三维对流扩散方程非等距网格上的四阶紧致格式及其多重网格方法

提出了数值求解三维变系数对流扩散方程非等距网格上的四阶精度19点紧致差分格式,为了提高求解效率,采用多重网格方法求解高精度格式所形成的大型代数方程组。数值实验结果表明本文方法对于不同的网格雷诺数问题,在精确性、稳定性和减少计算工作量方面均明显优于7点中心差分格式。     

局部时间步长间断有限元方法求解三维欧拉方程

使用间断有限元方法求解三维流体力学方程.空间剖分采用非结构四面体网格,为了克服显格式在单元网格尺寸差别较大时计算效率低下的问题,在格式中采用局部时间步长技术(LTS),即控制方程在空间、时间上积分得到一种单步格式,既可以局部计算每个单元又避免了Runge-Kutta高精度格式处理三维问题时存储量过大的问题.为了提高流体力学方程计算精度,在计算单元边界的数值流通量时使用任意高阶精度方法(ADER).数值算例表明格式稳定有效.     

解三维扩散方程的积分内插法格式

研究三维扩散方程的数值模拟.在非正规六面体网格上,使用积分内插法建立扩散方程差分格式,涉及到27个相邻网格,适用于大变形网格上带间断系数的拟线性扩散方程的计算.叙述差分格式的建立,推导通量流和网格顶点温度的计算公式,给出了数值试验结果.     

用多块多网格方法数值模拟三维粘性流动

本文给出了一个模拟三维粘性流动的数值方法．该方法来用高分辨率 ＴＶＤ Ｌａｘ－Ｗｅｎｄｒｏｆｆ格式求解三维雷诺平均Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程，使用Ｂａｌｄｗｉｎ－Ｌｏｍａｘ模型估计湍流粘性系数，用多重网格技术加速收敛，采用多块结构化网格处理复杂的物理域．文中给出了叶轮机械多个叶片排和透平排汽缸内的全三维粘性流动的数值结果．     

三维紧致修正方法和圆筒内自然对流数值模拟

本文发展了差分法求解流动与换热问题的三维非均分网格7点紧致格式,并利用延迟修正方法将其与SIMPLE算法相结合形成了一种三维紧致修正方法。利用所发展的紧致修正方法对圆筒内同心开缝圆筒的三维自然对流与换热问题进行了数值模拟,所获得的数值结果与实验结果一致。采用Richardson方法证实所发展的三维紧致修正方法大约具有4阶精度。进一步的数值计算表明,在特征参数Ra数大于一定值时,由圆筒内同心开缝圆筒的三维自然对流问题简化的二维模型数值结果与实验结果逐渐加大,用三维模型才能得到比较可靠的结果。     

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。     

液舱横荡非结构网格高阶格式数值模拟

提出一种液舱横荡数值模拟的方法,将气液两相交界面视为物理间断,通过高阶精度离散格式捕捉间断.根据NVD(Normalized Variable Diagram)实现非结构化网格上高精度离散格式,建立固定网格上自由表面运动模拟方法.在开发的非结构网格有限体积法求解器GTEA(General Transport Equation Analyzer)基础上,实现上述方法.首先对经典的溃坝过程进行模拟,并与文献结果对比验证方法和程序的可信度.对二维矩形液舱在不同激振频率时的横荡进行数值计算,并与实验以及商业软件CFX计算结果进行比较.结果表明方法和软件可以模拟自由面的翻卷、破碎运动现象,对距自由面较深点处流体载荷的计算结果与实验值符合较好,与商业软件CFX相比,在相同计算网格下,算法可以更好的计算次峰值,验证方法正确可行.     

低压蒸汽透平排汽缸内三维粘性流动的数值模拟

采用三维粘性流场求解软件Fine／TURBO[1]对低压蒸汽透平下游排汽缸内的复杂流动进行数值模拟。计算中使用了Jameson的Runge-Kutta中心差分格式[2]和Baldwin-Lomax的代数湍流模型、计算结果同部分实验结果进行了对比,表明数值模拟揭示了排汽缸内复杂的旋涡结构,以及影响排气缸内压力恢复和总压损失的主要因素。     

采用高分辨率高精度格式求解跨音压气机转子内三维粘性流场

采用一种新型的ＬＵ隐式算法及４－５阶高分辨率ＴＶＤ格式求解了叶轮机械内的三维Ｎ－Ｓ方程和ｑ－ｗ湍流模型。应用几何法与ＰＤＦ方法混合的网格生成程序,得到了ＲＯＴＲＯ３７跨音速压气机转子内的Ｈ型网格,并将数值模拟的结果同ＬＤＶ实验值进行了比较。     

平面叶栅气膜冷却流动的数值模拟

为了能够准确地对透平叶栅气膜冷却效率进行数值预测,本文采用了FNM形式的结构化网格,对一个平面叶栅中的气膜冷却流场进行了数值模拟。计算中采用了包括LU-SGS-GE隐式格式和改良型高精度、高分辨率的MUSCL TVD格式的时间推进算法求解三维RANS方程以及低Reynolds数q-ω双方程湍流模型。计算结果表明本文采用的模型及方法在低吹风比的条件下可以较准确地对气膜冷却效率进行数值预测。     

A geometrical predictor–corrector advection scheme and its application to the volume fraction function

We present a multidimensional Eulerian advection method for interfacial and incompressible flows in two-dimensional Cartesian geometry. In the scheme we advect the grid nodes backwards along the streamlines to compute the pre-images of the grid lines. These pre-images are approximated by continuous, piecewise-linear lines. The enforcement of the discrete version of the incompressibility constraint is a very important issue to determine correctly the flux polygons and to reduce considerably the integration, discretization and interpolation numerical errors. The proposed method compares favorably with other previous multidimensional advection methods as long as the initial interface line is well reconstructed. Conversely, we show that when the interface is very fragmented the total numerical error is completely dominated by the reconstruction error and in these conditions it is very difficult to assess which advection scheme is the most reliable one.     

Integration of some examples of geodesic flows via solvable structures

Solvable structures are particularly useful in the integration by quadratures of ordinary differential equations. Nevertheless, for a given equation, it is not always possible to compute a solvable structure. In practice, the simplest solvable structures are those adapted to an already known system of symmetries. In this paper we propose a method of integration which uses solvable structures suitably adapted to both symmetries and first integrals. In the variational case, due to Noether theorem, this method is particularly effective as illustrated by some examples of integration of the geodesic flows.     

NS方程的各向异性笛卡尔网格法研究

将近年发展起来的用于Euler方程求解的具有局部均匀网格总体非结构特性的笛卡尔网格法推广到NS方程的求解。为了与流场的各向异性相适应、减少网格点数量,提出了一种各向异性网格加密法。另外还研究了分级笛卡尔网格对内点格式稳定性的影响和插值固体边界条件的稳定性。数值结果表明各向异性笛卡尔网格法相对于传统的各向同性网格方法能大量节省网格点数量而且与后者具有同样的精度。     

A splitting extrapolation for solving nonlinear elliptic equations with d-quadratic finite elements

Nonlinear elliptic partial differential equations are important to many large scale engineering and science problems. For this kind of equations, this article discusses a splitting extrapolation which possesses a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than Richardson extrapolation. According to the problems, some domain decompositions are constructed and some independent mesh parameters are designed. Multi-parameter asymptotic expansions are proved for the errors of approximations. Based on the expansions, splitting extrapolation formulas are developed to compute approximations with high order of accuracy on a globally fine grid. Because these formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems.     

A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion

We develop numerical methods for solving partial differential equations (PDE) defined on an evolving interface represented by the grid based particle method (GBPM) recently proposed in [S. Leung, H.K. Zhao, A grid based particle method for moving interface problems, J. Comput. Phys. 228 (2009) 7706–7728]. In particular, we develop implicit time discretization methods for the advection–diffusion equation where the time step is restricted solely by the advection part of the equation. We also generalize the GBPM to solve high order geometrical flows including surface diffusion and Willmore-type flows. The resulting algorithm can be easily implemented since the method is based on meshless particles quasi-uniformly sampled on the interface. Furthermore, without any computational mesh or triangulation defined on the interface, we do not require remeshing or reparametrization in the case of highly distorted motion or when there are topological changes. As an interesting application, we study locally inextensible flows governed by energy minimization. We introduce tension force via a Lagrange multiplier determined by the solution to a Helmholtz equation defined on the evolving interface. Extensive numerical examples are also given to demonstrate the efficiency of the proposed approach.     

Nested Cartesian grid method in incompressible viscous fluid flow

In this work, the local grid refinement procedure is focused by using a nested Cartesian grid formulation. The method is developed for simulating unsteady viscous incompressible flows with complex immersed boundaries. A finite-volume formulation based on globally second-order accurate central-difference schemes is adopted here in conjunction with a two-step fractional-step procedure. The key aspects that needed to be considered in developing such a nested grid solver are proper imposition of interface conditions on the nested-block boundaries, and accurate discretization of the governing equations in cells that are with block-interface as a control-surface. The interpolation procedure adopted in the study allows systematic development of a discretization scheme that preserves global second-order spatial accuracy of the underlying solver, and as a result high efficiency/accuracy nested grid discretization method is developed. Herein the proposed nested grid method has been widely tested through effective simulation of four different classes of unsteady incompressible viscous flows, thereby demonstrating its performance in the solution of various complex flow–structure interactions. The numerical examples include a lid-driven cavity flow and Pearson vortex problems, flow past a circular cylinder symmetrically installed in a channel, flow past an elliptic cylinder at an angle of attack, and flow past two tandem circular cylinders of unequal diameters. For the numerical simulations of flows past bluff bodies an immersed boundary (IB) method has been implemented in which the solid object is represented by a distributed body force in the Navier–Stokes equations. The main advantages of the implemented immersed boundary method are that the simulations could be performed on a regular Cartesian grid and applied to multiple nested-block (Cartesian) structured grids without any difficulty. Through the numerical experiments the strength of the solver in effectively/accurately simulating various complex flows past different forms of immersed boundaries is extensively demonstrated, in which the nested Cartesian grid method was suitably combined together with the fractional-step algorithm to speed up the solution procedure.     

Multigrid Solution of the Incompressible Navier–Stokes Equations for Density-Stratified Flow past Three-Dimensional Obstacles

The steady incompressible Navier–Stokes equations in three dimensions are solved for neutral and stably stratified flow past three-dimensional obstacles of increasing spanwise width. The continuous equations are approximated using a finite volume discretisation on staggered grids with a flux-limited monotonic scheme for the advective terms. The discrete equations which arise are solved using a nonlinear multigrid algorithm with up to four grid levels using the SIMPLE pressure correction method as smoother. When at its most effective the multigrid algorithm is demonstrated to yield convergence rates which are independent of the grid density. However, it is found that the asymptotic convergence rate depends on the choice of the limiter used for the advective terms of the density equation, and some commonly used schemes are investigated. The variation with obstacle width of the influence of the stratification on the flow field is described and the results of the three-dimensional computations are compared with those of the corresponding computation of flow over a two-dimensional obstacle (of effectively infinite width). Also given are the results of time-dependent computations for three-dimensional flows under conditions of strong static stability when lee-wave propagation is present and the multigrid algorithm is used to compute the flow at each time step.     

A non-uniform grid for triangular differential quadrature

The triangular differential quadrature method based on a non-uniform grid is proposed in the paper. Explicit expressions of the non-uniform grid point coordinates are given and the weighting coefficients of the triangular differential quadrature method are determined with the aid of area coordinates. Two typical examples are presented to testify the effectiveness of the non-uniform grid. It is shown that rapid convergence is achieved under the non-uniform grid in comparison with those from the uniform grid with the same order of approximation.     

Parallel re-initialization of level set functions on distributed unstructured tetrahedral grids

Level set functions are employed to track interfaces in various application areas including simulation of two-phase flows and image segmentation. Often, a re-initializing algorithm is incorporated to transform a numerically instable level set function to a signed distance function. In this note, we present a parallel algorithm for re-initializing level set functions on unstructured, three-dimensional tetrahedral grids. The main idea behind this new domain decomposition approach is to combine a parallel brute-force re-initializing algorithm with an efficient way to compute distances between the interface and grid points. Time complexity and error analysis of the algorithm are investigated. Detailed numerical experiments demonstrate the accuracy and scalability on up to 128 processes.     

High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition

The direct numerical simulation of receptivity, instability and transition of hypersonic boundary layers requires high-order accurate schemes because lower-order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields. The main limiting factor in the application of high-order schemes to practical boundary-layer flow problems is the numerical instability of high-order boundary closure schemes on the wall. This paper presents a family of high-order non-uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary-layer transition. By using an appropriate grid stretching, and clustering grid points near the boundary, high-order schemes with stable boundary closures can be obtained. The order of the schemes ranges from first-order at the lowest, to the global spectral collocation method at the highest. The accuracy and stability of the new high-order numerical schemes is tested by numerical simulations of the linear wave equation and two-dimensional incompressible flat plate boundary layer flows. The high-order non-uniform-grid schemes (up to the 11th-order) are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge. The steady and unsteady results show that the new high-order schemes are stable and are able to produce high accuracy for computations of the nonlinear two-dimensional Navier–Stokes equations for the wall bounded supersonic flow.