OVERLAPPING CONTROL VOLUME APPROACH FOR CONVECTION–DIFFUSION PROBLEMS

This paper introduces a finite volume method to solve 2D steady state convection–diffusion problems on structured non-orthogonal grids. Overlapping control volumes (OCV) are used to discretize the physical domain and the governing equations are solved without transformation. An isoparametric formulation is used to compute diffusion and for upwinding. Four test problems are solved using this and other schemes. The modelling of diffusion in OCV seems very effective even on distorted meshes. The convection modelling in OCV is found to be second-order-accurate, like QUICK, on regular meshes. Although its accuracy is slightly inferior to the latter on rectangular grids, its faster convergence gives it a better overall performance. On non-orthogonal grids, OCV gives better accuracy for a large and practical range of Peclet numbers than does QUICK applied to the transformed equations using the conventional five-point diffusion modelling. The results obtained also demonstrate that the scheme reduces false diffusion to a considerable extent in comparison with the power-law scheme.     

对流扩散方程的摄动有限体积(PFV)方法及讨论

在有限体积(FV)方法的重构近似中,引入数值摄动处理,即把界面数值通量摄动展开成网格间距的幂级数,并利用积分方程自身的性质求出幂级数的系数,同时获得高精度迎风和中心型摄动有限体积(PFV)格式.对标量输运方程给出积分近似为二阶、重构近似为二、三和四阶迎风和中心型PFV格式,这些PFV格式的结构形式及使用基点数与一阶迎风格式完全一致,迎风PFV格式满足对流有界准则;二阶和四阶中心PFV格式对网格Peclet数的任意值均为正型格式,比常用的二阶中心格式优越.用一维标量输运和方腔流动算例说明PFV格式的优良性能,并把PFV方法与性质相近的摄动有限差分(PFD)方法及相关的高精度方法作了对比分析.     

A study of upstream-weighted high-order differencing for approximation to flow convection

This paper is concerned with a number of upstream-weighted second- and third-order difference schemes. Also considered are the conventional upwind and central difference schemes for comparison. It commences with a general difference equation which unifies all the given first-, second- and third-order schemes. The various schemes are evaluated through the use of the general equation. The unboundedness and accuracy of the solutions by the difference schemes are assessed via various analyses: examination of the coefficients of the difference equation, Taylor series truncation error analysis, study of the upstream connection to numerical diffusion, single-cell analysis. Finally, the difference schemes are tested on one- and two-dimensional model problems. It is shown that the high-order schemes suffer less from the problem of numerical diffusion than the first-order upwind difference scheme. However, unboundedness cannot be avoided in the solutions by these schemes. Among them the linear upwind difference scheme presents the best compromise between numerical diffusion and solution unboundedness.     

Colocated schemes for the incompressible Navier–Stokes equations on non‐smooth grids for two‐dimensional problems

The accuracy of colocated finite volume schemes for the incompressible Navier–Stokes equations on non‐smooth curvilinear grids is investigated. A frequently used scheme is found to be quite inaccurate on non‐smooth grids. In an attempt to improve the accuracy on such grids, three other schemes are described and tested. Two of these are found to give satisfactory results. Copyright © 2000 John Wiley & Sons, Ltd.     

Perturbation finite volume method for convective-diffusion integral equation

A perturbation finite volume (PFV) method for the convective-diffusion integral equation is developed in this paper. The PFV scheme is an upwind and mixed scheme using any higher-order interpolation and second-order integration approximations, with the least nodes similar to the standard three-point schemes, that is, the number of the nodes needed is equal to unity plus the face-number of the control volume. For instance, in the two-dimensional (2-D) case, only four nodes for the triangle grids and five nodes for the Cartesian grids are utilized, respectively. The PFV scheme is applied on a number of 1-D linear and nonlinear problems, 2-D and 3-D flow model equations. Comparing with other standard three-point schemes, the PFV scheme has much smaller numerical diffusion than the first-order upwind scheme (UDS). Its numerical accuracies are also higher than the second-order central scheme (CDS), the power-law scheme (PLS) and QUICK scheme. The project supported by the National Natural Science Foundation of China (10272106, 10372106)     

Analysis of fluid flow in constricted tubes and ducts using body-fitted non-staggered grids

A finite volume method for the calculation of laminar and turbulent fluid flows inside constricted tubes and ducts is described. The selected finite volume method is based on curvilinear non-orthogonal co-ordinates (body-fitted co-ordinates) and a non-staggered grid arrangement. The grids are either generated by transfinite interpolation or an elliptic grid generator. The method is employed for calculation of laminar flows through a tube, a converging-diverging duct and different constricted tubes by both a two- and a three-dimensional computer program. In addition, turbulent flow through an axisymmetric constricted tube is calculated. Both the power law scheme and the second-order upwind scheme are used. The calculated results are compared with the experimental data and with other numerical solutions.     

An overlapping control volume method for the Navier–Stokes equations on non‐staggered grids

An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two‐dimensional incompressible Navier–Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non‐staggered grid arrangement. The problem of pressure–velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second‐order accuracy for both diffusion‐ and convection‐dominated flows. There is little loss of accuracy on grids that are moderately non‐orthogonal. Copyright © 1999 John Wiley & Sons, Ltd.     

Fluid flow and heat transfer test problems for non-orthogonal grids: Bench-mark solutions

Four problems of fluid flow and heat transfer were designed in which non-orthogonal, boundary-fitted grids were to be used. These are proposed to serve as test cases for testing new solution methods. This paper presents solutions of the test problems obtained by using a multigrid finite volume method with grids of up to 320 × 320 control volumes. Starting from zero fields, iterations were performed until the sum of the absolute residuals had fallen seven orders of magnitude, thus ensuring that the variable values did not change to six most significant digits. By comparing the solutions for successive grids at moderate Reynolds and Rayleigh numbers, the discretization errors were estimated to be lower than 0·1%. The results presented in this paper may thus serve for comparison purposes as bench-mark solutions.     

The effect of using non-orthogonal boundary-fitted grids for solving the shallow water equations

The purpose of this paper is to examine the effects of using non-orthogonal boundary-fitted grids for the numerical solution of the shallow water equations. Two geometries with well known analytical solutions are introduced in order to investigate the accuracy of the numerical solutions. The results verify that a reasonable departure from orthogonality can be allowed when the rate of change of cell areas is kept sufficiently small (i.e. it is not necessary to create a strictly orthogonal grid when the grid is sufficiently smooth).     

A simple high‐resolution advection scheme

A simple, robust, mass‐conserving numerical scheme for solving the linear advection equation is described. The scheme can estimate peak solution values accurately even in regions where spatial gradients are high. Such situations present a severe challenge to classical numerical algorithms. Attention is restricted to the case of pure advection in one and two dimensions since this is where past numerical problems have arisen. The authors' scheme is of the Godunov type and is second‐order in space and time. The required cell interface fluxes are obtained by MUSCL interpolation and the exact solution of a degenerate Riemann problem. Second‐order accuracy in time is achieved via a Runge–Kutta predictor–corrector sequence. The scheme is explicit and expressed in finite volume form for ease of implementation on a boundary‐conforming grid. Benchmark test problems in one and two dimensions are used to illustrate the high‐spatial accuracy of the method and its applicability to non‐uniform grids. Copyright © 2007 John Wiley & Sons, Ltd.     

Treatment of numerical diffusion in strong convective flows

A three-dimensional extension of the QUICK scheme adapted for the finite volume method and non-uniform grids is presented to handle convection-diffusion problems for high Peclet numbers and steep gradients. The algorithm is based on three-dimensional quadratic interpolation functions in which the transverse curvature terms are maintained and the diagonal dominance of the coefficient matrix is preserved. All formulae are explicitly given in an appendix. Results obtained with the classical upwind (UDS), the simplified QUICK (transverse terms neglected) and the present full QUICK schemes are given for two benchmark problems, one two-dimensional, steady state and the other three-dimensional, unsteady state. Both QUICK schemes are shown to give superior solutions compared with the UDS in terms of accuracy and efficiency. The full QUICK scheme performs better than the simplified QUICK, giving even for coarse grids acceptable results closer to the analytical solutions, while the computational time is not affected much.     

Consistent hybrid finite volume/element formulations: model and complex viscoelastic flows

The accuracy and consistency of a new cell‐vertex hybrid finite element/volume scheme are investigated for viscoelastic flows. Finite element (FE) discretization is employed for the momentum and continuity equation, with finite volume (FV) applied to the constitutive law for stress. Here, the interest is to explore the consequences of utilizing conventional cell‐vertex methodology for an Oldroyd‐B model and to demonstrate resulting drawbacks in the presence of complex source terms on structured and unstructured grids. Alternative strategies worthy of consideration are presented. It is demonstrated how high‐order accuracy may be achieved in steady state by respecting consistency in the formulation. Both FE and FV spatial discretizations are embedded in the scheme, with FV triangular sub‐cells referenced within parent triangular finite elements. Both model and complex flow problems are selected to quantify and assess accuracy, appealing to analysis and experimental validation. The test problem is that of steady sink flow, a pure extensional flow, which reflects some of the numerical difficulties involved in solving more generalized viscoelastic flows, where both source and flux terms may contribute equally to stress propagation. In addition, a complex transient filament‐stretching flow is chosen to compute the evolution of stress fields within liquid bridges. Shortcomings of the various stress upwinding schemes are discussed in this context, whilst dealing with such free‐surface type problems. Here, stress fluctuation distribution alone is advocated, and a Lax‐scheme is found to deliver accuracy and stability to the computational results, comparing well with the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Implementation of some higher-order convection schemes on non-uniform grids

A generalized formulation is applied to implement the quadratic upstream interpolation (QUICK) scheme, the second-order upwind (SOU) scheme and the second-order hybrid scheme (SHYBRID) on non-uniform grids. The implementation method is simple. The accuracy and efficiency of these higher-order schemes on non-uniform grids are assessed. Three well-known bench mark convection-diffusion problems and a fluid flow problem are revisited using non-uniform grids. These are: (1) transport of a scalar tracer by a uniform velocity field; (2) heat transport in a recirculating flow; (3) two-dimensional non-linear Burgers equations; and (4) a two-dimensional incompressible Navier-Stokes flow which is similar to the classical lid-driven cavity flow. The known exact solutions of the last three problems make it possible to thoroughly evaluate accuracies of various uniform and non-uniform grids. Higher accuracy is obtained for fewer grid points on non-uniform grids. The order of accuracy of the examined schemes is maintained for some tested problems if the distribution of non-uniform grid points is properly chosen.     

Recent Development on the Conservation Property of Chimera

An estimate on the conservation error due to the non-conservative data interpolation scheme for overset grids is given in this paper. It is shown that the conservation error is a first-order term if second-order conservative schemes are employed for the Chimera grids and if discontinuities are located away from overlapped grid interfaces. Therefore in the limit of global grid refinement, valid numerical solutions should be obtained with a data interpolation scheme. In one demonstration case the conservation error in the original Chimera scheme was shown to affect flow even without discontinuities on coarse to medium grids. The conservative Chimera scheme was shown to give significantly better solutions than the original Chimera scheme on these grids with other factors being the same.     

Efficient discretization of pressure‐correction equations on non‐orthogonal grids

An alternative discretization of pressure‐correction equations within pressure‐correction schemes for the solution of the incompressible Navier–Stokes equations is introduced, which improves the convergence and robustness properties of such schemes for non‐orthogonal grids. As against standard approaches, where the non‐orthogonal terms usually are just neglected, the approach allows for a simplification of the pressure‐correction equation to correspond to 5‐point or 7‐point computational molecules in two or three dimensions, respectively, but still incorporates the effects of non‐orthogonality. As a result a wide range (including rather high values) of underrelaxation factors can be used, resulting in an increased overall performance of the underlying pressure‐correction schemes. Within this context, a second issue of the paper is the investigation of the accuracy to which the pressure‐correction equation should be solved in each pressure‐correction iteration. The scheme is investigated for standard test cases and, in order to show its applicability to practical flow problems, for a more complex configuration of a micro heat exchanger. Copyright © 2003 John Wiley & Sons, Ltd.     

Simulation of interface and free surface flows in a viscous fluid using adapting quadtree grids

A new adaptive quadtree method for simulating laminar viscous fluid problems with free surfaces and interfaces is presented in this paper. The Navier–Stokes equations are solved with a SIMPLE‐type scheme coupled with the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) (Numerical prediction of two fluid systems with sharp interfaces, Ph.D. Thesis, Imperial College of Science, Technology and Medicine, London, 1997) volume of fluid (VoF) method and PLIC reconstruction of the volume fraction field during refinement and derefinement processes. The method is demonstrated for interface advection cases in translating and shearing flow fields and found to provide high interface resolution at low computational cost. The new method is also applied to simulation of the collapse of a water column and the results are in excellent agreement with other published data. The quadtree grids adapt to follow the movement of the free surface, whilst maintaining a band of the smallest cells surrounding the surface. The calculation is made on uniform and adapting quadtree grids and the accuracy of the quadtree calculation is shown to be the same as that made on the equivalent uniform grid. Copyright © 2004 John Wiley & Sons, Ltd.     

基于GPU加速的三阶有限体积格式管道瞬变流求解模型

为高效和高精度求解长距离输水系统瞬变流变化过程,应用三阶ENO有限体积格式求解一维管道非恒定流方程组,基于Lax-Friedrichs通量裂分法重构界面通量,上下游界面采用虚拟网格技术并结合交叉管网边界条件建立了一套高效和高精度求解管道瞬变流水锤波的数值模型。引入GPU加速技术,实现对大型输水系统的高效计算。通过特征线法、一阶及二阶Godunov有限体积格式对模型进行验证,结果表明,三阶ENO格式在极低的Courant数时也能保持较好的间断捕捉性能且无非物理振荡。同时,对Courant数的高度不敏感性,使得模型划分网格时具有高度的灵活性并能显著提高计算速度。应用GPU加速技术,发现模型在较多网格数时有明显的加速效果,且加速效果随网格数增多而显著。本文模型可为长距离输水系统非恒定瞬变过程的高效精准快速模拟预测提供理论支撑。     

A numerical method to simulate turbulent cavitating flows

The objective of this paper is to develop a numerical method for simulating multiphase cavitating flows on unstructured grids. The multiphase medium is represented using a homogeneous mixture model that assumes thermal equilibrium between the liquid and vapor phases. We develop a predictor–corrector approach to solve the governing Navier–Stokes equations for the liquid/vapor mixture, together with the transport equation for the vapor mass fraction. While a non-dissipative and symmetric scheme is used in the predictor step, a novel characteristic-based filtering scheme with a second order TVD filter is developed for the corrector step to handle shocks and material discontinuities in non-ideal gases and mixtures. Additionally, a sensor based on vapor volume fraction is proposed to localize dissipation to the vicinity of discontinuities. The scheme is first validated for simple one dimensional canonical problems to verify its accuracy in predicting jump conditions across material discontinuities and shocks. It is then applied to two turbulent cavitating flow problems – over a hydrofoil using RANS and over a wedge using LES. Our results show that the simulations are in good agreement with experimental data for the above tested cases, and that the scheme can be successfully applied to both RANS and LES methodologies.     

Three‐dimensional instability of axisymmetric flows: solution of benchmark problems by a low‐order finite volume method

Several problems on three‐dimensional instability of axisymmetric steady flows driven by convection or rotation or both are studied by a second‐order finite volume method combined with the Fourier decomposition in the periodic azimuthal direction. The study is focused on the convergence of the critical parameters with mesh refinement. The calculations are done on the uniform and stretched grids with variation of the stretching. Converged results are reported for all the problems considered and are compared with the previously published data. Some of the calculated critical parameters are reported for the first time. The convergence studies show that the three‐dimensional instability of axisymmetric flows can be computed with a good accuracy only on fine enough grids having about 100 nodes in the shortest spatial direction. It is argued that a combination of fine uniform grids with the Richardson extrapolation can be a good replacement for a grid stretching. It is shown once more that the sparseness of the Jacobian matrices produced by the finite volume method allows one to enhance performance of the Newton and Arnoldi iteration procedures by combining them with a direct sparse linear solver instead of using the Krylov‐subspace‐based iteration methods. Copyright © 2006 John Wiley & Sons, Ltd.     

An Adaptive Grid Eulerian Method for the Computation of Free Surface Flows

A numerical scheme for the prediction of free surface flows is presented and investigated. The method is based on an adaptive grid Eulerian finite-volume method, where non-orthogonal boundary-fitted moving grids are employed to follow the free surface. The underlying flow solver consists in a pressure-correction scheme of SIMPLE type with multigrid acceleration, which is iteratively combined with the moving grid technique. Several numerical examples are considered to illustrate the capabilities of the approach.