bluuebunny/crossref_metadata_2025 huggingface dataset:
A semi‐implicit collocation method: Application to thermal convection in 2D compressible fluids
Authors: Gauthier Serge
Published: August 2005

A semi‐implicit pseudo‐spectral collocation method using a third‐order Runge‐Kutta numerical scheme for the full Navier‐Stokes equations is described. The Courant‐Friedrichs‐Lewy condition is overcome by the implicit handling of a diffusive term, as suggested by Harned and Kerner. All such terms are solved with an iterative scheme in the Fourier space. Simulation of thermal convection in 2D compressible fluids is made by expanding variables on a Fourier‐Chebyshev basis. We give some examples of sub‐ and supersonic steady solutions in the case where the heat flux at the upper boundary is governed by a black body.
The application of the method of characteristics for the simulation of nearly horizontal flow in two and three spatial dimensions
Authors: Matsoukis P. F. C.
Published: August 2005

Certain free surface flows exhibit in nature negligible vertical accelerations and as a result the pressure within the fluid remains hydrostatic. The method of characteristics is developed as a solution technique for the integration of the partial differential equations describing this kind of flow. The equations are integrated over the depth to provide a two‐dimensional model which is then tested and validated by comparing its results with tide‐induced flows occurring in a number of cases where either analytical or observational data are available for comparison. On the basis of the results of the 2D model, a finite difference 3D model is developed which provides the values of the unknown velocities u, v and w along the three axes x, y and z. This combined 2D–3D model is verified by applying it in cases of wind‐induced flow inside closed or open basins for which the classical Ekman solution may be used as a testing means.
A stable algorithm for bed friction in three‐dimensional shallow sea modal models
Authors: Davies A. M. , Aldridge J. N.
Published: August 2005

The mathematical formulation of a three‐dimensional shallow sea model using a modal expansion in the veitical is briefly described.

The importance of the time discretization of the vertical diffusion term and bottom friction term is discussed in some detail. Both stability theory and numerical calculations show the importance of time centring or evaluating the modal form of the viscosity term at the higher time step in order to develop a numerically efficient algorithm. Similar analysis and calculations show that in shallow water it is essential to time centre or evaluate bottom friction at the higher time step. In the case of linear bottom friction it is shown that this condition can be readily accomplished. However, using a quadratic friction formulation (a more physically realistic form), this cannot be readily achieved. A new algorithm is presented whereby a stable solution can be obtained even in shallow water using quadratic bottom friction.
An improved finite difference scheme for a Newtonian jet swell problem
Authors: Yu Tsai‐An , Liu Ta‐Jo
Published: August 2005

The finite difference scheme developed by Liu et al. for the Newtonian jet swell problem has been improved: an algebraic approach has been adopted for the numerical mapping; a new formulation for free surface iteration has been proposed; the discrete flow equations have been solved by a combination of the successive line underrelaxation method and the Picard method. With these modifications we are capable of achieving more accurate numerical solutions and a substantial saving in computing time.
A simple and efficient BEM algorithm for planar cavity flows
Authors: Wrobel L. C.
Published: August 2005

This paper presents a boundary element formulation for solution of planar Riabouchinsky cavity flow problems. An iterative procedure for adjusting the free surface position is developed and shown to be stable and convergent. Numerical results are compared with finite difference and finite element solutions, showing the superior accuracy of the BEM models.
Prediction of three‐dimensional general shape extrudates by an implicit iterative scheme
Authors: Legat Vincent , Marchal Jean‐Marie
Published: August 2005

This paper presents a numerical technique for solving three‐dimensional free surface problems in extrusion applications. The method is fully implicit in the sense that a Newton‐Raphson scheme is applied on all variables, and geometrically general. In particular, the die section shape may be complex and contains multiple corners: very few restrictions apply on the mesh generation because the method does not require the nodes to be located on straight lines (spines). A clear distinction is introduced between the directions associated with the kinematic condition and the remeshing rules. As a difference with respect to earlier publications, these concepts are handled separately. Only Stokes problems are solved in this paper and we have not introduced surface tension. Therefore corners in the die section propagate discontinuities in the extrudate shape, an a method for relocating corners without losing the quadratic convergence of the scheme is presented. Data structures used for the implementation are briefly discussed.

We present results on the extrusion of various profiles, including a rectangular die (a benchmark problem) and various complex sections containing multiple corners.
Finite element modelling of natural‐convection‐controlled change of phase
Authors: Usmani A. S. , Lewis R. W. , Seetharamu K. N.
Published: August 2005

The problem of phase change in the presence of natural convection has been investigated. A model has been proposed based on the treatment of the release/absorption of latent heat as a heat source/sink in combination with the standard Galerkin finite element method with a primitive variable formulation on a fixed grid. To demonstrate the capabilities of the model, three cases of phase change of an aluminium alloy in the presence of natural convection arc considered, i.e. solidification, melting and combined solidification and melting. The solidification of water in a square cavity is modelled as another example, taking into account the density extremum, and the results are compared with a previously published work.
Second‐order boundary element method calculations of hydrodynamic interactions between particles in close proximity
Authors: Chan Chiu Y. , Beris Antony N. , Advani Suresh G.
Published: August 2005

A second‐order boundary element technique was developed to simulate the 3D hydrodynamic interactions between multiple particles of arbitrary shape. This paper reports the results of an extensive validation procedure aimed at demonstrating the convergence characteristics of the technique, especially in cases where the particles are in close proximity. The quadratic elements are superior to the lower‐order elements in terms of accuracy, computer storage and CPU time required, thus resulting in a significant improvement in the overall computational efficiency. Superparametric discretization improves the accuracy over isoparametric discretization but lowers the convergence rate of the method. When the interparticle gap becomes very small (less than 1% of the particle radius), the numerical solution diverges owing to inaccurate determination of the element contributions in the gap region. An adaptive subdomain integration scheme was developed that dramatically improved the integration accuracy and provided convergent solutions for problems of very small gaps down to 0–01% of the particle diameter.
A finite difference method for 3D flows about bodies of complex geometry in rectangular co‐ordinate systems
Authors: Miyata Hideaki , Yamada Yoshihiro
Published: August 2005

A finite difference simulation method is developed for 3D flow about a body of complex geometry. The Navier–Stokes equation is approximated by a high‐order‐accurate difference scheme in the framework of rectangular co‐ordinate systems. The configuration of the 3D body is represented by use of both surface porosity and volume porosity and the no‐slip body boundary conditions are approximately implemented on the boundary cells. The validity of the method is demonstrated by a numerical test of flow past a sphere at a Reynolds number of 1000. The complicated structure of separated vortices is well revealed by this test computation. The versatility of the method is shown by application to an ocean‐engineering problem of flow about a bay with an island.
Boundary element method for long‐time water wave propagation over rapidly varying bottom topography
Authors: Nachbin A. , Papanicolaou G. C.
Published: August 2005

We study numerically the linear water wave equations for shallow channels with rapidly varying bottom topography. We do not use the shallow water approximation because it is not valid when the bottom is rapidly varying. We use the boundary element method because it allows accurate tracking of the surface waves for long times. We present the results of a range of numerical validation experiments and a comparison between propagation over a periodic and a random rough bottom topography.
A problem of heat and mass transfer: Proof of the existence condition by a finite difference method
Authors: Gaultier M. , Lezaun M. , Vadillo F.
Published: August 2005

We solve by a finite difference method a system of simultaneous non‐linear partial differential equations which modelizes the transfer of heat and mass when a fluid evaporates from the hot wall and condenses on the cold wall of an upright rectangular cavity. The need to verify a certain condition associating the physical parameters of the fluid for the existence of steady state solutions is proved.
Numerical problems in simulating tidal flows with a frictional‐velocity‐dependent eddy viscosity and the influence of stratification
Authors: Davies Alan M.
Published: August 2005

The paper deals with the accurate determination of tidal current profiles in both homogeneous and stratified regions when a no‐slip condition is used at the seabed with a flow‐dependent eddy viscosity related to the depth‐mean current or the bed frictional velocity.

Calculations show that it is essential to accurately resolve the high‐shear region which occurs at the bed and across the pycnocline/thermocline in the case of stratified flow. A computationally accurate and economic method of resolving these regions is demonstrated using the Galerkin method with a set of basis functions designed to accurately reproduce the high‐shear layers which occur in these regions.

With a flow‐independent eddy viscosity a stability analysis can be readily performed and an unconditionally stable algorithm developed. However, with a flow‐dependent viscosity, in particular a viscosity computed from the frictional velocity, a non‐linear numerical instability can occur. A method of maintaining numerical stability in this case is also described.

The importance of near‐bed resolution to the computed value of the frictional velocity is demonstrated and its influence on the total tidal velocity profile is illustrated by a number of idealized calculations using various eddy viscosity formulations.

The influence of stratification on the computed tidal profiles is shown in the latter part of the paper.
Stratified flow under an ice keel—a numerical approach
Authors: Jameel M. I. , Rowe R. D. , Topham D. R.
Published: August 2005

A numerical investigation of the flow of two immiscible stratified fluids under an isolated keel has been undertaken. The investigation utilized the two‐dimensional Euler equations for incompressible flow, and the solution of these equations has been obtained by using the well‐known finite volume marker and cell approach. Experimental drag‐force measurements are also presented for a family of two‐dimensional topographic models of fixed height with increasing surface slopes in a two‐layer density system. The range of flow speeds explored covers the Froude number range from subcritical to fully supercritical. The drag force measurements are augmented by detailed observations of the interface distortion. The results clearly show large drag increases arising from the internal wave systems generated in the stratified flow. Very good agreement has been found between the experimental and numerical results for both the interface shape between the two fluids and the drag force on a variety of keels.
Comparisons of TVD schemes for turbulent transonic projectile aerodynamics computations with a two‐equation model of turbulence
Authors: Lin Herng , Chieng Ching‐Chang
Published: August 2005

The development of a computer program to solve the axisymmetric full Navier‐‐Stokes equations with k‐ε two‐equation model of turbulence using various total variation diminishing (TVD) schemes is the primary interest of this study. The computations are performed for the turbulent, transonic, viscous flow over a projectile with/without supporting sting at zero angle of attack. The predicted results, as well as the convergence characteristics, by various TVD schemes are compared with each other. The results show that the TVD schemes of higher‐order accuracy do have influence on the regions of high gradients such as shock, base corner and base flow. However, the schemes of third‐order accuracy do not necessarily improve the agreement with measured data (which is not available on the base) than that of second‐order accuracy, but surely generate apparent different result of base flow. The supporting sting on the projectile base will complicate the base flow and the existence of the sting will slightly shift the shock location and slightly change the flow field after the shock. More iteration steps are needed to get the converged results in the computation for the projectile with sting.
Indented plate problem revisited
Authors: Cebeci Tuncer , Chen Hsun H. , Majeski Judith A.
Published: August 2005

An interactive boundary layer method, together with the en approach to the calculation of transition, has been used to investigate the flow over an indented surface for which previous calculations had led to numerical instabilities. The results show two possible reasons for these numerical difficulties. First, it is shown that the gradients of wall shear become very steep at larger Reynolds numbers, particularly in the vicinity of reattachment. Extremely fine numerical grids are required to resolve these gradients. Secondly, and perhaps of greater importance, transition is shown to occur within the region of recirculation for all Reynolds numbers except for the lowest ones. Thus, the calculated flows downstream of the transition locations are fictitious and may be expected to deviate from the corresponding real flows by increasing amounts as the Reynolds number becomes larger. Calculations involving laminar, transitional and turbulent flow have been performed and confirm this conjecture.
Comparison of several open boundary numerical treatments for laminar recirculating flows
Authors: Kobayashi M. H. , Pereira J. C. F. , Sousa J. M. M.
Published: August 2005

Several open boundary conditions (OBCs) are compared and evaluated in the framework of the SIMPLE algorithm using staggered and non‐staggered grid systems. The benchmark laminar flow test cases used for the OBC evaluation are Poiseuille‐Benard flow in a channel and stratified backward‐facing step flow. The investigated OBCs are linear explicit step space extrapolation, Orlanski's monochromatic wave, and pressure extrapolation. Orlanski's and pressure extrapolation open boundary treatment for unsteady and steady flows, respectively, yield little reflection and has proved to be adequate for engineering calculations.
Some special purpose preconditioners for conjugate gradient‐like methods applied to CFD
Authors: Strigberger J. , Baruzzi G. , Habashi W. , Fortin M.
Published: August 2005

Standard preconditioners such as incomplete LU decomposition perform well when used with conjugate gradient‐like iterative solvers such as GMRES for the solution of elliptic problems. However, efficient computation of convection‐dominated problems requires, in general, the use of preconditioners tuned to the particular class of fluid‐flow problems at hand. This paper presents three such preconditioners. The first is applied to the finite element computation of inviscid (Euler equations) transonic and supersonic flows with shocks and uses incomplete LU decomposition applied to a matrix with extra artificial dissipation. The second preconditioner is applied to the finite difference computation of unsteady incompressible viscous flow; it uses incomplete LU decomposition applied to a matrix to which a pseudo‐compressible term has been added. The third method and application are similar to the second, only the LU decomposition is replaced by Beam‐warming approximate factorization. In all cases, the results are in very good agreement with other published results and the new algorithms are found to be competitive with others; it is anticipated that the efficiency and robustness of conjugate‐gradient‐like methods will render them the method of choice as the difficulty of the problems that they are applied to is increased.
A decoupling numerical method for fluid flow
Authors: Liu Shijie , Masliyah Jacob H.
Published: August 2005

A first‐order non‐conforming numerical methodology, Separation method, for fluid flow problems with a 3‐point exponential interpolation scheme has been developed. The flow problem is decoupled into multiple one‐dimensional subproblems and assembled to form the solutions. A fully staggered grid and a conservational domain centred at the node of interest make the decoupling scheme first‐order‐accurate. The discretization of each one‐dimensional subproblem is based on a 3‐point interpolation function and a conservational domain centred at the node of interest. The proposed scheme gives a guaranteed first‐order accuracy. It is shown that the traditional upwind (or exponentially weighted upstream) scheme is less than first‐order‐accurate. The pressure is decoupled from the velocity field using the pressure correction method of SIMPLE. Thomas algorithm (tri‐diagonal solver) is used to solve the algebraic equations iteratively. The numerical advantage of the proposed scheme is tested for laminar fluid flows in a torus and in a square‐driven cavity. The convergence rates are compared with the traditional schemes for the square‐driven cavity problem. Good behaviour of the proposed scheme is ascertained.
Numerical simulation of an isolated vortex and shear flow interaction
Authors: Antropov I. V.
Published: August 2005

A numerical solution for the Navier‐Stokes equations in the unbounded region is considered for the interaction of an isolated vortex and shear flow. A Chebyshey collocation method in space and finite‐difference method for temporal discretization are used. The results of the numerical experiments for the interaction are discussed. It is shown that shear flow can both increase and decrease the vortex dissipation rate.
Numerical analysis of wall pressure and heat flux fluctuations in shock‐turbulent‐boundary‐layer interaction
Authors: Mehta R. C. , Sastri V. M. K.
Published: August 2005

The unsteady, compressible, Reynolds‐averaged Navier‐Stokes equations are solved numerically for an oblique shock‐wave‐induced turbulent boundary layer sepration. For the freestream Mach number 6 and the freestream Reynolds number 66·1 × 106 m−1, a time‐dependent computation is performed, using MacCormack's explicit‐implicit finite difference method with 82 × 42 grid points. A two‐layer eddy viscosity turbulence model is employed in conjunction with a relaxation modification. Comparisons of the mean wall pressure and the mean heat transfer coefficient with the available experimental results are made and the evaluation of unsteady data for surface pressure and heat flux fluctuations is presented. It is found that the fluctuations in heat flux have qualitatively the same features as those of wall pressure but are different quantitatively.
On the accuracy and efficiency of CFD methods in real gas hypersonics
Authors: Drikakis D. , Tsangaris S.
Published: August 2005

A study of viscous and inviscid hypersonic flows using generalized upwind methods is presented. A new family of hybrid flux‐splitting methods is examined for hypersonic flows. The hybrid method is constructed by the superposition of the flux‐vector‐splitting (FVS) method and second‐order artificial dissipation in the regions of strong shock waves. The conservative variables on the cell faces are calculated by an upwind extrapolation scheme to third‐order accuracy. A second‐order‐accurate scheme is used for the discretization of the viscous terms. The solution of the system of equations is achieved by an implicit unfactored method. In order to reduce the computational time, a local adaptive mesh solution (LAMS) method is proposed. The LAMS method combines the mesh‐sequencing technique and local solution of the equations. The local solution of either the Euler or the NAVIER‐STOKES equations is applied for the region of the flow field where numerical disturbances die out slowly. Validation of the Euler and NAVIER‐STOKES codes is obtained for hypersonic flows around blunt bodies. Real gas effects are introduced via a generalized equation of state.
Treatment of vortex sheets for the transonic full‐potential equation
Authors: Dang T. Q.
Published: August 2005

This paper summarizes a combined analytical‐computational technique which models vortex sheets in transonic potential‐flow methods. In this approach, the inviscid nature of discontinuities across vortex sheets is preserved by employing the step function to remove singularities at these surfaces. The location and strength of the vortex sheets are determined by satisfying the flow‐tangency boundary condition and the vorticity transport equation. The theory is formulated for the general three‐dimensional case, but its application is confined to the problem of computing slipstreams behind propellers with free‐vortex blading in axisymmetric flows.
Solution of free‐boundary problems using finite‐element/Newton methods and locally refined grids: Application to analysis of solidification microstructure
Authors: Tsiveriotis K. , Brown R. A.
Published: August 2005

A new method is presented for the solution of free‐boundary problems using Lagrangian finite element approximations defined on locally refined grids. The formulation allows for direct transition from coarse to fine grids without introducing non‐conforming basis functions. The calculation of elemental stiffness matrices and residual vectors are unaffected by changes in the refinement level, which are accounted for in the loading of elemental data to the global stiffness matrix and residual vector. This technique for local mesh refinement is combined with recently developed mapping methods and Newton's method to form an efficient algorithm for the solution of free‐boundary problems, as demonstrated here by sample calculations of cellular interfacial microstructure during directional solidification of a binary alloy.
Numerical solution of axisymmetric cavity flows using the boundary element method
Authors: Wrobel L. C.
Published: August 2005

This paper presents a formulation of the boundary element method (BEM) for solution of axisymmetric cavity flow problems. The governing equation is written in terms of Stokes' stream function, requiring a new fundamental solution to be found. The iterative procedure for adjusting the free‐surface position is similar to that used for planar cavity flows. Numerical results are compared with finite difference and finite element solutions, showing the robustness of the BEM model.
New numerical method for partial differential equations. 1: Application to the diffusion equation
Authors: Hatzikonstantinou P.
Published: August 2005

In this paper a new, highly accurate method called PH is presented for the numerical integration of partial differential equations. The method is applied for the solution of the one‐dimensional diffusion equation. Upon integrating the equation within a subdomain of space and time using the prismoidal approximation, a three‐point implicit scheme is obtained with a truncation error of order O(k4, h6), where k and h represent the time and space steps respectively. The method is stable under the condition s = αk/h2 ⩽ S(δ), where the function S(δ) increases as the parameter δ decreases from 1/12 to negative values. In practice the method behaves as unconditionally stable upon choosing an appropriate value for δ. A new formula is also adopted for the implementation of a Neumann boundary condition, introducing a truncation error of order O(h4). Numerical solutions are obtained incorporating Dirichlet and Neumann boundary conditions. The results prove that our method is far more accurate than any other‐implicit or explicit method.
Optimal estimation of eddy viscosity for a quasi‐three‐dimensional numerical tidal and storm surge model
Authors: Lardner R. W. , Das S. K.
Published: August 2005

It is shown that the eddy viscosity profile in a quasi‐three‐dimensional numerical tidal and storm surge model can be estimated by assimilation of velocity data from one or more current meters located on the same vertical line. The computational model used is a simplified version of the so‐called vertical/horizontal splitting algorithm proposed by Lardner and Cekirge. We have estimated eddy viscosity both as a constant and as a variable parameter. The numerical scheme consists of a two‐level leapfrog method to solve the depth‐averaged equations and a generalized Crank‐Nicolson scheme to compute the vertical profile of the velocity field. The cost functional in the adjoint scheme consists of two terms. The first term is a certain norm of the difference between computed and observed velocity data and the second term measures the total variation in the eddy viscosity function. The latter term is not needed when the data are exact for the model but is necessary to smooth out the instabilities associated with ‘noisy’ data. It is shown that a satisfactory minimization can be accomplished using either the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) quasi‐Newton algorithm or Nash's truncated Newton algorithm. Very effective estimation of eddy viscosity profiles is shown to be achieved even when the amount of data is quite small.
Calculation of separated flow around multi‐element aerofoils using conformal mapping
Authors: Cao Jianfa
Published: August 2005

A method is presented to calculate the low‐speed incompressible separated flow around multi‐element aerofoils. The geometries of multi‐element aerofoils in the physical plane are completely arbitrary and are transformed into multiple circles in the computational plane by a conformal mapping technique. Jacob's model, which distributes sources on the separated surfaces of multi‐element aerofoils to simulate the effects of separation, is adopted here. The position of the separated point and the pressure on the surfaces of multi‐element aerofoils are calculated by iteratively coupling the potential flow and boundary layer. The effects of the boundary layer are simulated by modification of the boundary condition. All iterative procedures converge rapidly as a result of using the fast Fourier transform (FFT) technique.
Aspects of the transpiration model for aerofoil design
Authors: Yiu K. F. C. , Stow P.
Published: August 2005

Transpiration is a technique in which extra non‐physical normal flows are created on an aerofoil surface in order to form a new streamline pattern such that the surface streamlines no longer follow the aerofoil surface under inviscid flow. The transpiration model is an important technique adopted in aerofoil design either to avoid mesh regeneration when aerofoil profile co‐ordinates are adjusted or to find shape corrections in inverse design methods. A first‐order approximation (with respect to the normal streamline displacement) to the transpiration model is commonly adopted; it is shown that this can be a poor approximation especially in regions of high curvature. In this paper more accurate approximations are developed to address this problem and improve the accuracy.
On estimating pressure build‐up in severely accelerated, shocked flow
Authors: Cooke Charlie H.
Published: August 2005

A new approach to the calculation of the high pressures characterizing the flow field in front of a piston undergoing severe acceleration over the short term is presented. In contrast with previous approaches where the computational domain is altered but which stop short of transforming velocities, here the problem is solved in an accelerating non‐Euclidean co‐ordinate system where the piston is stationary. The method is applied to a study of the problem of premature sabot separation. Through use of Harten's second‐order‐accurate TVD scheme, flow simulations are performed for both 1D and 3D axisymmetric geometries. The simple 1D model gives pressure profiles surprisingly close to those of the more physical 3D model.
A second‐order radiation boundary condition for the shallow water wave equations on two‐dimensional unstructured finite element grids
Authors: Johnsen Matthias , Lynch Daniel R.
Published: August 2005

A second‐order radiation boundary condition (RBC) is derived for 2D shallow water problems posed in ‘wave equation’ form and is implemented within the Galerkin finite element framework. The RBC is derived by matching the dispersion relation for the interior wave equation with an approximate solution to the exterior problem for outgoing waves. The matching is correct to second order, accounting for curvature of the wave front and the geometry. Implementation is achieved by using the RBC as an evolution equation for the normal gradient on the boundary, coupled through the natural boundary integral of the Galerkin interior problem. The formulation is easily implemented on non‐straight, unstructured meshes of simple elements. Test cases show fidelity to solutions obtained on extended meshes and improvement relative to simpler first‐order RBCs.
A vertex‐centred finite volume method with shock detection
Authors: Crumpton P. I. , Shaw G. J.
Published: August 2005

This paper introduces a vertex‐centred finite volume method for compressible viscous flow incorporating a new shock detection procedure. The discretization is designed to be robust and accurate on the highly stretched and curved meshes necessary for resolving turbulent boundary layers around the leading edge of an aerofoil. Details of the method are described for two‐dimensional problems and the natural extension of three‐dimensional multiblock meshes is discussed. The shock detection procedure is used to limit the range of the shock‐capturing dissipation specifically to regions containing shocks. For transonic turbulent flow this is shown to improve the boundary layer representation significantly.
Analysis of periodic 3D viscous flows using a quadratic discrete Galerkin boundary element method
Authors: Chan Chiu Y. , Beris Antony N. , Advani Suresh G.
Published: August 2005

A discrete Galerkin boundary element technique with a quadratic approximatión of the variables was developed to simulate the three‐dimensional (3D) viscous flow established in periodic assemblages of particles in suspensions and within a periodic porous medium. The Batchelor's unit‐cell approach is used. The Galerkin formulation effectively handles the discontinuity in the traction arising in flow boundaries with edges or corners, such as the unit cell in this case. For an ellipsoidal dilute suspension over the range of aspect ratio studied (1 to 54), the numerical solutions of the rotational velocity of the particles and the viscosity correction were found to agree with the analytic values within 0.2% and 2% respectively, even with coarse meshes. In a suspension of cylindrical particles the calculated period of rotation agreed with the experimental data. However, Burgers' predictions for the correction to the suspension viscosity were found to be 30% too low and therefore the concept of the equivalent ellipsoidal ratio is judged to be inadequate. For pressure‐driven flow through a fixed bed of fibres, the prediction on the permeability was shown to deviate by as much as 10% from the value calculated based on approximate permeability additivity rules using the corresponding values for planar flow past a periodic array of parallel cylinders. These applications show the versatility of the technique for studying viscous flows in complicated 3D geometries.
On the group velocity of symmetric and upwind numerical schemes
Authors: Karni Smadar
Published: August 2005

Dissipative numerical approximations to the linear advection equation are considered with respect to their behaviour in the limit of weak dissipation. The context is wave propagation under typical far‐field conditions where grids are highly stretched and waves are underresolved. Three classes of schemes are analysed: explicit two‐level (i) symmetric and (ii) upwind schemes of optimal accuracy are considered as well as (iii) (symmetric) Runge‐Kutta schemes. In the far‐field the dissipation of all schemes diminishes. Group speeds of high‐frequency modes assume the incorrect sign and may admit ‘backward’ wave propagation if waves are not damped. A fundamental difference arises between the symmetric and upwind cases owing to the different rates at which the dissipation diminishes. In the upwind case, while the amount of damping per time step diminishes, the accumulative damping remains exponential in time. In the symmetric case the accumulative damping tends to unity, yielding in practice non‐dissipative schemes. In this light, parasitic modes constitute much less of a problem in the upwind case than in the symmetric case. Numerical tests confirm these findings.
Large‐scale computational fluid dynamics by the finite element method
Authors: Habashi W. G. , Robichaud M. , Nguyen V.‐N. , Ghaly W. S. , Fortin M. , Liu J. W. H.
Published: August 2005

Solution methods are presented for the large systems of linear equations resulting from the implicit, coupled solution of the Navier‐Stokes equations in three dimensions. Two classes of methods for such solution have been studied: direct and iterative methods.

For direct methods, sparse matrix algorithms have been investigated and a Gauss elimination, optimized for vector‐parallel processing, has been developed. Sparse matrix results indicate that reordering algorithms deteriorate for rectangular, i.e. M × M × N, grids in three dimensions as N gets larger than M. A new local nested dissection reordering scheme that does not suffer from these difficulties, at least in two dimensions, is presented. The vector‐parallel Gauss elimination is very efficient for processing on today's supercomputers, achieving execution rates exceeding 2.3 Gflops the Cray YMP‐8 and 9.2 Gflops on the NEC on SX3.

For iterative methods, two approaches are developed. First, conjugate‐gradient‐like methods are studied and good results are achieved with a preconditioned conjugate gradient squared algorithm. Convergence of such a method being sensitive to the preconditioning, a hybrid viscosity method is adopted whereby the preconditioner has an artificial viscosity that is gradually lowered, but frozen at a level higher than the dissipation introduced in the physical equations. The second approach is a domain decomposition one in which overlapping domain and side‐by‐side methods are tested. For the latter, a Lagrange multiplier technique achieves reasonable rates of convergence.
Numerical boundary conditions for unsteady transonic flow calculations
Authors: Jiang Hong , Lee B. H. K. , Wong Yau Shu
Published: August 2005

In calculations of transonic flows it is necessary to limit the domain of computation to a size that is manageable by computers. At the boundary of the computational domain, boundary conditions are required to ensure a unique solution. Since wave solutions exist in the unsteady transonic flow field, incorrect boundary conditions may result in spurious reflections from the computational boundary. This may introduce errors into the solution. To prevent the spurious reflections, absorbing boundary conditions are often used on the computational boundary. In this paper we describe a method to derive absorbing boudary conditions for transonic calculations. We demonstrate both theoretically and numerically that the use of the absorbing boundary conditions will reduce the spurious reflections in the calculation.
A multiblock Navier–Stokes algorithm using equal‐order quadratic finite elements
Authors: Hill D. L. , Baskharone E. A.
Published: August 2005

A new multiblock pressure‐based finite element algorithm has been developed. This methodology implements quadratic interpolation for both the elemental velocity and pressure fields. A direct streamline upwinding scheme previously developedby the authors is used to model the non‐linear inertia effects. Details of the algorithm and its multiblock foundation are provided along with validating test cases. The results presented clearly demonstrate the accuracy of this new approach and the differences in the pressure field for an element using quadratic versus the traditional bi linear approximation of the pressure field.
A composite multigrid method for calculating unsteady incompressible flows in geometrically complex domains
Authors: Zang Y. , Street R. L.
Published: August 2005

A time‐accurate, finite volume method for solving the three‐dimensional, incompressible Navier‐Stokes equations on a composite grid with arbitrary subgrid overlapping is presented. The governing equations are written in a non‐orthogonal curvilinear co‐ordinate system and are discretized on a non‐staggered grid. A semi‐implicit, fractional step method with approximate factorization is employed for time advancement. Multigrid combined with intergrid iteration is used to solve the pressure Poisson equation. Inter‐grid communication is facilitated by an iterative boundary velocity scheme which ensures that the governing equations are well‐posed on each subdomain. Mass conservation on each subdomain is preserved by using a mass imbalance correction scheme which is secondorder‐accurate. Three test cases are used to demonstrate the method's consistency, accuracy and efficiency.
A staggered spectral element model with application to the oceanic shallow water equations
Authors: Iskandarani Mohamed , Haidvogel Dale B. , Boyd John P.
Published: August 2005

A staggered spectral element model for the solution of the oceanic shallow water equations is presented. We introduce and compare both an implicit and an explicit time integration scheme. The former splits the equations with the operator‐integration factor method and solves the resulting algebraic system with generalized minimum residual (GMRES) iterations. Comparison of the two schemes shows the performance of the implicit scheme to lag that of the explicit scheme because of the unpreconditioned implementation of GMRES. The explicit code is successfully applied to various geophysical flows in idealized and realistic basins, notably to the wind‐driven circulation in the North Atlantic Ocean. The last experiment reveals the geometric versatility of the spectral element method and the effectiveness of the staggering in eliminating sprious pressure modes when the flow is nearly non‐divergent.
Modelling of the turbulent thermal boundary layer implemented in CFD code N3S and application to flows in heated and cooled rooms
Authors: Delenne B. , Manzoni D. , Pot G.
Published: August 2005

This paper presents an efficient finite element method for solving the unsteady Navier–Stokes equations for turbulent incompressible flow coupled with thermal problems. This method has been implemented in the N3S code, developed at Electricité de France. The time discretization is first described. We precise then the Chorin and the ‘projected Uzawa’ algorithms used for the Stokes problem. Recent improvements concerning the optimization of finite element calculations are also detailed. The second part deals with the modelling of the thermal boundary layer used to simulate walls with fixed temperature in turbulent flows. The differences with other modelling suggested in the literature are discussed. The last part presents some applications. EDF is involved in the conception of heating or cooling systems and numerical methods constitute a very useful tool to study the movements of air in habitations. The calculations are validated by comparisons with measurements.
Numerical study of radiative ignition of pyrolysing solid fuels
Authors: Durbetaki P. , Phuoc T. X. , Mathur M. P. , Ekmann J. M.
Published: August 2005

A numerical model of radiative ignition of pyrolysing solid fuels is developed. The model is one‐dimensional and transient. The following mechanisms are simultaneously accounted for: (i) the surface heat and mass transport, (ii) the surface oxidation chemical reaction, (iii) the in‐depth pyrolysis, (iv) the gas‐phase heating by absorption of the radiation and by heat conduction/convection from the solid surface, and (v) the gas‐phase chemical reaction. The solutions are obtained numerically with the method of lines. Using lignite and bituminous coal for the simulations, the results confirm that the pyrolysis products absorb a significant amount of the external radiation. Predictions of the ignition times show that both the surface ignition time and the gas‐phase ignition time decrease rapidly with increasing radiation intensities. A good agreement between predictions and experiments is obtained.

A sensitivity analysis is also carried out with the key kinetic parameters. This analysis establishes an upper limit for surface and pyrolysis activation energies and a lower limit for gas‐phase activation energy. Within these limits, the radiative ignition of coals appears as an integration of two consecutive ignition modes: the surface ignition occurs first, which is then followed by the gas‐phase ignition. Beyond these limits, the single gas‐phase ignition mode is the only ignition mode to prevail.
Numerical modelling of confined flow past a cylinder of square cross‐section at various orientations
Authors: Li Guoping , Humphrey Joseph A. C.
Published: August 2005

Results are presented for the unsteady, two‐dimensional flow and heat transfer due to a square obstruction of diameter d located asymmetrically between the parallel sliding walls of a channel with length‐to‐height ratio W/H = 6·44. Analysis is based on the numerical solution of spatially and temporally second‐order accurate finite difference approximations of the transport equations expressed in curvilinear co‐ordinates. Laminar, constant property flow is assumed for obstruction configurations in which the blockage ratio is d/H = 0·192, the nearest‐wall distances are g/d = 0·2, 0·5 and 1, the orientation angles are α=0°, 10° and 20° and the Reynolds numbers are Re=100, 500, and 1000. Preparatory testing of the numerical procedure was performed for a variety of documented flows to verify its physiconumerical accuracy and obtain estimates of the residual grid‐dependent uncertainties in the variables calculated. Heat transfer, drag and lift coefficients and Strouhal numbers for the present flow were finally calculated to within 4%–7% of their grid‐dependent values using non‐uniformly spaced grids consisting of (x=99, y=55) nodes. Above a critical value of the Reynolds number, which depends on the geometrical parameters, the flow is characterized by alternate vortex shedding from the obstruction top and bottom surfaces. Streamline, vorticity and particle streakline plots provide qualitative impressions of the unsteady vortical flow. Especially noteworthy are the extremes in the lift coefficient which ranges from large positive values for an obstruction with g/d=0·2 and α=10° to negative values for one with g/d=0·5 and α=0°. Both the drag and lift coefficients as well as the Strouhal number exhibit non‐monotonic variations with respect to the parameters explored. Asymmetries in the obstruction location and orientation account for relatively large vortex‐induced periodic variations in heat transfer, especially along the wall nearest the obstruction. Notable differences are also predicted for the heat transfer coefficients of the individual obstruction surfaces as a function of the orientation angle.
Numerical solution of coupled burgers equations in inhomogeneous form
Authors: Shankar Rama , Singh T. V. , Bassaif Anwar A.
Published: August 2005

A finite difference scheme based on the operator‐splitting technique with cubic spline functions is derived for solving the two‐dimensional Burgers equations in ‘inhomogeneous’ form. The scheme is of first‐order accuracy in time and second‐order accuracy in space direction and is unconditionally stable. The numerical results are obtained with severe/moderate gradients in the initial and boundary conditions and the steady state solutions are plotted for different values of the parameters. It is concluded that the resulting scheme works very well even in the case of very severe gradient in the solution. Also, the general nature of the scheme provides a wider application in the solution of non‐linear problems.
Fractional step method for solution of incompressible Navier‐Stokes equations on unstructured triangular meshes
Authors: Despotis G. K. , Tsangaris S.
Published: August 2005

In this paper an implicit fractional step method for the solution of the two‐dimensional, time‐dependent, incompressible Navier‐Stokes equations is presented. The current method was developed for use on an unstructured grid made up of triangles. The basic principles of this method are that the evaluation of the time evolution is split into intermediate steps and that for the spatial discretization of the flow equations a finite volume discretization on an unstructured triangular mesh is used. The present approach has been used to simulate viscous, laminar flows for various Reynolds numbers in test cases such as a backward‐facing step, a square cavity and a channel with wavy boundaries.
A variable boundary model of storm surge flooding in generalized curvilinear grids
Authors: Shi Fengyan , Sun Wenxin
Published: August 2005

In this paper, generalized 2D shallow sea dynamic equations in movable curvilinear co‐ordinates are derived. Through a differential co‐ordinate transformation a self‐adaptive grid is proposed to treat a continuously deforming lateral boundary and a kinematical boundary condition is adopted. The self‐adaptive grid method (SAM) is used to simulate numerically the storm surge flooding in the Bohai Sea on 23 April 1969, which was one of the largest storm surge inundations in China.
A numerical study of flow over a confined backward‐facing step
Authors: Barton I. E.
Published: August 2005

Numerical solutions using the SIMPLE algorithms for laminar flow over a backward‐facing step are presented. Five differencing schemes were used: hybrid; quadratic upwind (QUICK); second‐order upwind (SOUD); central‐differencing and a novel scheme named second‐order upwind biased (SOUBD). The SOUBD scheme is shown to be part of a family of schemes which include the central‐differencing, SOUD and QUICK schemes for uniform grids. The results of the backward‐facing step problem are presented and are compared with other numerical solutions and experimental data to evaluate the accuracy of the differencing schemes. The accuracy of the differencing schemes was ascertained by using uniform grids of various grid densities. The QUICK, SOUBD and SOUD schemes gave very similar accurate results. The hybrid scheme suffered from excessive diffusion except for the finest grids and the central‐differencing scheme only converged for the finest grids.