the correct qualitative picture in most simulations, but the type of solution at short time (and hence grid refinement) presumably resolves the degeneracy in the azimuthal orientation of convection cells in flows driven by self-gravitation and (perhaps) centrifugal buoyancy. The final state in transient flows is a motionless isothermal fluid. However, residual motions can be nullified only in the limit of zero grid size in flows driven by centrifugal buoyancy (self-gravitation), while a fairly coarse grid is sufficient for this purpose in normal gravity-driven flows.     

Calculation of laminar recirculating flows using a local non-staggered grid refinement system

A grid-embedding technique for the solution of two-dimensional incompressible flows governed by the Navier-Stokes equations is presented. A finite volume method with collocated primitive variables is employed to ensure conservation at the interfaces of embedding grids as well as global conservation. The discretized equations are solved simultaneously for the whole domain, providing a strong coupling between regions of different refinement. The formulation presented herein is applicable to uniform or non-uniform Cartesian meshes. The method was applied to the solution of two scalar transport equations, to cavity flows driven by body and shear forces and to a sudden plane contraction flow. The numerical predictions are compared with the exact solutions when available and with experimental data. The results show that neither the convergence rate nor the stability of the method is affected by the presence of embedded grids. Embedded grids provide a better distribution of grid nodes over the computational domain and consequently the solution accuracy was improved. The grid-embedding technique proved also that significant savings in computing time could be achieved.     

A multigrid procedure for Cartesian ghost‐cell methods

This paper proposes a multigrid technique for Cartesian grid flow solvers. A recently developed ghost body‐cell method for inviscid flows is combined with a nested‐level local refinement procedure, which employs multigrid to accelerate convergence to steady state. Different from standard multigrid applications for body‐fitted grids, a fictitious residual needs to be defined in the ghost cells to perform a correct residual collection and thus to avoid possible stalling of the multigrid procedure. The efficiency of the proposed local refinement multigrid Cartesian method is demonstrated for the case of the inviscid subsonic flow past a circular body. Copyright © 2008 John Wiley & Sons, Ltd.     

A local grid refinement method for three‐dimensional turbulent recirculating flows

A local grid refinement method is presented and applied to a three‐dimensional turbulent recirculating flow. It is based on the staggered grid arrangement. The computational domain is covered by block‐structured subgrids of different refinement levels. The exchange of information between the subgrids is fully conservative and all grids are treated implicitly. This allows for a simultaneous solution of one variable in all grids. All variables are stored in one‐dimensional arrays. The solver selected for the solution of the discretised finite difference equations is the preconditioned bi‐conjugate gradient (Bi‐CG) method. For the case examined (turbulent flow around a surface‐mounted cube), it was found that the latter method converges faster than the line solver. The locally refined mesh improved the accuracy of the pressure distribution on cube faces compared with a coarse mesh and yielded the same results as a fine single mesh, with a 62% gain in computer time. Copyright © 1999 John Wiley & Sons, Ltd.     

CAD-consistent adaptive refinement using a NURBS-based discontinuous Galerkin method

This study concerns the development of a new method combining high-order computer-aided design (CAD)-consistent grids and adaptive refinement/coarsening strategies for efficient analysis of compressible flows. The proposed approach allows to use geometrical data from CAD without any approximation. Thus, the simulations are based on the exact geometry, even for the coarsest discretizations. Combining this property with a local refinement method allows to start computations using very coarse grids and then relies on dynamic adaption to construct suitable computational domains. The resulting approach facilitates interactions between CAD and computational fluid dynamics solvers and focuses the computational effort on the capture of physical phenomena, since geometry is exactly taken into account. The proposed methodology is based on a discontinuous Galerkin method for compressible Navier-Stokes equations, modified to use nonuniform rational B-Spline representations. Local refinement and coarsening are introduced using intrinsic properties of nonuniform rational B-Spline associated with a local error indicator. A verification of the accuracy of the method is achieved and a set of applications are presented, ranging from viscous subsonic to inviscid trans- and supersonic flow problems.     

Finite element analysis of incompressible material by residual energy balancing

Incompressibility is gradually introduced into the finite elements with the mesh refinement in such a way as to balance it with the residual discretization energy in order to ensure fastest convergence to the incompressible solution with best conditioned stiffness matrix to minimize round-off errors in the computations.     

样条有限条塑性铰法分析板梁的极限荷载

用样条有限条塑性铰法分析了板梁的极限荷载。首先对条单元以样条位移函数表达的总势能进行求导而推导了位移-荷载关系式。然后用塑性铰法推导了单元的塑性刚度矩阵。因此该方法兼具二者优点:样条有限条法的位移量少和塑性铰法形成塑性刚度矩阵的便利。它还可以考虑梁的初始缺陷,如残余应力和初弯曲。通过与相关的试验数据比较,证明该方法有效与可靠。     

Three‐dimensional instability of axisymmetric flows: solution of benchmark problems by a low‐order finite volume method

Several problems on three‐dimensional instability of axisymmetric steady flows driven by convection or rotation or both are studied by a second‐order finite volume method combined with the Fourier decomposition in the periodic azimuthal direction. The study is focused on the convergence of the critical parameters with mesh refinement. The calculations are done on the uniform and stretched grids with variation of the stretching. Converged results are reported for all the problems considered and are compared with the previously published data. Some of the calculated critical parameters are reported for the first time. The convergence studies show that the three‐dimensional instability of axisymmetric flows can be computed with a good accuracy only on fine enough grids having about 100 nodes in the shortest spatial direction. It is argued that a combination of fine uniform grids with the Richardson extrapolation can be a good replacement for a grid stretching. It is shown once more that the sparseness of the Jacobian matrices produced by the finite volume method allows one to enhance performance of the Newton and Arnoldi iteration procedures by combining them with a direct sparse linear solver instead of using the Krylov‐subspace‐based iteration methods. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical modeling of ideal incompressible fluid flow over a step

Results of calculations of fluid flow over a step located on a channel bottom are given. Numerical modeling is performed for the model of free-boundary potential flows of an ideal incompressible fluid using a finite-difference method with dynamically adaptive grids. The behavior of the free surface in the neighborhood of the step is studied as a function of the incident-flow velocity. The results are compared with experimental data. __________ Translated from PrikladnayaMekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 17–22, November–December, 2006.     

NESTED GRID METHODS FOR AN OCEAN MODEL: A COMPARATIVE STUDY

In this paper a comparison is carried out between three correction methods for multigrid local mesh refinement in oceanic applications: FIC, LDC and the direct method (DM) proposed by Spall and Holland. This study is based on a nested primitive equation model developed by Laugier on the basis of the code OPA (LODYC). The external barotropic problem is solved using any of the three local grid correction algorithms yielding an interactive nested grid model. The non-linear elliptic equation for the barotropic streamfunction tendency is solved on two nested grids, called the global and the zoom grid, that interact between themselves. The zoom grid is entirely embedded within the global domain with a horizontal grid step ratio of 3:1. The computation on the global grid supplies the boundary conditions for the zoom grid region and the fine grid fields are used to correct the global coarse solution. The three local correction methods are tested on two problems relevant to oceanic circulation phenomena proposed by Spall and Holland: a barotropic modon and an anticyclonic vortex. The results show that the nesting technique is a very efficient way to solve these problems in terms of a gain in precision compared with the required CPU time. The two-domain model with local mesh refinement allows one both to manage effectively the open boundary conditions for the local grid and to correct the global solution thanks to the zoom solution. In the case of the modon propagation the three local correction methods provide approximately the same results. For the baroclinic vortex it appears that the two iterative methods are more efficient than the direct one.     

A finite volume method for numerical grid generation

A novel method to generate body‐fitted grids based on the direct solution for three scalar functions is derived. The solution for scalar variables ξ, η and ζ is obtained with a conventional finite volume method based on a physical space formulation. The grid is adapted or re‐zoned to eliminate the residual error between the current solution and the desired solution, by means of an implicit grid‐correction procedure. The scalar variables are re‐mapped and the process is reiterated until convergence is obtained. Calculations are performed for a variety of problems by assuming combined Dirichlet–Neumann and pure Dirichlet boundary conditions involving the use of transcendental control functions, as well as functions designed to effect grid control automatically on the basis of boundary values. The use of dimensional analysis to build stable exponential functions and other control functions is demonstrated. Automatic procedures are implemented: one based on a finite difference approximation to the Cristoffel terms assuming local‐boundary orthogonality, and another designed to procure boundary orthogonality. The performance of the new scheme is shown to be comparable with that of conventional inverse methods when calculations are performed on benchmark problems through the application of point‐by‐point and whole‐field solution schemes. Advantages and disadvantages of the present method are critically appraised. Copyright © 1999 National Research Council of Canada.     

Multigrid third‐order least‐squares solution of Cauchy–Riemann equations on unstructured triangular grids

In this paper, a multigrid algorithm is developed for the third‐order accurate solution of Cauchy–Riemann equations discretized in the cell‐vertex finite‐volume fashion: the solution values stored at vertices and the residuals defined on triangular elements. On triangular grids, this results in a highly overdetermined problem, and therefore we consider its solution that minimizes the residuals in the least‐squares norm. The standard second‐order least‐squares scheme is extended to third‐order by adding a high‐order correction term in the residual. The resulting high‐order method is shown to give sufficiently accurate solutions on relatively coarse grids. Combined with a multigrid technique, the method then becomes a highly accurate and efficient solver. We present some results to demonstrate its accuracy and efficiency, including both structured and unstructured triangular grids. Copyright © 2006 John Wiley & Sons, Ltd.     

非结构动网格在可动边界问题中的应用研究

本文对用于非结构动网格生成的弹簧近似方法进行了研究。通过采用顶点弹簧方法,分析研究了弹簧倔强系数的取值,同时通过引入挤压倔强系数和边界修正,对标准弹簧近似方法进行了改进。改进后的方法可以大大提高网格变形能力和网格质量。应用本文发展的非结构动网格生成方法并通过耦合求解基于(ArbitraryLagrangian-Eulerian ALE)描述的Euler方程,模拟了谐和振动NACA0012翼型及M6机翼的跨音速绕流,计算结果与参考文献提供的结果及实验结果吻合良好。     

h-自适应边界元方法的插值残差计算及误差估计

本文提出了一种用于估计ｈ－自适应边界元过程解误差的新方法,这种方法基于ｈ－自适应边界元过程生成的离散网络,通过计算近似解的插值残差,以此作为误差估计的数据。此外,这种误差分析方法易于程序化,可以很方便地接入现有ｈ－自适应边界元计算机程序（简单细分或分层细分）,而对原程序不作大的改动。通过对二个经典的弹性静力学问题的分析表明：本文的方法能较好地估计边界元解的误差,并使ｈ－自适应边界程序的分析更加有效     

Jacobian-free Newton-Krylov subspace method with wavelet-based preconditioner for analysis of transient elastohydrodynamic lubrication problems with surface asperities

This paper presents an investigation into the effect of surface asperities on the over-rolling of bearing surfaces in transient elastohydrodynamic lubrication(EHL) line contact. The governing equations are discretized by the finite difference method. The resulting nonlinear system of algebraic equations is solved by the Jacobian-free Newtongeneralized minimal residual(GMRES) from the Krylov subspace method(KSM). The acceleration of the GMRES iteration is accomplished by a wavelet-based preconditioner.The profiles of the lubricant pressure and film thickness are obtained at each time step when the indented surface moves through the contact region. The prediction of pressure as a function of time provides an insight into the understanding of fatigue life of bearings.The analysis confirms the need for the time-dependent approach of EHL problems with surface asperities. This method requires less storage and yields an accurate solution with much coarser grids. It is stable, efficient, allows a larger time step, and covers a wide range of parameters of interest.     

非结构动网格在三维可动边界问题中的应用

研究用于非结构动网格的弹簧近似方法，采用顶点弹簧描述，导出并讨论了弹簧倔强系数的取值。通过引入边界修正和扭转效应修正，对标准弹簧近似方法进行了改进，转动翼型算例的结果表明，改进后的方法大大提高了网格变形能力和网格质量，应用该动网格方法耦合求解基于（Arbitrary Lagrangian-Eulerian,ALE）描述的三维Euler方程，模拟了作俯抑振动的矩形机翼绕流，计算结果与实验数据及文献计算结果十分一致，作为多个自由刚与流体耦合运动问题的简单例证，耦合刚体动力学方程，模拟了激波与双立方体的相互作用，得到了非定常流场结构，研究表明，基于弹簧近似的非结构动网格与有限体积流式流场解算器相结合，是模拟包含运动边界的非定常流动问题的有效方法。     

A parallel adaptive numerical method with generalized curvilinear coordinate transformation for compressible Navier–Stokes equations

A fourth‐order finite‐volume method for solving the Navier–Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. The method employs fourth‐order quadrature rules for evaluating face‐averaged fluxes. Our approach is freestream preserving, guaranteed by the way of computing the averages of the metric terms on the faces of cells. The standard Runge–Kutta marching method is used for time discretization. Solutions of a smooth flow are obtained in order to verify that the method is formally fourth‐order accurate when applying the nonlinear viscous operators on mapped grids. Solutions of a shock tube problem are obtained to demonstrate the effectiveness of adaptive mesh refinement in resolving discontinuities. A Mach reflection problem is solved to demonstrate the mapped algorithm on a non‐rectangular physical domain. The simulation is compared against experimental results. Future work will consider mapped multiblock grids for practical engineering geometries. Copyright © 2016 John Wiley & Sons, Ltd.     

Convergence properties of high-Reynolds-number separated flow calculations

The convergence properties of an iterative solution technique for the Reduced Navier–Stokes equations are examined for two-dimensional steady subsonic flow over bump and trough geometries. Techniques for decreasing the sensitivity to the initial pressure approximation, for fine meshes in particular, are investigated. They are shown to improve the robustness of the relaxation process and to decrease the computational work required to obtain a converged solution. A semi-coarsening multigrid technique that has previously been found to be particularly advantageous for high-Reynolds-number (Re) flows with flow separation and with highly stretched surface-normal grids is applied herein to further accelerate convergence. Solutions are obtained for the laminar flow over a trough that is more severe than has been considered to date. Sufficient axial grid refinement in this case leads to a shock-like reattachment and, for sufficiently large Re, to a local ‘divergence’ of the numerical computations. This ‘laminar flow breakdown’ appears to be related to an instability associated with high-frequency fine-grid modes that are not resolvable with the present modelling. This behaviour may be indicative of dynamic stall or of incipient transition. The breakdown or instability is shown to be controllable by suitable introduction of transition turbulence models or by laminar flow control, i.e. small amounts of wall suction. This lends further support to the hypothesis that the instability is of a physical rather than numerical character and suggests that full three-dimensional analysis is required to properly capture the flow behaviour. Another inference drawn from this investigation is that there is a need for careful grid refinement studies in high-Re flow computations, since coarser grids may yield oscillation-free solutions that cannot be obtained on finer grids.