Comparison of the GMRES and ORTHOMIN for the black oil model in porous media

This paper deals with the application of the GMRES algorithm to a three‐dimensional, three‐phase black oil model used in petroleum reservoir simulation. Comparisons between the GMRES and ORTHOMIN algorithms in terms of storage and total flops per restart step are given. Numerical results show that the GMRES is faster than the ORTHOMIN for large‐scale simulation problems. The GMRES uses only as much as 63% of the CPU time of the ORTHOMIN for some of the problems tested. Copyright © 2005 John Wiley & Sons, Ltd.     

A two‐dimensional finite volume morphodynamic model on unstructured triangular grids

We discuss the application of a finite volume method to morphodynamic models on unstructured triangular meshes. The model is based on coupling the shallow water equations for the hydrodynamics with a sediment transport equation for the morphodynamics. The finite volume method is formulated for the quasi‐steady approach and the coupled approach. In the first approach, the steady hydrodynamic state is calculated first and the corresponding water velocity is used in the sediment transport equation to be solved subsequently. The second approach solves the coupled hydrodynamics and sediment transport system within the same time step. The gradient fluxes are discretized using a modified Roe's scheme incorporating the sign of the Jacobian matrix in the morphodynamic system. A well‐balanced discretization is used for the treatment of source terms. We also describe an adaptive procedure in the finite volume method by monitoring the bed–load in the computational domain during its transport process. The method uses unstructured meshes, incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep bed gradients that may form in the approximate solution. Numerical results are shown for a test problem in the evolution of an initially hump‐shaped bed in a squared channel. For the considered morphodynamical regimes, the obtained results point out that the coupled approach performs better than the quasi‐steady approach only when the bed–load rapidly interacts with the hydrodynamics. Copyright © 2009 John Wiley & Sons, Ltd.     

Improved multislope MUSCL reconstruction on unstructured grids for shallow water equations

In shallow water flow and transport modeling, the monotonic upstream‐centered scheme for conservation laws (MUSCL) is widely used to extend the original Godunov scheme to second‐order accuracy. The most important step in MUSCL‐type schemes is MUSCL reconstruction, which calculate‐extrapolates the values of independent variables from the cell center to the edge. The monotonicity of the scheme is preserved with the help of slope limiters that prevent the occurrence of new extrema during reconstruction. On structured grids, the calculation of the slope is straightforward and usually based on a 2‐point stencil that uses the cell centers of the neighbor cell and the so‐called far‐neighbor cell of the edge under consideration. On unstructured grids, the correct choice for the upwind slope becomes nontrivial. In this work, 2 novel total variation diminishing schemes are developed based on different techniques for calculating the upwind slope and the downwind slope. An additional treatment that stabilizes the scheme is discussed. The proposed techniques are compared to 2 existing MUSCL reconstruction techniques, and a detailed discussion of the results is given. It is shown that the proposed MUSCL reconstruction schemes obtain more accurate results with less numerical diffusion and higher efficiency.     

Accuracy and convergence of element-by-element iterative solvers for incompressible fluid flows using penalty finite element model

The ability of two types of Conjugate Gradient like iterative solvers (GMRES and ORTHOMIN) to resolve large-scale phenomena as a function of mesh density and convergence tolerance limit is investigated. The flow of an incompressible fluid inside a sudden expansion channel is analysed using three meshes of 400, 1600 and 6400 bilinear elements. The iterative solvers utilize the element-by-element data structure of the finite element technique to store and maintain the data at the element level. Both the mesh density and the penalty parameter are found to influence the choice of the convergence tolerance limit needed to obtain accurate results. An empirical relationship between the element size, the penalty parameter, and the convergence tolerance is presented. This relationship can be used to predict the proper choice of the convergence tolerance for a given penalty parameter and element size.     

Multigrid convergence acceleration for implicit and explicit solution of Euler equations on unstructured grids

The multigrid method is one of the most efficient techniques for convergence acceleration of iterative methods. In this method, a grid coarsening algorithm is required. Here, an agglomeration scheme is introduced, which is applicable in both cell‐center and cell‐vertex 2 and 3D discretizations. A new implicit formulation is presented, which results in better computation efficiency, when added to the multigrid scheme. A few simple procedures are also proposed and applied to provide even higher convergence acceleration. The Euler equations are solved on an unstructured grid around standard transonic configurations to validate the algorithm and to assess its superiority to conventional explicit agglomeration schemes. The scheme is applied to 2 and 3D test cases using both cell‐center and cell‐vertex discretizations. Copyright © 2009 John Wiley & Sons, Ltd.     

Employment of the second‐moment turbulence closure on arbitrary unstructured grids

The paper presents a finite‐volume calculation procedure using a second‐moment turbulence closure. The proposed method is based on a collocated variable arrangement and especially adopted for unstructured grids consisting of ‘polyhedral’ calculation volumes. An inclusion of 23k in the pressure is analysed and the impact of such an approach on the employment of the constant static pressure boundary is addressed. It is shown that this approach allows a removal of a standard but cumbersome velocity–pressure –Reynolds stress coupling procedure known as an extension of Rhie‐Chow method (AIAA J. 1983; 21 : 1525–1532) for the Reynolds stresses. A novel wall treatment for the Reynolds‐stress equations and ‘polyhedral’ calculation volumes is presented. Important issues related to treatments of diffusion terms in momentum and Reynolds‐stress equations are also discussed and a new approach is proposed. Special interpolation practices implemented in a deferred‐correction fashion and related to all equations, are explained in detail. Computational results are compared with available experimental data for four very different applications: the flow in a two‐dimensional 180o turned U‐bend, the vortex shedding flow around a square cylinder, the flow around Ahmed Body and in‐cylinder engine flow. Additionally, the performance of the methodology is assessed by applying it to different computational grids. For all test cases, predictions with the second‐moment closure are compared to those of the k–εmodel. The second‐moment turbulence closure always achieves closer agreement with the measurements. A moderate increase in computing time is required for the calculations with the second‐moment closure. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation analysis for migration-accumulation of oil and water

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适合于求解边界元方程组的GMRES算法的实用化和并行化研究

为了将GMRES算法应用于大型边界元方程组的求解,采用预条件技术和重正交技术相结合的方法实现了该算法的实用化,然后在实用化的基础上针对迭代算法具有良好并行性的特点,研究了该算法在网络机群环境下的并行化技术。数值试验和分析表明所用的这些技术是行之有效的,对于提高求解速度和增大求解问题的规模是有意义的。     

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.     

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.     

Discretization of transport equations on 2D Cartesian unstructured grids using data from remote cells for the convection terms

This paper presents a new finite volume discretization methodology for the solution of transport equations on locally refined or unstructured Cartesian meshes. The implementation of the cell‐face values of the dependent variables enables the employment of data from remote cells and thus the use of higher‐order differencing schemes. It also results in simple and flux‐conservative multiple‐scale stencils for the discretization of the governing equations. The latter are finally cast into a generalized form that does not depend on the local mesh structure. The performance of the numerical model is demonstrated on some classical 2D problems using various gridding techniques and a bounded second‐order upwind scheme. A stable and efficient behaviour of the algorithm is observed in all test cases. The results indicate that the combination in the present model of both local grid refinement and second‐order discretization can produce substantially more accurate solutions than each of the above techniques alone, for the same computational effort. The method is also applicable to turbulent flows and can be easily extended to three‐dimensions. Copyright © 2003 John Wiley & Sons, Ltd.     

Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grids

The discontinuous Galerkin (DG) and spectral volume (SV) methods are two recently developed high‐order methods for hyperbolic conservation laws capable of handling unstructured grids. In this paper, their overall performance in terms of efficiency, accuracy and memory requirement is evaluated using a 2D scalar conservation laws and the 2D Euler equations. To measure their accuracy, problems with analytical solutions are used. Both methods are also used to solve problems with strong discontinuities to test their ability in discontinuity capturing. Both the DG and SV methods are capable of achieving their formal order of accuracy while the DG method has a lower error magnitude and takes more memory. They are also similar in efficiency. The SV method appears to have a higher resolution for discontinuities because the data limiting can be done at the sub‐element level. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

2‐D transmitral flows simulation by means of the immersed boundary method on unstructured grids

Interaction between computational fluid dynamics and clinical researches recently allowed a deeper understanding of the physiology of complex phenomena involving cardio‐vascular mechanisms. The aim of this paper is to develop a simplified numerical model based on the Immersed Boundary Method and to perform numerical simulations in order to study the cardiac diastolic phase during which the left ventricle is filled with blood flowing from the atrium throughout the mitral valve. As one of the diagnostic problems to be faced by clinicians is the lack of a univocal definition of the diastolic performance from the velocity measurements obtained by Eco–Doppler techniques, numerical simulations are supposed to provide an insight both into the physics of the diastole and into the interpretation of experimental data. An innovative application of the Immersed Boundary Method on unstructured grids is presented, fulfilling accuracy requirements related to the development of a thin boundary layer along the moving immersed boundary. It appears that this coupling between unstructured meshes and the Immersed Boundary Method is a promising technique when a wide range of spatial scales is involved together with a moving boundary. Numerical simulations are performed in a range of physiological parameters and a qualitative comparison with experimental data is presented, in order to demonstrate that, despite the simplified model, the main physiological characteristics of the diastole are well represented. Copyright © 2002 John Wiley & Sons, Ltd.     

The composite finite volume method on unstructured meshes for the two‐dimensional shallow water equations

A composite finite volume method (FVM) is developed on unstructured triangular meshes and tested for the two‐dimensional free‐surface flow equations. The methodology is based on the theory of the remainder effect of finite difference schemes and the property that the numerical dissipation and dispersion of the schemes are compensated by each other in a composite scheme. The composite FVM is formed by global composition of several Lax–Wendroff‐type steps followed by a diffusive Lax–Friedrich‐type step, which filters out the oscillations around shocks typical for the Lax–Wendroff scheme. To test the efficiency and reliability of the present method, five typical problems of discontinuous solutions of two‐dimensional shallow water are solved. The numerical results show that the proposed method, which needs no use of a limiter function, is easy to implement, is accurate, robust and is highly stable. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

A gas‐kinetic scheme for the simulation of turbulent flows on unstructured meshes

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart–Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of the gas‐kinetic scheme is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non‐manifold hybrid mesh data structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.     

An implementation of the Spalart–Allmaras DES model in an implicit unstructured hybrid finite volume/element solver for incompressible turbulent flow

In this paper, we describe an implicit hybrid finite volume (FV)/element (FE) incompressible Navier–Stokes solver for turbulent flows based on the Spalart–Allmaras detached eddy simulation (SA‐DES). The hybrid FV/FE solver is based on the segregated pressure correction or projection method. The intermediate velocity field is first obtained by solving the original momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE 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 centers and the auxiliary variable at vertices, making the current solver a staggered‐mesh scheme. The SA‐DES turbulence equation is solved after the velocity and the pressure fields have been updated at the end of each time step. The same matrix‐free FV method as the one used for momentum equations is used to solve the turbulence equation. The turbulence equation provides the eddy viscosity, which is added to the molecular viscosity when solving the momentum equation. In our implementation, we focus on the accuracy, efficiency and robustness of the SA‐DES model in a hybrid flow solver. This paper will address important implementation issues for high‐Reynolds number flows where highly stretched elements are typically used. In addition, some aspects of implementing the SA‐DES model will be described to ensure the robustness of the turbulence model. Several numerical examples including a turbulent flow past a flat plate and a high‐Reynolds number flow around a high angle‐of‐attack NACA0015 airfoil will be presented to demonstrate the accuracy and efficiency of our current implementation. Copyright © 2008 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.     

Evaluation of adaptive mesh refinement and coarsening for the computation of compressible flows on unstructured meshes

The compressible gas flows of interest to aerospace applications often involve situations where shock and expansion waves are present. Decreasing the characteristic dimension of the computational cells in the vicinity of shock waves improves the quality of the computed flows. This reduction in size may be accomplished by the use of mesh adaption procedures. In this paper an analysis is presented of an adaptive mesh scheme developed for an unstructured mesh finite volume upwind computer code. This scheme is tailored to refine or coarsen the computational mesh where gradients of the flow properties are respectively high or low. The refinement and coarsening procedures are applied to the classical gas dynamic problems of the stabilization of shock waves by solid bodies. In particular, situations where oblique shock waves interact with an expansion fan and where bow shocks arise around solid bodies are considered. The effectiveness of the scheme in reducing the computational time, while increasing the solution accuracy, is assessed. It is shown that the refinement procedure alone leads to a number of computational cells which is 20% larger than when alternate passes of refinement and coarsening are used. Accordingly, a reduction of computational time of the same order of magnitude is obtained. Copyright © 2005 John Wiley & Sons, Ltd.