A finite element model for free surface flows on fixed meshes

In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element models for such flows, is that we take advantage of the particularities of free surface flow, instead of considering it a particular case of two‐phase flow. The fact that a given free surface implies a known boundary condition on the interface, allows us to solve the Navier–Stokes equations on the fluid domain uncoupled from the solution on the rest of the finite element mesh. This, together with the use of enhanced integration allows us to model low Froude number flows accurately, something that is not possible with typical two‐phase flow models applied to free surface flow. Copyright © 2007 John Wiley & Sons, Ltd.     

Periodic solution of turbulent oscillating channel flows

The time-dependent turbulent Navier–Stokes equations are solved numerically by a finite element method with an algebraic eddy viscosity model (Baldwin–Lomax formulation) for oscillating turbulent channel flows. The method of averaging is used to analyse the resulting periodic motion of the fluid. Numerical results are obtained for various Strouhal numbers and relative amplitudes. A comparison is made between the numerical and published experimental results. It appears that for low relative amplitudes in a certain range of frequencies the agreement is satisfactory.     

Finite element and implicit Runge–Kutta implementation of an acoustics–convection upstream resolution algorithm for the time‐dependent two‐dimensional Euler equations

The second of a two‐paper series, this paper details a solver for the characteristics‐bias system from the acoustics–convection upstream resolution algorithm for the Euler and Navier–Stokes equations. An integral formulation leads to several surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi‐dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier–Stokes equations, an approximation that remains multi‐dimensional independently of the orientation of the reference axes and computational cells. The time‐dependent discrete equations are then integrated in time via an implicit Runge–Kutta procedure that in this paper is proven to remain absolutely non‐linearly stable for the spatially‐discrete Euler and Navier–Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics–convection upstream resolution algorithm generates essentially non‐oscillatory solutions for subsonic, transonic and supersonic flows, encompassing oblique‐ and interacting‐shock fields that converge within 40 time steps and reflect reference exact solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Retracted and replaced: A flow‐condition‐based interpolation finite element procedure for triangular grids

A flow‐condition‐based interpolation finite element scheme is presented for use of triangular grids in the solution of the incompressible Navier–Stokes equations. The method provides spatially isotropic discretizations for low and high Reynolds number flows. Various example solutions are given to illustrate the capabilities of the procedure. This article and been retracted and replaced. See retraction and replacement notice DOI: 10.1002/fld.1247 . Copyright © 2005 John Wiley & Sons, Ltd.     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

A marker‐and‐cell approach to free surface 2‐D multiphase flows

This work describes a methodology to simulate free surface incompressible multiphase flows. This novel methodology allows the simulation of multiphase flows with an arbitrary number of phases, each of them having different densities and viscosities. Surface and interfacial tension effects are also included. The numerical technique is based on the GENSMAC front‐tracking method. The velocity field is computed using a finite‐difference discretization of a modification of the Navier–Stokes equations. These equations together with the continuity equation are solved for the two‐dimensional multiphase flows, with different densities and viscosities in the different phases. The governing equations are solved on a regular Eulerian grid, and a Lagrangian mesh is employed to track free surfaces and interfaces. The method is validated by comparing numerical with analytic results for a number of simple problems; it was also employed to simulate complex problems for which no analytic solutions are available. The method presented in this paper has been shown to be robust and computationally efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

A study of time integration schemes for the numerical modelling of free surface flows

A semi‐discrete finite element methodology for the modelling of transient free surface flows in the context of Eulerian interface capturing is proposed. The focus of this study is put on the choice of an appropriate time integration strategy for the accurate modelling of the dynamics of free surfaces and of interfacial physics. It is composed of an adaptive time integration scheme for the Navier–Stokes equations, and of the implicit midpoint rule for the transport equation of the Eulerian marker variable. The adaptive scheme allows the automatic determination of a time‐step size that follows the physics of the problem under study, which facilitates the accurate modelling of stiff free surface flows. It is shown that the implicit midpoint rule reduces mass loss for each fluid. Various free surface flow problems are studied to verify and validate the proposed time integration strategy. Copyright © 2005 John Wiley & Sons, Ltd.     

Implementation of high‐order compact finite‐difference method to parabolized Navier–Stokes schemes

The numerical solution to the parabolized Navier–Stokes (PNS) and globally iterated PNS (IPNS) equations for accurate computation of hypersonic axisymmetric flowfields is obtained by using the fourth‐order compact finite‐difference method. The PNS and IPNS equations in the general curvilinear coordinates are solved by using the implicit finite‐difference algorithm of Beam and Warming type with a high‐order compact accuracy. A shock‐fitting procedure is utilized in both compact PNS and IPNS schemes to obtain accurate solutions in the vicinity of the shock. The main advantage of the present formulation is that the basic flow variables and their first and second derivatives are simultaneously computed with the fourth‐order accuracy. The computations are carried out for a benchmark case: hypersonic axisymmetric flow over a blunt cone at Mach 8. A sensitivity study is performed for the basic flowfield, including profiles and their derivatives obtained from the fourth‐order compact PNS and IPNS solutions, and the effects of grid size and numerical dissipation term used are discussed. The present results for the flowfield variables and also their derivatives are compared with those of other basic flow models to demonstrate the accuracy and efficiency of the proposed method. The present work represents the first known application of a high‐order compact finite‐difference method to the PNS schemes, which are computationally more efficient than Navier–Stokes solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

The calculation of incompressible separated turbulent boundary layers

The algebraic turbulent model of Baldwin and Lomax was incorporated into the incompressible full Navier–Stokes code FIDAP. This model was extensively tested in the past in finite difference codes. We believe that the incorporation of the model also into the finite element code has resulted in a practical method to compute a variety of separated turbulent 2D flows. Firstly, we use the model to compute the attached flow about an aerofoil. Next, the application of the model to separated flows is presented by computing the flows at high angles of attack up to maximum lift. It is shown that the model is capable of predicting separation, steady stall and C_Lmax. As a difficult test of the model we compute the laminar separation bubble development directly using the full Navier–Stokes finite element code. As far as we know, this approach has not yet been reported. The importance of using an appropriate upwinding is discussed. When possible, comparison of computed results with experiments is presented and the agreement is good.     

Computation of turbulent free‐surface flows around modern ships

This paper presents the calculated results for three classes of typical modern ships in modelling of ship‐generated waves. Simulations of turbulent free‐surface flows around ships are performed in a numerical water tank, based on the FINFLO‐RANS SHIP solver developed at Helsinki University of Technology. The Reynolds‐averaged Navier–Stokes (RANS) equations with the artificial compressibility and the non‐linear free‐surface boundary conditions are discretized by means of a cell‐centred finite‐volume scheme. The convergence performance is improved with the multigrid method. A free surface is tracked using a moving mesh technology, in which the non‐linear free‐surface boundary conditions are given on the actual location of the free surface. Test cases recommended are a container ship, a US Navy combatant and a tanker. The calculated results are compared with the experimental data available in the literature in terms of the wave profiles, wave pattern, and turbulent flow fields for two turbulence models, Chien's low Reynolds number k–εmodel and Baldwin–Lomax's model. Furthermore, the convergence performance, the grid refinement study and the effect of turbulence models on the waves have been investigated. Additionally, comparison of two types of the dynamic free‐surface boundary conditions is made. Copyright © 2003 John Wiley& Sons, Ltd.     

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.     

Boundary element analysis of driven cavity flow for low and moderate Reynolds numbers

A boundary element method for steady two‐dimensional low‐to‐moderate‐Reynolds number flows of incompressible fluids, using primitive variables, is presented. The velocity gradients in the Navier–Stokes equations are evaluated using the alternatives of upwind and central finite difference approximations, and derivatives of finite element shape functions. A direct iterative scheme is used to cope with the non‐linear character of the integral equations. In order to achieve convergence, an underrelaxation technique is employed at relatively high Reynolds numbers. Driven cavity flow in a square domain is considered to validate the proposed method by comparison with other published data. Copyright © 2001 John Wiley & Sons, Ltd.     

Time‐related element‐free Taylor–Galerkin method with non‐splitting decoupling process for incompressible steady flow

The time‐related element‐free Taylor–Galerkin method with non‐splitting decoupling process (EFTG‐NSD) is proposed for the simulation of steady flows. The goal of the present paper is twofold. One is to raise the efficiency of the time‐related methods for solving steady flow problems, and the other is to obtain a good stability. The EFTG‐NSD method, which uses the time‐related Navier–Stokes equations to describe steady flows, does not care about the intermediate process and obtains solution of steady flows through time marching. Different from the classical time‐related fractional step methods, the EFTG‐NSD method decouples the Navier–Stokes equations without any operator‐splitting and correction. Because the elimination of correction at each iteration step reduces the computation cost, the EFTG‐NSD method possesses higher computation efficiency. In addition, the EFTG‐NSD method has a good stability due to the use of the Taylor–Galerkin formula in time and space discretization. Furthermore, the method combining element‐free Galerkin method with Taylor–Galerkin method is an important supplement of the element‐free Galerkin method for solving flow problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Two‐level finite element method with a stabilizing subgrid for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions. The domain is discretized into a set of regular triangular elements and the finite‐dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method with a stabilizing subgrid of a single node is described, and its application to the Navier–Stokes equation is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems. The results show that the proper choice of the subgrid node is crucial in obtaining stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Copyright © 2008 John Wiley & Sons, Ltd.     

Vorticity–velocity formulation of the 3D Navier–Stokes equations in cylindrical co‐ordinates

A finite difference method is presented for solving the 3D Navier–Stokes equations in vorticity–velocity form. The method involves solving the vorticity transport equations in ‘curl‐form’ along with a set of Cauchy–Riemann type equations for the velocity. The equations are formulated in cylindrical co‐ordinates and discretized using a staggered grid arrangement. The discretized Cauchy–Riemann type equations are overdetermined and their solution is accomplished by employing a conjugate gradient method on the normal equations. The vorticity transport equations are solved in time using a semi‐implicit Crank–Nicolson/Adams–Bashforth scheme combined with a second‐order accurate spatial discretization scheme. Special emphasis is put on the treatment of the polar singularity. Numerical results of axisymmetric as well as non‐axisymmetric flows in a pipe and in a closed cylinder are presented. Comparison with measurements are carried out for the axisymmetric flow cases. Copyright © 2003 John Wiley & Sons, Ltd.     

An implicit two‐dimensional non‐hydrostatic model for free‐surface flows

An implicit finite volume model in sigma coordinate system is developed to simulate two‐dimensional (2D) vertical free surface flows, deploying a non‐hydrostatic pressure distribution. The algorithm is based on a projection method which solves the complete 2D Navier–Stokes equations in two steps. First the pressure term in the momentum equations is excluded and the resultant advection–diffusion equations are solved. In the second step the continuity and the momentum equation with only the pressure terms are solved to give a block tri‐diagonal system of equation with pressure as the unknown. This system can be solved by a direct matrix solver without iteration. A new implicit treatment of non‐hydrostatic pressure, similar to the lower layers is applied to the top layer which makes the model free of any hydrostatic pressure assumption all through the water column. This treatment enables the model to evaluate both free surface elevation and wave celerity more accurately. A series of numerical tests including free‐surface flows with significant vertical accelerations and nonlinear behaviour in shoaling zone are performed. Comparison between numerical results, analytical solutions and experimental data demonstrates a satisfactory performance. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical modelling of 3D stratified free surface flows: a case study of sediment dumping

A three‐dimensional numerical model has been developed to simulate stratified flows with free surfaces. The model is based on the Reynolds‐averaged Navier–Stokes (RANS) equations with variable fluid density. The equations are solved in a transformed σ‐coordinate system with the use of operator‐splitting method (Int. J. Numer. Meth. Fluids 2002; 38 :1045–1068). The numerical model is validated against the one‐dimensional diffusion problem and the two‐dimensional density‐gradient flow. Excellent agreements are obtained between numerical results and analytical solutions. The model is then used to study transport phenomena of dumped sediments into a water body, which has been modelled as a strongly stratified flow. For the two‐dimensional problem, the numerical results compare well with experimental data in terms of mean particle falling velocity and spreading rate of the sediment cloud for both coarse and medium‐size sediments. The model is also employed to study the dumping of sediments in a three‐dimensional environment with the presence of free surface. It is found that during the descending process an annulus‐like cloud is formed for fine sediments whereas a plate‐like cloud for medium‐size sediments. The model is proven to be a good tool to simulate strongly stratified free surface flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Three‐dimensional finite element modelling of coupled free/porous flows: applications to industrial and environmental flows

Conjunctive modelling of free/porous flows provides a powerful and cost‐effective tool for designing industrial filters used in the process industry and also for quantifying surface–subsurface flow interactions, which play a significant role in urban flooding mechanisms resulting from sea‐level rise and climate changes. A number of well‐established schemes are available in the literature for simulation of such regimes; however, three‐dimensional (3D) modelling of such flow systems still presents numerical and practical challenges. This paper presents the development of a fully 3D, transient finite element model for the prediction and quantitative analyses of the hydrodynamic behaviour encountered in industrial filtrations and environmental flows represented by coupled flows. The weak‐variational formulation in this model is based on the use of C⁰ continuous equal‐order Lagrange polynomial functions for velocity and pressure fields represented by 3D hexahedral finite elements. A mixed UVWP finite element scheme based on the standard Galerkin technique satisfying the Ladyzhenskaya–Babuska–Brezzi stability criterion through incorporation of an artificial compressibility term in the continuity equation has been employed for the solution of coupled partial differential equations. We prove that the discretization generates unified stabilization for both the Navier–Stokes and Darcy equations and preserves the geometrical flexibility of the computational grids. A direct node‐linking procedure involving the rearrangement of the global stiffness matrix for the interface elements has been developed by the authors, which is utilized to couple the governing equations in a single model. A variety of numerical tests are conducted, indicating that the model is capable of yielding theoretically expected and accurate results for free, porous and coupled free/porous problems encountered in industrial and environmental engineering problems representing complex filtration (dead‐end and cross‐flow) and interacting surface–subsurface flows. The model is computationally cost‐effective, robust, reliable and easily implementable for practical design of filtration equipments, investigation of land use for water resource availability and assessment of the impacts of climatic variations on environmental catastrophes (i.e. coastal and urban floods). The model developed in this work results from the extension of a multi‐disciplinary project (AEROFIL) primarily sponsored by the European aerospace industries for development of a computer simulation package (Aircraft Cartridge Filter Analysis Modelling Program), which was successfully utilized and deployed for designing hydraulic dead‐end filters used in Airbus A380.Copyright © 2012 John Wiley & Sons, Ltd.     

Evaluation of an energy relaxation method for the simulation of unsteady,viscous, real gas flows

This study investigates a new energy relaxation method designed to capture the dynamics of unsteady, viscous, real gas flows governed by the compressible Navier–Stokes equations. We focus on real gas models accounting for inelastic molecular collisions and yielding temperature‐dependent heat capacities. The relaxed Navier–Stokes equations are discretized using a mixed finite volume/finite element method and a high‐order time integration scheme. The accuracy of the energy relaxation method is investigated on three test problems of increasing complexity: the advection of a periodic set of vortices, the interaction of a temperature spot with a weak shock, and finally, the interaction of a reflected shock with its trailing boundary layer in a shock tube. In all cases, the method is validated against benchmark solutions and the numerical errors resulting from both discretization and energy relaxation are assessed independently. Copyright © 2004 John Wiley & Sons, Ltd.