A fast nested multi‐grid viscous flow solver for adaptive Cartesian/Quad grids

A nested multi‐grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body‐fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub‐division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell‐cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell‐centered finite volume discretization, Roe's flux splitting, a least‐squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell‐cutting. Several cycling strategies, such as the saw‐tooth, W‐ and V‐cycles, have been studies. The V‐cycle has been found to be the most efficient. In general, the multi‐grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.     

复杂无粘流场数值模拟的矩形/三角形混合网格技术

建立了一套模拟复杂无粘流场的矩形／三角形混合网格技术，其中三角形仅限于物面附近，发挥非结构网格的几何灵活性，以少量的网格模拟复杂外型；同时在以外的区域采用矩形结构网格，发挥矩形网格计算简单快速的优势，有效地克服全非结构网格计算方法需要较大内存量和较长ＣＰＵ时间的不足．混合网格系统由修正的四分树方法生成．将ＮＮＤ有限差分与ＮＮＤ有限体积格式有机地融合于混合网格计算，消除了全矩形网格模拟曲边界的台阶效应，同时保证了网格间的通量守恒．数值实验表明本方法在模拟复杂无粘流场方面的灵活性和高效性．     

A numerical study on a Cartesian-based body-fitted adaptive grid method

ABSTRACT

A hybrid Cartesian-based body-fitted adaptive grid method for compressible Navier–Stokes equations is implemented and investigated. In this method, the body-fitted structured grids are generated around the geometries, and the left regions are filled with Cartesian grids. To transfer the data between the different grids, the donor cell searching technique is adopted. An unstructured data-based finite volume update procedure is used, and least squares method is suggested to retain the second order in the overlap region. The moving shock waves with different speeds and vortex passing through the interfaces of the hybrid Cartesian grid are used to explore the accuracy and conservation. A new technique is presented to deal with the non-physical stagnation of slowly moving shock wave around the interface of grid. Numerical examples are presented to demonstrate the results. The three-dimensional extension has also been shown by a benchmark problem.     

动态混合网格生成及隐式非定常计算方法

建立了一种基于动态混合网格的非定常数值计算方法. 混合网格由贴体的四边形网格、外场 的多层次矩形网格和中间的三角形网格构成. 当物体运动时，贴体四边形网格随物体运动而 运动，而外场的矩形网格保持静止，中间的三角形网格随之变形；当物体运动位移较大，导 致三角形网格的质量降低，甚至导致网格相交时，在局部重新生成网格. 新网格上的物理量 由旧网格上的物理量插值而得. 为了提高计算效率，采用了双时间步和子迭代相结合的隐式 有限体积格式计算非定常Navier-Stokes方程. 子迭代采用高效的块LU-SGS方法. 利用该 方法数值模拟了NACA0012振荡翼型的无黏和黏性绕流，得到了与实验和他人计算相当一致 的结果.     

Numerical diffusion for flow‐aligned unstructured grids with application to estuarine modeling

The benefits of unstructured grids in hydrodynamic models are well understood but in many cases lead to greater numerical diffusion compared with methods available on structured grids. The flexible nature of unstructured grids, however, allows for the orientation of the grid to align locally with the dominant flow direction and thus decrease numerical diffusion. We investigate the relationship between grid alignment and diffusive errors in the context of scalar transport in a triangular, unstructured, 3‐D hydrodynamic code. Analytical results are presented for the 2‐D anisotropic numerical diffusion tensor and verified against idealized simulations. Results from two physically realistic estuarine simulations, differing only in grid alignment, show significant changes in gradients of salinity. Changes in scalar gradients are reflective of reduced numerical diffusion interacting with the complex 3‐D structure of the transporting flow. We also describe a method for utilizing flow fields from an unaligned grid to generate a flow‐aligned grid with minimal supervision. Copyright © 2013 John Wiley & Sons, Ltd.     

非结构混合网格高超声速绕流与磁场干扰数值模拟

对均匀磁场干扰下的二维钝头体无粘高超声速流场进行了基于非结构混合网格的数值模拟.受磁流体力学方程组高度非线性的影响及考虑到数值模拟格式的精度,目前在此类流场的数值模拟中大多使用结构网格及有限差分方法,因而在三维复杂外形及复杂流场方面的研究受到限制.本文主要探索使用非结构网格(含混合网格)技术时的数值模拟方法.控制方程为耦合了Maxwell方程及无粘流体力学方程的磁流体力学方程组,数值离散格式采用Jameson有限体积格心格式,5步Runge-Kutta显式时间推进.计算模型为二维钝头体,初始磁场均匀分布.对不同磁感应强度影响下的高超声速流场进行了数值模拟,并与有限的资料进行了对比,得到了较符合的结果.     

A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

In large‐scale shallow flow simulations, local high‐resolution predictions are often required in order to reduce the computational cost without losing the accuracy of the solution. This is normally achieved by solving the governing equations on grids refined only to those areas of interest. Grids with varying resolution can be generated by different approaches, e.g. nesting methods, patching algorithms and adaptive unstructured or quadtree gridding techniques. This work presents a new structured but non‐uniform Cartesian grid system as an alternative to the existing approaches to provide local high‐resolution mesh. On generating a structured but non‐uniform Cartesian grid, the whole computational domain is first discretized using a coarse background grid. Local refinement is then achieved by directly allocating a specific subdivision level to each background grid cell. The neighbour information is specified by simple mathematical relationships and no explicit storage is needed. Hence, the structured property of the uniform grid is maintained. After employing some simple interpolation formulae, the governing shallow water equations are solved using a second‐order finite volume Godunov‐type scheme in a similar way as that on a uniform grid. Copyright © 2010 John Wiley & Sons, Ltd.     

Shallow flow simulation on dynamically adaptive cut cell quadtree grids

A computationally efficient, high‐resolution numerical model of shallow flow hydrodynamics is described, based on dynamically adaptive quadtree grids. The numerical model solves the two‐dimensional non‐linear shallow water equations by means of an explicit second‐order MUSCL‐Hancock Godunov‐type finite volume scheme. Interface fluxes are evaluated using an HLLC approximate Riemann solver. Cartesian cut cells are used to improve the fit to curved boundaries. A ghost‐cell immersed boundary method is used to update flow information in the smallest cut cells and overcome the time step restriction that would otherwise apply. The numerical model is validated through simulations of reflection of a surge wave at a wall, a low Froude number potential flow past a circular cylinder, and the shock‐like interaction between a bore and a circular cylinder. The computational efficiency is shown to be greatly improved compared with solutions on a uniform structured grid implemented with cut cells. Copyright © 2006 John Wiley & Sons, Ltd.     

A robust algorithm for computing fluid flows on highly non‐smooth staggered grids

A new method for computing the fluid flow in complex geometries using highly non‐smooth and non‐orthogonal staggered grid is presented. In a context of the SIMPLE algorithm, pressure and physical tangential velocity components are used as dependent variables in momentum equations. To reduce the sensitivity of the curvature terms in response to coordinate line orientation change, these terms are exclusively computed using Cartesian velocity components in momentum equations. The method is then used to solve some fairly complicated 2‐D and 3‐D flow field using highly non‐smooth grids. The accuracy of results on rough grids (with sharp grid line orientation change and non‐uniformity) was found to be high and the agreement with previous experimental and numerical results was quite good. Copyright © 2008 John Wiley & Sons, Ltd.     

An efficient and accurate algebraic interface capturing method for unstructured grids in 2 and 3 dimensions: The THINC method with quadratic surface representation

We present in this paper an efficient and accurate volume of fluid (VOF) type scheme to compute moving interfaces on unstructured grids with arbitrary quadrilateral mesh elements in 2D and hexahedral elements in 3D. Being an extension of the multi‐dimensional tangent of hyperbola interface capturing (THINC) reconstruction proposed by the authors in Cartesian grid, an algebraic VOF scheme is devised for arbitrary quadrilateral and hexahedral elements. The interface is cell‐wisely approximated by a quadratic surface, which substantially improves the numerical accuracy. The same as the other THINC type schemes, the present method does not require the explicit geometric representation of the interface when computing numerical fluxes and thus is very computationally efficient and straightforward in implementation. The proposed scheme has been verified by benchmark tests, which reveal that this scheme is able to produce high‐quality numerical solutions of moving interfaces in unstructured grids and thus a practical method for interfacial multi‐phase flow simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

A sharp‐interface immersed boundary framework for simulations of high‐speed inviscid compressible flows

A new finite‐volume flow solver based on the hybrid Cartesian immersed boundary (IB) framework is developed for the solution of high‐speed inviscid compressible flows. The IB method adopts a sharp‐interface approach, wherein the boundary conditions are enforced on the body geometry itself. A key component of the present solver is a novel reconstruction approach, in conjunction with inverse distance weighting, to compute the solutions in the vicinity of the solid‐fluid interface. We show that proposed reconstruction leads to second‐order spatial accuracy while also ensuring that the discrete conservation errors diminish linearly with grid refinement. Investigations of supersonic and hypersonic inviscid flows over different geometries are carried out for an extensive validation of the proposed flow solver. Studies on cylinder lift‐off and shape optimisation in supersonic flows further demonstrate the efficacy of the flow solver for computations with moving and shape‐changing geometries. These studies conclusively highlight the capability of the proposed IB methodology as a promising alternative for robust and accurate computations of compressible fluid flows on nonconformal Cartesian meshes.     

A novel multidimensional solution reconstruction and edge‐based limiting procedure for unstructured cell‐centered finite volumes with application to shallow water dynamics

On unstructured meshes, the cell‐centered finite volume (CCFV) formulation, where the finite control volumes are the mesh elements themselves, is probably the most used formulation for numerically solving the two‐dimensional nonlinear shallow water equations and hyperbolic conservation laws in general. Within this CCFV framework, second‐order spatial accuracy is achieved with a Monotone Upstream‐centered Schemes for Conservation Laws‐type (MUSCL) linear reconstruction technique, where a novel edge‐based multidimensional limiting procedure is derived for the control of the total variation of the reconstructed field. To this end, a relatively simple, but very effective modification to a reconstruction procedure for CCFV schemes, is introduced, which takes into account geometrical characteristics of computational triangular meshes. The proposed strategy is shown not to suffer from loss of accuracy on grids with poor connectivity. We apply this reconstruction in the development of a second‐order well‐balanced Godunov‐type scheme for the simulation of unsteady two‐dimensional flows over arbitrary topography with wetting and drying on triangular meshes. Although the proposed limited reconstruction is independent from the Riemann solver used, the well‐known approximate Riemann solver of Roe is utilized to compute the numerical fluxes, whereas the Green–Gauss divergence formulation for gradient computations is implemented. Two different stencils for the Green–Gauss gradient computations are implemented and critically tested, in conjunction with the proposed limiting strategy, on various grid types, for smooth and nonsmooth flow conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical simulation of fluid flows using an unstructured finite volume method with adaptive tri‐tree grids

A tri‐tree grid generation procedure is developed together with a finite volume method on the unstructured grid for solving the Navier–Stokes equations. A hierarchic numbering system for the data structure is used. The grid is adapted by adding and removing cell elements dependent on the vorticity magnitude. A special treatment is developed to ensure good quality triangular elements around the cylinder boundary. The adopted finite volume method is based on the cell‐centred scheme. The pressure–velocity coupling is treated using the SIMPLE algorithm. A modified QUICK scheme for unstructured grids is derived. The developed method is used to simulate the flow past a single and multiple cylinders at low Reynolds number. The obtained results are in good agreement with the published data. Copyright © 2002 John Wiley & Sons, Ltd.     

A three‐dimensional hydrodynamic model for shallow waters using unstructured Cartesian grids

This paper presents a three‐dimensional unstructured Cartesian grid model for simulating shallow water hydrodynamics in lakes, rivers, estuaries, and coastal waters. It is a flux‐based finite difference model that uses a cut‐cell approach to fit the bottom topography and shorelines and, at the same time, has the flexibility of discretizing complex geometries with Cartesian grids that can be arbitrarily downsized in the two horizontal directions simultaneously. Because of the use of Cartesian grids, the grid generation is very simple and does not suffer the grid generation headache often seen in many other unstructured models, as the unstructured Cartesian grid model does not have any requirements on the orthogonality of the grids. The newly developed unstructured Cartesian grid model was validated against analytical solutions for a three‐dimensional seiching case in a rectangular basin, before it was compared with another three‐dimensional model named LESS3D for circulations and salinity transport processes in an idealized embayment that is driven by tides and freshwater inflows. Model tests show that the numerical procedure used in the unstructured Cartesian grid model is robust. Similar to other unstructured models, a variable grid size has resulted in a smaller number of grids required for a reasonable model simulation, which in turn reduces the CPU time used in the model run. Copyright © 2010 John Wiley & Sons, Ltd.     

Octree partitioning of hybrid grids for parallel adaptive viscous flow simulations

A parallel finite volume method for the Navier–Stokes equations with adaptive hybrid prismatic/tetrahedral grids is presented and evaluated in terms of parallel performance. A new method of domain partitioning for complex 3D hybrid meshes is also presented. It is based on orthogonal bisection of a special octree corresponding to the hybrid mesh. The octree is generated automatically and can handle any type of 3D geometry and domain connectivity. One important property of the octree-based partitioning that is exploited is the octree's ability to yield load-balanced partitions that follow the shape of the geometry. This biasing of the octree results in a reduced number of grid elements on the interpartition boundaries and thus fewer data to communicate among processors. Furthermore, the octree-based partitioning gives similar quality of partitions for very different geometries, while requiring minimal user interaction and little computational time. The partitioning method is evaluated in terms of quality of the subdomains as well as execution time. Viscous flow simulations for different geometries are employed to examine the effectiveness of the octree-based partitioning and to test the scalability of parallel execution of the Navier–Stokes solver and hybrid grid adapter on two different parallel systems, the Intel Paragon and the IBM SP2. © 1998 John Wiley & Sons, Ltd.     

基于复合叉树的自适应笛卡尔网格应用研究

采用复合叉树自适应笛卡尔网格和有限体积法求解三维Euler方程,在网格生成过程中,以模型几何外形、模型表面曲率为基础,构建了基于复合叉树的网格生成和加密方法。在流场计算过程中,又针对流场变化特征,建立了基于复合叉树的网格各向异性拆分模式,同时采用以中心差分为基础的Jameson有限体积法。通过对M6机翼在跨音速情况下的数值仿真,表明计算结果与风洞实验结果符合良好,同时也表明本算法具有高分辨率、节省机时,提高计算效率等特点。     

A simplified adaptive Cartesian grid system for solving the 2D shallow water equations

This paper presents a new simplified grid system that provides local refinement and dynamic adaptation for solving the 2D shallow water equations (SWEs). Local refinement is realized by simply specifying different subdivision levels to the cells on a background uniform coarse grid that covers the computational domain. On such a non‐uniform grid, the structured property of a regular Cartesian mesh is maintained and neighbor information is determined by simple algebraic relationships, i.e. data structure becomes unnecessary. Dynamic grid adaptation is achieved by changing the subdivision level of a background cell. Therefore, grid generation and adaptation is greatly simplified and straightforward to implement. The new adaptive grid‐based SWE solver is tested by applying it to simulate three idealized test cases and promising results are obtained. The new grid system offers a simplified alternative to the existing approaches for providing adaptive mesh refinement in computational fluid dynamics. Copyright © 2011 John Wiley & Sons, Ltd.     

Developing a unified FVE‐ALE approach to solve unsteady fluid flow with moving boundaries

In this study, an arbitrary Lagrangian–Eulerian (ALE) approach is incorporated with a mixed finite‐volume–element (FVE) method to establish a novel moving boundary method for simulating unsteady incompressible flow on non‐stationary meshes. The method collects the advantages of both finite‐volume and finite‐element (FE) methods as well as the ALE approach in a unified algorithm. In this regard, the convection terms are treated at the cell faces using a physical‐influence upwinding scheme, while the diffusion terms are treated using bilinear FE shape functions. On the other hand, the performance of ALE approach is improved by using the Laplace method to improve the hybrid grids, involving triangular and quadrilateral elements, either partially or entirely. The use of hybrid FE grids facilitates this achievement. To show the robustness of the unified algorithm, we examine both the first‐ and the second‐order temporal stencils. The accuracy and performance of the extended method are evaluated via simulating the unsteady flow fields around a fixed cylinder, a transversely oscillating cylinder, and in a channel with an indented wall. The numerical results presented demonstrate significant accuracy benefits for the new hybrid method on coarse meshes and where large time steps are taken. Of importance, the current method yields the second‐order temporal accuracy when the second‐order stencil is used to discretize the unsteady terms. Copyright © 2009 John Wiley & Sons, Ltd.     

Hybrid grid generation for viscous flow analysis

Cartesian grid with cut‐cell method has drawn attention of CFD researchers owing to its simplicity. However, it suffers from the accuracy near the boundary of objects especially when applied to viscous flow analysis. Hybrid grid consisting of Cartesian grid in the background, body‐fitted layer near the object, and transition layer connecting the two is an interesting alternative. In this paper, we propose a robust method to generate hybrid grid in two‐dimensional (2D) and three‐dimensional (3D) space for viscous flow analysis. In the first step, body‐fitted layer made of quadrangles (in 2D) or prisms (in 3D) is created near the object's boundary by extruding front nodes with a speed function depending on the minimum normal curvature obtained by quadric surface fitting. To solve global interferences effectively, a level set method is used to find candidates of colliding cells. Then, axis‐aligned Cartesian grid (quadtree in 2D or octree in 3D) is filled in the rest of the domain. Finally, the gap between body‐fitted layer and Cartesian grid is connected by transition layer composed of triangles (in 2D) or tetrahedrons (in 3D). Mesh in transition layer is initially generated by constrained Delaunay triangulation from sampled points based on size function and is further optimized to provide smooth connection. Our approach to automatic hybrid grid generation has been tested with many models including complex geometry and multi‐body cases, showing robust results in reasonable time. Copyright © 2012 John Wiley & Sons, Ltd.