bluuebunny/crossref_metadata_2025 huggingface dataset:
A vector upstream differencing scheme for problems in fluid flow involving significant source terms in steady‐state linear systems
Authors: Lillington J. N.
Published: August 2005

This paper considers a finite difference scheme for modelling the convection/diffusion equation in strongly convective flow regimes including circumstances in which significant source terms are present.

The main objective is to provide an alternative approach to central and/or upwind difference methods which for various reasons are unsatisfactory. To illustrate the main features of the scheme, an assessment of its accuracy is made by means of a Taylor expansion analysis and a study of its performance in two model problems. As a demonstration of its generality for use in large‐scale practical problems, some numerical results are presented for the prediction of the temperature distribution in a flow through a partially blocked heated rod bundle.

The main conclusions are that in almost all practical circumstances results obtained using the scheme are not susceptible to false diffusion or spatial oscillations, which are, respectively, the inherent weaknesses in many upwind and central difference scheme formulations, and in general its use results in improved overall accuracy.
The cause and cure (?) of the spurious pressures generated by certain FEM solutions of the incompressible Navier‐Stokes equations: Part 1
Authors: Sani R. L. , Gresho P. M. , Lee R. L. , Griffiths D. F.
Published: August 2005

The spurious pressures and ostensibly acceptable velocities which sometimes result from certain FEM approximate solutions of the incompressible Navier–Stokes equations are explained in detail. The concept of pressure modes, physical and spurious, pure and impure, is introduced and their effects on discretized solutions is analysed, in the context of both mixed interpolation and penalty approaches. Pressure filtering schemes, which are capable of recovering useful pressures from otherwise polluted numerical results, are developed for two particular elements in two‐dimensions and one element in three‐dimensions. Implications regarding the effect of spurious pressure modes on accuracy and ultimate convergence with mesh refinement are discussed and a list of unanswered questions presented. Sufficient numerical examples are discussed to corroborate the theory presented herein.
On higher order multigrid methods with application to a geothermal reservoir model
Authors: Gary John
Published: August 2005

This paper considers the multigrid iterative method applied to the solution of finite difference approximations to a linear second‐order self‐adjoint elliptic equation. It represents an extension of work by Dinar and Brandt. We compare two methods to obtain fourth‐order convergence. The first is local error extrapolation developed by Brandt, the second is iterative improvement developed by Lindberg. This work considers non‐separable problems, but only on a rectangular domain with Dirichlet boundary conditions. We consider test cases with non‐smooth (i.e. discontinuous second derivatives) as well as smooth solutions. We also apply the multigrid method to an elliptic equation with non‐separable coefficients which occurs in a geothermal model. In this case an analysis of the error fails to show any advantage in a fourth‐order difference scheme over a second‐order scheme. However, we do demonstrate that the multigrid iteration performs well on this problem. Also, this example shows that the multigrid iteration can be combined with iterative improvement to create an efficient fourth‐order method for a non‐separable elliptic equation which is coupled with a marching equation. Other work has found an advantage in this fourth‐order scheme for a similar geothermal model.
Selective lumping finite element method for shallow water flow
Authors: Kawahara Mutsuto , Hirano Hirokazu , Tsubota Khoji , Inagaki Kazuo
Published: August 2005

A finite element method for solving shallow water flow problems is presented. The standard Galerkin method is employed for spatial discretization. The numerical integration scheme for the time variation is the explicit two step scheme, which was originated by the authors and their co‐workers. However, the original scheme has been improved to remove the erroneous artifical damping effect. Since the improved scheme employs a combination of lumped and unlumped coefficients, the scheme is referred to as a selective lumping scheme. Stability conditions and accuracy are investigated by considering several numerical examples. The method has been applied to the tidal flow in Osaka Bay and Yatsushiro Bay.
General considerations of numerical stability and accuracy in inviscid, compressible flow calculations employing primitive variables
Authors: Bosman C. , Ahrabian D. , Kahrom M.
Published: August 2005

The numerical stability of a number of computation schemes currently used for three‐dimensional, inviscid, compressible flow is analysed using one‐dimensional Fourier analysis. Whereas Reference 1 analysed schemes which were modified to render them amenable to simple analysis, the present work analyses the stability of schemes as actually used by Highton,3 Ahrabian,1 Denton2 and Spalding.6 The use of current values of the variables as they become available is shown to bring a general improvement to stability margin. The manner of damping introduced by the time marching formulation is shown to be deleterious to modifications which reduce truncation error. Staggered grid schemes can be formulated to second order accuracy with better stability margin than the corresponding first order scheme. While unstaggered grid schemes can be formulated to second order error and remain stable, their stability margin becomes very small. Agreement of the theory with numerical experiments continues to be of a high order for both one and three‐dimensional disturbances.
On penalty function methods in the finite‐element analysis of flow problems
Authors: Reddy J. N.
Published: August 2005

In this paper the penalty function method is reviewed in the general context of solving constrained minimization problems. Mathematical properties, such as the existence of a solution to the penalty problem and convergence of the solution of a penalty problem to the solution of the original problem, are studied for the general case. Then the results are extended to a penalty function formulation of the Stokes and Navier‐Stokes equations. Conditions for the equivalence of two penalty‐finite element models of fluid flow are established, and the theoretical error estimates are verified in the case of Stokes's problem.
Conditions along the boundaries of bladed zones within the flow tracts of turbines
Authors: Lurie K. A. , Fedorov A. V. , Klimovich V. I.
Published: August 2005

Hamilton's variational principle is applied to derive a system of conditions which expresses the balance of momentum and energy of an ideal gas across the selvadges

The reviewer of this paper confesses to be unfamiliar with the term 'selvadge(s)' which appears frequently herein. Clearly, however, it (variously) means 'boundary', 'edge', 'projection of edge line' (meridional or axial), 'locus of leading edges', etc.

of bladed zones within the flow tracts of turbines. This system provides the background for a correct formulation of optimal design problems for turbines and compressors. The exposition follows the model of a large number of blades when the basic equations can be averaged over the azimuthal co‐ordinate.

An analysis is given of the obtained conditions and a computational algorithm described.
The simulation of secondary flow effects in turbulent non‐circular passage flows
Authors: Rapley C. W.
Published: August 2005

A finite difference method has been developed to predict the overall features of the local mean flow in fully developed turbulent non‐circular passage flows. The main transport effects of secondary flow have been identified and simulated with diffusion transport in a simple way which eliminates solution of the cross‐plane momentum and continuity equations and produces a compact calculation method. Predictions are presented for four different passage shapes and are discussed in relation to experimental measurements and predictions from other more complex methods. Although some minor details were not predicted, the main effects of secondary flow on the mean flow were found to have been quite well simulated, yielding predictions that are in reasonable overall agreement with experiment.
General hyperbolic difference formulas for linear and quasilinear hyperbolic equations
Authors: Cushman John H. , Huang Chi‐Hua
Published: August 2005

The method of non‐standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth‐order accuracy in space and time (O(Δtk, Δxk)) was developed for one‐dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth‐order accuracy.

Numerical results are presented for the linear and non‐linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.
Numerical analysis of a moving boundary problem in coastal hydrodynamics
Authors: Gopalakrishnan T. C. , Tung C. C.
Published: August 2005

A finite element model to tackle the moving boundary problem of wave run‐up on moderately steep slopes is developed. The special aspects considered in this study are (1) the modification of shallow water equations to accommodate the effect of vertical accelerations and (2) the use of Lagrangian acceleration coupled with an element that adapts itself to the moving boundary closely. The pressure term in the one‐dimensional momentum equation is derived using the Eulerian equation in the vertical direction. This takes care of the vertical accelerations which are significant during the motion of a wave on moderately steep slopes. The element near the boundary is allowed to change its dimension so that the fluid boundary is closely followed. Such a flexible element precludes the need for approximation of the variables with regard to the indefinite position of the boundary. This element is split into two when its dimension becomes unduly large compared to the unchanging elements. The need for such a splitting is shown by an examination of the entries in the global matrix. Results of water profile as a wave runs up a structure are given. A brief history of the work on similar problems is outlined.
A comparative study of central and upwind difference schemes using the primitive variables
Authors: Timin Tsai , Esmail M. Nabil
Published: August 2005

The use of the velocity‐pressure formulation of the Navier‐Stokes equations for the numerical solution of fluid flow problems is favoured for free‐surface problems, more involved flow configurations, and three‐dimensional flows. Many engineering problems involve such features in addition to strong inertial effects. The computational instabilities arising from central‐difference schemes for the convective terms of the governing equations impose serious limitations on the range of Reynolds numbers that can be investigated by the numerical method. Solutions for higher Reynolds numbers Re > 1000 could be reached using upwind‐difference schemes. A comparative study of both schemes using a method based on the primitive variables is presented. The comparison is made for the model problem of the driven flow in a square cavity. Using a central scheme stable solutions of the pressure and velocity fields were obtained for Reynolds numbers up to 5000. The streamfunction and vorticity fields were calculated from the resulting velocity field and compared with previous solutions. It is concluded that total upwind differencing results in a considerable change in the flow pattern due to the false diffusion. For practical calculations, by a proper choice of a small amount of partial upwind differencing the vorticity diffusion near the walls and the global features of the solutions are not sigificantly altered.
Fundamental solutions and numerical methods for flow problems
Authors: Wu J. C.
Published: August 2005

An approach for the numerical solution of flow problems based on the concept of fundamental solutions of differential equations is described. This approach uses the finite element methodology but does not rely on the concept of variational principle or that of residuals. The approach is shown to be well‐suited for many types of flow problems. Various applications of this approach are discussed in this paper, with particular emphasis placed on the solution of potential flows and viscous flows containing appreciable regions of separation.
A Dorodnitsyn finite element formulation for laminar boundary layer flow
Authors: Fletcher C. A. J. , Fleet R. W.
Published: August 2005

The Dorodnitsyn boundary later formulation is given a finite element interpretation and found to generate very accurate and economical solutions when combined with an implicit, non‐iterative marching scheme in the downstream direction. The algorithm is of order (Δ2u, Δx) whether linear or quadratic elements are used across the boundary layer. Solutions are compared with a Dorodnitsyn spectral formulation and a conventional finite difference formulation for three Falkner‐Skan pressure gradient cases and the flow over a circular cylinder. With quadratic elements the Dorodnitsyn finite element formulation is approximately five times more efficient than the conventional finite difference formulation.
A method for computing three dimensional flows using non‐orthogonal boundary‐fitted co‐ordinates
Authors: Maliska C. R. , Raithby G. D.
Published: August 2005

For three‐dimensional fluid flows in complex geometries, it is convenient to make predictions using a non‐orthogonal boundary‐fitted mesh. The present paper describes an economical method of solving the equations of motion for two and three dimensional problems using such meshes. The locations on the mesh at which the depenent variables are calculated, and the methods used to solve the equations, are key issues in the development of a successful algorithm; these are discussed in the present paper. Results obtained when the proposed method is applied to several problems are also described. The method is intended for flows in which compressibility effects do not dominate.
A single cell high order scheme for the convection‐diffusion equation with variable coefficients
Authors: Gupta Murli M. , Manohar Ram P. , Stephenson John W.
Published: August 2005

A new finite difference scheme for the convection‐diffusion equation with variable coefficients is proposed. The difference scheme is defined on a single square cell of size 2h over a 9‐point stencil and has a truncation error of order h4. The resulting system of equations can be solved by iterative methods. Numerical results of some test problems are given.
Numerical predictions of flows over backward‐facing steps
Authors: Hackman L. P. , Raithby G. D. , Strong A. B.
Published: August 2005

Predictions are reported for two‐dimensional, steady, incompressible flows over rearward‐facing steps for both laminar and turbulent conditions. The standard k‐ϵ turbulence model was used for the turbulent flow. Attention was focused on obtaining accurate solutions to the differential equations. It is concluded that some of the serious discrepancies that have occurred between prediction and observation, and attributed in earlier studies to the inadequacy of the turbulence model, may have been due to the inaccuracy of the solution.
A mixed algorithm for the calculation of rapidly varying fluid flows: The impulsively started circular cylinder
Authors: Rinaldo Andrea , Giorgini Aldo
Published: August 2005

A mixed discrete Fourier transform‐Finite difference algorithm is developed and used for the calculation of rapidly changing viscous fluid flows past a circular cylinder. The numerical approach has been designed to overcome certain difficulties arising for high Reynolds number simulations. The foremost advantage of the technique lies in its fast calculations of the convolution sums portraying the convective terms of the governing equations. Third‐order spatial discretizations and fourth‐order time marching are implemented.

New schemes are proposed for the boundary conditions at the solid wall and at large distances. The techniques are tested on a case study with other schemes (summarized by Roache1) in order to obtain an optimal choice. Definite indications on the stability and accuracy of boundary condition schemes are achieved. Support for the statement of dominant importance of boundary conditions is also given.

A comparison of computational results with experimental data is presented for the case study of the flow past an impulsively started cylinder at Reynolds number 20.

The time development of the symmetrical zone of recirculation, which is formed at an early stage of the flow, has been studied for 300 ≤ Re ≤ 9500 by means of the proposed algorithm. Computational results, comparisons with experimental data2 and discussion of upper limits of validity of the procedure will be presented in a companion paper.
Stream function finite element solution of Stokes flow with penalties
Authors: Carey Graham F. , Utku Mehmet
Published: August 2005

A finite element method for solution of the stream function formulation of Stokes flow is developed. The method involves complete cubic non‐conforming (C0) triangular Hermite elements. This element fails the patch test. To correct the element and produce a convergent method we employ a penalty method to weakly enforce the desired continuity constraint on the normal derivative across the inter‐element boundaries. Successful use of the method is demonstrated to require reduced integration of the inter‐element penalty with a 1‐point Gauss rule. Error estimates relate the optimal choice of penalty parameter to mesh size and are corroborated by numerical convergence studies. The need for reduced integration is interpreted using rank relations for an associated hybrid method.
On the critical problem of F. D. pressure treatment for laminar flows confined by permeable walls
Authors: Albanese R. , Grasso F. , Meola C.
Published: August 2005

In the present work the viscous (low Reynolds) flow in plane ducts confined by permeable walls has been studied. A simple model of the filtrating walls has been used, with the normal velocity component proportional to the pressure jump across the wall, resulting in a non‐standard boundary value Navier‐Stokes problem.

A critical analysis of the appropriate boundary condition and pressure problem has led to the conclusions of employing a simple explicit finite volume approach, and of avoiding the use of higher order finite difference schemes. In this paper a special emphasis on the structure of the involved computational matrices has been given to illustrate the chosen algorithm. The latter yields a steady state solution that is second order accurate in space, and it has an accuracy in time of order ≤ Δt (the time step), due to the explicit treatment of the velocity boundary conditions along the membrane. The model has been tested to study the effects of the inlet/outlet conditions, Reynolds number and filtrating wall constant.
A numerical technique for three‐dimensional compressible boundary layers
Authors: Wadia Aspi Rustom
Published: August 2005

The non‐linear two‐point boundary value problem for three‐dimensional compressible boundary layers is solved through the application of a boundary value technique for a range of parameters characterizing the nature of stagnation point flows. The analytical boundary conditions, at infinity, are applied at the edge of the computational mesh with iterations on the size of the domain. The solutions obtained show excellent agreement with the established similarity solutions for three‐dimensional flows. The present method has the potential advantage of yielding the wall values of f″w, g″w and θ′w as a part of the solution, contrary to the previously used ‘shooting’ methods. The algorithm is computationally simple and numerically stable and extremely suitable for engineering design applications.
The group velocity of some numerical schemes
Authors: Cathers B. , O'Connor B. A.
Published: August 2005

The performances of various numerical schemes used to model hyperbolic/parabolic equations have been studied by the calculation of their numerical group velocities. Numerical experiments conducted with one dimensional linear and quadratic Lagrangian finite elements with a Crank‐Nicolson finite differencing in time confirm the results of the analysis. The group velocity analysis supplements the well‐known amplitude and phase portraits introduced by Leendertse1 and helps explain the occurrence and behaviour of numerical oscillations in both finite difference and finite element schemes.
Application of the two‐equation k‐ϵ turbulence model to a two‐dimensional, steady, free surface flow problem with separation
Authors: Chapman Raymond S. , Kuo Chin Y.
Published: August 2005

An application of the depth‐integrated k‐ϵ turbulence model is presented for separated flow in a wide, shallow, rectangular channel with an abrupt expansion in width. The well‐known numerical problems associated with the use of upwind and central finite differences for convection are overcome by the adoption of the spatially third‐order accurate QUICK finite difference technique. Results show that modification of the depth‐integrated k‐ϵ turbulence closure model for streamline curvature leads to significant improvement in the agreement between model predictions and experimental measurements.
Numerical and experimental study of driven flow in a polar cavity
Authors: Fuchs L. , Tillmark N.
Published: August 2005

A numerical and an experimental study of the flow of an incompressible fluid in a polar cavity is presented. The experiments included flow visualization, in two perpendicular planes, and quantitative measurements of the velocity field by a laser Doppler anemometer. Measurements were done for two ranges of Reynolds numbers; about 60 and about 350. The stream function‐vorticity form of the governing equations was approximated by upwind or central finite‐differences. Both types of finite‐difference approximations were solved by a multi‐grid method. Numerical solutions were computed on a sequence of grids and the relative accuracy of the solutions was studied. Our most accurate numerical solutions had an estimated error of 0.1 per cent and 1 per cent for Re = 60 and Re = 350, respectively. It was also noted that the solution to the second order finite difference equations was more accurate, compared to the solution to the first order equations, only if fine enough meshes were used. The possibility of using extrapolations to improve accuracy was also considered. Extrapolated solutions were found to be valid only if solutions computed on fine enough meshes were used. The numerical and the experimental results were found to be in very good agreement.
A method for predicting unsteady potential flow about an aerofoil
Authors: Vezza M. , Galbraith R. A. McD.
Published: August 2005

A new model is presented for the calculation of the incompressible, inviscid flow around an arbitrary aerofoil undergoing unsteady motion. The technique was developed from the steady flow algorithm of Leishman and Galbraith1 in which use was made of a linear distribution of panel vorticity. The procedure is in the same class as that of Basu and Hancock2 but, because of the particular approach to the manner of specifying the shed vorticity, only a set of linear simultaneous equations needs be solved, unlike the method of Reference 2, complicated by the necessary solution of a quadratic.

A brief history of unsteady flow modelling is given in the introduction, followed by the mathematical details of the current method. Results are presented and discussed for a number of cases which clearly illustrate relevant characteristics of unsteady flow.
Finite element analysis of incompressible laminar boundary layer flows
Authors: Kim Sang‐Wook , Payne F. R.
Published: August 2005

A numerical procedure was developed to solve the two‐dimensional and axisymmetric incompressible laminar boundary layer equations using the semi‐discrete Galerkin finite element method. Linear Lagrangian, quadratic Lagrangian, and cubic Hermite interpolating polynomials were used for the finite element discretization; the first‐order, the second‐order backward difference approximation, and the Crank‐Nicolson method were used for the system of non‐linear ordinary differential equations; the Picard iteration and the Newton‐Raphson technique were used to solve the resulting non‐linear algebraic system of equations. Conservation of mass is treated as a constraint condition in the procedure; hence, it is integrated numerically along the solution line while marching along the time‐like co‐ordinate. Among the numerical schemes tested, the Picard iteration technique used with the quadratic Lagrangian polynomials and the second‐order backward difference approximation case turned out to be the most efficient to achieve the same accuracy. The advantages of the method developed lie in its coarse grid accuracy, global computational efficiency, and wide applicability to most situations that may arise in incompressible laminar boundary layer flows.
Added mass for fluid‐structure vibration problems
Authors: Jennings Alan
Published: August 2005

The problem of computing the vibration modes of a structure vibrating in a fluid is examined with specific application to ship hulls. In particular, methods which take proper account of the three‐dimensional nature of the water movements are described. Fluid singularities involving either a line doublet on the intersection of the water surface with the plane of symmetry or distributed sources on retracted boundaries are particularly effective at modelling the fluid flow and appear to give better numerical efficiencies than finite element or boundary element methods. The effects of the extra coefficients in the mass matrix arising from added mass on the various methods of eigensolution are discussed.
A coupled solution of the vorticity–velocity formulation of the incompressible Navier–Stokes equations
Authors: Farouk Bakhtier , Fusegi Toru
Published: August 2005

A relatively novel formulation of the Navier‐Stokes equations is used for obtaining solutions of two dimensional incompressible fluid flow and convective heat transfer problems. A vorticity transport equation along with two Poisson equations for the velocity components and the energy equation are solved by a finite difference scheme. A coupled solution procedure is used for solving simultaneously the dependent variables along a line, using a block tridiagonal matrix algorithm. The formulation is found to be stable and has features that may be desirable for solving a wide variety of flow and heat transfer problems.
A three‐dimensional simulation of wind flow around buildings
Authors: Hanson T. , Summers D. M. , Wilson C. B.
Published: August 2005

The steady Navier–Stokes equation is solved to simulate the wind‐flow environment of three‐dimensional configurations of buildings. The method assumes an incident wind described by a power‐law velocity profile. A new method for controlling the two‐part nested solution iteration is introduced. The simulation is compared to some published wind‐tunnel measurements.
Numerical solution of the St. Venant equations with the MacCormack finite‐difference scheme
Authors: Garcia Reinaldo , Kahawita Rene A.
Published: August 2005

This paper describes the use of the MacCormack explicit time‐spilitting scheme in the development of a two‐dimensional (in plan) hydraulic simulation model that solves the St. Venant equations. Various tests devised to assess the performance of the method have been performed and the results are reported.

Finally, two industrial applications of the model are presented. The method has been found to be computationally efficient and warrants further development.
An efficient algorithm for solving the incompressible fluid flow equations
Authors: Theodossiou V. M. , Sousa A. C. M.
Published: August 2005

The present paper reports on a modified pressure implicit predictor corrector type scheme for solving the flow governing equations, in which a consistent formulation is combined with a multi‐grid solver for the pressure correction. In addition a parabolic sublayer (PSL) approach for the treatment of the flow in the vicinity of solid walls is critically evaluated in terms of accuracy and computational efficiency. The lid‐driven cavity flow is chosen as the test case and results are presented for Reynolds numbers ranging from 100 to 1000. Predictions with the proposed scheme indicate substantial computational savings and fairly good agreement when compared with previous work. The PSL approach reduces the computing time, but with increasing Reynolds numbers the accuracy of the solutions tends to deteriorate.
Magnetohydrodynamic flow in an infinite channel
Authors: Sezgin Münevver
Published: August 2005

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed to finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the nonconducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.
A boundary collocation method for the solution of a flow problem in a complex three‐dimensional porous medium
Authors: Levin D. , Tal A.
Published: August 2005

A flow problem in a complex three‐dimensional domain with a free surface and mixed‐type boundary conditions is solved by the boundary collocation method. The solution is expressed as a combination of source functions distributed all around the domain close to the boundary, plus a special basis function to take care of a corner singularity. The resulting procedure is compared with the boundary integral elements method and is found to be simpler and more flexible to implement and faster to compute.
Internal fe approximation of spaces of divergence‐free functions in three‐dimensional domains
Authors: Křižek Michal , Neittaanmäuki Pekka
Published: August 2005

The space of divergence‐free vector functions with vanishing normal flux on the boundary is approximated by subspaces of finite elements having the same property. An easy way of generating basis functions in these subspaces is shown.
Numerical solution of the transport equation for passive contaminants in three‐dimensional complex terrains
Authors: Glekas J. , Bergeles G. , Athanassiadis N.
Published: August 2005

This paper presents a numerical solution of the transport equation for heat and species in complex three‐dimensional spaces. The solution domain of the equation is transformed into a cube, as also is the governing equation; the resultant equation is solved in the transformed space via a finite difference technique. The validity of the developed computer code is tested by predicting test cases for which either analytical or reliable experimental results exist. Results are also presented for the rate of convergence of the method and the computer storage requirements, from which the validity, the flexibility and the economy of the developed method are proved for flows in real three‐dimensional complex terrains.
The consistent Galerkin FEM for computing derived boundary quantities in thermal and or fluids problems
Authors: Gresho Philip M. , Lee Robert L. , Sani Robert L. , Maslanik Miriam K. , Eaton Brian E.
Published: August 2005

A consistent, accurate and reasonably simple method of obtaining derived quantities when the conventional Galerkin finite element method (GFEM) is used to obtain the primary quantities is defined and demonstrated, both theoretically and numerically.
Application of the centre implicit method for investigation of pressure transients in pipelines
Authors: Tan J. K. , Ng K. C. , Nathan G. K.
Published: August 2005

This paper investigates the application of the centre implicit method for the determination of the pressure transient in a pipeline, and compares the results with those obtained using the method of characteristics and an experimental investigation. The study shows that there are unique values for the stability criterion (ratio of the linear and time increments) and the artificial viscosity term (a damping factor) used in the numerical computation. The time step and the number of nodes required for the accuracy of the method have been considered. The centre implicit method can be readily adapted to transient flow with variable wave speed provided the established conditions are used.
Periodic solutions of rigid body–viscous flow interaction
Authors: Pattani P. G. , Olson M. D.
Published: August 2005

This paper describes the work on extending the finite element method to cover interactions between a viscous flow and a moving body. The problem configuration of interest is that of an arbitrarily shaped body undergoing a simple harmonic motion in an otherwise undisturbed incompressible fluid. The finite element modelling is based on a primitive variables representation of the Navier‐Stokes equations using curved isoparametric elements. The non‐linear boundary conditions on the moving body are obtained using Taylor series expansion to approximate the velocities at the fixed finite element grid points. The method of averaging is used to analyse the resulting periodic motion of the fluid. The stability of the periodic solutions is studied by introducing small perturbations and applying Floquet theory. Numerical results are obtained for several example body shapes and compared with published experimental results. Good agreement is obtained for the basic non‐linear phenomenon of steady streaming.
Three‐dimensional hydrodynamics on finite elements. Part I: Linearized harmonic model
Authors: Lynch Daniel R. , Werner Francisco E.
Published: August 2005

The linearized three‐dimensional hydrodynamic equations are solved numerically for periodic motions, subject to a linear slip condition at the bottom. The structure of the linearized equations allows an exact uncoupling of the horizontal and vertical computations, so that they may be achieved sequentially rather than simultaneously, and without iteration. The solution strategy involves simple horizontal C° finite elements for the description of free surface elevation. Vertical variations in velocity may be treated analytically for some special variations of viscosity with depth; more generally the finite element method is employed with one‐dimensional linear elements. Because of the uncoupling, the entire three‐dimensional solution scales as a two‐dimensional vertically averaged problem. The limiting two‐dimensional problem may be solved as a Helmholtz‐type problem for elevation alone, using established techniques.

Solutions for test problems are compared with known analytic solutions. Some simple gridding rules are established for the vertical discretization. Finally, a field application is shown involving the tidal response of the Lake Maracaibo (Venezuela) system.
Fast finite difference solution for steady‐state Navier‐Stokes equations using the BID method
Authors: Mittal R. C. , Sharma P. K.
Published: August 2005

A first biharmonic boundary value problem is obtained by combining the coupled steady‐state Navier‐Stokes equations in their stream‐function‐vorticity formulation. This biharmonic boundary value problem is solved by a fast biharmonic solver developed by the authors wherein the idea of preconditioned conjugate gradient method is used. The biharmonic driver (BID) method using this solver has been found fast converging, and produces accurate results up to moderately large Reynolds numbers. Also, the mesh size does not affect the convergence rate.
Computer simulation of turbulence in internal combustion engines
Authors: Shah P. , Markatos N. C.
Published: August 2005

The paper presents two‐ and three‐dimensional computations of the in‐cylinder turbulent flow in a diesel engine. The mathematical formulation is presented first, with emphasis on the modifications made to the standard k‐ε model of turbulence, to account for rapid compression/expansion, and on the k‐w model also used in the computations. Then, the results of two‐and three‐dimensional transient calculations are presented and compared with experimental data. It is realized that two‐dimensional computations may be of little value to real engines, which would probably require three‐dimensional analyses. However, two‐dimensional studies are still useful in allowing the testing of new ideas easily and economically. It is concluded that the standard k‐ε model may lead to poor predictions when used for internal combustion (IC) engine simulations, and that the modified model leads to more reasonable length‐scale distributions, and it improves significantly the overall agreement of velocity predictions with experiment. The effect of the k‐ε modification is apparent in both the two‐ and three‐dimensional simulations. It is also demonstrated that the k‐w model provides better turbulence predictions than the unmodified k‐ε model, for the cases considered, and that a similar modification of the k‐w model, to account for rapid compression/expansion, might improve its predictions even further.
Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier‐Stokes equations
Authors: Hughes Thomas J. R.
Published: August 2005

SUPG methods were originally developed for the scalar advection‐diffusion equation and the incompressible Navier‐Stokes equations. In the last few years successful extensions have been made to symmetric advective‐diffusive systems and, in particular, the compressible Euler and Navier‐Stokes equations. New procedures have been introduced to improve resolution of discontinuities and thin layers. In this paper a brief overview is presented of recent progress in the development and understanding of SUPG methods.
Forces on oscillating bodies in viscous fluid
Authors: Pattani P. G. , Olson M. D.
Published: August 2005

This paper describes a method for determining the fluid forces on oscillating bodies in viscous fluid when the corresponding flow problem has been solved using the finite element method. These forces are characterized by the concept of added mass, added damping and added force. Numerical results are obtained for several example body shapes. Comparison is made with exact analytical results and other finite element results for the limiting cases of Stoke's flow and inviscid flow, and good agreement is obtained. The results for finite values of the body amplitude parameter β show the appearance of added force from the steady streaming component of the flow for asymmetric bodies. Results are also obtained for the associated flow where the fluid remote from a fixed body is oscillating.
Some fast 3D finite element solvers for the generalized Stokes problem
Authors: Cahouet J. , Chabard J.‐P.
Published: August 2005

This paper is devoted to a comparison of various iterative solvers for the Stokes problem, based on the preconditioned Uzawa approach. In the first section the basic equations and general results of gradient‐like methods are recalled. Then a new class of preconditioners, whose optimality will be shown, is introduced. In the last section numerical experiments and comparisons with multigrid methods prove the quality of these schemes, whose discretization is detailed.
Space conservation law in finite volume calculations of fluid flow
Authors: Demirdžić I. , Perić M.
Published: August 2005

In the numerical solutions of fluid flow problems in moving co‐ordinates, an additional conservation equation, namely the space conservation law, has to be solved simultaneously with the mass, momentum and energy conservation equations. In this paper a method of incorporating the space conservation law into a finite volume procedure is proposed and applied to a number of test cases. The results show that the method is efficient and produces accurate results for all grid velocities and time steps for which temporal accuracy suffices. It is also demonstrated, by analysis and test calculations, that not satisfying the space conservation law in a numerical solution procedure introduces errors in the form of artificial mass sources. These errors can be made negligible only by choosing a sufficiently small time step, which sometimes may be smaller than required by the temporal discretization accuracy.
The stability of spectral schemes for the large eddy simulation of channel flows
Authors: Leslie D. C. , Gao S.
Published: August 2005

In a spectral LES code it is not possible to treat the actual eddy viscosity implicitly. We have therefore examined the effect on stability of adding a constant pseudo‐viscosity to the implicit term and subtracting it from the explicit term: stability limits have been derived theoretically and verified computationally for two different treatments of the explicit term. We have also studied the effect of a stochastic temporal variation of the eddy viscosity.
Onset of transition in boundary layers
Authors: Herbert Thorwald
Published: August 2005

Laminar‐turbulent transition in boundary layers involves a cascade of weak and strong instabilities. In the model considered here the first instability occurs with respect to two‐dimensional TS waves and causes streamwise, nearly periodic concentrations of vorticity. Linear stability analysis of this periodic flow leads to Floquet systems of equations. These systems support different classes of three‐dimensional disturbances which may initiate different routes to transition. Numerical solutions by use of accurate spectral methods reveal the spectrum of eigenmodes, growth rates and disturbance velocities. The characteristics of this secondary instability are in good agreement with results of experiments and computer simulations of transition. Non‐linear self‐interaction of the rapidly growing three‐dimensional disturbances can sustain or enhance the vital periodic accumulations of spanwise vorticity once their amplitude exceeds some threshold. This feedback loop is considered to be the key to the transition process. Owing to the broad‐band nature of secondary instability, however, the prediction of transition in practice requires additional insight into the ‘natural’ disturbance background. The sensitivity of the transition process to initial data in a broad band of frequencies and spanwise wave numbers poses new challenges for non‐linear theories and numerical simulations.
The application of multigrid to Navier‐Stokes simulation of 3D flow in axial and radial flow turbomachinery
Authors: Dawes W. N.
Published: August 2005

This paper describes the implementation of a 3D Navier‐Stokes solver within the framework of a multigrid strategy. The ability of multigrid to improve and sustain code performance over a range of mesh sizes and a variety of difficult flow problems is investigated. The code is applied to the study of the shock‐induced boundary layer separation on a channel hump, the flow in a 2D cascade of compressor blades, the secondary flow development in a linear cascade of gas turbine blades and the Eckardt backswept centrifugal impeller. In each case multigrid leads to significantly improved performance.
On finite element approximations of the streamfunction‐vorticity and velocity‐vorticity equations
Authors: Gunzburger Max D. , Peterson Janet S.
Published: August 2005

We consider finite element methods for vorticity formulations of viscous incompressible flows. In two‐dimensional settings the familiar streamfunction‐vorticity formulation is examined. We focus on its accuracy, especially when using low‐order elements, and on its use with a variety of boundary conditions and in multiply connected domains. In three dimensions the velocity‐vorticity formulation is shown to be preferable, and a promising algorithm using this formulation is presented. We close by considering the recovery of the pressure field once the streamfunction or velocity fields are known. In particular we describe and analyse an algorithm for recovering the pressure which is based on well known methods for the primitive variable formulation and which requires no boundary conditions on the pressure at solid walls.
Petrov‐Galerkin methods on multiply connected domains for the vorticity‐stream function formulation of the incompressible Navier‐Stokes equations
Authors: Tezduyar T. E. , Glowinski R. , Liou J.
Published: August 2005

In this paper we present streamline‐upwind/Petrov‐Galerkin finite element procedures for two‐dimensional fluid dynamics computations based on the vorticity‐stream function formulation of the incompressible Navier‐Stokes equations. We address the difficulties associated with the convection term in the vorticity transport equation, lack of boundary condition for the vorticity at no‐slip boundaries, and determination of the value of the stream function at the internal boundaries for multiply connected domains. The proposed techniques, implemented within the framework of block‐iteration methods, have successfully been applied to various problems involving simply and multiply connected domains.
Simple high‐accuracy resolution program for convective modelling of discontinuities
Authors: Leonard B. P.
Published: August 2005

For steady multi‐dimensional convection, the QUICK scheme has several attractive properties. However, for highly convective simulation of step profiles, QUICK produces unphysical overshoots and a few oscillations, and this may cause serious problems in non‐linear flows. Fortunately, it is possible to modify the convective flux by writing the ‘normalized’ convected control‐volume face value as a function of the normalized adjacent upstream node value, developing criteria for monotonic resolution without sacrificing formal accuracy. This results in a non‐linear functional relationship between the normalized variables, whereas standard methods are all linear in this sense. The resulting Simple High‐Accuracy Resolution Program (SHARP) can be applied to steady multi‐dimensional flows containing thin shear or mixing layers, shock waves and other frontal phenomena. This represents a significant advance in modelling highly convective flows of engineering and geophysical importance. SHARP is based on an explicit, conservative, control‐volume flux formulation, equally applicable to one‐, two‐, or three‐dimensional elliptic, parabolic, hyperbolic or mixed‐flow regimes. Results are given for the bench‐mark purely convective oblique‐step test. The monotonic SHARP solutions are compared with the diffusive first‐order results and the non‐monotonic predictions of second‐ and third‐order upwinding.