不可压缩粘性流动的CBS有限元解法

对于二维不可压缩粘性流动,首先通过坐标变换的方式得到了的不含对流项的NS方程,并给出了CBS有限元方法求解的一般过程。结合一类同时含有压力和速度的出口边界条件,对方腔顶盖驱动流、后向台阶绕流和圆柱绕流进行了计算。所得结果与基准解符合良好,验证了CBS算法对于定常、非定常粘性不可压缩流动问题的可行性和所用出口边界条件的无反射特性。特别的,对于圆柱绕流,Re=100时非定常升、阻力系数及漩涡脱落等非定常都得到了较好地模拟,为一进步研究自激振动等更加复杂的非定常流动问题奠定了基础。     

配置点谱方法-人工压缩法(SCM-ACM)求解不可压缩流体流动

开发了配置点谱方法SCM（spectral collocation method）与人工压缩法ACM（artificial compressibility method）相结合的方法SCM-ACM,用于求解不可压缩粘性流动问题。选取典型的方腔顶盖驱动流为研究测试对象,首先建立人工压缩格式的控制方程,其次采用SCM离散控制方程的空间偏微分项,推导出矩阵形式的代数方程,最后测试了SCM-ACM代码的有效性。结果显示,SCM-ACM能够有效求解不可压缩流动问题,并继承了谱方法的指数收敛特性,且具有ACM求解过程简单及易于实施的特点。     

Optimized incompressible smoothed particle hydrodynamics methods and validations

The solution of the Poisson's equation used by the incompressible smoothed particle hydrodynamics (ISPH) methods for estimating the pressure field is expensive in CPU time. The CPU time, consumed by the inversion of the operator ∇(1/ρ∇) and the estimation of the right hand side of the Poisson's equation, increases with the number N of particles used in a purely Lagrangian framework. In this work, this default of ISPH methods is overcome by solving the Poisson's equation on a Cartesian grid. This SPH-mesh coupling is equivalent to the particle in cell method. In a first step, in order to analyze its efficiency, the optimized version of two ISPH methods (divergence free and density invariant) is compared with the standard weakly compressible SPH method through two benchmarks of incompressible bidimensional flows characterized by the Reynolds number Re, Lamb-Oseen vortex (10 ≤Re≤ 100) and lid-driven cavity flow (100 ≤Re≤ 1000). In a second step, the numerical results obtained by the three SPH methods are compared to laboratory experimental data of a dam break flow in order to show the performance of the SPH-mesh coupling in a practical and complex flow problem. As in the configuration of the experimental setup, the numerical results are obtained for a Reynolds number Re = 3.8 10⁶.     

NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE FLOW BY A NEW FORMULATION

A new mathematical formulation, called the pseudovorticity–velocity formulation, of the three-dimensional incompressible Navier–Stokes equations is presented as an alternative to the vorticity–velocity approach. For the model lid-driven cavity flow problem in two and three dimensions, combined with an explicit mixed spectral /finite different numerical scheme the proposed formulation is found to be efficient and very accurate as compared with the results available in the literature. In particular, the simulation results demonstrate an attractive feature of the present formulation compared with the vorticity–velocity approach, namely that the divergence-free condition of the velocity field can always be achieved on a non-staggered mesh.     

Lattice Boltzmann simulation of two-sided lid-driven flow in a staggered cavity

In this article, the lattice Boltzmann method is employed in order to explore incompressible fluid flow inside a two-sided lid-driven staggered cavity. Results of the lattice Boltzmann simulation for antiparallel motion of lids are compared with the data from existing literature. For parallel motion of lids, the characteristics of flow pattern for a variety of Re numbers (50–3200) are presented. An asymmetric steady-state flow pattern for parallel motion of lids is obtained.     

SUPG finite element method based on penalty function for lid-driven cavity flow up to $$Re = 27500$$

A streamline upwind/Petrov–Galerkin(SUPG)finite element method based on a penalty function is proposed for steady incompressible Navier–Stokes equations.The SUPG stabilization technique is employed for the formulation of momentum equations. Using the penalty function method, the continuity equation is simplified and the pressure of the momentum equations is eliminated. The lid-driven cavity flow problem is solved using the present model. It is shown that steady flow simulations are computable up to Re = 27500, and the present results agree well with previous solutions. Tabulated results for the properties of the primary vortex are also provided for benchmarking purposes.     

A mesh-free radial basis function–based semi-Lagrangian lattice Boltzmann method for incompressible flows

In this paper, a local radial basis function–based semi-Lagrangian lattice Boltzmann method (RBF-SL-LBM) is proposed. This is a mesh-free method that can be used for the simulation of incompressible flows. In this method, the collision step is performed locally, which is the same as in the standard LBM. In the meanwhile, the steaming step is solved in a semi-Lagrangian framework. The distribution functions at the departure points, which may be not the grid points in general, are computed by the local radial basis function interpolation. Several numerical tests are conducted to validate the present method, including the lid-driven cavity flow, the steady and unsteady flow past a circular cylinder, and the flow past an NACA0012 airfoil. The present results are in good agreement with those published in the previous literature, which demonstrates the capability of RBF-SL-LBM for the simulation of incompressible flows.     

高阶谱元区域分解算法求解定常方腔驱动流

主要利用Jacobian-free的Newton-Krylov方法求解定常不可压缩Navier-Stokes方程,将基于高阶谱元法的区域分解Stokes算法的非定常时间推进步作为Newton迭代的预处理,回避了传统Newton方法Jacobian矩阵的显式装配,节省了程序内存,同时降低了Newton迭代线性系统的条件数,且没有非线性对流项的隐式求解,大大加快了收敛速度。对有分析解的Kovasznay流动的计算结果表明,本高阶谱元法在空间上有指数收敛的谱精度,且对定常解的Newton迭代是二次收敛的。本文模拟了二维方腔顶盖一致速度驱动流,同基准解符合得很好,表明本文方法是准确可靠的。本文还考虑了Re=800时方腔顶盖正弦速度驱动流,除得到已知的一个稳定对称解和一对稳定非对称解外,还获得了一对新的不稳定的非对称解。     

Aspect ratio effects on three-dimensional incompressible flow in a two-sided non-facing lid-driven parallelepiped cavity

A numerical study of the three-dimensional fluid flow has been carried out to determine the effects of the transverse aspect ratio, Ay, on the flow structure in two-sided non-facing lid-driven cavities. The flow is complex, unstable and can undergo bifurcation. The numerical method is based on the finite volume method and multigrid acceleration. Computations have been investigated for several Reynolds numbers and various aspect ratio values. At a fixed Reynolds number, Re=500, the three-dimensional flow characteristics are analyzed considering four transverse aspect ratios, Ay=1,0.75,0.5 and 0.25. It is observed that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation. An analysis of the flow evolution shows that, at Ay=0.75, the flow bifurcates to a periodic regime at (Re=600) with a frequency f=0.093 less than the predicted value in the cubical cavity. A correlation is established when Ay=0.5 and gives the critical Reynolds number value. At Ay=0.25, the periodic regime occurs at high Re value beyond 3500, after which the flow becomes chaotic. It is shown that, when increasing Ay over the unit, the flow in the cavity exhibits a complex behavior. The kinetic energy transmission from the driven walls to the cavity center is reduced at low Ay values.     

Fourth‐order compact formulation of Navier–Stokes equations and driven cavity flow at high Reynolds numbers

A new fourth‐order compact formulation for the steady 2‐D incompressible Navier–Stokes equations is presented. The formulation is in the same form of the Navier–Stokes equations such that any numerical method that solve the Navier–Stokes equations can easily be applied to this fourth‐order compact formulation. In particular, in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601 × 601. Using this formulation, the steady 2‐D incompressible flow in a driven cavity is solved up to Reynolds number with Re = 20 000 fourth‐order spatial accuracy. Detailed solutions are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Study of the effect of the non-orthogonality for non-staggered grids—the theory

An investigation has been conducted to determine the effect of the grid non-orthogonality on the convergence behavior of two-dimensional lid-driven cavity flows. The relevant theory is presented in this article. In the present work, the contravariant velocity fluxes are used as the dependent variables on non-orthogonal, non-staggered grids. The momentum equations retain a strongly conservative form. Two practices for treating the momentum interpolation method in general curvilinear co-ordinates are presented. In each practice, the momentum interpolation formulations with and without velocity underrelaxation factor are considered. The discretization equations are solved using the SIMPLE, SIMPLEC and SIMPLER algorithms. © 1998 John Wiley & Sons, Ltd.     

A face-based monolithic approach for the incompressible magnetohydrodynamics equations

A novel numerical algorithm has been developed to solve the incompressible resistive magnetohydrodynamics equations in a fully coupled form. The numerical method is based on the face-centered unstructured finite volume approximation, where the velocity and magnetic field vector components are defined at the center of edges/faces; meanwhile, the pressure term is defined at element centroid. In order to enforce a divergence-free magnetic field, the gradient of a scalar Lagrange multiplier is introduced into the induction equation. A special attention will be given to satisfy the continuity equation and the Gauss' law for magnetism within each element and the summation of the equations can be exactly reduced to the domain boundary. The first modification to the original algorithm involves the evaluation of the convective fluxes over the two neighboring elements, where the discrete continuity equations are exactly satisfied. The second modification is based on the neglecting electric field term from the Lorentz force in two dimensions. The resulting large-scale algebraic linear equations are solved in a fully coupled manner using the one- and two-level restricted additive Schwarz preconditioners to avoid any time step restrictions forced by stability requirements. The spatial convergence of the algorithm is confirmed by solving the Hartmann flow, and then the algorithm is applied to the classical lid-driven cavity and backward facing step benchmark problems in two and three dimensions. The lid-driven cavity flow calculations at relatively high Stuart numbers indicate the perfect braking effect of the magnetic field in two dimensions.     

Fluid flow due to combined convection in lid-driven enclosure having a circular body

The present work is aimed to study mixed convection heat transfer characteristics for a lid-driven air flow within a square enclosure having a circular body. Flows are driven by the left lid, which slides in its own plane constant velocity. This wall is isothermal and it moves up or down in y direction while the other walls remain stationary. The horizontal walls are adiabatic. The cavity is differentially heated and the left wall is maintained at a higher temperature than the right wall. Three different temperature boundary conditions were applied for the inner cylinder as adiabatic, isothermal or conductive. The computation is carried out for wide ranges of Richardson numbers, diameter of inner cylinder and center and location of the inner cylinder. It was found that the most effective parameter on flow field and temperature distribution is the orientation of the moving lid. The circular body can be a control parameter for heat and fluid flow. An interesting obtained result that the thermal conductivity becomes insignificant for small values of diameter of the circular body.     

纳米尺度圆柱绕流尾迹区流动形式模拟研究

采用非平衡分子动力学模拟方法,对微尺度低{Re}数下的圆柱绕流问题进行了研究,模拟结果表明:当{Re}<12时,圆柱下游形成对称、无分离的定常流;当{Re}>20时,圆柱下游形成周期性交替出现的对称涡;当12相似文献   

Numerical solutions of 2‐D steady incompressible driven cavity flow at high Reynolds numbers

Numerical calculations of the 2‐D steady incompressible driven cavity flow are presented. The Navier–Stokes equations in streamfunction and vorticity formulation are solved numerically using a fine uniform grid mesh of 601 × 601. The steady driven cavity flow solutions are computed for Re ? 21 000 with a maximum absolute residuals of the governing equations that were less than 10^?10. A new quaternary vortex at the bottom left corner and a new tertiary vortex at the top left corner of the cavity are observed in the flow field as the Reynolds number increases. Detailed results are presented and comparisons are made with benchmark solutions found in the literature. Copyright © 2005 John Wiley & Sons, Ltd.     

Development of an Artificial Compressibility Methodology with Implicit LU-SGS Method

A numerical method has been developed to solve the steady and unsteady incompressible Navier-Stokes equations in a two-dimensional, curvilinear coordinate system. The solution procedure is based on the method of artificial compressibility and uses a third-order flux-difference splitting upwind differencing scheme for convective terms and second-order center difference for viscous terms. A time-accurate scheme for unsteady incompressible flows is achieved by using an implicit real time discretization and a dual-time approach, which introduces pseudo-unsteady terms into both the mass conservation equation and momentum equations. An efficient fully implicit algorithm LU-SGS, which was originally derived for the compressible Eulur and Navier-Stokes equations by Jameson and Toon [1], is developed for the pseudo-compressibility formulation of the two dimensional incompressible Navier-Stokes equations for both steady and unsteady flows. A variety of computed results are presented to validate the present scheme. Numerical solutions for steady flow in a square lid-driven cavity and over a backward facing step and for unsteady flow in a square driven cavity with an oscillating lid and in a circular tube with a smooth expansion are respectively presented and compared with experimental data or other numerical results.     

Fhree-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method, a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper. Numerical characteristics of the scheme are studied by the Fourier analysisl Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convectiondiffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.     

Coupling of discrete-element method and smoothed particle hydrodynamics for liquid-solid flows

Particle based methods can be used for both the simulations of solid and fluid phases in multiphase medium, such as the discrete-element method for solid phase and the smoothed particle hydrodynamics for fluid phase. This paper presents a computational method combining these two methods for solid-liquid medium. The two phases are coupled by using an improved model from a reported Lagrangian-Eulerian method. The technique is verified by simulating liquid-solid flows in a two-dimensional lid-driven cavity.     

Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method,a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper.Numerical characteristics of the scheme are studied by the Fourier analysis. Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node,the proposed scheme is explicit and can achieve arbitrary order of accuracy in space.Application examples for the convection- diffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given.It is found that the proposed compact scheme is not only simple to implement and economical to use,but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.     

ADAPTIVE GRID REFINEMENT USING CELL-LEVEL AND GLOBAL IMBALANCES

A methodology for local solution-adaptive mesh refinement in computational fluid dynamics (CFD) using cell-level and global kinetic energy balances is formulated and tested. Results are presented for two two-dimensional steady incompressible laminar benchmark problems: a lid-driven cavity (Reynolds number Re=1000) and a backward-facing step (Re=400). It is demonstrated that local kinetic energy imbalance correlates with local solution accuracy, that normalized global imbalance is an appropriate criterion for halting mesh refinement and that a specified level of accuracy is realized at lower computational effort using local refinement compared with a uniform finer mesh. © 1997 by John Wiley and Sons, Ltd.