Comparison of spectral and finite element methods applied to the study of the core‐annular flow in an undulating tube

A Galerkin/finite element and a pseudo‐spectral method, in conjunction with the primitive (velocity‐pressure) and streamfunction‐vorticity formulations, are tested for solving the two‐phase flow in a tube, which has a periodically varying, circular cross section. Two immiscible, incompressible, Newtonian fluids are arranged so that one of them is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). The physical and flow parameters are such that the interface between the two fluids remains continuous and single‐valued. This arrangement is usually referred to as Core‐Annular flow. A non‐orthogonal mapping is used to transform the uneven tube shape and the unknown, time dependent interface to fixed, cylindrical surfaces. With both methods and formulations, steady states are calculated first using the Newton–Raphson method. The most dangerous eigenvalues of the related linear stability problem are calculated using the Arnoldi method, and dynamic simulations are carried out using the implicit Euler method. It is shown that with a smooth tube shape the pseudo‐spectral method exhibits exponential convergence, whereas the finite element method exhibits algebraic convergence, albeit of higher order than expected from the relevant theory. Thus the former method, especially when coupled with the streamfunction‐vorticity formulation, is much more efficient. The finite element method becomes more advantageous when the tube shape contains a cusp, in which case the convergence rate of the pseudo‐spectral method deteriorates exhibiting algebraic convergence with the number of the axial spectral modes, whereas the convergence rate of the finite element method remains unaffected. Copyright © 2002 John Wiley & Sons, Ltd.     

Three‐dimensional boundary element analysis of drop deformation in confined flow for Newtonian and viscoelastic systems

An adaptive (Lagrangian) boundary element approach is proposed for the general three‐dimensional drop deformation in confined flow. The adaptive method is stable as it includes remeshing capabilities of the deforming interface between drop and suspending fluid, and thus can handle large deformations. Both drop and surrounding fluid are viscous incompressible and can be Newtonian or viscoelastic. A boundary‐only formulation is implemented for fluids obeying the linear Jeffrey's constitutive equation. Similarly to the formulation for two‐dimensional Newtonian fluids (Khayat RE, Luciani A, Utracki LA. Boundary element analysis of planar drop deformation in confined flow. Part I. Newtonian fluids. Engineering Analysis of Boundary Elements 1997; 19 : 279), the method requires the solution of two simultaneous integral equations on the interface between the two fluids and the confining solid boundary. Although the problem is formulated for any confining geometry, the method is illustrated for a deforming drop as it is driven by the ambient flow inside a cylindrical tube. The accuracy of the method is assessed by comparison with the analytical solution for two‐phase radial spherical flow, leading to good agreement. The influence of mesh refinement is examined for a drop in simple shear flow. Copyright © 2000 John Wiley & Sons, Ltd.     

Weakly-imposed Dirichlet boundary conditions for non-Newtonian fluid flow

We propose a new formulation for weakly imposing Dirichlet boundary conditions in non-Newtonian fluid flow. It is based on the Gerstenberger–Wall formulation for Newtonian fluids [1], but extended to non-Newtonian fluids. It uses a stabilization term in the weak form that is independent from the actual fluid model used, except for an adjustable parameter κ, having the physical dimension of a viscosity. The new formulation is tested, combined with an extended finite element method, for the flow past a cylinder between two walls using both a generalized Newtonian and a viscoelastic fluid. It is shown that the convergence is optimal for the generalized Newtonian fluid by comparing with a converged boundary-fitted solution using traditional strong boundary conditions. Also the solution of the viscoelastic fluid compares very well with a traditional solution using a boundary-fitted mesh and strong Dirichlet boundary conditions. For both fluid models we also test various values of the κ parameter and it turns out that a value equal to the zero-shear-viscosity gives good results. But, it is also shown that a wide range of κ values can be chosen without sacrificing accuracy.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

A Hybrid Boundary/Finite Element Method for Simulating Viscous Flows and Shapes of Droplets in Electric Fields

A mixed boundary element and finite element numerical algorithm for the simultaneous prediction of the electric fields, viscous flow fields, thermal fields and surface deformation of electrically conducting droplets in an electrostatic field is described in this paper. The boundary element method is used for the computation of the electric potential distribution. This allows the boundary conditions at infinity to be directly incorporated into the boundary integral formulation, thereby obviating the need for discretization at infinity. The surface deformation is determined by solving the normal stress balance equation using the weighted residuals method. The fluid flow and thermal fields are calculated using the mixed finite element method. The computational algorithm for the simultaneous prediction of surface deformation and fluid flow involves two iterative loops, one for the electric field and surface deformation and the other for the surface tension driven viscous flows. The two loops are coupled through the droplet surface shapes for viscous fluid flow calculations and viscous stresses for updating the droplet shapes. Computing the surface deformation in a separate loop permits the freedom of applying different types of elements without complicating procedures for the internal flow and thermal calculations. Tests indicate that the quadratic, cubic spline and spectral boundary elements all give approximately the same accuracy for free surface calculations; however, the quadratic elements are preferred as they are easier to implement and also require less computing time. Linear elements, however, are less accurate. Numerical simulations are carried out for the simultaneous solution of free surface shapes and internal fluid flow and temperature distributions in droplets in electric fields under both microgravity and earthbound conditions. Results show that laser heating may induce a non-uniform temperature distribution in the droplets. This non-uniform thermal field results in a variation of surface tension along the surface of the droplet, which in turn produces a recirculating fluid flow in the droplet. The viscous stresses cause additional surface deformation by squeezing the surface areas above and below the equator plane.     

Parallelization of a vorticity formulation for the analysis of incompressible viscous fluid flows

A parallel computer implementation of a vorticity formulation for the analysis of incompressible viscous fluid flow problems is presented. The vorticity formulation involves a three‐step process, two kinematic steps followed by a kinetic step. The first kinematic step determines vortex sheet strengths along the boundary of the domain from a Galerkin implementation of the generalized Helmholtz decomposition. The vortex sheet strengths are related to the vorticity flux boundary conditions. The second kinematic step determines the interior velocity field from the regular form of the generalized Helmholtz decomposition. The third kinetic step solves the vorticity equation using a Galerkin finite element method with boundary conditions determined in the first step and velocities determined in the second step. The accuracy of the numerical algorithm is demonstrated through the driven‐cavity problem and the 2‐D cylinder in a free‐stream problem, which represent both internal and external flows. Each of the three steps requires a unique parallelization effort, which are evaluated in terms of parallel efficiency. Copyright © 2002 John Wiley & Sons, Ltd.     

Transient deformation of an inviscid inclusion in a viscoelastic extensional flow

The transient deformation of a bubble in a viscoelastic extentional flow is analyzed by means of a finite element algorithm for viscoelastic moving boundary problems. Using the Oldroyd-B constitutive model, we find that bubbles in a viscoelastic fluid deform to the same steady-state configurations as bubbles in a Newtonian fluid at equal values of the far-field extensional stresses (corresponding to different stretch rates). Vapor bubbles in a developed extensional flow collapse more readily in the viscoelastic liquid than bubbles in Newtonian fluids because of the large compressive stresses associated with the viscoelastic liquid.     

A stress‐based least‐squares finite‐element model for incompressible Navier–Stokes equations

In this paper we present a stress‐based least‐squares finite‐element formulation for the solution of the Navier–Stokes equations governing flows of viscous incompressible fluids. Stress components are introduced as independent variables to make the system first order. Continuity equation becomes an algebraic equation and is eliminated from the system with suitable modifications. The h and p convergence are verified using the exact solution of Kovasznay flow. Steady flow past a large circular cylinder in a channel is solved to test mass conservation. Transient flow over a backward‐facing step problem is solved on several meshes. Results are compared with that obtained using vorticity‐based first‐order formulation for both benchmark problems. Copyright © 2007 John Wiley & Sons, Ltd.     

A method for solving the factorized vorticity-stream function equations by finite elements

A new finite element method for solving the time-dependent incompressible Navier-Stokes equations with general boundary conditions is presented. The two second-order partial differential equations for the vorticity and the stream function are factorized, apart from the non-linear advection term, by eliminating the coupling due to the double specification on the stream function at (a part of) the boundary. This is achieved by reducing the no-slip boundary conditions to projection integral conditions for the vorticity field and by evaluating the relevant quantities involved according to an extension of the method of Glowinski and Pironneau for the biharmonic problem. Time integration schemes and iterative algorithms are introduced which require the solution only of banded linear systems of symmetric type. The proposed finite element formulation is compared with its finite difference equivalent by means of a few numerical examples. The results obtained using 4-noded bilinear elements provide an illustration of the superiority of the finite element based spatial discretization.     

Simultaneous solution of the Navier-Stokes and elastic membrane equations by a finite element method

Fluid flow through a significantly compressed elastic tube occurs in a variety of physiological situations. Laboratory experiments investigating such flows through finite lengths of tube mounted between rigid supports have demonstrated that the system is one of great dynamical complexity, displaying a rich variety of self-excited oscillations. The physical mechanisms responsible for the onset of such oscillations are not yet fully understood, but simplified models indicate that energy loss by flow separation, variation in longitudinal wall tension and propagation of fluid elastic pressure waves may all be important. Direct numerical solution of the highly non-linear equations governing even the most simplified two-dimensional models aimed at capturing these basic features requires that both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. To accomplish this, previous algorithms have decoupled the solid and fluid mechanics, solving for each separately and converging iteratively on a solution which satisfies both. This paper describes a finite element technique which solves the incompressible Navier-Stokes equatikons simultaneously with the elastic membrane equations on the flexible boundary. The elastic boundary position is parametized in terms of distances along spines in a manner similar to that which has been used successfully in studies of viscous free surface flows, but here the membrane curvature equation rather than the kinematic boundary condition of vanishing normal velocity is used to determine these diatances and the membrane tension varies with the shear stresses exerted on it by the fluid motions. Bothy the grid and the spine positions adjust in response to membrane deformation, and the coupled fluid and elastic equations are solved by a Newton-Raphson scheme which displays quadratic convergence down to low membrane tensions and extreme states of collapse. Solutions to the steady problem are discussed, along with an indication of how the time-dependent problem might be approached.     

Fundamental solutions of the streamfunction–vorticity formulation of the Navier–Stokes equations

A new boundary element procedure is developed for the solution of the streamfunction–vorticity formulation of the Navier–Stokes equations in two dimensions. The differential equations are stated in their transient version and then discretized via finite differences with respect to time. In this discretization, the non-linear inertial terms are evaluated in a previous time step, thus making the scheme explicit with respect to them. In the resulting discretized equations, fundamental solutions that take into account the coupling between the equations are developed by treating the non-linear terms as in homogeneities. The resulting boundary integral equations are solved by the regular boundary element method, in which the singular points are placed outside the solution domain.     

A simple and efficient BEM algorithm for planar cavity flows

This paper presents a boundary element formulation for solution of planar Riabouchinsky cavity flow problems. An iterative procedure for adjusting the free surface position is developed and shown to be stable and convergent. Numerical results are compared with finite difference and finite element solutions, showing the superior accuracy of the BEM models.     

Finite element solutions of axisymmetric Euler equations for an incompressible and inviscid fluid

In this paper we present a finite element method for the numerical solution of axisymmetric flows. The governing equations of the flow are the axisymmetric Euler equations. We use a streamfunction angular velocity and vorticity formulation of these equations, and we consider the non-stationary and the stationary problems. For industrial applications we have developed a general model which computes the flow past an annular aerofoil and a duct propeller. It is able to take into account jumps of angular velocity and vorticiy in order to model the flow in the presence of a propeller. Moreover, we compute the complete flow around the after-body of a ship and the interaction between a ducted propeller and the stern. In the stationary case we have developed a simple and efficient version of the characteristics/finite element method. Numerical tests have shown that this last method leads to a very fast solver for the Euler equations. The numerical results are in good agreement with experimental data.     

A sharp interface Cartesian grid method for viscous simulation of shocked particle-laden flows

A Cartesian grid-based sharp interface method is presented for viscous simulations of shocked particle-laden flows. The moving solid–fluid interfaces are represented using level sets. A moving least-squares reconstruction is developed to apply the no-slip boundary condition at solid–fluid interfaces and to supply viscous stresses to the fluid. The algorithms developed in this paper are benchmarked against similarity solutions for the boundary layer over a fixed flat plate and against numerical solutions for moving interface problems such as shock-induced lift-off of a cylinder in a channel. The framework is extended to 3D and applied to calculate low Reynolds number steady supersonic flow over a sphere. Viscous simulation of the interaction of a particle cloud with an incident planar shock is demonstrated; the average drag on the particles and the vorticity field in the cloud are compared to the inviscid case to elucidate the effects of viscosity on momentum transfer between the particle and fluid phases. The methods developed will be useful for obtaining accurate momentum and heat transfer closure models for macro-scale shocked particulate flow applications such as blast waves and dust explosions.     

FEM‐BEM coupling for the exterior Stokes problem with non‐conforming finite elements and an application to small droplet deformation dynamics

A non‐conforming, discontinuous Galerkin finite element–boundary element coupling procedure is presented for the exterior planar Stokes problem. The novel coupled formulation is developed using that for the conforming case as a guide to the introduction of extra mortar variables used to couple a discontinuous interior finite element solution with a continuous exterior boundary element solution. Convergence results for the new scheme are presented, for a range of different interior penalties, on computational domains discretized with regular structured meshes. To illustrate an application, the excitations required to model two‐phase droplet deformations in an extensional flow, under simple surface tension, with the new scheme are also presented. For a selection of different drop viscocities and exterior flows, with and without a rotational component, the progression to a steady‐state deformation of initially undeformed circular drops is calculated and the results compared with those from both a conforming FEM‐BEM equivalent scheme and from a small perturbation analysis where available. Copyright © 2011 John Wiley & Sons, Ltd.     

Investigation of the flow and heat transfer of viscous reacting fluids in long tubes

Heat transfer and resistance in the case of laminar flow of inert gases and liquids in a circular tube were considered in [1–4], the justification of the use of boundary-layer type equations for investigating two-dimensional flows in tubes being provided in [4]. The flow of strongly viscous, chemically reacting fluids in an infinite tube has been investigated analytically and numerically in the case of a constant pressure gradient or constant flow rate of the fluid [5–8]. An analytic analysis of the flow of viscous reacting fluids in tubes of finite length was made in [9, 10]. However, by virtue of the averaging of the unknown functions over the volume of the tube in these investigations, the allowance for the finite length of the tube reduced to an analysis of the influence of the time the fluid remains in the tube on the thermal regime of the flow, and the details of the flow and the heat transfer in the initial section of the tube were not taken into account. In [11], the development of chemical reactions in displacement reactors were studied under the condition that a Poiseuille velocity profile is realized and the viscosity does not depend on the temperature or the concentration of the reactant; in [12], a study was made of the regimes of an adiabatic reactor of finite length, and in [13] of the flow regimes of reacting fluids in long tubes in the case of a constant flow rate. The aim of the present paper is to analyze analytically and numerically in the two-dimensional formulation the approach to the regimes of thermal and hydrodynamic stabilization in the case of the flow of viscous inert fluids and details of the flow of strongly viscous reacting fluids.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 17–25, January–February, 1930.     

Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows

A hybrid finite volume/element method is analysed through the computation of creeping flows of viscoelastic fluids in plane 4:1 sharp and rounded-corner contraction geometries. Simulations are presented for three models: a constant viscosity Oldroyd-B fluid, and Phan-Thien/Tanner (PTT) shear thinning fluids of exponential and linear approximation form. A Taylor–Galerkin/pressure-correction scheme is implemented as the base time-stepping framework. The momentum equations are solved by a finite element method, whilst the constitutive equations are solved by a finite volume approach. Mesh convergence is analysed via refinement around the contraction to capture boundary layers and flow structure. Pressure drop is shown to increase with flow rate for a fixed fluid. For the Oldroyd-B model, singular behaviour is reported in the main stress component as one approaches the corner in the rounded, as with the sharp geometry. Velocity components display an asymptotic trend with a positive slope. Higher values of Weissenberg numbers (We) are reached with these finite volume schemes compared to their finite element counterparts, attributing this to superior accuracy properties.     

A three-dimensional fictitious domain method for incompressible fluid flow problems

A new Galerkin finite element method for the solution of the Navier–Stokes equations in enclosures containing internal parts which may be moving is presented. Dubbed the virtual finite element method, it is based upon optimization techniques and belongs to the class of fictitious domain methods. Only one volumetric mesh representing the enclosure without its internal parts needs to be generated. These are rather discretized using control points on which kinematic constraints are enforced and introduced into the mathematical formulation by means of Lagrange multipliers. Consequently, the meshing of the computational domain is much easier than with classical finite element approaches. First, the methodology will be presented in detail. It will then be validated in the case of the two-dimensional Couette cylinder problem for which an analytical solution is available. Finally, the three-dimensional fluid flow inside a mechanically agitated vessel will be investigated. The accuracy of the numerical results will be assessed through a comparison with experimental data and results obtained with a standard finite element method. © 1997 John Wiley & Sons, Ltd.     

Extremum problems of boundary control for steady equations of thermal convection

An inverse extremum problem of boundary control for steady equations of thermal convection is considered. The cost functional in this problem is chosen to be the root-mean-square deviation of flow velocity or vorticity from the velocity or vorticity field given in a certain part of the flow domain; the control parameter is the heat flux through a part of the boundary. A theorem on sufficient conditions on initial data providing the existence, uniqueness, and stability of the solution is given. A numerical algorithm of solving this problem, based on Newton’s method and on the finite element method of discretization of linear boundary-value problems, is proposed. Results of computational experiments on solving extremum problems, which confirm the efficiency of the method developed, are discussed.