# Showing papers in "International Journal for Numerical Methods in Fluids in 1994"

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TL;DR: In this paper, the authors have pointed out that many Riemann solvers contain subtle flaws that can cause spurious solutions to be computed, and identified one mechanism that might thwart attempts to produce very high-resolution simulations.

Abstract: The aims of this paper are threefold: to increase the level of awareness within the shock-capturing community of the fact that many Godunov-type methods contain subtle flaws that can cause spurious solutions to be computed; to identify one mechanism that might thwart attempts to produce very-high-resolution simulations; and to proffer a simple strategy for overcoming the specific failings of individual Riemann solvers.

740 citations

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TL;DR: In this paper, the authors used the control volume method to solve the conservation equations for laminar and turbulent flows for a series of Rayleigh numbers (Ra) reaching values up to 1010.

Abstract: Numerical simulations have been undertaken for the benchmark problem of natural convection flow in a square cavity. The control volume method is used to solve the conservation equations for laminar and turbulent flows for a series of Rayleigh numbers (Ra) reaching values up to 1010. The k-ϵ model has been used for turbulence modelling with and without logarithmic wall functions. Uniform and non-uniform (stretched) grids have been employed with increasing density to guarantee accurate solutions, especially near the walls for high Ra-values. ADI and SIP solvers are implemented to accelerate convergence. Excellent agreement is obtained with previous numerical solutions, while some discrepancies with others for high Ra-values may be due to a possibly different implementation of the wall functions. Comparisons with experimental data for heat transfer (Nusselt number) clearly demonstrates the limitations of the standard k-ϵ model with logarithmic wall functions, which gives significant overpredictions.

385 citations

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TL;DR: The two minisymposia on OBCs that are summarized in this paper had the objective of finding the best OBC's for a small subset of two-dimensional test problems, but the contributions obtained probably raised many more questions/issues than were resolved.

Abstract: The incompressible Navier-Stokes equations—and their thermal convection and stratified flow analogue, the Boussinesq equations—possess solutions in bounded domains only when appropriate/legitimate boundary conditions (BCs) are appended at all points on the domain boundary. When the boundary—or, more commonly, a portion of it—is not endowed with a Dirichlet BC, we are faced with selecting what are called open boundary conditions (OBCs), because the fluid may presumably enter or leave the domain through such boundaries. The two minisymposia on OBCs that are summarized in this paper had the objective of finding the best OBCs for a small subset of two-dimensional test problems. This objective, which of course is not really well-defined, was not met (we believe), but the contributions obtained probably raised many more questions/issues than were resolved—notable among them being the advent of a new class of OBCs that we call FBCs (fuzzy boundary conditions).

235 citations

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TL;DR: In this article, the authors proposed a finite volume method for the simulation of free surface flow and transport in rivers, estuaries and seas using sigma-transformed grids.

Abstract: Nowadays the simulation of free surface flow and transport in rivers, estuaries and seas is often based upon three-dimensional modelling systems. Most of these three-dimensional modelling systems use sigma co-ordinates in the vertical. By the use of the sigma transformation the water column can be divided into the same number of layers independently of the water depth. Especially for steep bottom slopes combined with vertical stratification of the density, sigma-transformed grids impose numerical problems for the accurate approximation of horizontal gradients. This paper deals with algorithms for the approximation in sigma co-ordinates of the horizontal diffusive fluxes of temperature and salinity and for the approximation of the horizontal pressure gradients. The approximation of the horizontal diffusive fluxes is based upon a finite volume method. The approximation of the pressure gradients is directly related to the approximation of the diffusive fluxes. Artificial vertical diffusion and artificial flow due to truncation errors are minimized. The method described in this paper is not hampered by the so-called ‘hydrostatic consistency condition’. This will be illustrated by numerical experiments.

200 citations

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TL;DR: In this article, an analytical solution to the incompressible Navier-Stokes equations is presented, which is fully three-dimensional vector solutions involving all three Cartesian velocity components, each of which depends non-trivially on all three co-ordinate directions.

Abstract: SUMMARY Unsteady analytical solutions to the incompressible Navier-Stokes equations are presented. They are fully three-dimensional vector solutions involving all three Cartesian velocity components, each of which depends non-trivially on all three co-ordinate directions. Although unlikely to be physically realized, they are well suited for benchmarking, testing and validation of three-dimensional incompressible Navier-Stokes solvers. The use of such a solution for benchmarking purposes is described.

195 citations

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TL;DR: In this paper, a monotonic scheme for the approximation of steady scalar transport is formulated and implemented within a collocated finite-volume/pressure-correction algorithm for general turbulent flows in complex geometries.

Abstract: A new monotonic scheme for the approximation of steady scalar transport is formulated and implemented within a collocated finite-volume/pressure-correction algorithm for general turbulent flows in complex geometries. The scheme is essentially a monotonic implementation of the quadratic QUICK interpolation and uses a continuous and compact limiter to secure monotonicity. The principal purpose is to allow an accurate and fully bounded, hence stable, approximation of turbulence convection in the context of two-equation eddy viscosity and Reynolds stress transport modelling of two- and three-dimensional flows, both subsonic and transonic. Among other benefits, this capability permits an assessment to be made of the adequacy of approximating turbulence convection with first-order upwind schemes in conjunction with higher-order formulations for mean-flow properties—a widespread practice. The performance characteristics of the bounded scheme are illustrated by reference to computations for scalar transport, for a transonic flow in a Laval nozzle, for one separated laminar flow and for two separated turbulent flows computed with a non-linear RNG model and full Reynolds stress closure.

191 citations

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TL;DR: In this article, a numerical simulation is presented to predict the free surface and its interactions with heat transfer and cure for flow of a shear-thinning resin through the fibre preform.

Abstract: A numerical simulation is presented to predict the free surface and its interactions with heat transfer and cure for flow of a shear-thinning resin through the fibre preform the flow part of the simulation is based on the finite element/control volume method. Since the traditional control volume approach produces an error associated with a mass balance inconsistency, a new method which overcomes this issue is proposed, the element control volume method.
The heat transfer and cure analysis in the simulation are based on the finite difference/control volume method. Since heat conduction is dominant in the through-thickness direction and most of the heat convection is in-plane, heat transfer and cure are solved in fully three-dimensional form. A simple concept of the boundary condition constant is introduced which models a realistic mould configuration with a heating element located at a distance behind the mould wall. The varying viscosity throughout the mould associated with the strain rate, temperature and degree of cure distribution may be accounted for in calculating the mould-filling pattern. This introduces a two-way coupling between momentum and energy transport in fibrous media during mould filling.

174 citations

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TL;DR: In this article, the open boundary conditions for the incompressible Navier-Stokes equations are given from a weak formulation in velocity-pressure variables, and some natural boundary conditions involving the traction or pseudotraction and inertial terms are established.

Abstract: SUMMARY The aim of this paper is to give open boundary conditions for the incompressible Navier-Stokes equations. From a weak formulation in velocity-pressure variables, some natural boundary conditions involving the traction or pseudotraction and inertial terms are established. Numerical experiments on the flow behind a cylinder show the efficiency of these conditions, which convey properly the vortices downstream. Comparisons with other boundary conditions for the velocity and pressure are also performed.

121 citations

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TL;DR: In this article, a p-version least square finite element formulation (LSFEF) was proposed for non-Newtonian fluid flow under isothermal and non-isothermal conditions.

Abstract: This paper presents a p- version least squares finite element formulation (LSFEF) for two-dimensional, incompressible, non-Newtonian fluid flow under isothermal and non-isothermal conditions. The dimensionless forms of the diffential equations describing the fluid motion and heat transfer are cast into a set of first-order differential equations using non-Newtonian stresses and heat fluxes as auxiliary variables. The velocities, pressure and temperature as well as the stresses and heat fluxes are interpolated using equal-order, C0-continuous, p-version hierarchical approximation functions. The application of least squares minimization to the set of coupled first-order non-linear partial differential equations results in finding a solution vector {δ} which makes the partial derivatives of the error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search.
The paper presents the implementation of a power-law model for the non-Newtonian Viscosity. For the non-isothermal case the fluid properties are considered to be a function of temperature. Three numerical examples (fully developed flow between parallel plates, symmetric sudden expansion and lid-driven cavity) are presented for isothermal power-law fluid flow. The Couette shear flow problem and the 4:1 symmetric sudden expansion are used to present numerical results for non-isothermal power-law fluid flow. The numerical examples demonstrate the convergence characteristics and accuracy of the formulation.

108 citations

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TL;DR: In this paper, a new characteristic-based method for the solution of the 2D laminar incompressible Navier-Stokes equations is presented, where the primitives variables (pressure and velocity components) are defined as functions of their values on the characteristics.

Abstract: 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.

105 citations

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TL;DR: An efficient numerical method is presented for solving the equations of motion for viscous fluids that is almost mesh-independent and the scaling of computing time with mesh size is close to the optimum.

Abstract: SUMMARY An efficient numerical method is presented for solving the equations of motion for viscous fluids. The equations are discretized on the basis of unstructured finite element meshes and then solved by direct iteration. Advective fluxes are temporarily fixed at each iteration to provide a linearized set of coupled equations which are then also solved by iteration using a fully implicit algebraic multigrid (AMG) scheme, A rapid convergence to machine accuracy is achieved that is almost mesh-independent. The scaling of computing time with mesh size is therefore close to the optimum.

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TL;DR: The results indicate that multigrid with smoothing based on incomplete factorization is more efficient than the other methods, but typically by no more than a factor of two.

Abstract: SUMMARY Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizatiom a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the dkretization. In this paper we compare the performance of four such methods, namely variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual and multigrid methods, for solving several two-dimensional model problems. The results indicate that multigrid with smoothing based on incomplete factorization is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantage of being independent of iteration parameters.

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TL;DR: In this paper, a fourth-order-accurate pressure gradient calculation has been implemented and installed in SPEM, a three-dimensional primitive equation ocean model, in order to minimize these pressure gradient errors.

Abstract: In stratified three-dimensional models the use of a boundary-fitted vertical co-ordinate is known to produce errors in the horizontal pressure gradient calculation near steep topography. The error is due to the splitting of the horizontal pressure gradient term in each of the momentum equations into two parts and the subsequent incomplete cancellation of the truncation errors of those parts. In order to minimize these pressure gradient errors, a fourth-order-accurate pressure gradient calculation has been implemented and installed in SPEM, a three-dimensional primitive equation ocean model. The stability and accuracy of the new scheme are compared with those of the original second-order-accurate model in a series of calculations of unforced flow in the vicinity of an isolated seamount. The new scheme is shown to have much smaller pressure gradient errors over a wide range of parameter space as well as a greater parametric domain of numerical stability.

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TL;DR: In this article, two approaches which employ the finite element method to solve for large-scale coupled, incompressible flows through adjacent porous and open domains are developed and evaluated in a model for the spontaneous ignition of coal stockpiles.

Abstract: SUMMARY Two approaches which employ the finite element method to solve for large-scale, coupled, incompressible flows through adjacent porous and open domains are developed and evaluated in a model for the spontaneous ignition of coal stockpiles. Both formulations employ the Navier-Stokes equations do describe flow in the open region; two different descriptions, Darcy's law and the Brinkman equation, are employed to model flows within the porous region. The formulation which uses Darcy's law employs the BeaversJoseph slip condition and a novel implementation of the interfacial conditions. The other approach invokes the Brinkman equation: this considerably simplifies the implementation of matching conditions at the interface between the porous and open fluid domains, but also results in velocity boundary layers in the porous region adjacent to this interface which can be difficult to resolve numerically. A direct comparison of model results shows that the Darcy-slip formulation produces solutions which are more accurate and more economical to compute than those obtained using the Brinkman formulation.

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TL;DR: In this article, a control-volume-based finite element method (CVFEM) for axisymmetric, two-dimensional, incompressible fluid flow and heat transfer in irregular-shaped domains is presented.

Abstract: The formulation of a control-volume-based finite element method (CVFEM) for axisymmetric, two-dimensional, incompressible fluid flow and heat transfer in irregular-shaped domains is presented. The calculation domain is discretized into torus-shaped elements and control volumes. In a longitudinal cross-sectional plane, these elements are three-node triangles, and the control volumes are polygons obtained by joining the centroids of the three-node triangles to the mid-points of the sides. Two different interpolation schemes are proposed for the scalar-dependent variables in the advection terms: a flow-oriented upwind function, and a mass-weighted upwind function that guarantees that the discretized advection terms contribute positively to the coefficients in the discretized equations. In the discretization of diffusion transport terms, the dependent variables are interpolated linearly. An iterative sequential variable adjustment algorithm is used to solve the discretized equations for the velocity components, pressure and other scalar-dependent variables of interest. The capabilities of the proposed CVFEM are demonstrated by its application to four different example problems. The numerical solutions are compared with the results of independent numerical and experimental investigations. These comparisons are quite encouraging.

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TL;DR: In this article, a finite element technique was used to solve the incompressible Navier-Stokes equatikons simultaneously with the elastic membrane equations on the flexible boundary, and the coupled fluid and elastic equations were solved by a Newton-Raphson scheme which displays quadratic convergence down to low membrane tensions and extreme states of collapse.

Abstract: 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.

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TL;DR: The multigrid strategy presented here is adapted from the generalized finite volume agglomerationMultigrid algorithm developed recently for the solution of the Euler equations, and focuses on Poisson's equation.

Abstract: We are interested in solving second-order PDEs with multigrid and unstructured meshes. The multigrid strategy we present here is adapted from the generalized finite volume agglomeration multigrid algorithm we have developed recently for the solution of the Euler equations. We now focus on Poisson's equation. A strategy is defined by introducing a correction factor for the diffusive terms, and some illustrating results are given.

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TL;DR: In this paper, the Navier-Stokes equations are cast as a set of first-order equations involving viscous stresses as auxiliary variables and the primary and auxiliary variables are interpolated using equal-order C0 continuity, p-version hierarchical approximation functions.

Abstract: A p-version least squares finite element formulation for non-linear problems is applied to the problem of steady, two-dimensional, incompressible fluid flow. The Navier-Stokes equations are cast as a set of first-order equations involving viscous stresses as auxiliary variables. Both the primary and auxiliary variables are interpolated using equal-order C0 continuity, p-version hierarchical approximation functions. The least squares functional (or error functional) is constructed using the system of coupled first-order non-linear partial differential equations without linearization, approximations or assumptions. The minimization of this least squares error functional results in finding a solution vector {δ} for which the partial derivative of the error functional (integrated sum of squares of the errors resulting from individual equations for the entire discretization) with respect to the nodal degrees of freedom {δ} becomes zero. This is accomplished by using Newton's method with a line search. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.

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TL;DR: In this paper, the magnetohydrodynamic flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with an external magnetic field applied transverse to the flow has been investigated.

Abstract: SUMMARY The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with an external magnetic field applied transverse to the flow has been investigated. The walls parallel to the applied magnetic field are conducting while the other two walls which are perpendicular to the field are insulators. The boundary element method (BEM) with constant elements has been used to cast the problem into the form of an integral equation over the boundary and to obtain a system of algebraic equations for the boundary unknown values only. The solution of this integral equation presents no problem as encountered in the solution of the singular integral equations for interior methods. Computations have been carried out for several values of the Hartmann number (1 < M < 10). It is found that as M increases, boundary layers are formed close to the insulated boundaries for both the velocity and the induced magnetic field and in the central part their behaviours are uniform. Selected graphs are given showing the behaviours of the velocity and the induced magnetic field.

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TL;DR: In this paper, the Lagrange-Galerkin mixed finite element approximation of the NavierStokes and continuity equations is considered and the iterative solution of such a system, compared with the performance of the one-level preconditioned conjugate residual method for indefinite matrices with that of a more traditional two-level pressure correction approach.

Abstract: SUMMARY The linear system arising from a Lagrange-Galerkin mixed finite element approximation of the NavierStokes and continuity equations is symmetric indefinite and has the same block structure as a system arising from a mixed finite element discretization of a Stokes problem. This paper considers the iterative solution of such a system, comparing the performance of the one-level preconditioned conjugate residual method for indefinite matrices with that of a more traditional two-level pressure correction approach. Asymptotic estimates for the amount of work involved in each method are given together with the results of related numerical experiments.

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TL;DR: The essential features of the finite element side-based scheme and the ID TVD approach are described and their numerical implementation is discussed and the solutions of some inviscid flows, obtained by advancing explicitly in time, are presented.

Abstract: The Galerkin finite element method is used as the basis for the construction of schemes for the solution of the two-dimensional compressible Euler equations on unstructured triangular grids. The use of a side-based data structure readily allows for the construction of a local (structured) stencil and the incorporation of a high-resolution shock-capturing method formulated within the TVD concept. The essential features of the finite element side-based scheme and the ID TVD approach are described and their numerical implementation is discussed. The choice of limiters and the support for their computation are analysed and the solutions of some inviscid flows, obtained by advancing explicitly in time, are presented.

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TL;DR: In this article, a numerical method for solving the free surface flow around a ship at forward speed in calm water is presented, where the fluid is assumed to be Newtonian and the Reynolds-averaged Navier-Stokes equations are solved by a finite difference method.

Abstract: We present here a numerical method for solving the free surface flow around a ship at forward speed in calm water. The fluid is assumed to be Newtonian and the Reynolds-averaged Navier-Stokes equations are solved by a finite difference method. Modelization of turbulence is achieved by the algebraic model proposed by Baldwin and Lomax. Fully non-linear free surface conditions are satisfied in the model and a method to avoid the incompatibility between free surface conditions and no-slip conditions at the waterline is proposed. Numerical results obtained for a Wigley hull are compared with experimental results

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TL;DR: In this article, the authors describe techniques for discretizing problems with horizontal wells in a three-dimensional-tetrahedral mesh, and the effectiveness of nonlinear flux limiters for reducing numerical dispersion is discussed.

Abstract: Numerical simulation of steam flush for clean-up of non-aqueous phase liquid (NAPL) contaminated groundwater sites involves solution of the multiphase, multicomponent subsurface flow equations. This paper describes techniques for discretizing problems with horizontal wells in a three-dimensiontetrahedral mesh. The effectiveness of non-linear flux limiters for reducing numerical dispersion is discussed. Primary variable selection and thermodynamic state transition rules will also be compared. Some example results for several steam flush scenarios will be presented.

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TL;DR: In this article, the Navier-Stokes equations and two turbulence equations are solved in a loosely coupled manner using a multigrid strategy on unstructured meshes, and a variety of aerodynamic flows are solved for, initializing all quantities with uniform freestream values.

Abstract: The system of equations consisting of the full Navier-Stokes equations and two turbulence equations was solved for in the steady state using a multigrid strategy on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multistage Runge-Kutta time stepping scheme with a stability bound local time step, while the turbulence equations are advanced in a point-implicit scheme with a time step which guarantees stability and positively. Low Reynolds number modifications to the original two equation model are incorporated in a manner which results in well behaved equations for arbitrarily small wall distances. A variety of aerodynamic flows are solved for, initializing all quantities with uniform freestream values, and resulting in rapid and uniform convergence rates for the flow and turbulence equations.

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TL;DR: In this paper, the transient development of cooling-induced flow in a triangular domain filled with water is studied by means of numerical simulation as a model for flow developing in the littoral region of lakes or coasts.

Abstract: The transient development of cooling-induced flow in a triangular domain filled with water is studied by means of numerical simulation as a model for flow developing in the littoral region of lakes or coasts. The domain is fitted in polar co-ordinates; solutions are obtained for different values of the independent parameters of the model, which are the Rayleigh number (Ra), the Prandtl number (Pr) and the slope of the domain (S). Within the ranges examined, as Ra is increased, different regimes of the developing flow are observed; these are found qualitatively to be insignificantly influenced by changes in S, whereas the flow is found to be quantitatively insensitive to Pr for high enough values of Pr. Several interesting features of the flow are depicted and integral values useful in the analysis of flow in lakes are extracted.

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TL;DR: In this article, a marker-and-cell method for the simulation of impacts of fluid free surfaces with solid boundaries is presented, which allows the intentional treatment of impact by making more appropriate adjustments of the tentative internal velocities associated with such cells.

Abstract: Deficiencies associated with the simulation of impacts of fluid free surfaces with solid boundaries by use of marker-and-cell methods are identified and addressed. New procedures are introduced that affect the movement of markers in cells adjacent to a solid boundary, the flags of the cells that comprise a solid boundary and the pressure boundary condition for a cell in which impact occurs. Combined with fundamental changes in the sequence of steps in the computational cycle, these new procedures allow the intentional treatment of impact. As a result, improved estimates are obtained of the pressure associated with the cells adjacent to a boundary along which impact occurs. Consequently, more appropriate adjustments are made of the tentative internal velocities associated with such cells. In addition, a special procedure is presented for the adjustment of the tentative internal velocity between two surface cells. Finally, a new cell type termed a corner cell is defined and a procedure for its treatment is presented. Numerical examples are included to illustrate the previous deficiencies associated with the simulation of impact as well as the effectiveness of the new methods presented in this paper. Validation of the new methods is achieved by comparison with experimental results for spillage over a containment dike.

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TL;DR: In this article, the stream function-vorticity equations for incompressible two-dimensional flows are uncoupled and solved in sequence by the finite element method, and the vorticity at no-slip boundaries is evaluated in the framework or the streamfunction equation.

Abstract: The streamfunction-vorticity equations for incompressible two-dimensional flows are uncoupled and solved in sequence by the finite element method. The vorticity at no-slip boundaries is evaluated in the framework or the streamfunction equation. The resulting scheme achieves convergence, even for very high values of the Reynolds number, without the traditional need for upwinding. The stability and accuracy of the approach are demonstrated by the solution or two well-known benchmark problems: flow in a lid-driven cavity at Re≤10,000 and flow over a backward-facing step at Re=800

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TL;DR: In this paper, a fully coupled method for the solution of incompressible Navier-Stokes equations is investigated, which uses a fully implicit time discretization of momentum equations, the standard linearization of convective terms, a cell-centred colocated grid approach and a block-nanodiagonal structure of the matrix of nodal unknowns.

Abstract: SUMMARY A fully coupled method for the solution of incompressible Navier-Stokes equations is investigated here. It uses a fully implicit time discretization of momentum equations, the standard linearization of convective terms, a cell-centred colocated grid approach and a block-nanodiagonal structure of the matrix of nodal unknowns. The Method is specific in the interpolation used for the flux reconstruction problem, in the basis iterative method for the fully coupled system and in the acceleration means that control the global efficiency of the procedure. The performance of the method is discussed using lid-driven cavity problems, both for two and three-dimensionai geometries, for steady and unsteady flows.

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TL;DR: Newton's method is applied to the finite volume approximation for the steady state heat transfer, fluid flow and unknown interfaces in a floating molten zone, where the stream function/vorticity and temperature formulation of the Navier-Stokes and energy equations and their associated boundary conditions are written in generalized curvilinear co-ordinates and conservative law form with the Boussinesq approximation as discussed by the authors.

Abstract: Newton's method is applied to the finite volume approximation for the steady state heat transfer, fluid flow and unknown interfaces in a floating molten zone. The streamfunction/vorticity and temperature formulation of the Navier–Stokes and energy equations and their associated boundary conditions are written in generalized curvilinear co-ordinates and conservative law form with the Boussinesq approximation. During Newton iteration the ILU(0) preconditioned GMRES matrix solver is applied for solving the linear system, where the sparse Jacobian matrix is estimated by finite differences. Nearly quadratic convergence of the method is observed. Sample calculations are reported for sodium nitrate, a high-Prandtl-number material (Pr = 9.12). Both natural convection and thermocapillary flow as well as an overall mass balance constraint in the molten zone are considered. The effects of convection and heat input on the flow patterns, zone position and interface shapes are illustrated. After the lens effect due to the molten zone is considered, the calculated flow patterns and interface shapes are compared with the observed ones and are found to be in good agreement.

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TL;DR: In this article, the authors considered symmetric and antisymmetric periodic boundary conditions for flows governed by the incompressible Navier-Stokes equations and applied them to finite element code using the penalty function formulation.

Abstract: SUMMARY In this paper we consider symmetric and antisymmetric periodic boundary conditions for flows governed by the incompressible Navier-Stokes equations. Classical periodic boundary conditions are studied as well as symmetric and antisymmetric periodic boundary conditions in which there is a pressure difference between inlet and outlet. The implementation of this type of boundary conditions in a finite element code using the penalty function formulation is treated and also the implementation in a finite volume code based on pressure correction. The methods are demonstrated by computation of a flow through a staggered tube bundle.