bluuebunny/crossref_metadata_2025 huggingface dataset:
Upwind basis finite elements for convection‐dominated problems
Authors: Steffler P. M.
Published: August 2005

Finite elements using higher‐order basis functions in the spirit of the QUICK method for convection‐dominated fluid flow and transport problems are introduced and demonstrated. Instead of introducing new internal degrees of freedom, completeness is achieved by including functions based on nodal values exterior and upwind to the element domain. Applied with linear test functions to the weak statements for convection‐dominated problems, a family of Petrov–Galerkin finite elements is developed. Quadratic and cubic versions are demonstrated for the one‐dimensional convection–diffusion test problem. Elements of up to seventh degree are used for local solution refinement. The behaviour of these elements for one‐dimensional linear and non‐linear advection is investigated. A two‐dimensional quadratic upwind element is demonstrated in a streamfunction–vorticity formulation of the Navier–Stokes equations for a driven cavity flow test problem. With some minor reservations, these elements are recommended for further study and application.
A point implicit unstructured grid solver for the euler and Navier–Stokes equations
Authors: Thareja Rajiv R. , Stewart James R. , Hassan Obey , Morgan Ken , Peraire Jaime
Published: August 2005

An upwind finite element technique that uses cell‐centred quantities and implicit and/or explicit time marching has been developed for computing hypersonic laminar viscous flows using adaptive triangular grids. The approach is an extension to unstructured grids of the LAURA algorithm due to Gnoffo. A structured grid of quadrilaterals is laid out near a solid surface. For inviscid flows the method is stable at Courant numbers of over 100000. A first‐order basic scheme and a higher‐order flux‐corrected transport (FCT) scheme have been implemented. This technique has been applied to the problem of predicting type III and IV shock wave interactions on a cylinder, with a view to simulating the pressure and heating rate augmentation caused by an impinging shock on the leading edge of a cowl lip of an engine inlet. The predictions of wall pressure and heating rates compare very well with experimental data. The flow features are distinctly captured with a sequence of adaptively generated grids.
A Chebyshev collocation method for the Navier–Stokes equations with application to double‐diffusive convection
Authors: Ehrenstein U. , Peyret R.
Published: August 2005

A Chebyshev collocation method for solving the unsteady two‐dimensional Navier–Stokes equations in vorticity–streamfunction variables is presented and discussed. The discretization in time is obtained through a class of semi‐implicit finite difference schemes. Thus at each time cycle the problem reduces to a Stokes‐type problem which is solved by means of the influence matrix technique leading to the solution of Helmholtz‐type equations with Dirichlet boundary conditions. Theoretical results on the stability of the method are given. Then a matrix diagonalization procedure for solving the algebraic system resulting from the Chebyshev collocation approximation of the Helmholtz equation is developed and its accuracy is tested. Numerical results are given for the Stokes and the Navier–Stokes equations. Finally the method is applied to a double‐diffusive convection problem concerning the stability of a fluid stratified by salinity and heated from below.
A finite element method for analysis of fluid flow, heat transfer and free interfaces in Czochralski crystal growth
Authors: Sackinger P. A. , Brown R. A. , Derby J. J.
Published: August 2005

A finite element algorithm is presented for simultaneous calculation of the steady state, axisymmetric flows and the crystal, melt/crystal and melt/ambient interface shapes in the Czochralski technique for crystal growth from the melt. The analysis is based on mixed Lagrangian finite element approximations to the velocity, temperature and pressure fields and isoparametric approximations to the interface shape. Galerkin's method is used to reduce the problem to a non‐linear algebraic set, which is solved by Newton's method. Sample solutions are reported for the thermophysical properties appropriate for silicon, a low‐Prandtl‐number semiconductor, and for GGG, a high–Prandtl–number oxide material. The algorithm is capable of computing solutions for both materials at realistic values of the Grashof number, and the calculations are convergent with mesh refinement. Flow transitions and interface shapes are calculated as a function of increasing flow intensity and compared for the two material systems. The flow pattern near the melt/gas/crystal tri‐junction has the asymptotic form predicted by an inertialess analysis assuming the meniscus and solidification interfaces are fixed.
An efficient upwind/relaxation algorithm for the Euler and Navier–Stokes equations
Authors: Zha Ge‐Cheng , Liu Dao‐Zhi , Ma Tie‐You
Published: August 2005

An efficient Euler and full Navier–Stokes solver based on a flux splitting scheme is presented. The original Van Leer flux vector splitting form is extended to arbitrary body‐fitted co‐ordinates in the physical domain so that it can be used with a finite volume scheme. The block matrix is inverted by Gauss–Seidel iteration. It is verified that the often used reflection boundary condition will produce incorrect flux crossing the wall and cause too large numerical dissipation if flux vector splitting is used. To remove such errors, an appropriate treatment of wall boundary conditions is suggested. Inviscid and viscous steady transonic internal flows are analysed, including the case of shock‐induced boundary layer separation.
Galerkin finite element simulation of convection driven by rotation and gravitation
Authors: Ladeinde Foluso , Torrance K. E.
Published: August 2005

The performance of the Galerkin finite element method when applied to time‐dependent convection involving rotation, self‐gravitation and the normal gravity field in a horizontal cylinder is discussed in this paper. The governing equations, the parameters of the problem and our implementation of the numerical schemes are presented. The accuracy, spatial scale of resolution, flexibility and robustness of the resulting code show the element method as a valuable tool for research in this area or in related problems in astrophysical fluid dynamics. The numerical difficulties in large‐amplitude flows are associated with an error‐control scheme for time integration and the ‘short‐time’ wiggles in transient Dirichlet problems. Coarse grids give the correct qualitative picture in most simulations, but the type of solution at short time (and hence grid refinement) presumably resolves the degeneracy in the azimuthal orientation of convection cells in flows driven by self‐gravitation and (perhaps) centrifugal buoyancy. The final state in transient flows is a motionless isothermal fluid. However, residual motions can be nullified only in the limit of zero grid size in flows driven by centrifugal buoyancy (self‐gravitation), while a fairly coarse grid is sufficient for this purpose in normal gravity‐driven flows.
Iterative stabilization of the bilinear velocity–constant pressure element
Authors: Fortin M. , Boivin S.
Published: August 2005

Some finite element approximations of incompressible flows, such as those obtained with the bilinear velocity–constant pressure element (Q1−P0), are well known to be unstable in pressure while providing reasonable results for the velocity.

We shall see that there exists a subspace of piecewise constant pressures that leads to a stable approximation. The main drawback associated with this subspace is the necessity of assembling groups of elements, the so‐called ‘macro‐elements’, which increases dramatically the bandwidth of the system.

We study a variant of Uzawa's method which enables us to work in the desired subspace without increasing the bandwidth of the system. Numerical results show that this method is efficient and can be made to work at a low extra cost. The method can easily be generalized to other problems and is very attractive in three‐dimensional cases.
A control volume finite difference method for buoyant flow in three‐dimensional curvilinear non‐orthogonal co‐ordinates
Authors: Yang H. Q. , Yang K. T. , Lloyd J. R.
Published: August 2005

This paper presents a control‐volume‐based finite difference method in non‐orthogonal curvilinear coordinates on a local basis in which the vectors and tensors are all based on the general curvilinear coordinates for buoyant flow calculations in arbitrary three‐dimensional geometries. The governing equations are transformed from Cartesian co‐ordinates into generalized curvilinear co‐ordinates. After integrating the set of equations for the control volumes, the finite difference equations are then formulated by a proper treatment of the heat flux and stress tensors and by incorporating the QUICK scheme for the convective terms. The solution procedure then follows the one for three‐dimensional Cartesian co‐ordinates.

Examples are given in problems of natural convection in such three‐dimensional enclosures as parallelepiped enclosures and horizontal closed cylinders with differentially heated ends. In the latter case, important applications have been found in crystal growth by means of chemical vapour deposition in a cylindrical ampoule, in which uniform heat fluxes along the two ends are required in order to produce high‐quality crystals. Special attention is given to the insertion of baffles in the cylinder to improve the recirculating flow patterns near the two ends.
Self‐adaptive hierarchic finite element solution of the one‐dimensional unsaturated flow equation
Authors: Abriola Linda M. , Lang John R.
Published: August 2005

The advantages associated with the use of self‐adaptive methods for the solution of problems which require the prediction of a frontal position in time are well known. In this paper a self‐adaptive finite element solution for the non‐linear unsaturated flow equation is developed using hierarchic p‐version enrichment of the interpolating space. Additional computational advantages are demonstrated for an iteration scheme in which iterations after enrichment are performed only over a subdomain. Numerical solutions are presented for a one‐dimensional infiltration scenario.
High‐order methods for differential equations with large first‐derivative terms
Authors: Dekema Sandra Kay , Schultz David H.
Published: August 2005

A method is developed to solve elliptic singular perturbation problems. Examples are presented in one and two dimensions for both linear and non‐linear problems. In particular, examples are presented for fluid flow problems with boundary layers. In the one‐dimensional case an approximating equation is developed using just three points. The method first presented is a fourth‐order approximation but is extended to become a higher‐order method. Results are included for the fourth‐, sixth‐, eighth‐ and tenth‐order methods.

The results are first compared with results found by Segal in an article about elliptic singular perturbation problems. The elliptic singular perturbation problems are compared with a method by Il'in and also with central and backward difference schemes from Segal's article. There was only one case where the results in Segal's paper were as accurate as the results presented in this paper. However, in this case the method used by Segal did not give accurate values for a second problem presented. The results are also compared with results given by Spalding and by Christie.

The method of this paper was also tested on the solution of some non‐linear diffusion equations with concentration‐dependent diffusion coefficients. The results were superior to results presented by Lee and by Schultz. Finally, the method is extended to several two‐dimensional problems.

The method developed in this paper is accurate, easy to use and can be generalized to other problems.
On the reversed flow and oscillating wake in an asymmetrically heated channel
Authors: Chang T. S. , Lin T. F.
Published: August 2005

The detailed processes of flow reversal in a buoyancy‐induced flow through a one‐side‐heated vertical channel of finite height were simulated numerically. It is of interest to note that the wake above the heated plate is oscillatory at high Rayleigh number and there exists a minimum in the transient variation of the average Nusselt number. Additionally, the predicted steady average Nusselt number and induced flow rate are correlated by empirical equations.
Numerical simulation of three‐dimensional transonic flows
Authors: Sahu Jubaraj , Steger Joseph L.
Published: August 2005

The three‐dimensional flow over a projectile has been computed using an implicit, approximately factored, partially flux‐split algorithm. A simple composite grid scheme has been developed in which a single grid is partitioned into a series of smaller grids for applications which require an external large memory device such as the SSD of the CRAY X‐MP/48 or multi‐tasking. The accuracy and stability of the composite grid scheme have been tested by numerically simulating the flow over an ellipsoid at an angle of attack and comparing the solution with a single‐grid solution. The flow field over a projectile at M = 0·96, 1·1 and 4° angle of attack has been computed using a fine grid and compared with experiment.
An application of the vorticity–vector potential method to laminar cube flow
Authors: Raul Robin , Bernard Peter S. , Buckley Frank T.
Published: August 2005

Three‐dimensional incompressible laminar flow around a cube is investigated using the vorticity–vector potential formulation of the equations of motion. Numerical solutions to á semi‐implicit finite difference approximation to the vorticity transport equation coupled to discrete Poisson equations for the scalar and vector potentials are obtained using an eight‐colour SOR algorithm. Calculations are done at a range of Reynolds numbers from 10 to 100. The predicted drag and other flow characteristics are found to agree well with experimental results, including those from a cube drop experiment performed for the present study.
Solving numerically the Navier–Stokes equations on parallel systems
Authors: Lin Avi
Published: August 2005

A new general approach for numerically computing flow fields on parallel computing environments is presented, discussed and analysed. The hierarchy presented here is based on a parallel split of operators. A portion of the theory is presented together with its application to two‐ and three‐dimensional flows. This strategy is applied to a two‐dimensional problem for which a specific parallel split, called a parabolized split, is given. The parallel algorithm that results from this split is analysed, leading to reasonably good parallel efficiency, which is close to 50%. Actual experiments lead to similar conclusions. This parallel strategy can also be used together with other parallel computing algorithms, such as domain decomposition, to give an optimal‐type parallel algorithm for the Navier–Stokes equations.
Unsymmetric conjugate gradient methods and sparse direct methods in finite element flow simulation
Authors: Howard D. , Connolley W. M. , Rollett J. S.
Published: August 2005

A series of numerical experiments on the Cray XMP/48 and on the Cray 2 investigate the robustness and economy of direct and unsymmetric conjugate gradient (CG) type methods for the solution of matrix systems arising from a 3D FEM discretization of fluid flow problems. Computations on a Boussinesq flow model problem with either ILU preconditioned or unpreconditioned unsymmetric CG methods are presented. Such experiments seem to indicate that the unpreconditioned BICG method is robust for moderately non‐linear incompressible Navier–Stokes FEM discretizations and that the ILU preconditioned BICG method is very robust and more economic than an unsymmetric frontal solver when the generous memory of the Cray 2 is exploited to store both the matrix and its preconditioner. We cover some of the programming aspects of direct and iterative methods on a supercomputer and find that direct methods have advantages: the crucial CPU‐consuming area of code is compact but overwhelming, and its percentage of total CPU usage is independent of the spectral properties of the matrix involved. An optimal implementation of the unsymmetric CG method is more difficult because its work is related to the spectral distribution of the matrix considered and because there is no single portion of the code that overwhelmingly dominates the CPU usuage.
On the accuracy of finite difference and modal methods for computing tidal and wind wave current profiles
Authors: Davies A. M.
Published: August 2005

This paper deals with the comparative accuracy of using finite difference grids or a modal representation through the vertical in modelling tidally or wind wave induced current profiles.

A point model is used in the vertical, with a no‐slip condition at the sea bed. In the finite difference approach the high‐shear bottom layer is resolved using either a regular grid on a logarithmic or log‐linear transformed co‐ordinate or an irregular grid, varying in such a manner as to retain second‐order accuracy. The accuracy of these various grid schemes is considered in detail.

The relative merits of using either the Crank‐Nicolson or Dufort‐Frankel time integration methods are considered; in the case of a fine grid in a high‐viscosity region, some numerical problems are found with the Dufort‐Frankel method.

An alternative approach to using a finite difference grid in the vertical, namely a modal (spectral) method, is described. The form of the modes is such that they can accurately resolve the high‐shear bottom boundary layer.

Calculations show that the thickness of the bottom boundary layer in relation to the total water depth is important in determining the choice of grid transform and rates of convergence of solutions using finite difference or modal methods. However, for the majority of problems the modal solution is numerically attractive owing to its computational efficiency and the ease with which solution algorithms based upon it can be coded in vectorizable form suitable for the new generation of vector computers.

The influence of viscosity profile, its time variation and water depth upon tidally induced or wave induced currents is considered. Calculations suggest that near‐bed measurements of tidal flow in shallow water together with associated modelling would enable appropriate formulations of eddy viscosity to be determined. Similar measurements, though using a laboratory flume, would be appropriate for wind wave problems.
Finite difference solution of a Newtonian jet swell problem
Authors: Liu Ta‐Jo , Yu Tsai‐An , Cheng Shu‐Hue
Published: August 2005

A finite difference technique has been developed to study the Newtonian jet swell problem. The streamfunction and vorticity were used as dependent variables to describe the jet flow. The boundary‐fitted co‐ordinate transformation method was adopted to map the flow geometry into a rectangular domain. The standard finite difference method was then applied for solving the flow equations. The location of the jet free surface was updated by the kinematic boundary condition, and an adjustable parameter was included in the free‐surface iteration. We could obtain numerical solutions for the Reynolds number as high as 100, and the differences between the present study and previous finite element simulations on the jet swell ratio are less than 5%.
Assessment of TVD schemes for inviscid and turbulent flow computation
Authors: Chen Ming‐Hsiung , Hsu Chen‐Chi , Shyy Wei
Published: August 2005

A systematic study has been conducted to assess the performance of the TVD schemes for practical flow computation. The viewpoint adopted here is to treat the TVD schemes as a combination of the standard central difference scheme with numerical dissipation terms. The controlled amount of numerical dissipation modifies the computed fluxes to ensure that the solution is oscillation‐free. Four variants of TVD schemes, two with upwind dissipation terms and two with symmetric dissipation terms, have been studied and compared with the conventional Beam‐Warming scheme for inviscid and turbulent axisymmetric flow computations. The results obtained show that all four variants can accurately resolve the shock and flow profiles with fewer grid points than the Beam‐Warming scheme. The convergence rates of the TVD schemes are also substantially superior to that of the Beam‐Warming scheme. The combination of high accuracy, good robustness and improved computational efficiency offered by the TVD schemes makes them attractive for computing high‐speed flow with shocks. In terms of the relative performances it is found that the symmetric schemes converge slightly faster but that the upwind schemes are less sensitive to the number of grid points being employed.
Numerical prediction of turbulent flow over a two‐dimensional ridge
Authors: Mouzakis F. N. , Bergels G. C.
Published: August 2005

Predictions are presented of the two‐dimensional turbulent flow over a triangular ridge. The time‐averaged Reynolds equations are written in an orthogonal curvilinear co‐ordinate system and transformed to finite difference form after discretization in physical space. Turbulence is simulated by the two‐equation κ‐ε model of turbulence. In the first part of the paper the basics of the numerical method are presented and in the second part comparisons are made between predictions and available laboratory data. Therefore the validity and reliability of the method as well as its flexibility in treating complex recirculating flows are assessed. Results of engineering significance are presented of the effect of the ridge slope on the length of the recirculation region and on the overspeed factor on top of the ridge.
The application of finite element analysis to the solution of Stokes wave diffraction problems
Authors: Clark P. J. , Bettess P. , Hearn G. E. , Downie M. J.
Published: August 2005

A new finite‐element based method of calculating non‐linear wave loads on offshore structures in extreme seas is presented in this paper. The diffraction wave field is modelled using Stokes wave theory developed to second order. Wave loads and free surface elevations are obtained for fixed surface‐piercing structures by solving a boundary value problem for the second‐order velocity potential. Special attention has been given to the radiation condition for the second‐order diffraction field. Results are presented for three test examples, the vertical cylinders of Kim and Yue and of Chakrabarti, and an elliptic cylinder. These results demonstrate that early problems with the application of second‐order theory arising from inadequate radiation conditions have been overcome.
An axisymmetric model of separated flow about a sphere using discrete vortices
Authors: Lee D. K. , Downie M. J. , Bettess P.
Published: August 2005

A procedure for the calculation of the starting flow around a sphere in a uniform stream is presented. The flow field is simulated by a flow of ideal fluid with embedded vorticity. With the assumption that the flow remains symmetric, the vorticity field is approximated by a number of discrete circular line vortices. The image vortices to satisfy the boundary condition for the normal component of velocity on the surface of the sphere are determined by Butler's sphere theorem. The Stokes streamfunction is used for the field description. The motion of vortices is tracked by the vortex‐in‐cell method, the cells being formed by square grids.
A consistent boundary element method for free surface hydrodynamic calculations
Authors: Medina Daniel E. , Liggett James A. , Birchwood Richard A. , Torrance Kenneth E.
Published: August 2005

Free surface phenomena are described by equations that exhibit two types of non‐linearities. The first is inherent to the equations themselves and the second is caused by the application of boundary conditions at a free surface at an unknown location. Numerical calculations usually do not specifically recognize the second non‐linearity, nor treat it in a fashion consistent with the more obvious non‐linearities in the boundary conditions. A consistent formulation is introduced in the present paper. The field equation is integrated and the free surface boundary conditions are applied on the unknown geometry by means of appropriate series expansions. The consistent formulation introduces improvements in accuracy and computing speed. The method is demonstrated on several hydrodynamic free surface problems and an error analysis is included.
A 3D finite element method for the simulation of thermoconvective flows and its performances on a vector‐parallel computer
Authors: Carriere Philippe , Jeandel Denis
Published: August 2005

A three‐dimensional finite element method for the simulation of thermoconvective flows is presented. Vector‐parallel performances of some preconditioned conjugate gradient methods are compared for solving both large linear systems and the Stokes problem. As significant examples, numerical experiments on the steady two‐ and three‐dimensional Rayleigh‐Bénard convection at high Prandtl number are reported.
A preconditioned conjugate gradient Uzawa‐type method for the solution of the stokes problem by mixed Q1–P0 stabilized finite elements
Authors: Vincent C. , Boyer R.
Published: August 2005

We study the behaviour of a conjugate gradient Uzawa‐type method for a stabilized finite element approximation of the Stokes problem. Many variants of the Uzawa algorithm have been described for different finite elements satisfying the well‐known Inf‐Sup condition of Babuška and Brezzi, but it is surprising that developments for unstable ‘low‐order’ discretizations with stabilization procedures are still missing. Our paper is presented in this context for the popular (so‐called) Q1–P0 element.

First we show that a simple stabilization technique for this element permits us to retain the property of a convergence factor bounded independently of the discretization mesh size. The second contribution of this work deals with the construction of a less costly preconditioner taking full advantages of the block diagonal structure of the stabilization matrix. Its efficiency is supported by 2D and. 3D numerical results.
Laminar fluid convection between concentric and eccentric heated horizontal rotating cylinders for low‐Prandtl‐number fluids
Authors: Lee T. S.
Published: August 2005

Numerical experiments are performed to study rotational effects on the mixed convection of low‐Prandtlnumber fluids enclosed between the annuli of concentric and eccentric horizontal cylinders. The inner cylinder is assumed to be heated and rotating. The rotational Reynolds number considered is in the range where the effect of Taylor vortices is negligible. The Prandtl number of the fluid considered is in the range 0·01–1·0. The Rayleigh number considered is up to 106. A non‐uniform mesh transformation technique coupled with the introduction of ‘false transient’ parameters to the vorticity and streamfunction‐vorticity expressions was used to solve the governing set of equations. Results show that when the inner cylinder is made to rotate, the multicellular flow patterns observed in stationary cylindrical annuli subside in a manner depending on the Prandtl number of the fluids. Eventually the flow tends toward a uniform flow similar to that of a solid body rotation. For a fixed Rayleigh number and with a Prandtl number of the order of 1·0, when the inner cylinder is made to rotate, the mean Nusselt number is observed to decrease throughout the flow. For lower Prandtl number of the order 0·1–0·01 the mean Nusselt number remained fairly constant when the inner cylinder was made to rotate. The mean Nusselt numbers obtained were also compared with available data from other investigators.
Non‐linear interaction of water waves with submerged obstacles
Authors: Yeung Ronald W. , Vaidhyanathan M.
Published: August 2005

The interaction of two‐dimensional water waves with a fixed submerged cylinder is studied using a finite difference scheme with boundary‐fitted co‐ordinates. A mixed Eulerian–Lagrangian (MEL) formulation is used to satisfy the fully non‐linear free surface conditions. The diffraction of small‐amplitude water waves by a cylinder is examined for various wavelengths and amplitudes of the incident wave. Fourier analyses of the incident and diffracted waves are performed to determine their spectra. An example of a large‐amplitude wave breaking over a cylinder is also studied. The non‐linear numerical solutions are compared with those of experiments and linear theory where appropriate.
A multigrid method for steady incompressible navier‐stokes equations based on flux difference splitting
Authors: Dick E. , Linden J.
Published: August 2005

The steady Navier–Stokes equations in primitive variables are discretized in conservative form by a vertex‐centred finite volume method Flux difference splitting is applied to the convective part to obtain an upwind discretization. The diffusive part is discretized in the central way.

In its first‐order formulation, flux difference splitting leads to a discretization of so‐called vector positive type. This allows the use of classical relaxation methods in collective form. An alternating line Gauss–Seidel relaxation method is chosen here. This relaxation method is used as a smoother in a multigrid method. The components of this multigrid method are: full approximation scheme with F‐cycles, bilinear prolongation, full weighting for residual restriction and injection of grid functions.

Higher‐order accuracy is achieved by the flux extrapolation method. In this approach the first‐order convective fluxes are modified by adding second‐order corrections involving flux limiting. Here the simple MinMod limiter is chosen. In the multigrid formulation the second‐order discrete system is solved by defect correction.

Computational results are shown for the well known GAMM backward‐facing step problem and for a channel with a half‐circular obstruction.
An extended κ–ϵ finite element model
Authors: Jaeger Marc , Dhatt Gouri
Published: August 2005

An extended κ–ϵ model (to include low‐Reynolds‐number regions) employing weighting functions is presented. Wall functions for the near‐wall zones are developed giving correct boundary values for the Shear stress and κ–ϵ. A finite element model using a penalty formulation for incompressible turbulent flow is applied to Solve a flow between two plates. Results with mesh boundaries situated in the near‐wall region and a: the wall are compared with measured values.
A new decoupled finite element algorithm for viscoelastic flow. Part 1: Numerical algorithm and sample results
Authors: Swarbrick S. J. , Nassehi V.
Published: August 2005

This paper presents an algorithm for two‐dimensional Steady viscoelastic flow Simulation in which the Solution of the momentum and continuity equations is decoupled from that of the constitutive equations. The governing equations are discretized by the finite element method, with 3 × 3 element subdivision for the stress field approximation. Non‐consistent Streamline upwinding is also used. Results are given for flow through a converging channel and through an abrupt planar 4:1 contraction.
On the accurate prediction of the wall‐normal velocity in compressible boundary‐layer flow
Authors: Pruett C. David
Published: August 2005

We consider a problem which arises in the numerical solution of the compressible two‐dimensional or axisymmetric boundary‐layer equations. Numerical methods for the compressible boundary‐layer equations are facilitated by transformation from the physical (x, y) plane to a computational (ξ, η) plane in which the evolution of the flow is ‘slow’ in the time‐like ξ direction. The commonly used Levy‐Lees transformation results in a computationally well‐behaved problem, but it complicates interpretation of the solution in physical space. Specifically, the transformation is inherently non‐linear, and the physical wall‐normal velocity is transformed out of the problem and is not readily recovered. Conventional methods extract the wall‐normal velocity in physical space from the continuity equation, using finite‐difference techniques and interpolation procedures. The present spectrally accurate method extracts the wall‐normal velocity directly from the transformation itself, without interpolation, leaving the continuity equation free as a check on the quality of the solution. The present method for recovering wall‐normal velocity, when used in conjunction with a highly accurate spectral collocation method for solving the compressible boundary‐layer equations, results in a discrete solution which satisfies the continuity equation nearly to machine precision. As demonstration of the utility of the method, the boundary layers of three prototypical high‐speed flows are investigated and compared: the flat plate, the hollow cylinder, and the cone. An important implication for classical linear stability theory is also briefly discussed.
Numerical experiments on the stability of leading edge boundary layer flow: A two‐dimensional linear study
Authors: Theofilis Vassilios
Published: August 2005

A numerical study is performed in order to gain insight to the stability of the infinite swept attachment line boundary layer. The basic flow is taken to be of the Hiemenz class with an added cross‐flow giving rise to a constant thickness boundary layer along the attachment line. The full Navier‐Stokes equations are solved using an initial value problem approach after two‐dimensional perturbations of varying amplitude are introduced into the basic flow.

A second‐order‐accurate finite difference scheme is used in the normal‐to‐the‐wall direction, while a pseudospectral approach is employed in the other directions; temporally, an implicit Crank‐Nicolson scheme is used. Extensive use of the efficient fast Fourier transform (FFT) algorithm has been made, resulting in substantial savings in computing cost.

Results for the two‐dimensional linear regime of perturbations are in very good agreement with past numerical and theoretical investigations, without the need for specific assumptions used by the latter, thus establishing the generality of our method.
Computation of fluid flow with a parallel multigrid solver
Authors: Schreck E. , Perić M.
Published: August 2005

A finite volume numerical method for the prediction of fluid flow and heat transfer in simple geometries was parallelized using a domain decomposition approach. The method is implicit, uses a colocated arrangement of variables and is based on the SIMPLE algorithm for pressure‐velocity coupling. Discretization is based on second‐order central difference approximations. The algebraic equation systems are solved by the ILU method of Stone.1 To accelerate the convergence, a multigrid technique was used. The efficiency was examined on three different parallel computers for laminar flow in a pipe with an orifice and natural convection in a closed cavity. It is shown that the total efficiency is made up of three major factors: numerical efficiency, parallel efficiency and load‐balancing efficiency. The first two factors were thoroughly investigated, and a model for predicting the parallel efficiency on various computers is presented. Test calculations indicate reasonable total efficiency and favourable dependence on grid size and the number of processors.
Development of a hybrid k‐ε turbulence model for swirling recirculating flows under moderate to strong swirl intensities
Authors: Chang Keh‐Chin , Chen Ching‐Shun
Published: August 2005

A hybrid k‐ε turbulence model, based on the concept that the modification of anisotropic effects should not be made in the flow regions inherent to small streamline curvatures, has been developed and examined with the swirling recirculating flows, with the swirl levels ranging from 0·6 to 1·23 in abrupt pipe expansion. A fairly satisfactory agreement of model predictions with the experimental data shows that this hybrid k‐ε model can perform better simulation of swirling recirculating flows as compared to the standard k‐ε model and the modified k‐ε model proposed by Abujelala and Lilley.
Surface waves propagation in shallow water: A finite element model
Authors: Do Carmo J. S. Antunes , Santos F. J. Seabra , Barthélemy E.
Published: August 2005

A two‐dimensional (in‐plane) numerical model for surface waves propagation based on the non‐linear dispersive wave approach described by Boussinesq‐type equations, which provide an attractive theory for predicting the depth‐averaged velocity field resulting from that wave‐type propagation in shallow water, is presented. The numerical solution of the corresponding partial differential equations by finite‐difference methods has been the subject of several scientific works. In the present work we propose a new approach to the problem: the spatial discretization of the system composed by the Boussinesq equations is made by a finite element method, making use of the weighted residual technique for the solution approach within each element. The model is validated by comparing numerical results with theoretical solutions and with results obtained experimentally.
Incompressible low‐speed‐ratio flow in non‐uniform distensible tubes
Authors: Reuderink P. J. , van de Vosse F. N. , van Steenhoven A. A. , van Dongen M. E. H. , Janssen J. D.
Published: August 2005

The fluid flow in distensible tubes is analysed by a finite element method based on an uncoupled solution of the equations of wall motion and fluid flow. Special attention is paid to the choice of proper boundary conditions. Computations were made for sinusoidal flow in a distensible uniform tube with the Womersley parameter α = 5, and a ratio between tube radius and wavelenth from 0·0001 to 0·5. The agreement between the numerical results and Womersley's analytic solution depends on the speed ratio between fluid and wave velocity, and is fair for speed ratios up to 0·05. The analysis of the flow field in a distensible tube with a local inhomogeneity revealed a marked influence of wave phenomena and wall motion on the velocity profiles.
Numerical solution of the generalized Serre equations with the MacCormack finite‐difference scheme
Authors: Do Carmo J. S. Antunes , Santos F. J. Seabra , Almeida A. B.
Published: August 2005

This paper describes a two‐dimensional numerical model to solve the generalized Serre equations. In order to solve the system equations, written in the conservative form, we use an explicit finite‐difference method based on the MacCormack time‐splitting scheme. The numerical method and the computational model are validated by comparing one‐ and two‐dimensional numerical solutions with theoretical and experimental results. Finally, the two‐dimensional model (in a horizontal plane) is tested in a domain with complicated boundary conditions.
Modelling large‐bubble formation at submerged orifices
Authors: Bugg J. D. , Rowe R. D.
Published: August 2005

The formation of large gas bubbles at submerged orifices is investigated numerically with a two‐dimensional, transient, finite difference model using a volume fraction specification to track the movement of the gas‐liquid interface. Experimentally observed features of large‐bubble formation such as the initial toroidal shape of the bubbles and the penetration of liquid down the pipe centreline are well predicted by the model. The expected oscillatory nature of growth is also observed. The bubble departure volume corresponds to experiments and to the model of Davidson and Schuler. At present the simulations do not extend far enough to investigate multiple‐bubble ejection and important bubble‐to‐bubble interactions during growth and after departure.
A comparison of coarse and fine grain parallelization strategies for the simple pressure correction algorithm
Authors: Lewis A. J. , Brent A. D.
Published: August 2005

The primary aim of this work was to determine the simplest and most effective parallelization strategy for control‐volume‐based codes solving industrial problems. It has been found that for certain classes of problems, the coarse‐grain functional decomposition strategy, largely ignored due to its limited scaling capability, offers the potential for significant execution speed‐ups while maintaining the inherent structure of traditional serial algorithms. Functional decomposition requires only minor modification of the existing serial code to implement and, hence, code portability across both concurrent and serial computers is maintained. Fine‐grain parallelization strategies at the ‘DO loop’ level are also easy to implement and largely preserve code portability. Both coarse‐grain functional decomposition and fine‐grain loop‐level parallelization strategies for the SIMPLE pressure correction algorithm are demonstrated on a Silicon Graphics 4D280S eight CPU shared memory computer system for a highly coupled, transient two‐dimensional simulation involving melting of a metal in the presence of thermal‐buoyancy‐driven laminar convection. Problems requiring the solution of a larger number of transport equations were simulated by including further scalar variables in the calculation. While resulting in slight degradation of the convergence rate, the functional decomposition strategy exhibited higher parallel efficiencies and yielded greater speed‐ups relative to the original serial code. Initially, this strategy showed a significant degradation in convergence rate due to an inconsistency in the parallel solution of the pressure correction equation. After correcting for this inconsistency, the maximum speed‐up for 16 dependent variables was a factor of 5·28 with eight processors, representing a parallel efficiency of 67%. Peak efficiency of 76% was achieved using five processors to solve for 10 dependent variables.
A collocated finite volume method for predicting flows at all speeds
Authors: Demirdžić I. , Lilek Ž. , Perić M.
Published: August 2005

An existing two‐dimensional method for the prediction of steady‐state incompressible flows in complex geometry is extended to treat also compressible flows at all speeds. The primary variables are the Cartesian velocity components, pressure and temperature. Density is linked to pressure via an equation of state. The influence of pressure on density in the case of compressible flows is implicitly incorporated into the extended SIMPLE algorithm, which in the limit of incompressible flow reduces to its well‐known form. Special attention is paid to the numerical treatment of boundary conditions. The method is verified on a number of test cases (inviscid and viscous flows), and both the results and convergence properties compare favourably with other numerical results available in the literature.
Direct simulation of turbulent flow and heat transfer in a channel. Part I: Smooth walls
Authors: Rutledge Jeffrey , Sleicher Charles A.
Published: August 2005

Interest in the use of supercomputers for the direct numerical calculation of turbulence prompts the development of efficient numerical techniques so that calculation at higher Reynolds numbers might be made. This paper presents an efficient pseudo‐spectral technique, similar to but different from others that have recently appeared, for the calculation of momentum and heat transfer to a constant‐property, turbulent fluid in a two‐dimensional channel with walls at different, uniform temperature. The code uses no empiricism, although periodic boundary conditions are used for fluctuating quantities in the streamwise and spanwise directions.

Calculations were made for a Prandtl number of 0·72 and Reynolds number based on friction velocity and channel half‐height of 180 or 2800 based on channel half‐height and average velocity. Calculations of mean velocity profile, turbulence intensities, skewness, flatness, Reynolds stress and eddy diffusivity of heat near a wall compare favourably with experimental results. Representative contour plots of the temperature field near the wall and of the spanwise and streamwise two‐point velocity correlations are given.

Deficiencies are that the calculation requires many hours on a fast computer with a large high‐speed memory and that the grid size in each direction for appropriate resolution is approximately proportional to the square of the Reynolds number and to the Prandtl number raised to some power greater than one.
Co‐located equal‐order control‐volume finite element method for two‐dimensional axisymmetric incompressible fluid flow
Authors: Masson C. , Saabas H. J. , Baliga B. R.
Published: August 2005

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.
Treatment of numerical diffusion in strong convective flows
Authors: Arampatzis G. , Assimacopoulos D. , Mitsoulis E.
Published: August 2005

A three‐dimensional extension of the QUICK scheme adapted for the finite volume method and non‐uniform grids is presented to handle convection‐diffusion problems for high Peclet numbers and steep gradients. The algorithm is based on three‐dimensional quadratic interpolation functions in which the transverse curvature terms are maintained and the diagonal dominance of the coefficient matrix is preserved. All formulae are explicitly given in an appendix. Results obtained with the classical upwind (UDS), the simplified QUICK (transverse terms neglected) and the present full QUICK schemes are given for two benchmark problems, one two‐dimensional, steady state and the other three‐dimensional, unsteady state. Both QUICK schemes are shown to give superior solutions compared with the UDS in terms of accuracy and efficiency. The full QUICK scheme performs better than the simplified QUICK, giving even for coarse grids acceptable results closer to the analytical solutions, while the computational time is not affected much.
Investigation of flux formulae in transonic shock wave/turbulent boundary layer interaction
Authors: Drikakis D. , Durst F.
Published: August 2005

The aim of the present study is to examine the accuracy and improvement of various numerical methods in the solution of the transonic shock/turbulent boundary layer interaction problem and to show that a significant source of numerical inaccuracies in turbulent flows is not only the inadequacy of the turbulence model but also the numerical discretization. Comparisons between a Riemann solver and a flux‐vector‐splitting method as well as between various numerical high‐order extrapolation schemes with corresponding experimental results are presented.