ALE-FEM for Two-Phase Flows

We present a finite element method for the flow of two immiscible incompressible fluids in two and three dimensions. Thereby the presence of surface active agents (surfactants) on the interface is allowed, which alter the surface tension. The model consists of the incompressible Navier-Stokes equations for velocity and pressure and a convection-diffusion equation on the interface for the distribution of the surfactant. A moving grid technique is applied to track the interface, on that account a Arbitrary-Lagrangian-Eulerian (ALE) formulation of the Navier-Stokes equation is used. The surface tension force is incorporated directly by making use of the Laplace-Beltrami operator technique [5]. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition II

Summary We consider mixed finite element approximations of the stationary, incompressible Navier-Stokes equations with slip boundary condition simultaneously approximating the velocity, pressure, and normal stress component. The stability of the schemes is achieved by adding suitable, consistent penalty terms corresponding to the normal stress component and to the pressure. A new method of proving the stability of the discretizations allows, us to obtain optimal error estimates for the velocity, pressure, and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition. The schemes can easily be implemented into existing finite element codes for the Navier-Stokes equations with standard Dirichlet boundary conditions.     

Finite element calculations of flow past a spherical bubble rising on the axis of a cylindrical tube

The velocity and pressure fields of a Newtonian fluid with homogeneous and constant physical properties flowing around a sphere on the axis of a cylindrical tube with no slip, free slip and partial slip at the sphere surface and no slip at the cylinder wall have been calculated by solving the Navier-Stokes equations and the continuity equation using the finite element technique with the penalty function method. Terminal rise velocities of spherical air bubbles in water have been calculated as function of the bubble radius and some conclusions have been drawn about the nature of the interface. Finally, the influence of the presence of a cylindrical wall on the drag force has been determined and a new empirical equation is derived for the wall correction factor for a sphere rising with free slip at its surface at low Reynolds number.     

A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS I： SPATIAL DISCRETIZATION

In this article we consider a two-level finite element Galerkin method using mixed finite elements for the two-dimensional nonstationary incompressible Navier-Stokes equations. The method yields a $H^1$-optimal velocity approximation and a $L_2$-optimal pressure approximation. The two-level finite element Galerkin method involves solving one small, nonlinear Navier-Stokes problem on the coarse mesh with mesh size $H$, one linear Stokes problem on the fine mesh with mesh size $h << H$. The algorithm we study produces an approximate solution with the optimal, asymptotic in $h$, accuracy.     

Spectral element solution of steady incompressible viscous free-surface flows

In this paper we present a new approach for the solution of the steady incompressible Navier-Stokes equations in a domain bounded in part by a free surface. In the spatial discretization procedure, a Legendre spectral element method is used to generate the discrete equations. For effective solution of the set of algebraic equations, the geometry is decoupled from the fluid velocity and pressure. In addition, two different algorithms are proposed depending on the importance of surface tension effects. Numerical results are presented to demonstrate the effectiveness of the proposed algorithms.     

A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows

This paper reports a new meshless Integrated Radial Basis Function Network (IRBFN) approach to the numerical simulation of interfacial flows in which the two-way interaction between a moving interface and the ambient viscous flow is fully investigated. When an interface between two immiscible fluids moves, not only its position and shape but also the flow variables (i.e. velocity field and pressure) change due to the presence of surface tension along the moving interface. The velocity field of the ambient flow, on the other hand, causes the interface to move and deform as a result of momentum transport between the two immiscible fluids on both sides of the interface. Numerical investigations of such a two-way interaction is reported in this paper where the level set method is used in combination with high-order projection schemes in the meshless framework of the IRBFN method. Numerical investigations on the meshless projection schemes are performed with typical benchmark incompressible viscous flow problems for verification purposes. The approach is then demonstrated with the numerical simulation of two bubbles moving, stretching and merging in an incompressible ambient fluid under the action of buoyancy force.     

A combined finite element and Marker and cell method for solving Navier-Stokes equations

Summary A new method is developed to approximate Navier-Stokes equations for incompressible fluids on polygonal domains. This method is based on a simple finite element approximation adapted to the Marker and Cell technique. Its main originality lies in the fact that the components of the velocity are approximated on distinct interlaced networks. The error analysis shows that this method is of order one.     

Modeling of two-phase flows with surface tension by finite pointset method (FPM)

A meshfree method for two-phase immiscible incompressible flows including surface tension is presented. The continuum surface force (CSF) model is used to include the surface tension force. The incompressible Navier–Stokes equation is considered as the mathematical model. Application of implicit projection method results in linear second-order partial differential equations for velocities and pressure. These equations are then solved by the finite pointset method (FPM), which is a meshfree and Lagrangian method. The fluid is represented as finite number of particles and the immiscible fluids are distinguished by the color of each particle. The interface is tracked automatically by advecting the color functions for each particle. Two test cases, Laplace's law and the Rayleigh–Taylor instability in 2D have been presented. The results are found to be consistent with the theoretical results.     

Stability of a finite element method for 3D exterior stationary Navier-Stokes flows

We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary.     

Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows Nonlinear model

Formulated in terms of velocity, pressure and the extra stress tensor, the incompressible Navier-Stokes is discretized by stabilized finite element method. The stabilized method proposed is analyzed for the full nonlinear model, and is applicable to various combinations of interpolation functions, including the simplest equal-order linear and bilinear elements.     

A Two-Grid Finite Element Method for Time-Dependent Incompressible Navier-Stokes Equations with Non-Smooth Initial Data

We analyze here, a two-grid finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size $H$ and solving a Stokes problem on a fine grid of size $h, h <相似文献   

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

A Nonconforming Finite Element Method of Streamline Diffusion Type for the Incompressible Navier-Stokes Equations

A nonconforming finite element method of streamline diffusion type for solving the stationary and incompressible Navier-Stokes equation is considered. Velocity field and pressure field are approximated by piecewise linear and piecewise constant functions, respectively. The existence of solutions of the discrete problem and the strong convergence of a subsequence of discrete solutions are established. Error estimates are presented for the uniqueness case.     

A numerical method for mass conservative coupling between fluid flow and solute transport

We present a new coupled discretization approach for species transport in an incompressible fluid. The Navier-Stokes equations for the flow are discretized by the divergence-free Scott-Vogelius element on barycentrically refined meshes guaranteeing LBB stability. The convection-diffusion equation for species transport is discretized by the Voronoi finite volume method. In accordance to the continuous setting, due to the exact integration of the normal component of the flow through the Voronoi surfaces, the species concentration fulfills discrete global and local maximum principles. Besides of the numerical scheme itself, we present important aspects of its implementation. Further, for the case of homogeneous Dirichlet boundary conditions, we give a convergence proof for the coupled scheme. We report results of the application of the scheme to the interpretation of limiting current measurements in an electrochemical flow cell with cylindrical shape.     

An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting.     

Study of the contraction flow using grid generation technique

We study a model procedure to solve the incompressible Navier-Stokes equations on the flow inside contraction geometry. The governing equations are expressed in the primitive variable formulation. A rectangular computational plane is arises by elliptic grid generation technique. The numerical solution is based on a technique of automatic numerical generation of a curvilinear coordinate system. By transformed the governing equation into computational plane. The time dependent momentum equations are solved explicitly for the velocity field using the explicit marching procedure, the continuity equation is applied at each grid point in the solution of pressure equation, while the successive over relaxation (SOR) method is used for the Neumann problem for pressure. We will apply the technique on several irregular-shape.     

A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media

The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure‐velocity equation and the concentration equation. In this article, we present a mixed finite volume element method for the approximation of pressure‐velocity equation and a discontinuous Galerkin finite volume element method for the concentration equation. A priori error estimates in L^∞(L²) are derived for velocity, pressure, and concentration. Numerical results are presented to substantiate the validity of the theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations

Summary The Lagrange-Galerkin method is a numerical technique for solving convection — dominated diffusion problems, based on combining a special discretisation of the Lagrangian material derivative along particle trajectories with a Galerkin finite element method. We present optimal error estimates for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation. The method is shown to be nonlinearly stable.     

MAXIMUM NORM ERROR ESTIMATES OF CROUZEIX-RAVIART NONCONFORMING FINITE ELEMENT APPROXIMATION OF NAVIER-STOKES PROBLEM

1. IntroductionThere are many research works on finite element approximation of Navier-Stokesproblem in the case of lower Reynold number, by using the so-called velocity--pressuremixed by Teman [26], the optimal results were also obtained. The other nonconforming finiteelement schemes for Navie--Stokes problem may be found in [4,8,9,14,15,23,26]. But sofar, maximum norm error estimates for any nonconforming finite element schemes werenot considered.Recently, the quasi--optimal maximum norm …     

不可压混流驱动问题的最小二乘混合元方法

1引言考虑多孔介质中两相不可压缩可混溶渗流驱动问题,它是由一组非线性耦合的椭园型压力方程和抛物型浓度方程组成：dVV。—一山人V什）gVV却）一q,VEn,（．1）＆,,。＿．、。。—一。x）＿＋u·grade－dlv（D（u）grade）一（1－c）q－,xEn,tEJ,（1．2）＆”－－’”””‘”－”””——－’——,、—’一其中a（）一a（x,c）一是（x）／卢（c）,J一［0,Ti,DcyR‘为水平油藏区域．方程式（1．l）一（1．2）中各物理量的意义如下：广为流体压力,c为流体的浓度,u为流体的Darer速度,叶为源汇项,／一—。x（q,O）,…