Anisotropy favoring triangulation CVD(MPFA) finite‐volume approximations

A family of flux‐continuous, control‐volume distributed multi‐point flux approximation schemes CVD (MPFA) have been developed for solving the general geometry‐permeability tensor pressure equation on structured and unstructured grids (Comput. Geo. 1998; 2 : 259–290, Comput. Geo. 2002; 6 : 433–452). The locally conservative schemes are applicable to the diagonal and full‐tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full‐tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization. Improved numerical convergence for the family of CVD(MPFA) schemes for specified quadrature points has been observed for lower anisotropy ratios for both structured and unstructured grids in two dimensions. However, for strong full‐tensor anisotropy fields the quadrilateral schemes can induce strong spurious oscillations in the numerical solution. This paper motivates and demonstrates the benefit of using anisotropy favoring triangulation for treating such cases. Test examples involving strong full‐tensor anisotropy fields are presented in 2‐D and 3‐D, which show that the family of schemes on anisotropy favoring triangulation (prisms in 3‐D) yield well‐resolved pressure fields with little or no spurious oscillations. Copyright © 2010 John Wiley & Sons, Ltd.     

Non‐linear flux‐splitting schemes with imposed discrete maximum principle for elliptic equations with highly anisotropic coefficients

Families of flux‐continuous, locally conservative, finite‐volume schemes have been developed for solving the general tensor pressure equation of petroleum reservoir simulation on structured and unstructured grids. The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization. Improved convergence using the quadrature parameterization has been established for the family of flux‐continuous schemes. When applied to strongly anisotropic full‐tensor permeability fields the schemes can fail to satisfy a maximum principle (as with other FEM and finite‐volume methods) and result in spurious oscillations in the numerical pressure solution. This paper presents new non‐linear flux‐splitting techniques that are designed to compute solutions that are free of spurious oscillations. Results are presented for a series of test‐cases with strong full‐tensor anisotropy ratios. In all cases the non‐linear flux‐splitting methods yield pressure solutions that are free of spurious oscillations. Copyright © 2010 John Wiley & Sons, Ltd.     

The effects of control‐volume distributed multi‐point flux approximation (CVD‐MPFA) on upscaling—A study using the CVD‐MPFA schemes

A family of flux‐continuous, locally conservative, finite‐volume schemes has been developed for solving the general geometry‐permeability tensor (petroleum reservoir‐simulation) pressure equation on structured and unstructured grids and are control‐volume distributed (textit Comput. Geo. 1998; 2 :259–290; Comput. Geo. 2002; 6 :433–452). The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir‐simulation schemes (two‐point flux approximation) when applied to full tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization (Int. J. Numer. Meth. Fluids 2006; 51 :1177–1203). Improved convergence (for two‐ and three‐dimensional formulation) using the quadrature parameterization has been observed for the family of flux‐continuous control‐volume distributed multi‐point flux approximation (CVD‐MPFA) schemes (Ph.D. Thesis, University of Wales, Swansea, U.K., 2007). In this paper family of flux‐continuous (CVD‐MPFA) schemes are used as a part of numerical upscaling procedure for upscaling the fine‐scale grid information (permeability) onto a coarse grid scale. A series of data‐sets (SPE, 2001) are tested where the upscaled permeability tensor is computed on a sequence of grid levels using the same fixed range of quadrature points in each case. The refinement studies presented involve:

• (i) Refinement comparison study: In this study, permeability distribution for cells at each grid level is obtained by upscaling directly from the fine‐scale permeability field as in standard simulation practice.
• (ii) Refinement study with renormalized permeability: In this refinement comparison, the local permeability is upscaled to the next grid level hierarchically, so that permeability values are renormalized to each coarser level. Hence, showing only the effect of increased grid resolution on upscaled permeability, compared with that obtained directly from the fine‐scale solution.
• (iii) Refinement study with invariant permeability distribution: In this study, a classical mathematical convergence test is performed. The same coarse‐scale underlying permeability map is preserved on all grid levels including the fine‐scale reference solution.

The study is carried out for the discretization of the scheme in physical space. The benefit of using specific quadrature points is demonstrated for upscaling in this study and superconvergence is observed. Copyright © 2010 John Wiley & Sons, Ltd.     

Positive‐definite q‐families of continuous subcell Darcy‐flux CVD(MPFA) finite‐volume schemes and the mixed finite element method

A new family of locally conservative cell‐centred flux‐continuous schemes is presented for solving the porous media general‐tensor pressure equation. A general geometry‐permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control‐volume distributed subcell flux‐continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994; Comput. Geosci. 1998; 2 :259–290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive‐definite (SPD) discretization matrix is obtained for the base member (q=1) of the formulation. The physical‐space schemes are shown to be non‐symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non‐symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical‐space and subcell‐space q‐families of schemes. M‐matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical‐space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell‐wise constant tensor schemes and that subcell tensor approximation using the control‐volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested. Copyright © 2007 John Wiley & Sons, Ltd.     

Convergence of MPFA on triangulations and for Richards' equation

Spatial discretization of transport and transformation processes in porous media requires techniques that handle general geometry, discontinuous coefficients and are locally mass conservative. Multi‐point flux approximation (MPFA) methods are such techniques, and we will here discuss some formulations on triangular grids with further application to the nonlinear Richards equation. The MPFA methods will be rewritten to mixed form to derive stability conditions and error estimates. Several MPFA versions will be shown, and the versions will be discussed with respect to convergence, symmetry and robustness when the grids are rough. It will be shown that the behavior may be quite different for challenging cases of skewness and roughness of the simulation grids. Further, we apply the MPFA discretization approach for the Richards equation and derive new error estimates without extra regularity requirements. The analysis will be accompanied by numerical results for grids that are relevant for practical simulation. Copyright © 2008 John Wiley & Sons, Ltd.     

Geometrical interpretation of the multi‐point flux approximation L‐method

In this paper, we first investigate the influence of different Dirichlet boundary discretizations on the convergence rate of the multi‐point flux approximation (MPFA) L‐method by the numerical comparisons between the MPFA O‐ and L‐method, and show how important it is for this new method to handle Dirichlet boundary conditions in a suitable way. A new Dirichlet boundary strategy is proposed, which in some sense can well recover the superconvergence rate of the normal velocity. In the second part of the work, the MPFA L‐method with homogeneous media is studied. A systematic concept and geometrical interpretations of the L‐method are given and illustrated, which yield more insight into the L‐method. Finally, we apply the MPFA L‐method for two‐phase flow in porous media on different quadrilateral grids and compare its numerical results for the pressure and saturation with the results of the two‐point flux approximation method. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

An internal penalty discontinuous Galerkin method for simulating conjugate heat transfer in a closed cavity

Using the discontinuous Galerkin (DG) method for conjugate heat transfer problems can provide improved accuracy close to the fluid‐solid interface, localizing the data exchange process, which may further enhance the convergence and stability of the entire computation. This paper presents a framework for the simulation of conjugate heat transfer problems using DG methods on unstructured grids. Based on an existing DG solver for the incompressible Navier‐Stokes equation, the fluid advection‐diffusion equation, Boussinesq term, and solid heat equation are introduced using an explicit DG formulation. A Dirichlet‐Neumann partitioning strategy has been implemented to achieve the data exchange process via the numerical flux of interface quadrature points in the fluid‐solid interface. Formal h and p convergence studies employing the method of manufactured solutions demonstrate that the expected order of accuracy is achieved. The algorithm is then further validated against 3 existing benchmark cases, including a thermally driven cavity, conjugate thermally driven cavity, and a thermally driven cavity with conducting solid, at Rayleigh numbers from 1000 to 100 000. The computational effort is documented in detail demonstrating clearly that, for all cases, the highest‐order accurate algorithm has several magnitudes lower error than first‐ or second‐order schemes for a given computational effort.     

基于混合网格多维梯度重构的热流预测方法研究

高超声速气动热环境的数值计算对算法和网格的敏感度极高. 随着高超声速飞行器外形日益复杂, 生成高质量的结构网格时间成本呈指数增加, 难以满足工程应用的需求. 非结构/混合网格因具有很强的复杂外形适应能力, 为了缩短任务周期, 有必要在非结构/混合网格上开展高精度的气动热环境数值计算方法研究. 梯度重构方法是影响非结构/混合网格热流计算精度的重要因素之一. 本文通过引入多维梯度重构方法, 发展了基于常规的非结构/混合网格的高精度热流计算方法, 对典型的高超声速Benchmark算例(二维圆柱)进行了模拟, 并与气动力计算广泛采用的Green-Gauss类方法和最小二乘类方法进行了对比. 计算结果表明, 多维梯度重构方法能有效提高非结构/混合网格热流预测精度, 其鲁棒性和收敛性更好. 最后将多维梯度重构方法应用于常规混合网格的三维圆柱和三维双椭球绕流问题, 得到了与实验值吻合较好的热流计算结果, 展现了良好的应用前景.      

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.     

动网格生成技术及非定常计算方法进展综述

对应用于飞行器非定常运动的数值计算方法(包括动态网格技术和相应的数值离散格式)进行了综述.根据网格拓扑结构的不同,重点论述了基于结构网格的非定常计算方法和基于非结构/混合网格的非定常计算方法,比较了各种方法的优缺点.在基于结构网格的非定常计算方法中,重点介绍了刚性运动网格技术、超限插值动态网格技术、重叠动网格技术、滑移动网格技术等动态结构网格生成方法,同时介绍了惯性系和非惯性系下的控制方程,讨论了非定常时间离散方法、动网格计算的几何守恒律等问题.在基于非结构/混合网格的非定常计算方法中,重点介绍了重叠非结构动网格技术、重构非结构动网格技术、变形非结构动网格技术以及变形/重构耦合动态混合网格技术等方法,以及相应的计算格式,包括非定常时间离散、几何守恒律计算方法、可压缩和不可压缩非定常流动的计算方法、各种加速收敛技术等.在介绍国内外进展的同时,介绍了作者在动态混合网格生成技术和相应的非定常方法方面的研究与应用工作.     

多维迎风方法在非结构/混合网格热流计算中的应用研究

非结构/混合网格具有极强的几何灵活性,在复杂外形飞行器的气动力特性数值模拟中已得到广泛应用,但目前还难以准确地预测气动热环境。本文从非结构/混合网格热流计算的三个需求出发,选取了多维迎风方法,并与其他方法进行了对比研究。以二维圆柱高超声速绕流这一Benchmark典型问题为例,对比研究了多维迎风方法和几种广泛使用的无粘通量格式(Roe格式、Van Leer格式和AUSMDV格式)对混合网格热流计算精度的影响。结果表明,多维迎风方法在热流计算精度、鲁棒性以及收敛性方面表现良好。最后,将多维迎风方法应用于常规混合网格上的圆柱和钝双锥绕流问题,均得到了较好的热流计算结果,为非结构/混合网格热流计算在复杂高超飞行器中的应用奠定了基础。     

A new TVD method for advection simulation on 2D unstructured grids

For advection schemes with flux limiters derived on one‐dimensional grids to two‐dimensional (2D) unstructured triangular ones. In this method, the variables located normal to this face are taken into more account to compute the flux, by means of defining required nodes along the line through the center point of the considered face and perpendicular to it. Besides, the new method adopts the improved total variation diminishing schemes in, which consider the face position between the two neighboring cells as well as the size differences of the related cells; therefore, it fits 2D unstructured grids well. Compared with the present flux limiting methods for 2D unstructured grids, a higher accuracy and efficiency and a good monotonicity are achieved by the new method.Copyright © 2012 John Wiley & Sons, Ltd.     

The Lagrange–Galerkin method for the two‐dimensional shallow water equations on adaptive grids

The weak Lagrange–Galerkin finite element method for the two‐dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized using triangular elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the non‐linearities introduced by the advection operator of the fluid dynamics equations. An additional fortuitous consequence of using Lagrangian methods is that the resulting spatial operator is self‐adjoint, thereby justifying the use of a Galerkin formulation; this formulation has been proven to be optimal for such differential operators. The weak Lagrange–Galerkin method automatically takes into account the dilation of the control volume, thereby resulting in a conservative scheme. The use of linear triangular elements permits the construction of accurate (by virtue of the second‐order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent Courant–Friedrich–Lewy (CFL) condition of Lagrangian methods) schemes on adaptive unstructured triangular grids. Lagrangian methods are natural candidates for use with adaptive unstructured grids because the resolution of the grid can be increased without having to decrease the time step in order to satisfy stability. An advancing front adaptive unstructured triangular mesh generator is presented. The highlight of this algorithm is that the weak Lagrange–Galerkin method is used to project the conservation variables from the old mesh onto the newly adapted mesh. In addition, two new schemes for computing the characteristic curves are presented: a composite mid‐point rule and a general family of Runge–Kutta schemes. Results for the two‐dimensional advection equation with and without time‐dependent velocity fields are illustrated to confirm the accuracy of the particle trajectories. Results for the two‐dimensional shallow water equations on a non‐linear soliton wave are presented to illustrate the power and flexibility of this strategy. 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.     

Numerical analysis of conservative unstructured discretisations for low Mach flows

Unstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell‐centred velocity reconstructions, the standard first‐order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd–even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable‐density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy‐preserving scheme is applied to the momentum equations, namely, the symmetry‐preserving scheme. Furthermore, a new approach to define the far‐neighbouring nodes of the quadratic upstream interpolation for convective kinematics scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self‐igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable‐density flows. Furthermore, the quadratic upstream interpolation for convective kinematics scheme shows close to second‐order behaviour on unstructured meshes, and the symmetry‐preserving is reliably used in all computations. Copyright © 2016 John Wiley & Sons, Ltd.     

A Cell-Centred CVD-MPFA Finite Volume Method for Two-Phase Fluid Flow Problems with Capillary Heterogeneity and Discontinuity

A novel finite volume method is presented that is applicable to discontinuous capillary pressure fields. The method is developed within the control-volume distributed multi-point flux approximation (CVD-MPFA) framework (Edwards and Rogers in Comput Geosci 02(04):259–290, 1998; Friis et al. in SIAM J Sci Comput 31(02):1192–1220, 2008). Results are computed on structured and unstructured grids that demonstrate the ability of the method to resolve flow in the presence of a discontinuous capillary pressure field for diagonal and full-tensor permeability fields. In addition to an upwind approximation for the saturation equation flux, the importance of upwinding on capillary pressure flux via a hybrid formulation is shown.

     

Multigrid solutions of the Euler and Navier–Stokes equations on unstructured grids

A computationally efficient multigrid algorithm for upwind edge‐based finite element schemes is developed for the solution of the two‐dimensional Euler and Navier–Stokes equations on unstructured triangular grids. The basic smoother is based upon a Galerkin approximation employing an edge‐based formulation with the explicit addition of an upwind‐type local extremum diminishing (LED) method. An explicit time stepping method is used to advance the solution towards the steady state. Fully unstructured grids are employed to increase the flexibility of the proposed algorithm. A full approximation storage (FAS) algorithm is used as the basic multigrid acceleration procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

The implementation of an aeronautical CFD flow code onto distributed memory parallel systems

The parallelization of an industrially important in‐house computational fluid dynamics (CFD) code for calculating the airflow over complex aircraft configurations using the Euler or Navier–Stokes equations is presented. The code discussed is the flow solver module of the SAUNA CFD suite. This suite uses a novel grid system that may include block‐structured hexahedral or pyramidal grids, unstructured tetrahedral grids or a hybrid combination of both. To assist in the rapid convergence to a solution, a number of convergence acceleration techniques are employed including implicit residual smoothing and a multigrid full approximation storage scheme (FAS). Key features of the parallelization approach are the use of domain decomposition and encapsulated message passing to enable the execution in parallel using a single programme multiple data (SPMD) paradigm. In the case where a hybrid grid is used, a unified grid partitioning scheme is employed to define the decomposition of the mesh. The parallel code has been tested using both structured and hybrid grids on a number of different distributed memory parallel systems and is now routinely used to perform industrial scale aeronautical simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method for the Navier–Stokes equations

The foundations of a new discontinuous Galerkin method for simulating compressible viscous flows with shocks on standard unstructured grids are presented in this paper. The new method is based on a discontinuous Galerkin formulation both for the advective and the diffusive contributions. High‐order accuracy is achieved by using a recently developed hierarchical spectral basis. This basis is formed by combining Jacobi polynomials of high‐order weights written in a new co‐ordinate system. It retains a tensor‐product property, and provides accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by appropriately lowering the basis order and performing h‐refinement around discontinuities. Convergence results are shown for analytical two‐ and three‐dimensional solutions of diffusion and Navier–Stokes equations that demonstrate exponential convergence of the new method, even for highly distorted elements. Flow simulations for subsonic, transonic and supersonic flows are also presented that demonstrate discretization flexibility using hp‐type refinement. Unlike other high‐order methods, the new method uses standard finite volume grids consisting of arbitrary triangulizations and tetrahedrizations. Copyright © 1999 John Wiley & Sons, Ltd.