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.     

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids.An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex.For the temporal integration of the momentum equations,an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term.The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM).The momentum interpolation is used to damp out the spurious pressure wiggles.The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure.The classic test cases,the lid-driven cavity flow,the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with the published benchmark solutions.     

Simulation of shallow flows over variable topographies using unstructured grids

Simulation of shallow flows over variable topographies is a challenging case for most available shock‐capturing schemes. This problem arises because the source terms and flux gradients are not balanced in the numerical computations. Treatments for this problem generally work well on structured grids, but they are usually too expensive, and most of them are not directly applicable to unstructured grids. In this paper we propose two efficient methods to treat the source terms without upwinding and to satisfy the compatibility condition on unstructured grids. In the first method, the calculation of the bed slope source term is performed by employing a compatible approximation of water depth at the cell interfaces. In the second one, different components of the bed slope term are considered separately and a compatible discretization of the components is proposed. The present treatments are applicable for most schemes including the Roe's method without changing the performance of the original scheme for smooth topographies. Copyright © 2006 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 finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

An unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

基于非结构/混合网格的高阶精度格式研究进展

尽管以二阶精度格式为基础的计算流体力学(CFD) 方法和软件已经在航空航天飞行器设计中发挥了重要的作用, 但是由于二阶精度格式的耗散和色散较大, 对于湍流、分离等多尺度流动现象的模拟, 现有成熟的CFD 软件仍难以给出满意的结果, 为此CFD 工作者发展了众多的高阶精度计算格式. 如果以适应的计算网格来分类, 一般可以分为基于结构网格的有限差分格式、基于非结构/混合网格的有限体积法和有限元方法,以及各种类型的混合方法. 由于非结构/混合网格具有良好的几何适应性, 基于非结构/混合网格的高阶精度格式近年来备受关注. 本文综述了近年来基于非结构/混合网格的高阶精度格式研究进展, 重点介绍了空间离散方法, 主要包括k-Exact 和ENO/WENO 等有限体积方法, 间断伽辽金(DG) 有限元方法, 有限谱体积(SV) 和有限谱差分(SD) 方法, 以及近来发展的各种DG/FV 混合算法和将各种方法统一在一个框架内的CPR (correctionprocedure via reconstruction) 方法等. 随后简要介绍了高阶精度格式应用于复杂外形流动数值模拟的一些需要关注的问题, 包括曲边界的处理方法、间断侦测和限制器、各种加速收敛技术等. 在综述过程中, 介绍了各种方法的优势与不足, 其间介绍了作者发展的基于"静动态混合重构" 的DG/FV 混合算法. 最后展望了基于非结构/混合网格的高阶精度格式的未来发展趋势及应用前景.     

Comparison of Galerkin and control volume finite element for advection–diffusion problems

The control volume finite element method (CVFEM) was developed to combine the local numerical conservation property of control volume methods with the unstructured grid and generality of finite element methods (FEMs). Most implementations of CVFEM include mass‐lumping and upwinding techniques typical of control volume schemes. In this work we compare, via numerical error analysis, CVFEM and FEM utilizing consistent and lumped mass implementations, and stabilized Petrov–Galerkin streamline upwind schemes in the context of advection–diffusion processes. For this type of problem, we find no apparent advantage to the local numerical conservation aspect of CVFEM as compared to FEM. The stabilized schemes improve accuracy and degree of positivity on coarse grids, and also reduce iteration counts for advection‐dominated problems. Published in 2005 by John Wiley & Sons, Ltd.     

Mass conservation in computational morphodynamics: uniform sediment and infinite availability

Computational morphodynamics in finite volume methods are based on the evaluation of the rate of bed level change in the vertices on the deforming bed. With the use of finite volume methods on collocated (unstructured) grids, the rate of bed level change needs to be interpolated from the mesh faces to the vertices. First, this work reviews two methods based on a vectorial shape of the bed evolution equation (no scalar contributions from storage, erosion and deposition) in terms of their mass conserving properties. Second, a method that allows for scalar contributions in the bed evolution equation (the Exner equation) is proposed for general, unstructured meshes, and an analytical derivation for the simple one‐dimensional problem on a non‐equidistantly discretised grid is considered. The solution is compared with the general two‐dimensional formulation. The two‐dimensional formulation leads to the formulation of a geometric sand sliding routine on unstructured grids. The newly proposed interpolation method and the sand sliding routine are tested, and mass conservation of the sediment is considered with special emphasis on the effect of the solution accuracy for the suspended sediment transport. Discussions on other interpolation methods and their mass conserving properties are given with a special focus of the distance weighted interpolation method directly available and easily applied in O penFOAM . Furthermore, effects from horizontal displacements of the vertices, explicit filtering of the evolving bed and morphological acceleration on global mass conservation, are discussed. Copyright © 2015 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 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.     

A formally second‐order cell centred scheme for convection–diffusion equations on general grids

We propose, in this paper, a finite volume scheme to compute the solution of the convection–diffusion equation on unstructured and possibly non‐conforming grids. The diffusive fluxes are approximated using the recently published SUSHI scheme in its cell centred version, that reaches a second‐order spatial convergence rate for the Laplace equation on any unstructured two‐dimensional/three‐dimensional grids. As in the MUSCL method, the numerical convective fluxes are built with a prediction‐limitation process, which ensures that the discrete maximum principle is satisfied for pure convection problems. The limitation does not involve any geometrical reconstruction, thus allowing the use of completely general grids, in any space dimension. Copyright © 2012 John Wiley & Sons, Ltd.     

High‐order strand grid methods for low‐speed and incompressible flows

A novel high‐order finite volume scheme using flux correction methods in conjunction with structured finite differences is extended to low Mach and incompressible flows on strand grids. Flux correction achieves a high order by explicitly canceling low‐order truncation error terms across finite volume faces and is applied in unstructured layers of the strand grid. The layers are then coupled together using a source term containing summation‐by‐parts finite differences in the strand direction. A preconditioner is employed to extend the method to low speed and incompressible flows. We further extend the method to turbulent flows with the Spalart–Allmaras model. Laminar flow test cases indicate improvements in accuracy and convergence using the high‐order preconditioned method, while turbulent body‐of‐revolution flow results show improvements in only some cases, perhaps because of dominant errors arising from the turbulence model itself. Copyright © 2016 John Wiley & Sons, Ltd.     

Pressure weighted upwinding for flow induced force predictions: application to iced surfaces

This paper addresses two main topics, namely the development of a pressure‐weighted upwinding method and its application to flow induced forces on iced cylinders. Although the near‐wall convective upwinding exhibits special applicability to iced surfaces, its capabilities extend more generally to other applications. By fully linking pressure and velocity at a sub‐element level near the wall, a higher order accuracy can be obtained. Also, a non‐physical de‐coupling between pressure and velocity can be prevented. The method is developed under the context of a control‐volume‐based finite element method for 2‐D, incompressible flows. Drag and lift coefficients are predicted, based on the pressure weighted upwinding near the wall. The numerical predictions are successfully compared against experimental data, including flow induced forces on iced cables. Copyright © 2004 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

基于非结构化同位网格的SIMPLE算法

通过基于非结构化网格的有限体积法对二维稳态Navier—Stokes方程进行了数值求解。其中对流项采用延迟修正的二阶格式进行离散；扩散项的离散采用二阶中心差分格式；对于压力-速度耦合利用SIMPLE算法进行处理；计算节点的布置采用同位网格技术，界面流速通过动量插值确定。本文对方腔驱动流、倾斜腔驱动流和圆柱外部绕流问题进行了计算，讨论了非结构化同位网格有限体积法在实现SIMPLE算法时，迭代次数与欠松弛系数的关系、不同网格情况的收敛性、同结构化网格的对比以及流场尾迹结构。通过和以往结果比较可知，本文的方法是准确和可信的。     

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.     

Finite volume scheme for the solution of fluid flow problems on unstructured non‐staggered grids

A recently developed non‐staggered methodology which uses the principle of applying fourth‐order dissipation to the governing pressure‐correction equation is developed so it can be applied to unstructured grids. A finite volume methodology is used for discretization. The fourth‐order dissipation term is found using second‐order gradient operators. This makes it straightforward to incorporate the dissipation term on unstructured grids. The new methodology is compared with solutions from a standard finite volume second‐order flow solver and is also tested for a standard laminar driven‐lid flow problem with grids systems that do not have a uniform structure. Finally, we demonstrate how the new methodology can be used to predict flow over a wavy boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

An aspect ratio agglomeration multigrid for unstructured grids

A robust aspect ratio‐based agglomeration algorithm to generate high quality of coarse grids for unstructured and hybrid grids is proposed in this paper. The algorithm focuses on multigrid techniques for the numerical solution of Euler and Navier–Stokes equations, which conform to cell‐centered finite volume special discretization scheme, combines vertex‐based isotropic agglomeration and cell‐based directional agglomeration to yield large increases in convergence rates. Aspect ratio is used as fusing weight to capture the degree of cell convexity and give an indication of cell stretching. Agglomeration front queue is established to propagate inward from the boundaries, which stores isotropic vertex and also high‐stretched cell marked with different flag according to aspect ratio. We conduct the present method to solve Euler and Navier–Stokes equations on unstructured and hybrid grids and compare the results with single grid as well as MGridGen, which shows that the present method is efficient in reducing computational time for large‐scale system equations. Copyright © 2013 John Wiley & Sons, Ltd.     

High‐order k‐exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids

This paper presents a family of High‐order finite volume schemes applicable on unstructured grids. The k‐exact reconstruction is performed on every control volume as the primary reconstruction. On a cell of interest, besides the primary reconstruction, additional candidate reconstruction polynomials are provided by means of very simple and efficient ‘secondary’ reconstructions. The weighted average procedure of the WENO scheme is then applied to the primary and secondary reconstructions to ensure the shock‐capturing capability of the scheme. This procedure combines the simplicity of the k‐exact reconstruction with the robustness of the WENO schemes and represents a systematic and unified way to construct High‐order accurate shock capturing schemes. To further improve the efficiency, an efficient problem‐independent shock detector is introduced. Several test cases are presented to demonstrate the accuracy and non‐oscillation property of the proposed schemes. The results show that the proposed schemes can predict the smooth solutions with uniformly High‐order accuracy and can capture the shock waves and contact discontinuities in high resolution. Copyright © 2011 John Wiley & Sons, Ltd.     

A finite volume cell‐centered Lagrangian hydrodynamics approach for solids in general unstructured grids

A finite volume cell‐centered Lagrangian hydrodynamics approach, formulated in Cartesian frame, is presented for solving elasto‐plastic response of solids in general unstructured grids. Because solid materials can sustain significant shear deformation, evolution equations for stress and strain fields are solved in addition to mass, momentum, and energy conservation laws. The total stress is split into deviatoric shear stress and dilatational components. The dilatational response of the material is modeled using the Mie‐Grüneisen equation of state. A predicted trial elastic deviatoric stress state is evolved assuming a pure elastic deformation in accordance with the hypo‐elastic stress‐strain relation. The evolution equations are advanced in time by constructing vertex velocity and corner traction force vectors using multi‐dimensional Riemann solutions erected at mesh vertices. Conservation of momentum and total energy along with the increase in entropy principle are invoked for computing these quantities at the vertices. Final state of deviatoric stress is effected via radial return algorithm based on the J‐2 von Mises yield condition. The scheme presented in this work is second‐order accurate both in space and time. The suitability of the scheme is evinced by solving one‐ and two‐dimensional benchmark problems both in structured grids and in unstructured grids with polygonal cells. Copyright © 2013 John Wiley & Sons, Ltd.