Numerical Solution of the Incompressible Navier-Stokes Equations with Singular Perturbation Considerations

A numerical method for coarse grids is proposed for the numerical solution of the incompressible Navier-Stokes equations. From singular perturbation considerations, we obtain partial differential equations and boundary conditions for the outer solution and the boundary layer correction. The former problem is solved with the finite difference method and the latter with the approximate method. Numerical experiments show that accurate outer flow and boundary flux result with little computational effort.     

Preconditioned implicit solution of linear hyperbolic equations with adaptivity

This paper describes a method for solving hyperbolic partial differential equations using an adaptive grid: the spatial derivatives are discretised with a finite volume method on a grid which is structured and partitioned into blocks which may be refined and derefined as the solution evolves. The solution is advanced in time via a backward differentiation formula. The discretisation used is second-order accurate and stable on Cartesian grids. The resulting system of linear equations is solved by GMRES at every time-step with the convergence of the iteration being accelerated by a semi-Toeplitz preconditioner. The efficiency of this preconditioning technique is analysed and numerical experiments are presented which illustrate the behaviour of the method on a parallel computer.     

Explicit-implicit difference scheme for the joint solution of the radiative transfer and energy equations by the splitting method

High-order accurate explicit and implicit conservative predictor-corrector schemes are presented for the radiative transfer and energy equations in the multigroup kinetic approximation solved together by applying the splitting method with respect to physical processes and spatial variables. The original system of integrodifferential equations is split into two subsystems: one of partial differential equations without sources and one of ordinary differential equations (ODE) with sources. The general solution of the ODE system and the energy equation is written in quadratures based on total energy conservation in a cell. A feature of the schemes is that a new approximation is used for the numerical fluxes through the cell interfaces. The fluxes are found along characteristics with the interaction between radiation and matter taken into account. For smooth solutions, the schemes approximating the transfer equations on spatially uniform grids are second-order accurate in time and space. As an example, numerical results for Fleck’s test problems are presented that confirm the increased accuracy and efficiency of the method.     

Lattice Boltzmann simulation for high-speed compressible viscous flows with a boundary layer

In this study, the lattice Boltzmann method is employed for simulating high-speed compressible viscous flows with a boundary layer. The coupled double-distribution-function lattice Boltzmann method proposed by Li et al. (2007) is employed because of its good numerical stability and non-free-parameter feature. The non-uniform mesh construction near the wall boundary in fine grids is combined with an appropriate wall boundary treatment for the finite difference method in order to obtain accurate spatial resolution in the boundary layer problem. Three typical problems in high-speed viscous flows are solved in the lattice Boltzmann simulation, i.e., the compressible boundary layer problem, shock wave problem, and shock boundary layer interaction problem. In addition, in-depth comparisons are made with the non-oscillatory and non-free-parameter dissipation (NND) scheme and second order upwind scheme in the present lattice Boltzmann model. Our simulation results indicate the great potential of the lattice Boltzmann method for simulating high-speed compressible viscous flows with a boundary layer. Further research is needed (e.g., better numerical models and appropriate finite difference schemes) because the lattice Boltzmann method is still immature for high-speed compressible viscous flow applications.     

用CT-VOF法在数值波试验槽中生成线性和非线性波

为渡水槽中波的模拟和传播提出了二维的数值模型．假设流动的流体为粘性、不可压缩的,并将Navier-Stokes方程和连续性方程作为控制方程．用标准的k-ε模型来模拟紊流流动;用交错网格的有限差分法,离散化Navier-Stokes方程;并用简化的标识和单元(SMAC)方法进行求解．使用活塞型波发生器生成并传播波;数值渡水槽的端部采用敞开式的边界条件．为了证明模型的有效性,进行了一些标准的试验,如顶盖驱动的方腔测试试验、单向的常速度场试验以及干燥河床上的溃坝试验．为了论证方法的性能及其精度,将所生成波的结果与已有波理论的结果进行比较．最后,采用群集技术(CT)生成网格,并提出最佳的网格生成条件．     

High resolution multi-moment finite volume method for supersonic combustion on unstructured grids

In this study, we present a novel numerical model for simulating detonation waves on unstructured grids. In contrast to the conventional finite volume method (FVM), two types of moment comprising the volume-integrated average (VIA) and the point value (PV) at the cell vertex are treated as the evolution variables for the reacting Euler equations. The VIA is computed based on a finite volume formulation of the flux form where the conventional Riemann problem is solved by the HLLC Riemann solver. The PV is updated in a point-wise manner by using the differential formulation where the Roe solver is used to compute the differential Riemann problems. In order to increase the accuracy around discontinuities, numerical oscillations and dissipations are reduced using the boundary variation diminishing algorithm. Convergence tests demonstrated that the proposed model could achieve third-order accuracy with unstructured grids for reacting Euler equations. The high resolution property of the proposed method was verified based on simulations of several detonation wave propagation problems in two and three dimensions. In particular, the current model could resolve the cellular structures with fewer degrees of freedom for the unstable oblique detonation wave problem. These fine structures may be smoothed out by the conventional FVM due to the excessive amount of numerical dissipation errors. Importantly, a simulation of stiff detonation waves showed that the proposed method could capture the correct position of the reaction front whereas the conventional FVMs produced spurious phenomena. Thus, the proposed model can obtain highly accurate solutions for detonation problems on unstructured grids, which is highly advantageous for real applications involving complex geometrical configurations.     

用SADI方法求解方形空腔中具有高Rayleigh数的非定常自然对流问题

本文用三次样条积分计算了在方形空腔中具有高Rayleigh数Ra=107和Ra=2×107的非定常自然对流问题。二维N-S方程和能量方程是在非均匀网格中用两个交替方向的三次样条公式进行计算的。文中简要讨论了过渡流动的主要特征,所得结果与理论予估值[1,2]吻合很好。Ra=107时的稳态结果与近期文献中的结果一致。     

对流扩散方程在成品油顺序输送混油分析中的应用

本文研究了对流与扩散对成品油顺序输送混油过程的影响;推导了紊流条件下,描述混油过程的对流占优的扩散方程;将该方程分解为纯对流方程和纯扩散方程,分别应用特征线法和差分法求解,数值计算结果和实际操作经验相符,能很好地解释混油的形成和发展.     

Low and moderate Reynolds number transient flow simulations using space filtered Navier-Stokes equations

In this study, low and moderate Reynolds number flow problems in the laminar range are solved numerically with grids that do not resolve all the significant scales of motion. Spatial averaging or filtering of the Navier-Stokes equations and Taylor series approximations to the filtered advective terms are used in order to account for the effects of the unresolved or subgrid scales on the resolved scales. Numerical experiments with a transient 2-D lid driven cavity flow problem, using a penalty method Galerkin finite element code, show that this approach enhances the momentum transfer properties of the numerical solution, eliminates 2Δx type oscillations, and enables the use of coarser grids. The significance and order of the terms that describe the interaction between the resolved and the subgrid scales is studied and the success of the series approximations to these terms is demonstrated. © 1994 John Wiley & Sons, Inc.     

Numerical simulation of tunnel fires using preconditioned finite volume schemes

This article is concerned with the numerical simulation of flows at low Mach numbers which are subject to the gravitational force and strong heat sources. As a specific example for such flows, a fire event in a car tunnel will be considered in detail. The low Mach flow is treated with a preconditioning technique allowing the computation of unsteady flows, while the source terms for gravitation and heat are incorporated via an operator splitting. It is shown that a first order discretization in space is not able to compute the buoyancy forces properly on reasonable grids. The feasibility of the method is demonstrated on several test cases.     

Numerical simulation of tunnel fires using preconditioned finite volume schemes

This article is concerned with the numerical simulation of flows at low Mach numbers which are subject to the gravitational force and strong heat sources. As a specific example for such flows, a fire event in a car tunnel will be considered in detail. The low Mach flow is treated with a preconditioning technique allowing the computation of unsteady flows, while the source terms for gravitation and heat are incorporated via an operator splitting. It is shown that a first order discretization in space is not able to compute the buoyancy forces properly on reasonable grids. The feasibility of the method is demonstrated on several test cases.     

基于Picard迭代的PN×PN-2谱元法求解定常不可压缩Navier-Stokes方程

该文给出了一种求解二维定常不可压缩Navier-Stokes方程的基于Picard线性化迭代的P_N×P_N-2谱元法.通过Picard线性化将不可压缩Navier-Stokes方程的求解转化为一系列线性的Stokes-type方程,再利用非交错网格的P_N×P_N-2谱元法计算每个迭代步的Stokes-type方程.为了消除伪压力模,压力离散比速度离散低两阶,非交错网格的应用使得方程的离散方便且不会带来相应的插值误差,从而保证了谱精度.通过此方法数值计算了有精确解的Stokes流动、Kovasznay流动和方腔顶盖驱动流,结果表明,迭代收敛非常快,误差收敛达到了谱精度收敛,并且避免了压力震荡的出现,表明了该文方法准确可靠.     

Numerical Solution of a Fluid Filtration Problem in a Fractured Medium by Using the Domain Decomposition Method

Under consideration are the numerical methods for simulation of a fluid flow in fractured porous media. The fractures are taken into account explicitly by using a discrete fracture model. The formulated single-phase filtering problem is approximated by an implicit finite element method on unstructured grids that resolve fractures at the grid level. The systems of linear algebraic equations (SLAE) are solved by the iterative methods of domain decomposition in the Krylov subspaces using the KRYLOVlibrary of parallel algorithms. The results of solving some model problem are presented. A study is conducted of the efficiency of the computational implementation for various values of contrast coefficients which significantly affect the condition number and the number of iterations required for convergence of the method.     

A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra

In this paper, we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell-centered finite difference scheme. A special non-symmetric quadrature rule is employed that yields a positive definite cell-centered system for the pressure by eliminating local velocities. The method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids, including hexahedra with non-planar faces. Theoretical and numerical results indicate first-order convergence for the pressure and face fluxes.     

Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems

It is well known that, spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques. In this paper, the pseudospectral Legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied. The technique is based on spectral collocation methods in which the trajectory, x(t), is approximated by the Nth degree interpolating polynomial, using Legendre-Gauss-Lobatto points as the collocation points, and Lagrange polynomials as trial functions. The integral involved in the formulation of the problem is approximated based on Legendre-Gauss-Lobatto integration rule, thereby reducing the problem to a nonlinear programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. Illustrative examples are included to confirm the convergence of the pseudospectral Legendre method. Moreover, a numerical experiment (on a nonsmooth problem) indicates that by applying a smoothing filter procedure to the pseudospectral Legendre approximation, one can recover the nonsmooth solution within spectral accuracy.     

Numerical simulation and visualization of vortical structure transformation in the flow past a sphere at an increasing degree of stratification

Based on numerical simulation and visualization, the vortex structure of the flow past a sphere moving uniformly and horizontally in a linearly (density) stratified viscous fluid with an increasing degree of stratification (with the internal Froude number Fr decreasing from infinity to 0.005) at Re = 100 is analyzed in detail for the first time. The classification of the flow regimes is refined. The direct numerical simulation is based on the method of splitting with respect to physical factors (MERANZH) with an explicit hybrid finite-difference scheme, which is second-order accurate in space, monotone, and has a minimal numerical viscosity and dispersion.     

A mixed hybrid Mortar method for solving flow in discrete fracture networks

We consider flow in discrete fracture networks made of 2D domains in intersection and solved with a mixed hybrid finite element method (MHFEM). The discretization within each fracture is performed in two steps: first, borders and intersections are discretized, second, based on these discretizations, a 2D mesh is built. Independent meshing process within each subdomain is of interest for practical use since it makes it easier to refine the chosen subdomains and to perform parallel computation. This article shows how MHFEM is well adapted for integrating a Mortar method to enforce the continuity of the fluxes and heads at the non-matching grids. Some numerical simulations are given to show the efficiency of the method in the case of a preferential orientation of the fractures where a comparison with the 2D solution is possible.     

Numerical modeling of turbulence-induced interfacial instability in two-phase flow with moving interface

The present paper introduces a new interfacial marker-level set method (IMLS) which is coupled with the Reynolds averaged Navier–Stokes (RANS) equations to predict the turbulence-induced interfacial instability of two-phase flow with moving interface. The governing RANS equations for time-dependent, axisymmetric and incompressible two-phase flow are described in both phases and solved separately using the control volume approach on structured cell-centered collocated grids. The transition from one phase to another is performed through a consistent balance of kinematic and dynamic conditions on the interface separating the two phases. The topological changes of the interface are predicted by applying the level set approach. By fitting a number of interfacial markers on the intersection points of the computational grids with the interface, the interfacial stresses and consequently, the interfacial driving forces are easily estimated. Moreover, the normal interface velocity, calculated at the interfacial markers positions, can be extended to the higher dimensional level set function and used for the interface advection process. The performance of linear and non-linear two-equation k–ε turbulence models is investigated in the context of the considered two-phase flow impinging problem, where a turbulent gas jet impinging on a free liquid surface. The numerical results obtained are evaluated through the comparison with the available experimental and analytical data. The nonlinear turbulence model showed superiority in predicting the interface deformation resulting from turbulent normal stresses. However, both linear and nonlinear turbulence models showed a similar behavior in predicting the interface deformation due to turbulent tangential stresses. In general, the developed IMLS numerical method showed a remarkable capability in predicting the dynamics of the considered two-phase immiscible flow problems and therefore it can be applied to quite a number of interface stability problems.