Turbulent flow computations on 3D unstructured grids

An edge-based finite element method is presented for the simulation of compressible turbulent flows on unstructured tetrahedral grids. A two equation k–ω turbulence model is employed and the standard Galerkin approach is used for spatial discretisation. Stabilisation of the resulting procedure is achieved by the addition of an appropriate diffusion. An explicit multistage time-stepping scheme is used to advance the solution in time to steady state. The performance of the algorithm is demonstrated for the simulation of a high Reynolds number transonic separated flow over a wing.     

Resolution of computational aeroacoustics problems on unstructured grids with a higher-order finite volume scheme

Computational fluid dynamics (CFD) has become increasingly used in the industry for the simulation of flows. Nevertheless, the complex configurations of real engineering problems make the application of very accurate methods that only work on structured grids difficult. From this point of view, the development of higher-order methods for unstructured grids is desirable. The finite volume method can be used with unstructured grids, but unfortunately it is difficult to achieve an order of accuracy higher than two, and the common approach is a simple extension of the one-dimensional case. The increase of the order of accuracy in finite volume methods on general unstructured grids has been limited due to the difficulty in the evaluation of field derivatives. This problem is overcome with the application of the Moving Least Squares (MLS) technique on a finite volume framework. In this work we present the application of this method (FV-MLS) to the solution of aeroacoustic problems.     

Convergence of two-dimensional staggered central schemes on unstructured triangular grids

In this paper, we present a convergence analysis of a two-dimensional central finite volume scheme on unstructured triangular grids for hyperbolic systems of conservation laws. More precisely, we show that the solution obtained by the numerical base scheme presents, under an appropriate CFL condition, an optimal convergence to the unique entropy solution of the Cauchy problem.     

A Lax-Wendroff type theorem for unstructured quasi-uniform grids

A well-known theorem of Lax and Wendroff states that if the sequence of approximate solutions to a system of hyperbolic conservation laws generated by a conservative consistent numerical scheme converges boundedly a.e. as the mesh parameter goes to zero, then the limit is a weak solution of the system. Moreover, if the scheme satisfies a discrete entropy inequality as well, the limit is an entropy solution. The original theorem applies to uniform Cartesian grids; this article presents a generalization for quasi-uniform grids (with Lipschitz-boundary cells) uniformly continuous inhomogeneous numerical fluxes and nonlinear inhomogeneous sources. The added generality allows a discussion of novel applications like local time stepping, grids with moving vertices and conservative remapping. A counterexample demonstrates that the theorem is not valid for arbitrary non-quasi-uniform grids.

     

Quasi-positive families of continuous Darcy-flux finite volume schemes on structured and unstructured grids

New families of flux-continuous control-volume distributed finite volume schemes are presented for the general full-tensor pressure equation arising in porous media and formulated for structured and unstructured grids. These schemes offer the practical advantage of being flux-continuous while only depending on one degree of freedom per control-volume, unlike rival approximations such as the Mixed Finite Element method. M-matrix bounds are presented, quasi QM-matrices are defined and an optimal quadrilateral scheme is identified. Anisotropy favoring triangulation is also shown to yield an optimal scheme. The new schemes prove to be relatively robust for the cases tested, including strongly anisotropic full tensor fields. Strong oscillations encountered with the earlier formulations, are removed or minimized.     

An unstructured finite volume time domain method for structural dynamics

An unstructured finite volume time domain method (UFVTDM) is proposed to simulate stress wave propagation, in which the original variables of displacement and stress are solved based on the dynamic equilibrium equations. An Euler explicit and unstructured finite volume method is used for time dependent and spacial terms respectively. The displacements are stored on the cell vertex and a vertex based finite volume method is formed with that integral surface and the stresses are as assumed to be uniform in the cell. The present UFVTDM has several features. (1) The governing equations are discretized with the finite volume method which naturally follows conservation laws. (2) It can handle complex engineering problem. (3) This method is also able to analyze the natural characteristics and the numerical experiment shows that it is very efficient. Several cases are used to show the capability of the algorithm.     

Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions

Summary. We prove convergence of a class of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. The result is applied to the discontinuous Galerkin method due to Cockburn, Hou and Shu. Received April 15, 1993 / Revised version received March 13, 1995     

On the design of an upwind scheme for compressible flow on general triangulations

We discuss the design features and mathematical background of an explicit upwind finite-volume method to simulate non-stationary flow of a compressible, inviscid fluid. One of the design goals was the rigorous mathematical justification of each ingredient of the method. The method itself contains elements from finite-difference methods as well as finite-element methods and is formulated in a finite volume framework. The use of well-known algorithmic ingredients in a new framework results in a robust time-accurate scheme. To be able to easily handle complex geometries as well as adaption algorithms a tringale-based formulation was chosen. Numerical tests for two-dimensional flow are presented.     

Multistep characteristic method for incompressible flow in porous media

In this paper we present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media. The discretization involves a three-level time scheme based on the characteristic method and a five-point finite difference scheme for space discretization. We prove that the convergence is of order O(h²+(Δt)²), which is in contrast to the convergence of order O(h+Δt) proved for a singlestep characteristic with the same space discretization. Numerical experiments demonstrate the stability and second-order convergence of the scheme.     

A reduced domain strategy for local mesh movement application in unstructured grids

Automatic control of mesh movement is mandatory in many fluid flow and fluid-solid interaction problems. This paper presents a new strategy, called reduced domain strategy (RDS), which enhances the efficiency of node connectivity-based mesh movement methods and moves the unstructured grid locally and effectively. The strategy dramatically reduces the grid computations by dividing the unstructured grid into two active and inactive zones. After any local boundary movement, the grid movement is performed only within the active zone. To enhance the efficiency of our strategy, we also develop an automatic mesh partitioning scheme. This scheme benefits from a new quasi-structured mesh data ordering, which determines the boundary of active zone in the original unstructured grid very easily. Indeed, the new partitioning scheme eliminates the need for sequential reordering of the original unstructured grid data in different mesh movement applications. We choose the spring analogy method and apply our new strategy to perform local mesh movements in two boundary movement problems including a multi-element airfoil with moving slat or deforming main body section. We show that the RDS is robust and cost effective. It can be readily employed in different node connectivity-based mesh movement methods. Indeed, the RDS provides a flexible local grid deformation tool for moving grid applications.     

Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems

We construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order PDEs in the state variable (e.g., pressure) and its flux (e.g., Darcy velocity). An important point is that no staggered grids or covolumes are used to stabilize the system. Only a single primary grid system is adopted, and the degrees of freedom are imposed on the interfaces. The approximate flux is sought in the lowest-order Raviart-Thomas space and the pressure field in the rotated- nonconforming space. Furthermore, we demonstrate that the present finite volume method can be interpreted as a rotated- nonconforming finite element method for the pressure with a simple local recovery of flux. Numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method.

     

The upwind finite difference fractional steps methods for two‐phase compressible flow in porous media

The upwind finite difference fractional steps methods are put forward for the two‐phase compressible displacement problem. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high‐order difference operators, and prior estimates, are adopted. Optimal order estimates in L² norm are derived to determine the error in the approximate solution. This method has already been applied to the numerical simulation of seawater intrusion and migration‐accumulation of oil resources. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 67–88, 2003     

Performance comparison of the NWF and DC methods for implementing High-Resolution schemes in a fully coupled incompressible flow solver

This paper reports on the use of the Normalized Weighting Factor (NWF) method and the Deferred Correction (DC) approach for the implementation of High Resolution (HR) convective schemes in an implicit, fully coupled, pressure-based flow solver. Four HR schemes are realized within the framework of the NWF and DC methods and employed to solve the following three laminar flow problems: (i) lid-driven flow in a square cavity, (ii) sudden expansion in a square cavity, and (iii) flow in a planar T-junction, over three grid systems with sizes of 10⁴, 5 × 10⁴, and 3 × 10⁵ control volumes. The merit of both approaches is demonstrated by comparing the computational costs required to solve these problems using the various HR schemes on the different grid systems. Whereas previous attempts to use the NWF method in a segregated flow solver failed to produce converged solutions, current results clearly demonstrate that both methods are suitable for utilization in a coupled flow solver. In terms of CPU efficiency, there is no global and consistent superiority of any method over another even though the DC method outperformed the NWF method in two of the three test problems solved.     

多孔介质中两相不可压缩流体的时间局部网格加密有限差分格式

引入Charent压力变量,对于多孔介质中两相不可压缩流体的非混溶驱动问题,其模型表现为耦合的非线性偏微分方程组,一个是压力方程,另一个为饱和度方程．文中考虑一维问题且假定达西速度“已知,建立了在时间上进行局部加密的有限差分格式,给出了饱和度的最大模误差估计．最后给出了数值算例．     

Combined higher order finite volume and finite element scheme for double porosity and non-linear adsorption of transport problem in porous media

In this work, a dual porosity model of reactive solute transport in porous media is presented. This model consists of a nonlinear-degenerate advection-diffusion equation including equilibrium adsorption to the reaction combined with a first-order equation for the non-equilibrium adsorption interaction processes. The numerical scheme for solving this model involves a combined high order finite volume and finite element scheme for approximation of the advection-diffusion part and relaxation-regularized algorithm for nonlinearity-degeneracy. The combined finite volume-finite element scheme is based on a new formulation developed by Eymard et al. (2010) [10]. This formulation treats the advection and diffusion separately. The advection is approximated by a second-order local maximum principle preserving cell-vertex finite volume scheme that has been recently proposed whereas the diffusion is approximated by a finite element method. The result is a conservative, accurate and very flexible algorithm which allows the use of different mesh types such as unstructured meshes and is able to solve difficult problems. Robustness and accuracy of the method have been evaluated, particularly error analysis and the rate of convergence, by comparing the analytical and numerical solutions for first and second order upwind approaches. We also illustrate the performance of the discretization scheme through a variety of practical numerical examples. The discrete maximum principle has been proved.     

A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions

In this paper we shall derive a posteriori error estimates in the -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions. This result is mainly based on some fundamental a priori error estimates published in a recent paper by C. Chainais-Hillairet. The theoretical results are confirmed by numerical experiments.

     

A mixed finite volume scheme for anisotropic diffusion problems on any grid

We present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution and of its gradient. The approximate solution is shown to converge to the continuous one as the size of the mesh tends to 0, and an error estimate is given. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids and for various diffusion matrices.