不可压粘流N－S方程的边界积分解法

对原变量的Ｎ－Ｓ方程进行一阶时间离散，采用共轭梯度法解除压强－速度的耦合．对所得的一系列Ｌａｐｌａｃｅ方程、Ｐｏｓｓｉｏｎ方程和Ｈｅｌｍｈｏｔｚ方程均进行边界积分法求解，首次得到了粘性Ｎ－Ｓ方程的边界积分表示式．圆柱的定常、非定常尾迹计算结果表明了本文方法的有效性．     

不可压粘流N－S方程的边界积分解法

对原变量的Ｎ－Ｓ方程进行一阶时间离散，采用共轭梯度法解除压强－速度的耦合．对所得的一系列Ｌａｐｌａｃｅ方程、Ｐｏｓｓｉｏｎ方程和Ｈｅｌｍｈｏｔｚ方程均进行边界积分法求解，首次得到了粘性Ｎ－Ｓ方程的边界积分表示式．圆柱的定常、非定常尾迹计算结果表明了本文方法的有效性．     

The Lagrangian approach of advective term treatment and its application to the solution of Navier—Stokes equations

A one-dimensional transport test applied to some conventional advective Eulerian schemes shows that linear stability analyses do not guarantee the actual performances of these schemes. When adopting the Lagrangian approach, the main problem raised in the numerical treatment of advective terms is a problem of interpolation or restitution of the transported function shape from discrete data. Several interpolation methods are tested. Some of them give excellent results and these methods are then extended to multi-dimensional cases. The Lagrangian formulation of the advection term permits an easy solution to the Navier-Stokes equations in primitive variables V, p, by a finite difference scheme, explicit in advection and implicit in diffusion. As an illustration steady state laminar flow behind a sudden enlargement is analysed using an upwind differencing scheme and a Lagrangian scheme. The importance of the choice of the advective scheme in computer programs for industrial application is clearly apparent in this example.     

A multigrid method for steady incompressible navier-stokes equations based on flux difference splitting

The steady Navier–Stokes equations in primitive variables are discretized in conservative form by a vertex-centred finite volume method Flux difference splitting is applied to the convective part to obtain an upwind discretization. The diffusive part is discretized in the central way. In its first-order formulation, flux difference splitting leads to a discretization of so-called vector positive type. This allows the use of classical relaxation methods in collective form. An alternating line Gauss–Seidel relaxation method is chosen here. This relaxation method is used as a smoother in a multigrid method. The components of this multigrid method are: full approximation scheme with F-cycles, bilinear prolongation, full weighting for residual restriction and injection of grid functions. Higher-order accuracy is achieved by the flux extrapolation method. In this approach the first-order convective fluxes are modified by adding second-order corrections involving flux limiting. Here the simple MinMod limiter is chosen. In the multigrid formulation the second-order discrete system is solved by defect correction. Computational results are shown for the well known GAMM backward-facing step problem and for a channel with a half-circular obstruction.     

A simplified marker and cell method for unsteady flows on non-staggered grids

The SMAC (simplified marker and cell) time-advancing method for solving the unsteady incompressible Navier-Stokes equations on non-staggered grids is developed in generalized co-ordinate systems. The primitive variable formulation uses Cartesian velocities and pressure, all defined at the centre of the control volume, as the dependent variables. A special elliptic flux correction at the faces of the finite volume is utilized in discretizing the continuity equation to suppress pressure oscillations. The test flows considered are a polar cavity flow starting from rest and the flow around a circular cylinder. The numerical results are compared with experimental results and results obtained by the well-known SIMPLEC and PISO methods. The comparisons show that the elliptic flux correction technique works well in suppressing pressure oscillations and that the SMAC method is more efficient than the SIMPLEC and PISO methods for both steady and unsteady flows.     

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.     

Numerical Analysis of Thermally Coupled Flow Problems with Interfaces and Phase-change Effects

In this work a fixed mesh finite element approach is presented to solve thermally coupled flow problems including moving interfaces between immiscible fluids and phase-change effects. The weak form of the full incompressible Navier-Stokes equations is obtained using a generalized streamline operator (GSO) technique that enables the use of equal order interpolation of the primitive variables of the problem: velocity, pressure and temperature. The interfaces are defined with a mesh of marker points whose motion is obtained applying a Lagrangian scheme. Moreover, a temperature-based formulation is considered to describe the phase-change phenomena. The proposed methodology is used in the analysis of a filling of a step mould and a gravity-driven flow of an aluminium alloy in an obstructed vertical channel.     

基于非结构化同位网格的SIMPLE算法

通过基于非结构化网格的有限体积法对二维稳态Navier—Stokes方程进行了数值求解。其中对流项采用延迟修正的二阶格式进行离散；扩散项的离散采用二阶中心差分格式；对于压力-速度耦合利用SIMPLE算法进行处理；计算节点的布置采用同位网格技术，界面流速通过动量插值确定。本文对方腔驱动流、倾斜腔驱动流和圆柱外部绕流问题进行了计算，讨论了非结构化同位网格有限体积法在实现SIMPLE算法时，迭代次数与欠松弛系数的关系、不同网格情况的收敛性、同结构化网格的对比以及流场尾迹结构。通过和以往结果比较可知，本文的方法是准确和可信的。     

Calculation of laminar recirculating flows using a local non-staggered grid refinement system

A grid-embedding technique for the solution of two-dimensional incompressible flows governed by the Navier-Stokes equations is presented. A finite volume method with collocated primitive variables is employed to ensure conservation at the interfaces of embedding grids as well as global conservation. The discretized equations are solved simultaneously for the whole domain, providing a strong coupling between regions of different refinement. The formulation presented herein is applicable to uniform or non-uniform Cartesian meshes. The method was applied to the solution of two scalar transport equations, to cavity flows driven by body and shear forces and to a sudden plane contraction flow. The numerical predictions are compared with the exact solutions when available and with experimental data. The results show that neither the convergence rate nor the stability of the method is affected by the presence of embedded grids. Embedded grids provide a better distribution of grid nodes over the computational domain and consequently the solution accuracy was improved. The grid-embedding technique proved also that significant savings in computing time could be achieved.     

求解二维Euler方程的时一空守恒格式

将作者原来得出的一维时－空守恒格式推广到了二维情形,得到了二维Ｅｕｌｅｒ方程的时．空守恒格式,并用几个典型算例进行了检验计算,结果表明：得到的二维时一空守恒格式保留了一维格式所有的优点,格式简单,通用性强,对微波等间断具有很高的分辨率．     

A characteristic-based method for incompressible flows

A new characteristic-based method for the solution of the 2D laminar incompressible Navier-Stokes equations is presented. For coupling the continuity and momentum equations, the artificial compressibility formulation is employed. The primitives variables (pressure and velocity components) are defined as functions of their values on the characteristics. The primitives variables on the characteristics are calculated by an upwind diffencing scheme based on the sign of the local eigenvalue of the Jacobian matrix of the convective fluxes. The upwind scheme uses interpolation formulae of third-order accuracy. The time discretization is obtained by the explicit Runge–Kutta method. Validation of the characteristic-based method is performed on two different cases: the flow in a simple cascade and the flow over a backwardfacing step.     

A comparative study of central and upwind difference schemes using the primitive variables

The use of the velocity-pressure formulation of the Navier-Stokes equations for the numerical solution of fluid flow problems is favoured for free-surface problems, more involved flow configurations, and three-dimensional flows. Many engineering problems involve such features in addition to strong inertial effects. The computational instabilities arising from central-difference schemes for the convective terms of the governing equations impose serious limitations on the range of Reynolds numbers that can be investigated by the numerical method. Solutions for higher Reynolds numbers Re > 1000 could be reached using upwind-difference schemes. A comparative study of both schemes using a method based on the primitive variables is presented. The comparison is made for the model problem of the driven flow in a square cavity. Using a central scheme stable solutions of the pressure and velocity fields were obtained for Reynolds numbers up to 5000. The streamfunction and vorticity fields were calculated from the resulting velocity field and compared with previous solutions. It is concluded that total upwind differencing results in a considerable change in the flow pattern due to the false diffusion. For practical calculations, by a proper choice of a small amount of partial upwind differencing the vorticity diffusion near the walls and the global features of the solutions are not sigificantly altered.     

Numerical simulation of steady-state and transient natural convection in an isothermal open cubic cavity

In this work the numeric results, of the steady-state and transient heat transfer by natural convection in a horizontal isothermal open cubic cavity are presented. The most important assumptions in the mathematical formulation are two, the flow is laminar and the Boussinesq approximation is valid. The conservation equations in primitive variables are solved using the finite volume method and the SIMPLEC algorithm. The advective terms are approximated by the SMART scheme and the diffusive terms are approximated using the central differencing scheme. The results are obtained for a Rayleigh number range from 10⁴ to 10⁷.The numerical model predicted flow instabilities and Nusselt number oscillations for high Rayleigh numbers.     

A high-order discontinuous Galerkin method for the incompressible Navier-Stokes equations on arbitrary grids

A high-order discontinuous Galerkin (DG) method is proposed in this work for solving the two-dimensional steady and unsteady incompressible Navier-Stokes (INS) equations written in conservative form on arbitrary grids. In order to construct the interface inviscid fluxes both in the continuity and in the momentum equations, an artificial compressibility term has been added to the continuity equation for relaxing the incompressibility constraint. Then, as the hyperbolic nature of the INS equations has been recovered, the local Lax-Friedrichs (LLF) flux, which was previously developed in the context of hyperbolic conservation laws, is applied to discretize the inviscid term. Unlike the traditional artificial compressibility method, in this work, the artificial compressibility is introduced only for the construction of the inviscid numerical fluxes; therefore, a consistent discretization of the INS equations is obtained, irrespective of the amount of artificial compressibility used. What is more, as the LLF flux can be obtained directly and straightforward, no numerical iteration for solving an exact Riemann problem is entailed in our method. The viscous term is discretized by the direct DG method, which was developed based on the weak formulation of the scalar diffusion problems on structured grids. The performance and the accuracy of the method are demonstrated by computing a number of benchmark test cases, including both steady and unsteady incompressible flow problems. Due to its simplicity in implementation, our method provides an attractive alternative for solving the INS equations on arbitrary grids.     

COVOLUME-DUAL VARIABLE METHOD FOR THERMALLY EXPANDABLE FLOW ON UNSTRUCTURED TRIANGULAR GRIDS

Finite difference like discretizations are developed for the time dependent Navier-Stokes equations and the thermal energy equation. The flow is assumed to be thermally expandable, that is, the density varies only with temperature. A new pointwise first order upwind scheme for convection is presented which is of nonnegative type. Also presented are new approaches to reconstructing the velocity vector field from the covolume primitive variables. The resulting difference equations reproduce linear flow fields.     

On general form of Navier-Stokes equations and implicit factored scheme

A general weak conservative form of Navier-Stokes equations expressed with respect to non-orthogonal Curvilinear coordinates and with primitive variables was obtained by using tensor analysis technique, where the contravariant and covariant velocity components were employed. Compared with the current coordinate transformation method, the established equations are concise and forthright, and they are more convenient to be used for solving problems in body-fitted curvilinear coordinate system. An implicit factored scheme for solving the equations is presented with detailed discussions in this paper. For n-dimensional flow the algorithm requires n-steps and for each step only a block tridiagonal matrix equation needs to be solved. It avoids inverting the matrix for large systems of equations and enhances the speed of arithmetic. In this study, the Beam- Warming’s implicit factored schceme is extended and developed in non-orthogonal curvilinear coordinate system.     

Operator-splitting methods for the incompressible Navier-Stokes equations on non-staggered grids. Part 1: First-order schemes

New implicit finite difference schemes for solving the time-dependent incompressible Navier-Stokes equations using primitive variables and non-staggered grids are presented in this paper. A priori estimates for the discrete solution of the methods are obtained. Employing the operator approach, some requirements on the difference operators of the scheme are formulated in order to derive a scheme which is essentially consistent with the initial differential equations. The operators of the scheme inherit the fundamental properties of the corresponding differential operators and this allows a priori estimates for the discrete solution to be obtained. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees boundedness of the solution. To derive the consistent scheme, special approximations for convective terms and div and grad operators are employed. Two variants of time discretization by the operator-splitting technique are considered and compared. It is shown that the derived scheme has a very weak restriction on the time step size. A lid-driven cavity flow has been predicted to examine the stability and accuracy of the schemes for Reynolds number up to 3200 on the sequence of grids with 21 × 21, 41 × 41, 81 × 81 and 161 × 161 grid points.     

Spectral Element Discretization of the Circular Driven Cavity. Part IV: The Navier-Stokes Equations

This paper deals with the spectral element discretization of the Navier-Stokes equations in a disk with discontinuous boundary data, which is known as the driven cavity problem. The numerical treatment does not involve any regularization of these data. Relying on a variational formulation in the primitive variables of velocity and pressure, we describe a discretization of these equations and derive error estimates in appropriate weighted Sobolev spaces. We propose an algorithm to solve the nonlinear discrete system and present numerical experiments to verify its efficiency.     

A variationally consistent finite element approach to the two‐fluid internal contact‐line problem

A new method for simulating incompressible viscous fluid flow involving moving internal contact lines is presented. The steady state interface shape is determined by a variationally consistent formulation of the surface tension contribution to the equations of motion adapted to the case of internal contact lines through the application of a global force balance compatibility condition that consistently removes the pressure indeterminacy. The Crouzeix–Raviart element is chosen to capture the pressure discontinuity at the two‐fluid interface. The resulting discrete equations are solved by an iterative procedure which displays fast convergence characteristics for small capillary numbers. Numerical results for the case of the steady movement of a fluid meniscus in a two‐dimensional channel are presented. Copyright © 2000 John Wiley & Sons, Ltd.