Parallel computation of incompressible flows with complex geometries

We present our numerical methods for the solution of large-scale incompressible flow applications with complex geometries. These methods include a stabilized finite element formulation of the Navier–Stokes equations, implementation of this formulation on parallel architectures such as the Thinking Machines CM-5 and the CRAY T3D, and automatic 3D mesh generation techniques based on Delaunay–Voronoi methods for the discretization of complex domains. All three of these methods are required for the numerical simulation of most engineering applications involving fluid flow. We apply these methods to the simulation of airflow past an automobile and fluid–particle interactions. The simulation of airflow past an automobile is of very large scale with a high level of detail and yielded many interesting airflow patterns which help in understanding the aerodynamic characteristics of such vehicles. © 1997 John Wiley & Sons, Ltd.     

The solution of two‐dimensional free‐surface problems using automatic mesh generation

A new method is described for the iterative solution of two‐dimensional free‐surface problems, with arbitrary initial geometries, in which the interior of the domain is represented by an unstructured, triangular Eulerian mesh and the free surface is represented directly by the piecewise‐quadratic edges of the isoparametric quadratic‐velocity, linear‐pressure Taylor–Hood elements. At each time step, the motion of the free surface is computed explicitly using the current velocity field and, once the new free‐surface location has been found, the interior nodes of the mesh are repositioned using a continuous deformation model that preserves the original connectivity. In the event that the interior of the domain must be completely remeshed, a standard Delaunay triangulation algorithm is used, which leaves the initial boundary discretisation unchanged. The algorithm is validated via the benchmark viscous flow problem of the coalescence of two infinite cylinders of equal radius, in which the motion is due entirely to the action of capillary forces on the free surface. This problem has been selected for a variety of reasons: the initial and final (steady state) geometries differ considerably; in the passage from the former to the latter, large free‐surface curvatures—requiring accurate modelling—are encountered; an analytical solution is known for the location of the free surface; there exists a large body of literature on alternative numerical simulations. A novel feature of the present work is its geometric generality and robustness; it does not require a priori knowledge of either the evolving domain geometry or the solution contained therein. Copyright © 1999 John Wiley & Sons, Ltd.     

A two‐phase adaptive finite element method for solid–fluid coupling in complex geometries

In this paper we present a method to solve the Navier–Stokes equations in complex geometries, such as porous sands, using a finite‐element solver but without the complexity of meshing the porous space. The method is based on treating the solid boundaries as a second fluid and solving a set of equations similar to those used for multi‐fluid flow. When combined with anisotropic mesh adaptivity, it is possible to resolve complex geometries starting with an arbitrary coarse mesh. The approach is validated by comparing simulation results with available data in three test cases. In the first we simulate the flow past a cylinder. The second test case compares the pressure drop in flow through random packs of spheres with the Ergun equation. In the last case simulation results are compared with experimental data on the flow past a simplified vehicle model (Ahmed body) at high Reynolds number using large‐eddy simulation (LES). Results are in good agreement with all three reference models. Copyright © 2010 John Wiley & Sons, Ltd.     

Mesh adaptation strategies for shallow water flow

This paper shows how the mesh adaptation technique can be exploited for the numerical simulation of shallow water flow. The shallow water equations are numerically approximated by the Galerkin finite element method, using linear elements for the elevation field and quadratic elements for the unit width discharge field; the time advancing scheme is of a fractional step type. The standard mesh refinement technique is coupled with the numerical solver; movement and elimination of nodes of the initial triangulation is not allowed. Two error indicators are discussed and applied in the numerical examples. The conclusion focuses the relevant advantages obtained by applying this adaptive approach by considering specific test cases of steady and unsteady flows. Copyright © 1999 John Wiley & Sons, Ltd.     

AUTOMATIC MESH GENERATION OF 3—D GEOMETRIC MODELS

In this paper the presentation of the ball-packing method is reviewed, and a scheme to generate mesh for complex 3-D geometric models is given, which consists of 4 steps: (1) create nodes in 3-D models by ball-packing method, (2) connect nodes to generate mesh by 3-D Delaunay triangulation, (3) retrieve the boundary of the model after Delaunay triangulation, (4) improve the mesh.     

A fully discrete ALE method over untwisted time–space control volumes

In this paper, a fully discrete high‐resolution arbitrary Lagrangian–Eulerian (ALE) method is developed over untwisted time–space control volumes. In the framework of the finite volume method, 2D Euler equations are discretized over untwisted moving control volumes, and the resulting numerical flux is computed using the generalized Riemann problem solver. Then, the fluid flows between meshes at two successive time steps can be updated without a remapping process in the classic ALE method. This remapping‐free ALE method directly couples the mesh motion into a physical variable update to reflect the temporal evolution in the whole process. An untwisted moving mesh is generated in terms of the vorticity‐free part of the fluid velocity according to the Helmholtz theorem. Some typical numerical tests show the competitive performance of the current method. Copyright © 2016 John Wiley & Sons, Ltd.     

高质量点集的快速局部网格生成算法

高效及高质量的局部网格生成算法是基于节点有限元并行方法设计的关键。泡泡布点算法能够在复杂区域上不经过人工干预生成高质量的节点集,本文提出了基于该方法所生成的节点集的快速局部网格生成算法。该算法充分利用泡泡布点方法提供的节点集及节点邻接链表信息,避免了桶数据结构的建立以及节点的局部搜索过程,只需应用Delaunay三角剖分的外接圆准则从中心节点的邻接链表中去除极少数的非卫星点,可快速地生成局部网格,比现有的局部网格生成算法更为快捷。算例结果表明,该算法高效可靠,生成网格与Delaunay三角剖分网格一致。     

三维约束Delaunay三角化的边界恢复和薄元消除方法

提出一种有效的三维约束Delaunay三角剖分的边界恢复算法，该算法综合了P．L．George算法和N．P．Weatherill算法的优点，通过将约束边和约束面加以恢复，保持了实体边界的完整性，解决了经典Delaunay算法不能剖分凹域的问题，从而实现了复杂三维实体的网格剖分。提出了一种简易而有效的消除薄元方法——薄元分解法，彻底解决了三维Delaunay三角剖分过程中所产生的薄元问题。实践证明，本文提出的边界恢复算法和薄元消除算法健壮有效，生成网格的质量高，并且易于实现。     

Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.     

任意平面区域的变尺寸有限元网格划分

利用任意平面域边界表述中的拓年要素定义网格的控制信息,通过引入节点间距函数来控制区域内网格尺寸变化,并将结合前沿生成法和Ｄｅｌａｕｎａｙ三角化方法的优点,优先处理前沿的最长边,尽可能在局部生成边长逐渐减小的Ｄｅｌａｕｎａｙ三角形,最终实验区域内网格的疏密过渡。     

An efficient implicit mesh‐less method for compressible flow calculations

A dual‐time implicit mesh‐less scheme is presented for calculation of compressible inviscid flow equations. The Taylor series least‐square method is used for approximation of spatial derivatives at each node which leads to a central difference discretization. Several convergence acceleration techniques such as local time stepping, enthalpy damping and residual smoothing are adopted in this approach. The capabilities of the method are demonstrated by flow computations around single and multi‐element airfoils at subsonic, transonic and supersonic flow conditions. Results are presented which indicate good agreements with other reliable finite‐volume results. The computational time is considerably reduced when using the proposed mesh‐less method compared with the explicit mesh‐less and finite‐volume schemes using the same point distributions. Copyright © 2010 John Wiley & Sons, Ltd.     

Development of a hybrid finite volume/element solver for incompressible flows

In this paper, we report our development of an implicit hybrid flow solver for the incompressible Navier–Stokes equations. The methodology is based on the pressure correction or projection method. A fractional step approach is used to obtain an intermediate velocity field by solving the original momentum equations with the matrix‐free implicit cell‐centred finite volume method. The Poisson equation derived from the fractional step approach is solved by the node‐based Galerkin finite element method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centres and the auxiliary variable at cell vertices, making the current solver a staggered‐mesh scheme. Numerical examples demonstrate the performance of the resulting hybrid scheme, such as the correct temporal convergence rates for both velocity and pressure, absence of unphysical pressure boundary layer, good convergence in steady‐state simulations and capability in predicting accurate drag, lift and Strouhal number in the flow around a circular cylinder. Copyright © 2007 John Wiley & Sons, Ltd.     

Mesh adaptation for large‐eddy simulations in complex geometries

Large‐eddy simulation (LES) consists in explicitly simulating the large scales of the fluid motion and in modeling the influence of the smallest scales. Thanks to the steady growth of computational resources, LES can now be used to simulate realistic systems with complex geometries. However, when LES is used in such complex geometries, an adequate mesh has to be determined to perform valid LES. In this work, a strategy is proposed to assess the quality of a given mesh and to adapt it locally. Two different criteria are used as mesh adaptation criteria. The first criterion is defined to ensure a correct discretization of the mean field, whereas the second criterion is defined to ensure enough explicit resolution of turbulent scales motions. The use of both criteria is shown in canonical flow cases. As a second part of this work, a numerical strategy for mesh adaptation in high‐performance computing context is proposed by coupling the flow solver, YALES2, and the remeshing library, MMG3D, for massively parallel computations. This coupling enables an efficient and parallel remeshing of grids alleviating any memory or performance issues encountered in sequential tools. This strategy is finally applied to the simulation of the isothermal flow in a complex meso‐combustor to demonstrate the applicability of the adaptation methodology to complex turbulent flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Three-dimensional mesh embedding for the Navier–Stokes equations using upwind control volumes

A numerical model for the compressible Navier–Stokes equations using local mesh embedding is presented. The model solves for three-dimensional turbulent flow using an algebraic mixing length model of turbulence. The technique of control volume upwinding is used to produce a novel treatment, whereby the hanging nodes on the mesh interfaces are left with null control volumes. This yields an efficient discretization scheme which ensures second-order accuracy, flux conservation and stability at the mesh interfaces, whilst retaining a simple interpolative treatment for the hanging nodes. The discrete flow equations are solved using the semi-implicit pressure correction method. The accuracy of the embedded mesh solver is demonstrated by modelling the three-dimensional flow through a cascade of turbine vanes at design and off-design conditions. Mesh embedding gives a saving of 48% in the number of nodes. The embedded mesh solutions compare well with fine structured mesh solutions and experimental measurements. The capability of the embedded mesh solver to perform solution adaptive calculations is demonstrated using a two-dimensional mid-height section of the cascade at the off-design flow conditions.     

Solution of incompressible flows with or without a free surface using the finite volume method on unstructured triangular meshes

An incompressible Navier–Stokes solver based on a cell‐centre finite volume formulation for unstructured triangular meshes is developed and tested. The solution methodology makes use of pseudocompressibility, whereby the convective terms are computed using a Godunov‐type second‐order upwind finite volume formulation. The evolution of the solution in time is obtained by subiterating the equations in pseudotime for each physical time step, with the pseudotime step set equal to infinity. For flows with a free surface the computational mesh is fitted to the free surface boundary at each time step, with the free surface elevation satisfying a kinematic boundary condition. A ‘leakage coefficient’, ε, is introduced for the calculation of flows with a free surface in order to control the leakage of flow through the free surface. This allows the assumption of stationarity of mesh points to be made during the course of pseudotime iteration. The solver is tested by comparing the output with a wide range of documented published results, both for flows with and without a free surface. The presented results show that the solver is robust. © 1999 John Wiley & Sons, Ltd.     

Computation of free‐surface flows with local mesh adaptation

This paper describes a modern free‐surface capturing strategy implemented in an unstructured finite‐volume viscous flow solver that can handle moving grids composed of arbitrary‐shaped control volumes. An adaptive mesh strategy is fully integrated in the code making it a single tool for dynamically maintaining a prescribed density of grid points around the steady or unsteady interface between air and water. The whole adaptive procedure is described in detail. The efficiency of the overall approach is examined on two‐ and three‐dimensional hydrodynamic applications. The adaptive strategy achieves interesting gains in terms of computational and human efforts compared to single‐mesh computations. Copyright © 2005 John Wiley & Sons, Ltd.     

Development of circular function‐based gas‐kinetic scheme (CGKS) on moving grids for unsteady flows through oscillating cascades

In this paper, the circular function‐based gas‐kinetic scheme (CGKS), which was originally developed for simulation of flows on stationary grids, is extended to solve moving boundary problems on moving grids. Particularly, the unsteady flows through oscillating cascades are our major interests. The main idea of the CGKS is to discretize the macroscopic equations by the finite volume method while the fluxes at the cell interface are evaluated by locally reconstructing the solution of the continuous Boltzmann Bhatnagar–Gross–Krook equation. The present solver is based on the fact that the modified Boltzmann equation, which is expressed in a moving frame of reference, can recover the corresponding macroscopic equations with Chapman–Enskog expansion analysis. Different from the original Maxwellian function‐based gas‐kinetic scheme, in improving the computational efficiency, a simple circular function is used to describe the equilibrium state of distribution function. Considering that the concerned cascade oscillating problems belong to cases that the motion of surface boundary is known a priori, the dynamic mesh method is suitable and is adopted in the present work. In achieving the mesh deformation with high quality and efficiency, a hybrid dynamic mesh method named radial basic functions‐transfinite interpolation is presented and applied for cascade geometries. For validation, several numerical test cases involving a wide range are investigated. Numerical results show that the developed CGKS on moving grids is well applied for cascade oscillating flows. And for some cases where nonlinear effects are strong, the solution accuracy could be effectively improved by using the present method. Copyright © 2017 John Wiley & Sons, Ltd.     

脆性材料破坏模拟的网格生成算法

脆性材料的破坏过程具有随机性,当前的网格生成算法没有充分考虑脆性材料破坏时裂纹扩展和碎块生成的随机性。在Persson网格生成算法与Delaunay随机网格剖分理论基础上,提出了一种可根据模拟需要动态控制网格品质的网格生成算法。通过对随机分布点的Delauna三角化,生成初始网格,然后将网格体系比拟为桁架结构,网格节点即为桁架节点。桁架节点在虚拟力作用下可动态调整位置,并最终达到整个体系受力平衡。对Persson 算法中的尺寸分布函数和收敛条件进行了修正,从而提高了收敛速度,并适用于任意形状对象的网格剖分。 基于VC++平台开发了算法程序。通过实例对算法进行了验证,表明算法能够满足脆性材料破碎模拟的需要。     

Calculation of compressible flows about complex moving geometries using a three‐dimensional Cartesian cut cell method

A three‐dimensional Cartesian cut cell method is described for modelling compressible flows around complex geometries, which may be either static or in relative motion. A background Cartesian mesh is generated and any solid bodies cut out of it. Accurate representation of the geometry is achieved by employing different types of cut cell. A modified finite volume solver is used to deal with boundaries that are moving with respect to the stationary background mesh. The current flow solver is an unsplit MUSCL–Hancock method of the Godunov type, which is implemented in conjunction with a cell‐merging technique to maintain numerical stability in the presence of arbitrarily small cut cells and to retain strict conservation at moving boundaries. The method is applied to some steady and unsteady compressible flows involving both static and moving bodies in three dimensions. Copyright © 2000 John Wiley & Sons, Ltd.     

A method for generating irregular computational grids in multiply connected planar domains

A method for generating irregular triangular computational grids in two-dimensional multiply connected domains is described. A set of points around each body is defined using a simple grid generation technique appropriate to the geometry of each body. The Voronoi regions associated with the resulting global point distribution are constructed from which the Delaunay triangulation of the set of points is thus obtained. The definition of Voronoi regions ensures that the triangulation produces triangles of reasonable aspect ratios given a grid point distribution. The approach readily accommodates local clustering of grid points to facilitate variable resolution of the domain. The technique is generally applicable and has been used with success in computing triangular grids in multiply connected planar domains. The suitability of such grids for flow calculations is demonstrated using a finite element method for solution of the inviscid transonic flow over two- dimensional high-lift aerofoil configurations.