bluuebunny/crossref_metadata_2025 huggingface dataset:
Investigation of solution of Navier–Stokes equations using a variational formulation
Authors: Ecer A. , Rout R. K. , Ward P.
Published: August 2005

A variational formulation for the solution of two dimensional, incompressible viscous flows has been developed by one of the authors.1 The main objective of the present paper is to demonstrate the applicability of this approach for the solution of practical problems and in particular to investigate the introduction of boundary conditions to the Navier‐Stokes equations through a variational formulation. The application of boundary conditions for typical internal and external flow problems is presented. Sample cases include flow around a cylinder and flow through a stepped channel.

Quadrilateral, bilinear isoparametric elements are utilized in the formulation. A single‐step, implicit, and fully coupled numerical integration scheme based on the variational principle is employed. Presented results include sample cases with different Reynolds numbers for laminar and turbulent flows. Turbulence is modelled using a simple mixing length model. Numerical results show good agreement with existing solutions.
Formulation of a linear three‐dimensional hydrodynamic sea model using a Galerkin‐eigenfunction method
Authors: Davies A. M.
Published: August 2005

The three dimensional linear hydrodynamic equations which describe wind induced flow in a sea are solved using the Galerkin method. A basis set of eigenfunctions is used in the calculation. These eigenfunctions are determined numerically using an expansion of B‐splines.

Using the Galerkin method the problem of wind induced flow in a rectangular basin is examined in detail. A no‐slip bottom boundary condition with a vertically varying eddy viscosity distribution is employed in the calculation. With a low (of order 1 cm2/s) value of viscosity at the sea bed there is high current shear in this region. Viscosities of the order of 1 cm2/s) value of viscosity at the sea bed there is high current shear in this region. Viscosities of the order of 1 cm2/s near the sea bed together with high current shear in this region are physically realistic and have been observed in the sea.

In order to accurately compute the eigenfunctions associated with large (of order 2000 cm2/s at the sea surface to 1 cm2/s at the sea bed) vertical variation of viscosity, an expansion of the order of thirty‐five B‐splines has to be used. The spline functions are distributed through the vertical so as to give the maximum resolution in the high shear region near the sea bed.

Calculations show that in the case of a no‐slip bottom boundary condition, with an associated region of high current shear near the sea bed, the Galerkin method with a basis set of the order of ten eigenfunctions (a Galerkin‐eigenfunction method) yields an accurate solution of the hydrodynamic equations. However, solving the same problem using the Galerkin method with a basis set of B‐splines, requires an expansion of the order of thirty‐five spline functions in order to obtain the same accuracy.

Comparisons of current profiles and time series of sea surface elevation computed using a model with a slip bottom boundary condition and a model with a no‐slip boundary condition have been made. These comparisions show that consistent solutions are obtained from the two models when a physically relistic coefficient of bottom friction is used in the slip model, and a physically realistic bottom roughness length and thickness of the bottom boundary layer are employed in the no‐slip model.
Evaluation of tidal residual currents in a wide estuary, using a finite element method
Authors: Jenkins A. D.
Published: August 2005

From the linearized, time‐independent, constant depth, shallow water tidal equations in an f‐plane for a two‐layer estuary, two independent modal Helmholtz equations are derived. These modal equations are solved using a fifth‐degree finite element technique. The first and second space derivatives of the complex modal tidal elevations, and thus the modal currents and their first derivatives, are evaluated directly from the solution at each node of the finite element mesh.

The Stokes drift, which is the major part of the residual tidal flow, is evaluated from these nodal values of the currents and their derivatives. Good agreement is obtained with the exact analytical solution for a wedge‐shaped estuary with a wedge angle of π/3, using a mesh of 64 equilateral triangles with sides approximately 1/10 of the wavelength 2πC2/σ of a Kelvin wave solution for the short‐wavelength mode.
Numerical solution of viscous flows using integral equation methods
Authors: Bush M. B. , Tanner R. I.
Published: August 2005

A formulation of the boundary element method for the solution of non‐zero Reynolds number incompressible flows in which the non‐linear terms are lumped together to form a forcing function is presented. Solutions can be obtained at low to moderate Reynolds numbers. The method was tested using the flow of a fluid in a two‐dimensional converging channel (Hamel flow) for which an exact solution is available. An axisymmetric formulation is demonstrated by examining the drag experienced by a sphere held stationary in uniform flow. Performance of the method was satisfactory. New results for an axisymmetric free jet at zero Reynolds number obtained using the boundary element method are also included. The method is ideal for this type of free‐surface problem.
Numerical solutions for solute transport in unconfined aquifers
Authors: Guvanasen V. , Volker R. E.
Published: August 2005

Two numerical methods for solving the problem of solute transport in unsteady flow in unconfined aquifers are studied. They are the method of characteristics (MOC) based on the finite difference method (FDM), and the finite element method (FEM). The FEM is further subdivided into four schemes: moving mesh, pseudo‐Lagrangian (FEM1); stationary mesh, pseudo‐Lagrangian (FEM2); pseudo saturated‐unsaturated, Eulerian (FEM3); and non‐stationary element, Eulerian (FEM4).

Experiments on a one‐dimensional flow case are performed to illustrate the schemes and to determine the effect of discretization on accuracy. In two‐dimensional flow the above methods are compared with experimental results from a sand box model. Results indicate that for a similar degree of accuracy, the FEM requires less computational effort than the MOC. Among the four FEM schemes, FEM4 appears to be most attractive as it is the most efficient and most convenient to apply.
Noise problems with ephemeral streams in aquifer models
Authors: Connorton B. J. , Wood W. L.
Published: August 2005

Previous papers12 have drawn attention to the sustained oscillations (‘noise’) in the solution by successive over‐relaxation of the equations from the finite difference approximation of regional groundwater flow including ephemeral streams. This paper shows that the trouble can be avoided by introducing an averaging step in the algorithm; the trouble can also be avoided by ‘under‐relaxation’ but this is far less efficient than averaging.
Natural convection of air in a square cavity: A bench mark numerical solution
Authors: De Vahl Davis G.
Published: August 2005

Details are given of the computational method used to obtain an accurate solution of the equations describing two‐dimensional natural convection in a square cavity with differentially heated side walls. Second‐order, central difference approximations were used. Mesh refnement and extrapolation led to solutions for 103⩽Ra⩽10 6 which are believed to be accurate to better than 1 per cent at the highest Rayleigh number and down to one‐tenth of that at the lowest value.
Angled jet flow model for a diesel engine intake process—random vortex method
Authors: Sung Nak Won , Laitone Jonathan A. , Patterson Donald J.
Published: August 2005

A numerical model is developed to study the interactions of multiple angled jet flows in the inlet port plane of the Detroit Diesel 6V‐92 two‐stroke engine cylinder. The random vortex method is used in two dimensions.

Results show axisymmetric swirl initially. As flow develops, the centre of the swirl moves to the mid‐radius region and begins to precess about the cylinder centre. The flow becomes progressively more chaotic as time progresses.
A method for solving the factorized vorticity‐stream function equations by finite elements
Authors: Quartapelle L. , Napolitano M.
Published: August 2005

A new finite element method for solving the time‐dependent incompressible Navier‐Stokes equations with general boundary conditions is presented. The two second‐order partial differential equations for the vorticity and the stream function are factorized, apart from the non‐linear advection term, by eliminating the coupling due to the double specification on the stream function at (a part of) the boundary. This is achieved by reducing the no‐slip boundary conditions to projection integral conditions for the vorticity field and by evaluating the relevant quantities involved according to an extension of the method of Glowinski and Pironneau for the biharmonic problem. Time integration schemes and iterative algorithms are introduced which require the solution only of banded linear systems of symmetric type. The proposed finite element formulation is compared with its finite difference equivalent by means of a few numerical examples. The results obtained using 4‐noded bilinear elements provide an illustration of the superiority of the finite element based spatial discretization.
A finite‐element study of the bénard problem using parameter‐stepping and bifurcation search
Authors: Jackson C. P. , Winters K. H.
Published: August 2005

The problem of fluid motion in a cavity with rigid sidewalls that is heated uniformly from below is studied by the finite‐element method. The techniques of parameter‐stepping and monitoring the determinant of the Jacobian matrix to find bifurcations are used. Results are presented for width‐to‐height ratios in the range 1 to 4, and for three different boundary conditions on the horizontal surfaces, namely both rigid, both free, and rigid bottom with free top. The non‐linear branches above the critical Rayleigh number are examined. Extensions to non‐Boussinesq flow are trivial.
Simulation of polymeric flows in the injection moulding process
Authors: Shen Shan‐Fu
Published: August 2005

Recent progress in the simulation of polymeric flows of two key problems in the injection moulding process, carried out by a team at Cornell University, is briefly described. For the filling of cooled thin cavities, the fluid is characterized by a power‐law viscosity with exponential temperature dependence, and interaction between the transient thermal boundary‐layer and the core flow in a domain with moving boundary is essential. The earlier procedure of Hieber and Shen is modified in two aspects: a boundary‐integral formulation replaces the finite‐element treatment of the pressure, and an ‘energy integral’ approach is used for the transient temperature. The second problem is the steady visco‐elastic flow in the juncture region where sudden changes of the geometry and large strain rates occur. The constitutive equation is postulated according to the Leonov model. The main features in the numerical implementation are: integration along a streamline to determine the elastic deformation tensors for a given velocity field, and finite‐element treatment (in time‐dependent form) of the pressure and fields for given stresses. In an example where the contraction ratio is 7:1, results for nominal Deborah number exceeding 100 show no numerical instability. (However, for this problem, the true Weissenberg number, i.e. the ratio of local first‐normal‐stress difference to shear stress turns out to be generally O(10).) The predictions also correlate very well with experimental birefringence measurements.
Finite element solutions for laminar and turbulent compressible flow
Authors: Srinivas K. , Fletcher C. A. J.
Published: August 2005

The time‐split finite element method is extended to compute laminar and turbulent flows with and without separation. The examples considered are the flows past trailing edges of a flat plate and a backward‐facing step. Eddy viscosity models are used to represent effects of turbulence. It is found that the time‐split method produces results in agreement with previous experimental and computational results. The eddy viscosity models employed are found to give accurate predictions in all regions of flow except downstream of reattachment.
Finite element simulation of dip coating, I: Newtonian fluids
Authors: Tanguy P. , Fortin M. , Choplin L.
Published: August 2005

A finite element simulation of the dip coating process based on a discretization of the continuum with discontinuous pressure elements is presented. The algorithm computes the flow field from natural boundary conditions while an extra condition provided by the existence of free surface is employed to displace the meniscus location towards the actual position. The process is iterative and uses a pseudo‐time stepping technique coupled to a cubic spline fitting of the free surface. Numerical predictions exhibit good agreement with experimental data for Newtonian fluids in the case of flat plate dip coating as well as in the case of wire dip coating.
A modified finite element method for solving the time‐dependent, incompressible Navier‐Stokes equations. Part 2: Applications
Authors: Gresho Philip M. , Chan Stevens T. , Lee Robert L. , Upson Craig D.
Published: August 2005

Three examples will be presented to demonstrate the performance of the scheme described in Part 1 of this paper.1 Two are isothermal (T = 0) and two‐dimensional, and one of these is steady and the other time‐dependent. The third example involves buoyancy effects, is time‐dependent and three‐dimensional, and is presented in less detail. The paper concludes with a short discussion and some conclusions from both Parts 1 and 2.
A new, reliable family of split‐operator methods for computing reacting flows
Authors: Greenberg J. B.
Published: August 2005

A new, reliable family of split‐operator methods is presented for the computation of reacting flows. The methods may be inconsistent or consistent depending upon their applicability. Analyses are carried out for flow combusting according to general multicomponent, multireaction linear or non‐linear chemical kinetic models, and conditions for guaranteed stability/convergence are established. A special chemical splitting parameter can also be embedded in the methods enabling accelerated convergence of an iterative finite‐difference solution. Computed results highlight the capabilities and properties of the methods.
On a family of Stokes flows
Authors: Zaturska M. B. , Banks W. H. H.
Published: August 2005

A family of simple Stokes flows involving sliding surfaces adjacent to surfaces at rest is considered. Principally, two specific flow configurations are investigated: (i) that arising when parts of the boundary of an infinitely long circular cylinder are rotating about the axis while other parts of the boundary are at rest, and (ii) the flow produced when a cap of a sphere is held at rest while the remainder of the sphere rotates about the symmetry axis. In each case computer plots of streamlines or constant velocity lines are presented to give a general impression of the resulting flow pattern.
Pressure method for the numerical solution of transient, compressible fluid flows
Authors: Casulli V. , Greenspan D.
Published: August 2005

In this paper the pressure method for incompressible fluid flow simulation is extended and applied to the numerical simulation of compressible fluid flow. The governing equations, obtained from the physical principles of conservation of momentum, mass and energy, are first studied from a characteristic point of view. Then they are discretized with a semi‐implicit finite difference technique in such a fashion that stability is achieved independently of the speed of sound. The resulting algorithm is fast, accurate and particularly efficient in subsonic flow calculations. As an example, the computer simulation of the von Kármán vortex street is described and discussed.
A comparison of mixed methods for solving the flow of a Maxwell fluid
Authors: Van Schaftingen J. J. , Crochet M. J.
Published: August 2005

The proficiency of available mixed methods for solving the flow of a Maxwell fluid is evaluated through their application to the same problem. The reasons for the usual degeneracy of the numerical results beyond some level of elasticity are investigated. The best‐performing technique is applied to the flow through an abrupt 4/1 contraction.
On the consistency of four different control surfaces used for finite area blade‐to‐blade flow calculations
Authors: Arts T.
Published: August 2005

The aim of this contribution is to investigate the consistency and order of accuracy of different control surfaces used for finite area blade‐to‐blade flow calculations. The following cases will be treated: a hexagonal element, a trapezoidal element, a bitrapezoidal element and a quadrilateral element. Finally, the consistency conditions will be discussed and compared with respect to a cascade flow application.
Towards generalization and optimization of implicit methods
Authors: Lin Avi
Published: August 2005

A general implicit (GI) method for solving iteratively the algebraic system arising from a finite difference approximation of an elliptic partial differential equation is formulated. Under certain assumptions this method can be reduced to the already known implicit techniques. It is shown that the GI method has a very special physical meaning when solving fluid flow problems. It is shown also how this method can be optimized to achieve the maximum rate of convergence. Finally it is shown how this new strategy is applied by solving some classical numerical fluid dynamics problems.
The parametrized strongly implicit method for solving elliptic difference equations
Authors: Lin Avi
Published: August 2005

A new method for solving elliptic difference equations is derived based on the strongly implicit method. This parametrized strongly implicit method has three free parameters which may be functions of the field's nodal point. The method has some resemblance to the SOR techniques, but in the present method the off‐diagonal entries are also over‐relaxed. The main application of this method is for transport equations such as those governing the fluid flow and heat transfer fields.
Adaptive grids in numerical fluid dynamics
Authors: Haase Werner , Misegades Kent , Naar Mathias
Published: August 2005

A solution adaptive grid (SAG) method which redistributes the nodal points of a function according to its curvature is presented. A single, user‐selected step parameter, P, is available for controlling the maximum step size, allowing the application of the technique to a wide variety of problems. Three test cases are cited: (1) the 1‐dimensional inviscid Burgers equation, (2) the Falkner‐Skan equation and (3) the finite‐volume form of the Navier‐Stokes equations for transonic aerofoil flows. In all three cases, significant solution improvement in terms of accuracy and convergence acceleration were achieved.
Numerical simulation of three‐dimensional flow in a cavity
Authors: Freitas Christopher J. , Street Robert L. , Findikakis Angelos N. , Koseff Jeffrey R.
Published: August 2005

Previous three‐dimensional simulations of the lid‐driven cavity flow have reproduced only the most general features of the flow. Improvements to a finite difference code, REBUFFS, have made possible the first completely successful simulation of the three‐dimensional lid‐driven cavity flow. The principal improvement to the code was the incorporation of a modified QUICK scheme, a higher‐order upwind finite difference formulation. Results for a cavity flow at a Reynolds number of 3200 have reproduced experimentally observed Taylor‐Görtler‐like vortices and other three‐dimensional effects heretofore not simulated. Experimental results obtained from a unique experimental cavity facility validate the calculated results.
An inviscid model of unsteady aerofoil flow with fixed upper surface separation
Authors: Vezza M. , Galbraith R. A. McD.
Published: August 2005

Presented in this paper is a new method for the prediction of unsteady, incompressible separated flow over a two‐dimensional aerofoil. The algorithm was developed from an existing unsteady potential flow model1 and makes use of an inviscid formulation for the flow field. The aerofoil is represented by vortex panels of linearly varying strength which are piecewise continuous at the corners. Discrete vortices with finite cores are used to model the separating shear layers.

Following a brief summary of unsteady separation modelling, the theoretical framework is presented and the subsequent numerical implementation is discussed in detail.

Results are given for flows which tend asymptotically to the steady state and conclusions are drawn regarding the usefulness of the method.
A note on the steady‐state advection‐diffusion equation
Authors: Ferreri J. C.
Published: August 2005

In this note we show that the numerical solution of the advection‐diffusion equation can be improved by considering the asymptotic behaviour of its analytical solution. This is accomplished by including a correction term based on the numerical differentiation of the asymptotic (Pe » 1, Pe being the Peclet number) solution. This correction forces the usual oscillations associated with centred schemes to disappear.
On numerical solutions of the time‐dependent Euler equations for incompressible flow
Authors: Saiac J.‐H.
Published: August 2005

This paper presents finite element methods for the non‐stationary Euler equations of a two dimensional inviscid and incompressible flow. For the time discretization, we compare numerical results obtained by the use of a leap‐frog scheme and a semi‐implicit scheme of order two.
A new method for computing solenoidal vector fields on arbitrary regions
Authors: Gustafson K. , Halasi K. , Young D. P.
Published: August 2005

We develop a new method for the efficient calculation of solenoidal vector fields on general regions. The method takes advantage of fast direct methods and uses boundary integral equations to satisfy boundary conditions. For the latter we give an effective scheme for computing far‐field boundary influences (based on discrete charges). Examples and numerical results are given. The method is applicable to incompressible Navier‐Stokes calculations.
Boundary conformed co‐ordinate systems for selected two‐dimensional fluid flow problems. Part I: Generation of BFGs
Authors: Häuser J. , Paap H. G. , Eppel D. , Sengupta S.
Published: August 2005

In this paper the generation of general curvilinear co‐ordinate systems for use in selected two‐dimensional fluid flow problems is presented. The curvilinear co‐ordinate systems are obtained from the numerical solution of a system of Poisson equations. The computational grids obtained by this technique allow for curved grid lines such that the boundary of the solution domain coincides with a grid line. Hence, these meshes are called boundary fitted grids (BFG). The physical solution area is mapped onto a set of connected rectangles in the transformed (computational) plane which form a composite mesh. All numerical calculations are performed in the transformed plane. Since the computational domain is a rectangle and a uniform grid with mesh spacings Δξ = Δη = 1 (in two‐dimensions) is used, the computer programming is substantially facilitated. By means of control functions, which form the r.h.s. of the Poisson equations, the clustering of grid lines or grid points is governed. This allows a very fine resolution at certain specified locations and includes adaptive grid generation. The first two sections outline the general features of BFGs, and in section 3 the general transformation rules along with the necessary concepts of differential geometry are given. In section 4 the transformed grid generation equations are derived and control functions are specified. Expressions for grid adaptation arc also presented. Section 5 briefly discusses the numerical solution of the transformed grid generation equations using sucessive overrelaxation and shows a sample calculation where the FAS (full approximation scheme) multigrid technique was employed. In the companion paper (Part II), the application of the BFG method to selected fluid flow problems is addressed.
Boundary conformed co‐ordinate systems for selected two‐dimensional fluid flow problems. Part II: Application of the BFG method
Authors: Häuser J. , Paap H. G. , Eppel D. , Sengupta S.
Published: August 2005

This paper gives the results of an application of the SWEs (shallow water equations) to a part of the Hamburg harbour area, which is a complex flow domain, using the BFG approach, outlined in Part I. The results of a grid doubling procedure generating the desired computational grid from a coarse initial mesh are also presented. A second class of problems which is addressed, demands time‐dependent co‐ordinate systems. The problems which are solved are the free surface problem for a moving wave which eventually breaks and for a wave which is reflected by the solid walls of a rectangular basin.
Numerical simulation of swirling flow in diffusers
Authors: Armfield S. W. , Fletcher C. A. J.
Published: August 2005

A reduced form of Navier–Stokes equations is developed which does not have the usual minimum axial step size restriction. The equations are able to predict accurately turbulent swirling flow in diffusers. An efficient single sweep implicit scheme is developed in conjunction with a variable grid size domain‐conforming co‐ordinate system. The present scheme indicates good agreement with experimental results for (1) turbulent pipe flow, (2) turbulent diffuser flow, (3) turbulent swirling diffuser flow. The strong coupling between the swirl and the axial velocity profiles outside of the boundary layer region is demonstrated.
Some new methods for numerically solving the compressible laminar boundary layer equations
Authors: Friel R. G.
Published: August 2005

Four new methods are presented: a method of accelerating the truncation error convergence, a fourth and fifth order difference scheme for discretizing the normal partial derivatives, high order quadrature formulae for integrating a stream function and a third order implicit scheme for treating the streamwise partial derivatives. These are seen to be effective in finding solutions to the boundary layer equations in which the step sizes are adaptively altered to meet an error bound.
Lagrangian finite element method for the analysis of two‐dimensional sloshing problems
Authors: Ramaswamy Balasubramaniam , Kawahara Mutsuto , Nakayama Tsukasa
Published: August 2005

A new version of a numerical algorithm for the Lagrangian treatment of incompressible fluid flows with free surfaces is developed. The novel features of the present method are the adoptions of the Lagrangian finite element method and the velocity correction technique. The use of the velocity correction approach makes the computational scheme extremely simple in algorithmic structure. Hence, the present method is particularly attractive for large‐scale problems. The techniques discussed here are applied to some two‐dimensional sloshing problems, which may indicate the versatility and effectiveness of the present method.
Solution to the orr–sommerfeld equation for liquid film flowing down an inclined plane: An optimal approach
Authors: Lam T. T. , Bayazitoglu Y.
Published: August 2005

The Orr–Sommerfeld equation is solved numerically for a layer of liquid film flowing down an inclined plane under the action of gravity using the sequential gradient‐restoration algorithm (SGRA) The method consists of solving the governing equation as it is a Bolza problem in the calculus of variations. The neutral stability curves, eigenvalues and eigenfunctions to the stability problem can be determined simultaneously during the process.
Finite element ground‐water models implemented on vector computers
Authors: Pelka W. , Peters A.
Published: August 2005

Finite element models, optimized for running on conventional serial computers, are not suitable to make use of the potentional high performance of today's vector or parallel computers. Also, automatic vectorization by the compiler or manual vectorization at the local level do not by far lead to the required and expected computational speed.

A global change of the overall program logic or a complete redesign becomes necessary, i.e. a completely new generation of program systems has to be created, considering the new hardware characteristics and abilities.

A new computer‐independent programming technique for finite element problems to be implemented on vector computers is proposed and the test results of scalar and vectorized program structures on a conventional serial and on a non‐conventional pipeline computer are discussed.
Review: Formulating efficient finite‐element codes for flows in regular domains
Authors: Staniforth A.
Published: August 2005

Many fluid flow problems of current interest occur in domains that are mappable to a rectangle or a box; conformal mappings are particularly useful in this regard. We are concerned here with the efficient solution of such problems using finite elements. The central issue is the element choice, and this issue is addressed in terms of operation counts, computer memory and I/0 requirements, and the extent to which code vectorization is possible. It is concluded that rectangular (box) elements generally lead to more efficient algorithms that triangular (tetrahedral) elements. A synthesis of algorithms, based on bilinear (trilinear) elements, is presented. The algorithms have the attributes of simplicity, accuracy, stability and straightforward incorporation of boundary conditions. For bilinear and trilinear elements, it is found that product and first‐derivative terms are well‐handled by the Galerkin FE method, but that it is advantageous to go outside of the Galerkin framework when treating second‐derivative terms. It is particularly important to consider the form of the governing equations, vis‐à‐vis the choice of staggered, non‐staggered and/or mixed‐order elements, and to choose an appropriate time scheme. The described techniques have been successfully applied to a variety of problems in regular domains, including the solution of the three‐dimensional time‐dependent hydrostatic primitive equations; these are stiff and include first and second derivative terms, non‐linearities and variable coefficients due to a conformal mapping.
Numerical simulation of the horizontal Bridgman growth. Part II: Three‐dimensional flow
Authors: Dupont S. , Marchal J. M. , Crochet M. J. , Geyling F. T.
Published: August 2005

We study the steady‐state three‐dimensional flow which occurs in a horizontal crucible of molten metal under the action of a horizontal temperature gradient. The geometry and the boundary conditions are similar to those encountered in the Bridgman growth process of semiconductor crystals. We find that three‐dimensional effects can have a dramatic influence upon the flow, which, before the onset of periodic disturbances, differs appreciably from its two‐dimensional counterpart. We also investigate the sensitivity of the flow to non‐symmetric disturbances.
Numerical studies of slow viscous rotating flow past a sphere—1
Authors: Raghavarao C. V. , Valli K. Pramada
Published: August 2005

The Navier‐Stokes equations for a steady, viscous rotating fluid, rotating about the z‐axis with angular velocity ω are linearized using the Stokes approximation. The linearized Navier‐Stokes equations governing the axisymmetric flow can be written as three coupled partial differential equations for the stream function, vorticity and rotational velocity components. One parameter, Reω = 2ωa2/v, enters the resulting equations. For Reω « 1, the coupled equations are solved by the Peaceman‐Rachford A.D.I. (Alternating Direction Implicit) method and the resulting algebraic equations are solved by the ‘method of sweeps’. Stream lines for ψ = 0·05, 0·2, 0·5 and magnitude of the vorticity vector z = 0·2 are plotted for Reω = 0·1, 0·3, 0·5. Correction to the Stokes drag due to the rotation of fluid is calculated.
Numerical solution of reaction–diffusion equations by compact operators and modified equation methods
Authors: Ramos J. I. , Shih T. I‐P.
Published: August 2005

A system of reaction–diffusion equations which governs the propagation of an ozone decomposition laminar flame in Lagrangian co‐ordinates is analysed by means of compact operators and modified equation methods. It is shown that the use of fourth‐order accurate compact operators yields very accurate solutions if sufficient numbers of grid points are located at the flame front, where very steep gradients of temperature and species concentrations exist. Modified equation methods are shown to impose a restriction on the time step under certain conditions. The solutions obtained by means of compact operators and modified equation methods are compared with solutions obtained by other methods; good agreement is obtained.
Application of local co‐ordinate systems to the solution of integral equations referred to plane fluid flow problems
Authors: Sulisz Wojciech
Published: August 2005

Two‐dimensional fluid flow problems expressed in terms of velocity potentials or stream functions are often summarized as boundary‐value problems for the Laplace or Poisson equations, or the homogeneous or non‐homogeneous biharmonic equations. Simple local co‐ordinate systems have been applied to the solution of integral equations associated with these boundary‐value problems. This procedure has been shown to be an efficient technique in the numerical solution of fluid flow problems.
Wave propagation analysis of difference schemes for hyperbolic equations: A review
Authors: Vichnevetsky Robert
Published: August 2005

There are spurious phenomena in the numerical approximation of the hyperbolic equations of fluid dynamics that may be investigated by invoking concepts which originate from wave propagation theory. Many of the significant results which have been obtained by pursuing this kind of analysis are reviewed in this paper by using as an illustration a family of implicit approximations of the simple linear advection equation. Included in this family of algorithms are the common six‐point implicit finite difference scheme, the linear finite element/Galerkin scheme and the ‘box’ method.

The phase and group velocities of sinusoidal solutions are brought into the analysis of the accuracy and of the spurious reflection or scattering phenomena which are created at computational boundaries and in non‐uniform grids. General properties become apparent in this Fourier/wave propagation approach to the analysis. One of these is in the form of an analogy with quantum mechanics. Another shows that certain energy norms of the errors are independent of time discretization, i.e. depend on space discretization alone.
A study of the motion in rotating containers using a boundary fitted co‐ordinate system
Authors: Raghunath R. , Sengupta S. , Haeuser J.
Published: August 2005

Numerical solutions are often inaccurate because conventional co‐ordinate systems do not represent the complex physical boundaries accurately. In the present work, the numerical solution of linear shallow water wave equations has been obtained by transforming the physical domain into a rectangular computational domain using elliptic differential operators. This work is part of a programme to develop three‐dimensional body‐fit grid systems for environmental flows. Solutions have been obtained for a cylindrical container and also a parabolic container. The initial conditions chosen are the ones for which analytical solutions exist. The numerical solutions compare well with analytical solutions.
A taylor weak‐statement algorithm for hyperbolic conservation laws
Authors: Baker A. J. , Kim J. W.
Published: August 2005

Finite element analysis, applied to computational fluid dynamics (CFD) problem classes, presents a formal procedure for establishing the ingredients of a discrete approximation numerical solution algorithm. A classical Galerkin weak‐statement formulation, formed on a Taylor series extension of the conservation law system, is developed herein that embeds a set of parameters eligible for constraint according to specification of suitable norms. The derived family of Taylor weak statements is shown to contain, as special cases, over one dozen independently derived CFD algorithms published over the past several decades for the high speed flow problem class. A theoretical analysis is completed that facilitates direct qualitative comparisons. Numerical results for definitive linear and non‐linear test problems permit direct quantitative performance comparisons.
Comparison between the traction and pressure‐imposed boundary conditions in finite element flow analysis
Authors: Yagawa Genki , Eguchi Yuzuru
Published: August 2005

A formulation is developed to impose pressure‐prescribed boundary conditions in the penalty finite element method. Some numerical experiments for the Poiseuille flow problem are performed to compare it with the conventional traction‐prescribed boundary condition. Also the incorrectness of the traction‐free outlet boundary condition for contained‐flows is studied with explanatory numerical examples. Discussion is focused on the inlet and outlet boundary conditions to simulate fully developed flows. Finally, the three‐dimensional flow in a bifurcated pipe is analysed with the proposed formulation.
Magnetohydrodynamic flow in a rectangular duct
Authors: Sezgin Münevver
Published: August 2005

The magnetohydrodynamic flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct, with an external magnetic field applied transverse to the flow, has been investigated. One of the duct's boundaries which is perpendicular to the magnetic field is taken 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 100. All the infinite series obtained are transformed to infinite integrals first and then to finite integrals which contain modified Bessel functions of the second kind. In this way, the difficulties associated with the computation of infinite integrals with oscillating integrands and slowly converging infinite series, the convergence of which is further affected for large values of M, have been avoided. It is found that, as M increases, boundary layers are formed near the non‐conducting boundaries and in the interface region, and a stagnant region is developed in front of the conducting boundary for velocity field. The maximm value of magnetic field takes place on the conducting part. These behaviours are shown on some graphs.
Use of the splitting scheme and multigrid method to compute flow separation
Authors: Qianshun Chang
Published: August 2005

The splitting difference scheme is used to study flow separation. Flows behind a circular cylinder are computed as a model problem. In view of the nature of the flow, the variables are transformed. The boundary condition for the pressure is given from an intermediate velocity. The free‐slip velocity boundary conditions on the rigid wall are given by interpolation. The multigrid algorithm is applied to the pressure iteration. We also choose better initial values for the model problem by means of the multigrid algorithm idea.
Fully developed flow in a curved pipe of arbitrary curvature ratio
Authors: Soh W. Y. , Berger S. A.
Published: August 2005

It is generally assumed in curved pipe flow analyses that the curvature ratio, δ, of the pipe is very small, in which case the flow depends on a single parameter, the Dean number. This is not the case if δ is not very small. To determine the importance of this effect we have numerically solved the full Navier‐Stokes equations, in primitive variable form, for arbitrary values of δ. A factored ADI finite‐difference scheme has been used, employing Chorin's artificial compressibility technique. The results show that the central‐difference calculation on a staggered grid is stable, without adding artificial damping terms, due to coupling between pressure and velocity. A spatially variable time step is used with a fixed Courant number.
An algorithm for the global volume conservation constraint in lake circulation
Authors: Dhatt Gouri , Leclerc Michel
Published: August 2005

We present an algorithm for introducing a global constraint of volume conservation in lake circulation problems. The algorithm is described for linear problems, and is then generalized to non‐linear cases. Numerical examples are presented to show the influence of water‐level conditions on convergence and to demonstrate the practical superiority of the global constraint algorithm for obtaining reliable convergent solutions.
Spectral methods and their implementation to solution of aerodynamic and fluid mechanic problems
Authors: Streett C. L.
Published: August 2005

Fundamental concepts underlying spectral collocation methods, especially pertaining to their use in the solution of partial differential equations, are outlined. Theoretical accuracy results are reviewed and compared with results from test problems. A number of practical aspects of the construction and use of spectral methods are detailed, along with several solution schemes which have found utility in applications of spectral methods to practical problems. Results from a few of the successful applications of spectral methods to problems of aerodynamic and fluid mechanic interest are then outlined, followed by a discussion of the problem areas in spectral methods and the current research under way to overcome these difficulties.
Adaptive finite element methods for high‐speed compressible flows
Authors: Oden J. T. , Strouboulis T. , Devloo Ph.
Published: August 2005

We describe an adaptive finite element algorithm for solving the unsteady Euler equations. The finite element algorithm is based on a Taylor/Galerkin formulation and uses a very fast and efficient data structure to refine and unrefine the grid in order to optimize the approximation. We give a general version of the method which can be applied to moving grids with sliding interfaces and we present the results for a transient supersonic calculation of rotor‐stator interaction.
Finite elements for exterior problems using integral equations
Authors: Nedelec Jean‐Claude
Published: August 2005

We present some integral methods for exterior problems for the Laplace equation. Then we give finite element approximations for these equations and some errors estimates. Finally, we indicate how these integral equations can be coupled with a usual finite element method on a bounded domain to solve an exterior non‐linear problem which is linear far away.
On a finite element CFD algorithm for compressible, viscous and turbulent aerodynamic flows
Authors: Baker A. J. , Kim J. W. , Freels J. D. , Orzechowski J. A.
Published: August 2005

This paper develops and analyses individual construction aspects of an efficient and accurate finite element algorithm for prediction of viscous and turbulent flow fields of impact in aerodynamics. The theoretical construction employs a Taylor weak statement (TWS) for coincident embedding of stability mechanisms within a classic Galerkin finite element formulation of semi‐discrete approximation error orthogonalization. A wide variety of the stabilizing mechanisms of independently derived CFD algorithms are contained within the TWS theory. An implicit construction that meets the requirement of efficient convergence to steady state is developed. The theoretical asymptotic error estimates of the TWS finite element algorithm for supersonic and viscous boundary layer flows are verified. Application to a three‐dimensional turbulent flow is cited.
Finite element computation of a turbulent flow over a two‐dimensional backward‐facing step
Authors: Autret A. , Grandotto M. , Dekeyser I.
Published: August 2005

This paper is devoted to the computation of turbulent flows by a Galerkin finite element method. Effects of turbulence on the mean field are taken into account by means of a k‐ϵ turbulence model. The wall region is treated through wall laws and, more specifically, Reichardt's law. An inlet profile for ϵ is proposed as a numerical treatment for physically meaningless values of k and ϵ. Results obtained for a recirculating flow in a two‐dimensional channel with a sudden expansion in width are presented and compared with experimental values.
The free surface flow of Newtonian and non‐Newtonian fluids trapped by surface tension
Authors: Tidd D. M. , Thatcher R. W. , Kaye A.
Published: August 2005

In this paper the position of the free surface of a swirling fluid held in by surface tension is calculated by the finite element method. A new locally mass‐conserving quadratic velocity, linear pressure triangular element is used to overcome non‐physical solutions produced by the well known Taylor‐Hood element.
Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow
Authors: Abdallah S. , Dreyer J.
Published: August 2005

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non‐staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.
Wave equation hydrodynamics on deforming elements
Authors: Sidén Gunnar L. D. , Lynch Daniel R.
Published: August 2005

The shallow water wave equation is derived in a general deforming co‐ordinate system. A weak form is developed which displays the natural boundary condition prominently and which may be implemented on C0 elements. A time‐stepping algorithm is implemented with clastic mapping of interior node motion. Lossless test cases show agreement with analytic solutions. A simple hypothetical test case shows intuitively good behaviour at length scales approaching those required of estuarine simulations.
Generation of three‐dimensional unstructured grids by the advancing‐front method
Authors: Löhner Rainald , Parikh Paresh
Published: August 2005

The generation of three‐dimensional unstructured grids using the advancing‐front technique is described. This technique has been shown to be effective for the generation of unstructured grids in two dimensions.1,2 However, its extension to three‐dimensional regions required algorithms to define the surface and suitable data structures that avoid excessive CPU‐time overheads for the search operations involved. After obtaining an initial triangulation of the surfaces, tetrahedra are generated by successively deleting faces from the generation front. Details of the grid generation algorithm are given, together with examples and timings.