基于非结构网格的高效求解方法研究

非结构网格的求解效率一直是计算流体力学工作者十分关注的问题。本文从一个新的角度分析了N-S(Euler/Navier-Stokes)方程求解效率的高低,表明计算效率不仅涉及时间离散的效率,空间离散和程序算法都与之息息相关。采用不同的计算状态,对目前非结构网格上广泛应用的LU-SGS、对称Gauss-Seidel和GMRES方法进行较详细地比较和分析,考查了空间离散的耗时对方程求解效率的影响。结果表明,LU-SGS方法的计算效率在所给的算例中均是最低的;在不考虑大量内存消耗时,GMRES算法求解Euler方程的效率较高,松耦合求解N-S方程时效率会有所降低;在大规模计算中,多次对称的Gauss-Seidel迭代方法应是较好的选择,特别是N-S方程的求解。     

Implicit schemes for unsteady Euler equations on triangular meshes

An implicit finite element method is presented for the solution of steady and unsteady inviscid compressible flows on triangular meshes under transonic conditions. The method involves a first-order time-stepping scheme with a finite element discretization that reduces to central differencing on a rectangular mesh. On a solid wall the slip condition is prescribed and the pressure is obtained from an approximation of the normal momentum equation. With this solver no artificial viscosity is added to ensure the success of the calculation. Numerical examples are given for steady and unsteady cases.     

Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids

A Godunov-type upwind finite volume solver of the non-linear shallow water equations is described. The shallow water equations are expressed in a hyperbolic conservation law formulation for application to cases where the bed topography is spatially variable. Inviscid fluxes at cell interfaces are computed using Roe's approximate Riemann solver. Second-order accurate spatial calculations of the fluxes are achieved by enhancing the polynomial approximation of the gradients of conserved variables within each cell. Numerical oscillations are curbed by means of a non-linear slope limiter. Time integration is second-order accurate and implicit. The numerical model is based on dynamically adaptive unstructured triangular grids. Test cases include an oblique hydraulic jump, jet-forced flow in a flat-bottomed circular reservoir, wind-induced circulation in a circular basin of non-uniform bed topography and the collapse of a circular dam. The model is found to give accurate results in comparison with published analytical and alternative numerical solutions. Dynamic grid adaptation and the use of a second-order implicit time integration scheme are found to enhance the computational efficiency of the model.     

Numerical solution of the unsteady Euler equations for airfoils using approximate boundary conditions

This paper presents an efficient numerical method for solving the unsteady Euler equations on stationary rectilinear grids. Boundary conditions on the surface of an airfoil are implemented by using their first-order expansions on the mean chord line. The method is not restricted to flows with small disturbances since there are no restrictions on the mean angle of attack of the airfoil. The mathematical formulation and the numerical implementation of the wall boundary conditions in a fully implicit time-accurate finite-volume Euler scheme are described. Unsteady transonic flows about an oscillating NACA 0012 airfoil are calculated. Computational results compare well with Euler solutions by the full boundary conditions on a body-fitted curvilinear grid and published experimental data. This study establishes the feasibility for computing unsteady fluid-structure interaction problems, where the use of a stationary rectilinear grid offers substantial advantages in saving computer time and program design since it does not require the generation and implementation of time-dependent body-fitted grids.     

Euler flow solutions for transonic shock wave–boundary layer interaction

Steady 2D Euler flow computations have been performed for a wind tunnel section, designed for research on transonic shock wave–boundary layer interaction. For the discretization of the steady Euler equations, an upwind finite volume technique has been applied. The solution method used is collective, symmetric point Gauss–Seidel relaxation, accelerated by non-linear multigrid. Initial finest grid solutions have been obtained by nested iteration. Automatic grid adaptation has been applied for obtaining sharp shocks. An indication is given of the mathematical quality of four different boundary conditions for the outlet flow. Two transonic flow solutions with shock are presented: a choked and a non-choked flow. Both flow solutions show good shock capturing. A comparison is made with experimental results.     

Multigrid solution of the Navier–Stokes equations on highly stretched grids

Relaxation-based multigrid solvers for the steady incompressible Navier–Stokes equations are examined to determine their computational speed and robustness. Four relaxation methods were used as smoothers in a common tailored multigrid procedure. The resulting solvers were applied to three two-dimensional flow problems, over a range of Reynolds numbers, on both uniform and highly stretched grids. In all cases the L₂ norm of the velocity changes is reduced to 10^?6 in a few 10's of fine-grid sweeps. The results of the study are used to draw conciusions on the strengths and weaknesses of the individual relaxation methods as well as those of the overall multigrid procedure when used as a solver on highly stretched grids.     

A second-order upwind finite-volume method for the Euler solution on unstructured triangular meshes

A scheme for the numerical solution of the two-dimensional (2D) Euler equations on unstructured triangular meshes has been developed. The basic first-order scheme is a cell-centred upwind finite-volume scheme utilizing Roe's approximate Riemann solver. To obtain second-order accuracy, a new gradient based on the weighted average of Barth and Jespersen's three-point support gradient model is used to reconstruct the cell interface values. Characteristic variables in the direction of local pressure gradient are used in the limiter to minimize the numerical oscillation around solution discontinuities. An Approximate LU (ALU) factorization scheme originally developed for structured grid methods is adopted for implicit time integration and shows good convergence characterisitics in the test. To eliminate the data dependency which prohibits vectorization in the inversion process, a black-gray-white colouring and numbering technique on unstructured triangular meshes is developed for the ALU factorization scheme. This results in a high degree of vectorization of the final code. Numerical experiments on transonic Ringleb flow, transonic channel flow with circular bump, supersonic shock reflection flow and subsonic flow over multielement aerofoils are calculated to validate the methodology.     

Multigrid solution of compressible turbulent flow on unstructured meshes using a two-equation model

The steady state solution of the system of equations consisting of the full Navier-Stokes equations and two turbulence equations has been obtained using a multigrid strategy on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multistage Runge-Kutta time-stepping scheme with a stability-bound local time step, while the turbulence equations are advanced in a point-implicit scheme with a time step which guarantees stability and positivity. Low-Reynolds-number modifications to the original two-equation model are incorporated in a manner which results in well-behaved equations for arbitrarily small wall distances. A variety of aerodynamic flows are solved, initializing all quantities with uniform freestream values. Rapid and uniform convergence rates for the flow and turbulence equations are observed.     

High‐order accurate p‐multigrid discontinuous Galerkin solution of the Euler equations

Discontinuous Galerkin (DG) methods have proven to be perfectly suited for the construction of very high‐order accurate numerical schemes on arbitrary unstructured and possibly nonconforming grids for a wide variety of applications, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods a p‐multigrid solution strategy has been developed, which is based on a semi‐implicit Runge–Kutta smoother for high‐order polynomial approximations and the implicit Backward Euler smoother for piecewise constant approximations. The effectiveness of the proposed approach is demonstrated by comparison with p‐multigrid schemes employing purely explicit smoothing operators for several 2D inviscid test cases. Copyright © 2008 John Wiley & Sons, Ltd.     

Perturbational finite volume method for the solution of 2-D navier-stokes equations on unstructured and structured colocated meshes

IntroductionThefinitevolume (FV)methodusestheintegralformoftheconservationequationasitsstartingpointandcanutilizeconvenientlydiversifiedgrids(structuredandunstructuredgrids)andissuitableforverycomplexgeometry ,whicharewhyitispopularwithengineeringandhasbeenwidelyusedinagreatvarietyofcommercialsoftwareofcomputationalfluiddynamics.Relativetothefiniteelement (FE)methodandthefinitedifferential (FD)method ,thedisadvantageofFVmethodisthatitisnothigheraccuracy .FVmethodisofsecondlevelapproximatio…     

用拟压缩性方法求解不可压Euler方程

用拟压缩性方法和Ｊａｍｅｓｏｎ的有限体积算法求解了二维和三维定常可可压Ｅｕｌｅｒ方程。分别采用显、隐式时间离散推进求解;分析了人工粘性的阶数对定常解收敛性的影响,应用该方法计算了单个翼型和翼身组合体的低速绕流,结果与实验吻合较好。     

Multigrid convergence acceleration for implicit and explicit solution of Euler equations on unstructured grids

The multigrid method is one of the most efficient techniques for convergence acceleration of iterative methods. In this method, a grid coarsening algorithm is required. Here, an agglomeration scheme is introduced, which is applicable in both cell‐center and cell‐vertex 2 and 3D discretizations. A new implicit formulation is presented, which results in better computation efficiency, when added to the multigrid scheme. A few simple procedures are also proposed and applied to provide even higher convergence acceleration. The Euler equations are solved on an unstructured grid around standard transonic configurations to validate the algorithm and to assess its superiority to conventional explicit agglomeration schemes. The scheme is applied to 2 and 3D test cases using both cell‐center and cell‐vertex discretizations. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimation of goal functional error arising from iterative solution of Euler equations

Estimation of the error arising in the cost (goal) functional due to stopping the iterative process is considered for a steady problem solved by temporal relaxation. The functional error is calculated using an iteration residual along with related adjoint parameters. Numerical tests demonstrate the applicability of this approach for the steady 2D Euler equations.     

2-D/3-D finite-element solution of the steady Euler equations for transonic lifting flow by stream vector correction

In this paper we study the validation of the new formulation (potential-stream vector) of the steady Euler equations in 2-D/3-D transonic lifting regime flow. This approach, which is based on the Helmholtz decomposition of a velocity vector field, is designed to extend the potential approximation of Euler equations for severe situations such as high transonic or rotational subsonic flows. Different results computed by a fixed point algorithm on the stream vector correction are shown and discussed by comparing them with those obtained by the full potential approach.     

UNCONDITIONAL STABLE SOLUTIONS OF THE EULER EQUATIONS FOR TWO－AND THREE－D WINGS IN ARBITRARY MOTION

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Symmetry techniques for the numerical solution of the 2D Euler equations at impermeable boundaries

The implementation of boundary conditions at rigid, fixed wall boundaries in inviscid Euler solutions by upwind, finite volume methods is considered. Some current methods are reviewed. Two new boundary condition procedures, denoted as the symmetry technique and the curvature-corrected symmetry technique are then presented. Their behaviour in relation to the problem of the subsonic flow about blunt and slender elliptic bodies is analysed. The subsonic flow inside the Stanitz elbow is then computed. The symmetry technique is proven to be as accurate as one of the current methods, second-order pressure extrapolation technique. Finally, for arbitrary curved geometries, dramatic advantages of the curvature-corrected symmetry technique over the other methods are shown. © 1998 John Wiley & Sons, Ltd.     

Improved artificial dissipation schemes for the Euler equations

The influence of artificial dissipation schemes on the accuracy and stability of the numerical solution of compressible flow is extensively examined. Using an implicit central difference factored scheme, an improved form of artificial dissipation is introduced which highly reduces the errors due to numerical viscosity. A function of the local Mach number is used to scale the amount of numerical damping added into the solution according to the character of the flow in several flow regimes. The resulting scheme is validated through several inviscid flow test cases.     

An adaptive least‐squares method for the compressible Euler equations

An adaptive least‐squares finite element method is used to solve the compressible Euler equations in two dimensions. Since the method is naturally diffusive, no explicit artificial viscosity is added to the formulation. The inherent artificial viscosity, however, is usually large and hence does not allow sharp resolution of discontinuities unless extremely fine grids are used. To remedy this, while retaining the advantages of the least‐squares method, a moving‐node grid adaptation technique is used. The outstanding feature of the adaptive method is its sensitivity to directional features like shock waves, leading to the automatic construction of adapted grids where the element edge(s) are strongly aligned with such flow phenomena. Using well‐known transonic and supersonic test cases, it has been demonstrated that by coupling the least‐squares method with a robust adaptive method shocks can be captured with high resolution despite using relatively coarse grids. Copyright © 1999 John Wiley & Sons, Ltd.     

多维迎风方法在非结构/混合网格热流计算中的应用研究

非结构/混合网格具有极强的几何灵活性,在复杂外形飞行器的气动力特性数值模拟中已得到广泛应用,但目前还难以准确地预测气动热环境。本文从非结构/混合网格热流计算的三个需求出发,选取了多维迎风方法,并与其他方法进行了对比研究。以二维圆柱高超声速绕流这一Benchmark典型问题为例,对比研究了多维迎风方法和几种广泛使用的无粘通量格式(Roe格式、Van Leer格式和AUSMDV格式)对混合网格热流计算精度的影响。结果表明,多维迎风方法在热流计算精度、鲁棒性以及收敛性方面表现良好。最后,将多维迎风方法应用于常规混合网格上的圆柱和钝双锥绕流问题,均得到了较好的热流计算结果,为非结构/混合网格热流计算在复杂高超飞行器中的应用奠定了基础。