An ‘assumed deviatoric stress–pressure–velocity’ mixed finite element method for unsteady,convective, incompressible viscous flow: Part I: Theoretical development

A formulation of a mixed finite element method for the analysis of unsteady, convective, incompressible viscous flow is presented in which: (i) the deviatoric-stress, pressure, and velocity are discretized in each element, (ii) the deviatoric stress and pressure are subject to the constraint of the homogeneous momentum balance condition in each element, a priori, (iii) the convective acceleration is treated by the conventional Galerkin approach, (iv) the finite element system of equations involves only the constant term of the pressure field (which can otherwise be an arbitrary polynomial) in each element, in addition to the nodal velocities, and (v) all integrations are performed by the necessary order quadrature rules. A fundamental analysis of the stability of the numerical scheme is presented. The method is easily applicable to 3-dimensional problems. However, solutions to several problems of 2-dimensional Navier-Stokes' flow, and their comparisons with available solutions in terms of accuracy and efficiency, are discussed in detail in Part II of this paper.     

Numerical analysis of density flows in adiabatic two-phase fluids through characteristic finite element method

The purpose of this study is to analyze the density flow in adiabatic two-phase fluids through the characteristic finite element method. The fluids are assumed to be liquids. The equations of conservations of mass and momentum for the adiabatic flows and the Birch–Murnaghan equation of state are employed as the governing equations. The employed finite element method is a combination of the characteristic method and the implicit method. The governing equations are divided into two parts: the advection part and the non-advection part. The characteristic method is applied to the advection part. The Hermite interpolation function, which is based on the complete third-order polynomial interpolation using triangular finite element is employed for the interpolation of both velocity and density. Using the discontinuity conditions, an interface translocation method can be derived. The interface of the two flow densities are interpolated through the third-order spline function, using which the curvature of the interface can be directly computed. For the numerical study, the development of density flow over the Tokyo bay is presented. It is detected out that high density area is abruptly diffused over the whole area. According to the differences in the two densities, various flow patterns are computed.     

带尾涡面的三维简化航天机亚音速有限元计算

本文采用Ｇａｌｅｒｋｉｎ有限元方法以全位势方程为控制方程计算了航天机三维简化模型的亚声速流动。为模拟实际流动,在后部加一尖劈形后体,并拖出一尾涡面。为局部超场 速区解的稳定,采用人工密度修正。对密度和环量进行双重迭代,得到了亚声速下三维航天机的压分分布和气动力系数。     

Periodic finite elements in two-layer tidal flow

The aim of this paper is to present the finite element method and its application to quasi-steady periodic two-layer tidal flow in estuaries and coastal seas. Formulating the weighted residual equations, using quadratic polynomials for velocity and linear polynomials for water elevation as interpolation functions and employing the periodic Galerkin method, the nonlinear simultaneous equations can be derived. The present method is used for the simulation analysis of the Niigata Port redevelopment planning.     

Finite element analysis of lift build-up due to aerofoil indicial motion

In this paper the problem of impulsively started aerofoil or suden change of incidence of an aerofoil in incompressible potential flow is investigated. The essence of solution lies in the representation of a timely and spatially varying wake in a largely irrotational potential flow field. This is achieved by representing the wake through velocity potential difference, which seems to be the only way of imposing a velocity difference condition in the finite element context with velocity potentials as the basic unknowns. Superposition is employed to meet various boundary conditions, which is justified by the linearity of the problem. The finite element solutions are compared with those from singularity method.     

Double diffusive convection of anomalous density fluids in a porous cavity

A numerical study has been performed to analyze the combined effect of temperature and species gradients on the buoyancy-driven natural convection flow of cold water near its density extremum contained in a porous cavity. The governing equations are descretized using the finite volume method. The results of the investigation are presented in the form of steady-state streamlines, velocity vectors, isotherms, and isoconcentrationlines. The results are discussed for different porosities, Darcy numbers, and Grashof numbers. The heat and mass transfer rates calculated are found to behave nonlinearly with hot wall temperature. The heat and mass transfer are increased with increasing Darcy number and porosity. It is found that the convective heat and mass transfer rate are greatly affected by the presence of density maximum.     

二维高亚音速Laval喷管流场的有限元计算

本文使用了八节点曲四边形等参元，通过变分有限元法，对二维高亚音速Ｌａｖａｌ喷管位势流场进行了计算，结果是满意的，并和一维结果进行了对比。文中还使用了有限元法特有的局部线化理论，取单元中心点密度去处理迭代过程中单元系数阵的计算，其结果与通常的有限元法计算相一致，但计算时间却大大减少了。     

Variable penalty method for finite element analysis of incompressible flow

A new scheme is applied for increasing the accuracy of the penalty finite element method for incompressible flow by systematically varying from element to element the sign and magnitude of the penalty parameter λ, which enters through ?.v + p/λ = 0, an approximation to the incompressibility constraint. Not only is the error in this approximation reduced beyond that achievable with a constant λ, but also digital truncation error is lowered when it is aggravated by large variations in element size, a critical problem when the discretization must resolve thin boundary layers. The magnitude of the penalty parameter can be chosen smaller than when λ is constant, which also reduces digital truncation error; hence a shorter word-length computer is more likely to succeed. Error estimates of the method are reviewed. Boundary conditions which circumvent the hazards of aphysical pressure modes are catalogued for the finite element basis set chosen here. In order to compare performance, the variable penalty method is pitted against the conventional penalty method with constant λ in several Stokes flow case studies.     

An ‘assumed deviatoric stress-pressure-velocity’ mixed finite element method for unsteady,convective, incompressible viscous flow: Part II: Computational studies

In Part I of this paper we presented a mixed finite element method, for solving unsteady, incompressible, convective flows, based on assumed ‘deviatoric stress–velocity–pressure’ fields in each element, which have the features: (i) the convective term is treated by the usual Galerkin technique; (ii) the unknowns in the global system of finite element equations are the nodal velocities, and the ‘constant term’ in the arbitrary pressure field over each element; and (iii) exact integrations are performed over each element. In this paper we present numerical studies, both for steady as well as unsteady cases, of the problems: (a) the driven cavity, (b) Jeffry–Hamel flow in a channel, (c) flow over a ‘backward’ or ‘downstream’ facing step, and (d) flow over a square step. All these problems are two-dimensional in nature, although certain 3-D solutions are to be presented in a separate paper. The present results are compared with those which are available in the literature and are based on alternative approaches to treat incompressibility and convective acceleration. The possible merits of the present method are thus pointed out.     

三维非定常可压跨声速有旋流动的统一变域变分理论

本文利用Ｃｌｅｂｓｃｈ变量,应用半反推法首次建立了三维非定常可压跨声有旋流体的广义变分原理及变分原理族,应用变域变分工具,几乎所有界面条件（包括激波、尾涡面条件等）都转化成了自然边界条件,这给有限元计算带来了很大的方便。     

A PRESSURE CORRECTION METHOD FOR UNSTRUCTURED MESHES WITH ARBITRARY CONTROL VOLUMES

A pressure correction procedure for general unstructured meshes is presented. It is a cell-centred, collocated finite volume method and the pressure–velocity coupling is treated using SIMPLEC. The cells can have an arbitrary number of grid points (cell vertices). In the present study the number of faces on the cells varies between three and six. The discretized equations are solved using either a symmetric Gauss–Seidel solver or a conjugate gradient solver with a preconditioner. The method is applied to three two-dimensional test cases in which the flow is incompressible and laminar. The extension to three dimensions as well as to turbulent flow using transport models is straightforward. It can also be extended to handle compressible flow.     

Finite element modelling of sheared flow effects on the radiation characteristics of acoustic sources in a circular duct

The recent interest in propeller noise generation, stimulated by development of new propeller types for commerical propjets, has generated a need for the ability of measure the noise characteristics of propellers. However, wind tunnel noise measurements are affected by reflections from the wind tunnel walls. Computer codes predicting the free-field noise of a propeller and its noise field in a circular wind tunnel allow validating the use of wind tunnel measurements to predict free-field noise characteristics. A wind tunnel contains flow which is uniform in the duct axial direction, but can vary in the radial direction. It can be shown that a third-order differential equation governs the acoustic pressure field for such a duct containing radially sheared subsonic flow. This third-order problem is then posed as a coupled pair of equations which are second-order in terms of acoustic density and first-order in terms of an artificial variable which represents the effects of the flow being sheared. It is shown that this form of the problem allows a natural extension of the existing numerical solution techniques for non-sheared flow. The sheared flow problem is presented, and a finite element method is developed to yield a solution for propeller-type acoustic forces. The finite element code and method are refined with numerical experiments, and results are presented for a specific propeller and duct geometry. Good agreement is shown between this method and an alternate approach to the sheared flow problem using a piecewise constant representation of the velocity in the boundary layer. This validates both the numerical methods.     

A calculation procedure for steady two-dimensional elliptic flows

A calculation procedure is presented for predicting steady two-dimensional elliptic flows. The method introduces a density correction concept and an algebraic equation for the velocity correction instead of the troublesome pressure correction equation in the SIMPLER procedure. Computations show that the method has the same rate of convergence as SIMPLER while saving about 20% computational effort per iteration. Although the method is described for steady two-dimensional situations, its extension to three-dimensional problems is very straightforward.     

Finite element,stream function-vorticity solution of secondary flow in the channels of a rod bundle

A finite element method has been applied to predict the overall features of the fully developed turbulent flow in the non-circular channels of a rod bundle. The finite element discretization is based on the conventional Galerkin method using an isoparametric quadrilateral element with mixed interpolation. The primary axial flow and turbulent kinetic energy distributions have been predicted for fully developed turbulent flow conditions right up to the wall. The secondary velocity is represented by the stream function-vorticity formulation and the no-slip boundary conditions are explicitly introduced in the nonlinear equations by a boundary vorticity formula. The Newton-Raphson method is applied to the stream function-vorticity equations and solved simultaneously by the frontal solution technique. A one-equation eddy viscosity model of turbulence and an algebraic stress transport model have been used to predict primary axial velocity, secondary velocities and turbulent kinetic energy. The predictions obtained for a central subchannel of an equilateral-triangular rod array with p/d= 1.3 are in reasonable agreement with experimental data.     

Finite element analysis of flow in a gas-filled rotating annulus

Linearized multidimensional flow in a gas centrifuge can be described away from the ends by Onsager's pancake equation. However a rotating annulus results in a slightly different set of boundary conditions from the usual symmetry at the axis of rotation. The problem on an annulus becomes ill-posed and requires some special attention. Herein we treat axially linear inner and outer rotor temperature distributions and velocity slip. An existence condition for a class of non-trivial, one-dimensional solutions is given. New exact solutions in the infinite bowl approximation have been derived containing terms that are important at finite gap width and non-vanishing velocity slip. The usual one-dimensional, axially symmetric solution is obtained as a limit. Our previously reported finite element algorithm has been extended to treat this new class of problems. Effects of gap width, temperature and slip conditions are illustrated. Lastly, we report on the compressible, finite length, circular Couette flow for the first time.     

NUMERICAL SIMULATION FOR NAVIER-STOKES EQUATIONS BY PROJECTION/CHARACTERISTIC-BASED OPERATOR-SPLITTING FINITE ELEMENT METHOD

提出了一种求解非定常不可压缩纳维-斯托克斯方程（N-S方程）的新型有限元法：基于投影法的特征线算子分裂有限元法.在每一个时间层上将N-S方程分裂成扩散项、对流项、压力修正项.对流项采用多步显式格式,且在每一个对流子时间步内采用更加精确的显式特征线-伽辽金法进行时间离散,空间离散采用标准伽辽金法.应用此算法对平面泊肃叶流、方腔流和圆柱绕流进行数值模拟,所得结果与基准解符合良好.尤其对于Re=10000的方腔流,给出了方腔中分离涡发展和运动的计算结果,并发现在该雷诺数下存在周期解,表明该算法能较好地模拟流体流动中的小尺度物理量以及流场中分离涡的运动.     

A finite element solution of the time-dependent incompressible Navier–Stokes equations using a modified velocity correction method

A finite element solution of the two-dimensional incompressible Navier–Stokes equations has been developed. The present method is a modified velocity correction approach. First an intermediate velocity is calculated, and then this is corrected by the pressure gradient which is the solution of a Poisson equation derived from the continuity equation. The novelty, in this paper, is that a second-order Runge–Kutta method for time integration has been used. Discretization in space is carried out by the Galerkin weighted residual method. The solution is in terms of primitive variables, which are approximated by polynomial basis functions defined on three-noded, isoparametric triangular elements. To demonstrate the present method, two examples are provided. Results from the first example, the driven cavity flow problem, are compared with previous works. Results from the second example, uniform flow past a cylinder, are compared with experimental data.     

Analysis of flow in the plate‐spiral of a reaction turbine using a streamline upwind Petrov–Galerkin method

The prediction of the flow field in a novel spiral casing has been accomplished. Hydraulic turbine manufacturers are considering the potential of using a special type of spiral casing because of the easier manufacturing process involved in its fabrication. These special spiral casings are known as plate‐spirals. Numerical simulation of complex three‐dimensional flow through such spiral casings has been accomplished using a finite element method (FEM). An explicit Eulerian velocity correction scheme has been deployed to solve the Reynolds‐average Navier–Stokes equations. The simulation has been performed to describe the flow in high Reynolds number (10⁶) regimes. For spatial discretization, a streamline upwind Petrov–Galerkin (SUPG) technique has been used. The velocity field and the pressure distribution inside the spiral casing reveal meaningful results. Copyright © 2000 John Wiley & Sons, Ltd.     

N-S方程基于投影法的特征线算子分裂有限元求解

提出了一种求解非定常不可压缩纳维-斯托克斯方程（N-S方程）的新型有限元法:基于投影法的特征线算子分裂有限元法.在每一个时间层上将N-S方程分裂成扩散项、对流项、压力修正项.对流项采用多步显式格式,且在每一个对流子时间步内采用更加精确的显式特征线-伽辽金法进行时间离散,空间离散采用标准伽辽金法.应用此算法对平面泊肃叶流、方腔流和圆柱绕流进行数值模拟,所得结果与基准解符合良好.尤其对于Re=10000的方腔流,给出了方腔中分离涡发展和运动的计算结果,并发现在该雷诺数下存在周期解,表明该算法能较好地模拟流体流动中的小尺度物理量以及流场中分离涡的运动.