bluuebunny/crossref_metadata_2025 huggingface dataset:
A stream function finite element solution for two‐dimensional natural convection with accurate representation of nusselt number variations near a corner
Authors: Betts P. L. , Haroutunian V.
Published: August 2005

A finite element stream function formulation is presented for the solution to the two‐dimensional double‐glazing problem. Laminar flow with constant properties is considered and the Boussinesq approximation used. A restricted variational principle is used, in conjunction with a triangular finite element of C1 continuity, to discretize the two coupled governing partial differential equations (4th order in stream function and second order in temperature). The resulting non‐linear system of equations is solved in a segregated (decoupled) manner by the Newton‐Raphson linearizing technique.

Results are produced for the standard test case of an upright square cavity. These are for Rayleigh numbers in the range 103−105, with a Prandtl number of 0.71. Comparisons are made with benchmark results presented at the 1981 International Comparison study in Venice. In the discussion of results, emphasis is placed on the variation of local Nusselt number along the isothermal walls, particularly near the corner. This reveals a noticeable source of error in the evaluation of the maximum Nusselt number by lower order discretization methods.
Simulation of transient gas flows in networks
Authors: Osiadacz A.
Published: August 2005

A technique is presented for calculating the transient flow in high pressure transportation systems where both simple systems (without compressors) and systems with compressors have been taken into consideration. A partial differential equation characterizing the dynamic gas flow through a pipeline and a numerical scheme for its solution are considered. A method of computing node pressures is also characterized.
An ‘assumed deviatoric stress‐pressure‐velocity’ mixed finite element method for unsteady, convective, incompressible viscous flow: Part II: Computational studies
Authors: Yang Chien‐Tung , Atluri Satya N.
Published: August 2005

In Part I of this paper we presented a mixed finite element method, for solving unsteady, incompressible, convective flows, based on assumed ‘deviatoric stress–velocity–pressure’ fields in each element, which have the features: (i) the convective term is treated by the usual Galerkin technique; (ii) the unknowns in the global system of finite element equations are the nodal velocities, and the ‘constant term’ in the arbitrary pressure field over each element; and (iii) exact integrations are performed over each element.

In this paper we present numerical studies, both for steady as well as unsteady cases, of the problems: (a) the driven cavity, (b) Jeffry–Hamel flow in a channel, (c) flow over a ‘backward’ or ‘downstream’ facing step, and (d) flow over a square step. All these problems are two‐dimensional in nature, although certain 3‐D solutions are to be presented in a separate paper. The present results are compared with those which are available in the literature and are based on alternative approaches to treat incompressibility and convective acceleration. The possible merits of the present method are thus pointed out.
Coating flow theory by finite element and asymptotic analysis of the navier‐stokes system
Authors: Kistler S. F. , Scriven L. E.
Published: August 2005

Coating flows are laminar free surface flows, preferably steady and two‐dimensional, by which a liquid film is deposited on a substrate. Their theory rests on mass and momentum accounting for which Galerkin's weighted residual method, finite element basis functions, isoparametric mappings, and a new free surface parametrization prove particularly well‐suited, especially in coping with the highly deformed free boundaries, irregular flow domains, and the singular nature of static and dynamic contact lines where fluid interfaces intersect solid surfaces. Typically, short forming zones of rapidly rearranging two‐dimensional flow merge with simpler asymptotic regimes of developing or developed flow upstream and downstream. The two‐dimensional computational domain can be shrunk in size by imposing boundary conditions from asymptotic analysis of those regimes or by matching to one‐dimensional finite element solutions of asymptotic equations.

The theory is laid out with special attention to conditions at free surfaces, contact lines, and open inflow and outflow boundaries. Efficient computation of predictions is described with emphasis on a grand Newton iteration that converges rapidly and brings other benefits. Sample results for curtain coating and roll coating flows of Newtonian liquids illustrate the power and effectiveness of the theory.
A natural interpolation formula for Prandtl's singular integrodifferential equation
Authors: Ioakimidis N. I.
Published: August 2005

Prandtl's singular integrodifferential equation and related equations appear in problems of aerofoil and propeller theory in fluid mechanics. Here a natural interpolation formula for the approximation to the unknwon function of Prandtl's equation when this is solved numerically by the direct quadrature method, based on the Gauss‐ and Lobatto‐Chebyshev quadrature rules, is proposed. This interpolation formula is analogous to Nyström's natural interpolation formula for Fredholm integral equations of the second kind and the corresponding formula for singular integral equations. Numerical applications of the same formula are also made.
Finite element method for unsteady MHD flow through pipes with arbitrary wall conductivity
Authors: Singh Bani , Lal Jia
Published: August 2005

A finite element method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady MHD flow through a pipe having arbitrarily conducting walls. Pipes of rectangular, circular and triangular sections have been taken for illustration. Computations have been carried out for different Hartmann numbers and wall conductivity at various time levels. It is found that if the wall conductivity increases, the flux through a section decreases. The same is the effect of increasing the Hartmann number. It is also observed that the steady state is approached at a faster rate for larger Hartmann numbers or larger wall conductivity. Selected graphs are given showing the behaviour of velocity, induced magnetic field and flux across a section.
Modelling fluid flow in fractured‐porous rock masses by finite‐element techniques
Authors: Baca R. G. , Arnett R. C. , Langford D. W.
Published: August 2005

One of the major difficulties of modelling fluid flow processes in hard‐rock geologies is the complex nature of the porosity systems. Hydraulic behaviour in these rock masses is characterized by both porous and fractured interflow zones. Traditionally, fractured‐porous rocks have been modelled as an equivalent porous medium or as a system of fractures separated by impermeable blocks. A new method is proposed that unifies these two approaches for modelling fluid flow processes in fractured‐porous media. The basic idea is to use a combination of isoparametric elements for the porous zones and line elements for the fractures. The coupling between the governing equations for each element type is achieved using the superposition principle. The effectiveness of the new approach is demonstrated by comparing numerical solutions with known solutions for problems of flow and solute transport in fractured‐porous media.
Solution of three‐dimensional viscous incompressible flows by a multi‐grid method
Authors: Fuchs L. , Zhao H.‐S.
Published: August 2005

The three‐dimensional Navier‐Stokes equations for viscous incompressible fluids are discretized on staggered or non‐staggered grids. The system of finite‐difference equations is solved by a multi‐grid method. The method and some possible sources of difficulties and their remedies are described. The numerical algorithm has been applied to the computations of flows in ducts for a range of Reynolds numbers up to 2000. As outflow boundary conditions, either the fully developed flow profile (Dirichlet condition) or parabolic conditions have been applied. The multi‐grid method has a fast rate of convergence (with both types of boundary conditions), and it is not sensitive to the number of mesh points and the Reynolds number. The numerical solution, using parabolic boundary conditions, is insensitive to the location of the outflow boundary, even for large Reynolds numbers, in contrast to the solution with Dirichlet boundary conditions.
Techniques of ray averaging
Authors: Southgate Howard N.
Published: August 2005

Ray methods are used in coastal and harbour wave disturbance investigations where the area to be modelled is large compared to the wavelength. The interpretation of forward‐plotted ray diagrams, once obtained, has always been a difficult problem. The technique described in this paper calculates wave amplitudes during the ray plotting process and requires only minor modifications to existing ray plotting programs. The idea is to superimpose a grid of square elements over the entire sea area under study, and to perform a spatial averaging of the rays crossing each square element. This ‘square‐averaging’ technique has a number of advantages. It smooths the rapid amplitude variations near caustics, calculates the interference of several wave trains, and generates amplitudes automatically in a square array covering the whole studied sea area. Two types of sensitivity tests are carried out. These tests are designed to determine the accuracy of the predicted wave amplitudes with respect to: (1) the square size per wavelength, and (2) the ray density. These two factors largely determine the computing storage, time and cost of a ray model. An upper limit on the square size per wavelength and a lower limit on the ray density are obtained.
Stability of viscous flow in a rotating porous medium in the form of an annulus: The small‐gap problem
Authors: Channabasappa M. N. , Ranganna G. , Rajappa B.
Published: August 2005

The paper deals with the linear stability analysis of laminar flow of a viscous fluid in a rotating porous medium in the form of an annulus bounded by two concentric circular impermeable cylinders. The usual no‐slip condition is imposed at both the boundaries. The resulting sixth order boundary value, eigenvalue problem has been solved numerically for the small‐gap case by the Runge‐Kutta‐Gill method, assuming that the marginal state is stationary. The results of computation reveal that the critical Taylor number increases with decreasing permeability of the medium. The problem is found to reduce to the case of ordinary viscous flow in the annulus obtained by Chandrasekhar,1 when the permeability parameter tends to zero.
Further numerical methods for the Falkner‐Skan equations: Shooting and continuation techniques
Authors: Laine Claudine , Reinhart Laure
Published: August 2005

We consider in this paper the numerical solution of the Falkner‐Skan differential equation, modelling under some similarity assumptions the boundary layer equation. We look for the extremal solution of this third order differential equation. The methods we use are basically the Newton method with a shooting process, which is coupled with a continuation method: they allow us to follow the solution arcs which contain regular and turning point solutions.
The stability of explicit Euler time‐integration for certain finite difference approximations of the multi‐dimensional advection–diffusion equation
Authors: Hindmarsh A. C. , Gresho P. M. , Griffiths D. F.
Published: August 2005

A comprehensive study is presented regarding the numerical stability of the simple and common forward Euler explicit integration technique combined with some common finite difference spatial discretizations applied to the advection‐diffusion equation. One‐dimensional results are obtained using both the matrix method (for several boundary conditions) and the classical von Neumann method of stability analysis and arguments presented showing that the latter is generally to be preferred, regardless of the type of boundary conditions. The less‐well‐known Godunov‐Ryabenkii theory is also applied for a particular (Robin) boundary condition. After verifying portions of the one‐dimensional theory with some numerical results, the stabilities of the two‐ and three‐dimensional equations are addressed using the von Neumann method and results presented in the form of a new stability theorem. Extension of a useful scheme from one dimension, where the pure advection limit is known variously as Leith's method or a Lax‐Wendroff method, to many dimensions via finite elements is also addressed and some stability results presented.
Three‐dimensional free‐surface suspended particles transport in the South Biscayne Bay, Florida
Authors: Miller Harvey P.
Published: August 2005

A three‐dimensional, time‐dependent free surface model has been developed which takes account of topographical and meterological parameters for application to suspended particles transport. This investigation of suspended particles transport is the most realistic in regard to treating the moving free‐surface in computing the hydrodynamic field. A unique mass‐conserving numerical model is used for solving the concentration equation by an explicit finite difference scheme. The paper presents a mathematical model which can be applied to surface water dispersion of particulates associated with dredging operations and land‐fill. The effects of settling velocity and bottom bed deposition rate are compared and discussed.
Finite element solution of the shallow water equations by a quasi‐direct decomposition procedure
Authors: Goussebaile J. , Hecht F. , Labadie G. , Reinhart L.
Published: August 2005

We describe in this paper some new methods for solving the shallow water flow problem; this problem is composed of two coupled non‐linear equations. First a fractional step method is used to decouple the difficulties due to the convection and the propagation. The diffusion propagation step amounts to solving two coupled linear equations on the water depth and the fluxes. We used then some quasi‐direct decomposition techniques to decouple the unknowns, leading to symmetric systems only. In addition some non‐trivial boundary conditions on the flux are examined. Particularly we are interested to get decoupled systems on the two components of the flux. Numerical tests are presented to enlighten the behaviour and the capabilities of the method. Finally an industrial example is treated on the Dunkirk harbour.
A Kantorovich computational method for free surface gravity flows
Authors: Toro E. F. , O'Carroll M. J.
Published: August 2005

Computing free surface gravity flows involves basically two coupled problems, namely, the location of the free surface position and the determination of the internal flow field (for assumed values H0 and Q of the total head and discharge, respectively).

Solution techniques are invariably based on iterative procedures, but those that iterate between the two coupled problems may become unstable.

In this paper we present a computational method in which the coupling is kept throughout the process of iteration. This is achieved by converting the coupled problems (by means of the Kantorovich method) into the single problem of finding a set of streamlines, including that of the free surface. These streamlines are moved (iteratively) to satisfy the stationary conditions of the governing variational principle.

The algorithm is very stable and converges rapidly. It is also easy to implement to solve various types of steady flows with a free surface under gravity.
Numerical study of thermal convection in rotating channel flow
Authors: Thangam Sivagnanam , Speziale Charles G.
Published: August 2005

A numerical study is conducted on the effect of sidewall heating in the pressure‐driven laminar flow of an incompressible viscous fluid through a rectangular channel that is subjected to a spanwise rotation. The time‐dependent Navier‐Stokes equations are solved along with the conservation equations for energy and mass by a finite‐difference technique. The effect of weak to moderate sidewall heating on the overall flow structure at different rotation rates is studied. It is observed that for weak sidewall heating, the secondary flow structure is quite similar to the corresponding isothermal case. However, when the sidewall heating is moderate, various types of secondary flow fields are found to occur depending on the magnitude of the rotation. The influence of rotational speed on the net heat transport for different levels of sidewall heating is also studied. It is found that when the sidewall heating is weak, the basic secondary flow structure for the non‐rotating case is of a unicellular form and an increase in the rotation speed leads to an increase in the net heat transfer due mainly to the rotationally driven transport of fluid from the high temperature to the low temperature region. On the other hand, when the sidewall heating is moderate so that the basic secondary flow structure for the non‐rotating case has a multicellular configuration, an increase in the rotation speed leads to a decrease in the heat transport due to the weakening of the shear layer near the hot wall.
On the solution for steady state and time dependent flows in certain channels with small wall curvature
Authors: Georgiou G. A. , Ellinas C. P.
Published: August 2005

An asymptotic scheme is presented for the solution of the steady state and time dependent stream functions for flows in symmetric curved walled channels. In this scheme a class of non‐linear Jeffery‐Hamel solutions appear at O(1), and thus provide the first approximation to the steady state stream function. This class of Jeffery‐Hamel solutions are evaluated by using a simple perturbation about Poiseuille flow.

The classic Orr‐Sommerfeld eigenproblem appears at O(1) in the asymptotic development of the time dependent stream function, but here there is a slow streamwise dependence. This eigenvalue problem, for a complex wave number, is solved using an algorithm which automatically provides an initial guess which is then used to iterate to the correct eigenvalue.

Higher order terms in the asymptotic development, for both the steady state and time dependent stream functions, are evaluated to provide a solution for the total stream function.
Finite element analysis of the steady Navier‐Stokes equations by a multiplier method
Authors: Mizukami Akira
Published: August 2005

In this paper a fully explicit finite element method (FEFEM) is presented for solving steady incompressible viscous flow problems. This full explicitness is achieved by combining the multiplier (or augmented Lagrangian) method with a pseudo‐time‐iteration method. FEFEM needs no global matrix at all and is of great advantage to large‐scale problems because they can be solved within the limit of core memory.

The optimum choice of a time increment and a penalty parameter is discussed and the driven cavity flow at a Reynolds number of 1000 is computed with a refined mesh (60 × 60 elements).
A three‐level generalized‐co‐ordinate group finite‐element method for compressible viscous flow
Authors: Srinivas K. , Fletcher C. A. J.
Published: August 2005

A consistent three‐level time‐split group finite‐element method, suitable for the computation of viscous compressible flows in irregular geometric domains, is described. Exploitation of the group12 formulation permits an accurate and economical algorithm to be developed in a generalized‐co‐ordinate (ζ,η) space. A variable sweep cycle is used to accelerate convergence to the steady state. The method is demonstrated by computing laminar and turbulent flow past a trailing edge. The method uses an algebraic eddy viscosity model to represent turbulence and produces results in close agreement with the experiments and computations of Viswanath et al8.
Computation of three‐dimensional flow using the Euler equations and a multiple‐grid scheme
Authors: Koeck Charles
Published: August 2005

The unsteady Euler equations are numerically solved using the finite volume one‐step scheme recently developed by Ron‐Ho Ni. The multiple‐grid procedure of Ni is also implemented. The flows are assumed to be homo‐enthalpic; the energy equation is eliminated and the static pressure is determined by the steady Bernoulli equation; a local time‐step technique is used. Inflow and outflow boundaries are treated with the compatibility relations method of ONERA. The efficiency of the multiple‐grid scheme is demonstrated by a two‐dimensional calculation (transonic flow past the NACA 12 aerofoil) and also by a three‐dimensional one (transonic lifting flow past the M6 wing). The third application presented shows the ability of the method to compute the vortical flow around a delta wing with leading‐edge separation. No condition is applied at the leading‐edge; the vortex sheets are captured in the same sense as shock waves. Results indicate that the Euler equations method is well suited for the prediction of flows with shock waves and contact discontinuities, the multiple‐grid procedure allowing a substantial reduction of the computational time.
Solution of shallow water equations for complex flow domains via boundary‐fitted co‐ordinates
Authors: Haeuser J. , Paap H.‐G. , Eppel D. , Mueller A.
Published: August 2005

Many problems of applied oceanography and environmental science demand the solution of the momentum, mass and energy equations on physical domains having curving coastlines. Finite‐difference calculations representing the boundary as a step function may give inaccurate results near the coastline where simulation results are of greatest interest for numerous applications. This suggests the use of methods which are capable of handling the problem of boundary curvature.

This paper presents computational results for the shallow water equations on a circular ring of constant depth, employing the concept of boundary fitted grids (BFG) for an accurate representation of the boundary. All calculations are performed on a rectangle in the transformed plane using a mesh with square grid spacing. Comparisons of the simulations of transient normal mode oscillations and analytic solutions are shown, demonstrating that this technique yields accurate results in situations (provided that there is a reasonable choice of grid) involving a curved boundary. The software developed allows application to any two‐dimensional area, regardless of the complexity of the geometry.

Simulation runs were made with two co‐ordinate systems. For the first system, the grid point distribution was obtained from polar co‐ordinates. For the second one, grid point positions were calculated numerically, solving Poisson's equation. It was found that small variations in the metric coefficients do not deteriorate the accuracy of the simulation results.

Moreover, comparisons of surface elevation and velocity components at grid points near the inner and outer radii obtained from an x−y Cartesian grid model with the BFG simulation were made. The former model produced inacccuracies at grid points near boundaries, and, owing to the large number of mesh points used to yield the necessary fine resolution, the computation time was found to be a factor of three higher.
The calculation of Green's functions in three dimensional hydrodynamic gravity wave problems
Authors: Pidcock M. K.
Published: August 2005

We have studied the problem of calculating Green's functions in three dimensional hydrodynamic gravity wave problems. A number of new expressions for these functions are presented for both finite and infinite depths. Various techniques for accelerating the convergence of some infinite series in these expressions are investigated and compared with the normal methods of evaluation. A significant improvement in the efficiency of the calculation is found using the results described in this paper.
Finite element modelling of stratified flow in estuaries and reservoirs
Authors: King Ian P.
Published: August 2005

Experiences relating to the application of finite element models for laterally averaged stratified flow are discussed and modifications to the basic approach are suggested that alleviate these difficulties. An example problem is used to demonstrate the revised model and to make a preliminary assessment of the hydrostatic pressure assumption when applied to reservoir analysis.
Finite element analysis of density flow using the velocity correction method
Authors: Kawahara Mutsuto , Ohmiya Kiyotaka
Published: August 2005

A finite element method is proposed for the analysis of density flow which is induced by a difference of density. The method employs the idea that density variation can be pursued by using markers distributed in the flow field. For the numerical integration scheme, the velocity correction method is successfully used, introducing a potential for the correction of velocity. This method is useful because one can use linear interpolation functions for velocity, pressure and potential based on the triangular finite element. The final equations can be formulated using the quasi‐explicit finite element method. A flume in a tank with sloping bottom has been analysed by the present method. The computed results show extremely good agreement with the experimental observations.
Inverse laminar boundary‐layer problems with assigned shear. The mechul function revisited
Authors: Kaufman K. C. , Hoffman G. H.
Published: August 2005

An inverse method is presented which accurately determines the pressure distribution for assigned wall shear in a two‐dimensional, laminar, incompressible boundary layer. The method reformulates the mechul function scheme of Cebeci and Keller to produce a stable solution in the marching direction and to increase accuracy in the normal direction. In the reformulation a modified pressure gradient parameter variation in the normal direction is used in conjunction with three‐point backward differences for streamwise derivatives and fourth‐order accurate splines for normal derivatives. The resulting spline‐finite difference equations are solved by Newton‐Raphson iteration together with partial pivoting. Numerical solutions are presented for self‐similar and non self‐similar flows and compared with published results.
Asymptotic finite element method for boundary value problems
Authors: Bar‐Yoseph Pinchas , Israeli Moshe
Published: August 2005

The Asymptotic Finite Element method for improvement of standard finite element solutions of perturbation equations by the addition of asymptotic corrections to the right hand side terms is presented. It is applied here to 1‐D and 2‐D diffusion–convection equations and to non‐linear similarity equations. Excellent results were obtained without the a priori use of special trial and test functions. Theoretical expectations were confirmed.
Improved averaging method for turbulent flow simulation. Part I: Theoretical development and application to Burgers' transport equation
Authors: Dakhoul Youssef M. , Bedford Keith W.
Published: August 2005

This is the first of two articles intended to develop, apply and verify a new method for averaging the momentum and mass transport equations for turbulence. The new method is based on Gaussian filtering in both the spatial and temporal domains. Application is made to the problem of momentum and scalar transport in a one‐dimensional transient Burgers' flow field. No actual calculations, with the averaged equations, are presented in this paper. However, an ‘exact’ solution of the one‐dimensional flow situation is presented as an economical tool for verifying the performance of the different turbulence models. In the second paper calculations are performed with the averaged one‐dimensional equations on coarse grids, and the results are compared to the exact or fully simulated data with a statistical verification procedure.
Unsteady compressible flow in long pipelines following a rupture
Authors: Flatt René
Published: August 2005

The unsteady frictional flow of a compressible fluid generated in a long pipeline after an accidental rupture is of considerable interest to the offshore gas industry. It answers several important questions concerning safety and pollution, e.g. the flow rate at the broken pipe end.

Laboratory tests cannot simulate the rather complex phenomenon satisfactorily. The problem is highly non‐linear and no general analytical solution is yet known.

In this study, based on computational fluid dynamics, the simplifying assumptions of isothermal and low Mach number flow often applied in the case of unsteady compressible flows in pipelines, have not been used.

Owing to the choking condition (Ma=1) which prevails for some time at the broken end. and the cumulative effect of friction over the 145 km long pipeline, we obtain (∂p/∂x)t→−∞. This analytically established singularity leads to numerical difficulties which seriously affect the accuracy. For short tubes (such as shock tubes) this negative feature is much less severe. Special procedures were necessary to keep the accuracy within the chosen limit of 1 per cent.
A discussion and comparison of numerical techniques used to solve the Navier–Stokes and Euler equations
Authors: Connell S. D. , Stow P.
Published: August 2005

This paper is intended to provide some background to a number of widely used methods for solving the Navier–Stokes and Euler equations. The difference between coupled and uncoupled iterative schemes is discussed together with methods for solving the equations. Methods covered include time marching (both explicit and implicit), pressure correction and a Newton–Raphson technique. The relationship between the methods is illustrated.
Comparison between two finite‐difference schemes for computing the flow around a cylinder
Authors: Borthwick A.
Published: August 2005

Separated flow past a circular cylinder is computed from two finite‐difference Navier–Stokes models. Stream functions are calculated using a successive‐over‐relaxation (SOR) procedure. Alternating‐direction‐implicil (ADI) and ‘upwind’ directional difference explicit (DDE) numerical schemes for solving the vorticity‐transport equation are compared. The ‘upwind’ differencing technique produces artificial viscosity which damps the wake and suppresses vortex shedding. It is shown to be unreliable and so the ADI approach is recommended.
Treatment of non‐linearities in the numerical solution of the incompressible Navier–Stokes equations
Authors: Galpin P. F. , Raithby G. D.
Published: August 2005

The solution of the full non‐linear set of discrete fluid flow equations is usually obtained by solving a sequence of linear equations. The type of linearization used can significantly affect the rate of convergence of the sequence to the final solution. The first objective of the present study was to determine the extent to which a full Newton–Raphson linearization of all non‐linear terms enhances convergence relative to that obtained using the ‘standard’ incompressible flow linearization. A direct solution procedure was employed in this evaluation. It was found that the full linearization enhances convergence, especially when grid curvature effects are important.

The direct solution of the linear set is uneconomical. The second objective of the paper was to show how the equations can be effectively solved by an iterative scheme, based on a coupled‐equation line solver, which implicitly retains all the inter‐equation couplings. This solution method was found to be competitive with the highly refined segregated solution methods that represent the current state‐of‐the‐art.
Performance of a multigrid calculation procedure in three‐dimensional sudden expansion flows
Authors: Vanka S. P.
Published: August 2005

The performance of a recently developed calculation procedure for steady incompressible flows is assessed in a variety of three‐dimensional sudden expansion type flows representative of those encountered in several types of industrial equipment. The calculation procedure, called here BLIMM (for block‐implicit multigrid method), is based on a coupled solution of the three‐dimensional momentum and continuity equations in primitive variables, using the multigrid technique. Different Reynolds numbers and finite difference grids are considered for each flow situation. The rates of convergence and the computational times are reported for each case.
Use of pseudo‐concentrations to follow creeping viscous flows during transient analysis
Authors: Thompson Erik
Published: August 2005

Creeping viscous flows are followed through finite element meshes by use of pseudo‐concentrations which define material position. The concentrations, assumed to be transported only by convection, serve as material markers. Illustrations are presented related to industrial forming processes and the slow deformation of geological structures.
A numerical method for the calculation of incompressible, steady, separated flows around aerofoils
Authors: Tzabiras G. , Dimas A. , Loukakis T.
Published: August 2005

The present work deals with the numerical calculation of the incompressible turbulent flow around aerofoils. An orthogonal curvilinear grid of ‘C’ type is used for the solution of the time averaged equations and Reynolds stresses are modelled according to the κ‐ε turbulence model. PISO and SIMPLE algorithms are used to solve the strongly coupled system of the derived finite volume equations and convergence is improved by applying the method of variable local underrelaxation factors. Comparisons between the calculated and measured pressure distributions are presented for NACA 0012 and NACA 4412 wing sections. The formation of separation bubbles according to calculations is also shown.
Porous aerofoil analysis using viscous‐inviscid coupling at transonic speeds
Authors: Olling Charles R. , Dulikravich George S.
Published: August 2005

Viscous‐inviscid interaction is used to compute steady two‐dimensional, transonic flows for solid and porous aerofoils. A full‐potential code was coupled with both a laminar/transition/turbulent integral boundary‐layer/turbulent wake code and the finite‐difference boundary‐layer code using the semi‐inverse methods of Carter and Wigton. The coupling was performed using the transpiration coupling concept, thus allowing for analysis of porous aerofoils with passive physical transpiration. The computations confirm experimental findings that passive physical transpiration can lead to a lower drag coefficient and a higher lift coefficient, a weaker shock and elimination of shock‐induced separation. Nevertheless, it is very important that the extent of the porous region and permeability factor distribution of the porous region are chosen carefully if these improvements are to be achieved.
A hodograph‐based method for the design of shock‐free cascades
Authors: Hassan A. A. , Dulikravich G. S.
Published: August 2005

A hodograph‐based method, originally developed by the first author for the design of shock‐free aerofoils, has been modified and extended to allow for the design of shock‐free compressor blades. In the present procedure, the subsonic and supersonic regions of the flow are decoupled, allowing the solution of either an elliptic or a hyperbolic‐type partial differential equation for the stream function. The coupling of both regions of the flow is carried out along the sonic line which adjoins both regions. For the subcritical portion of the flow considered here, the pressure distribution is prescribed in addition to the upstream and downstream flow conditions. For the supercritical portion of the flow, the stream function on the sonic line is given instead of the supercritical pressure distribution which is found as part of the solution. In the special hodograph variables used, the equation for the stream function is solved iteratively using a second‐order accurate line relaxation procedure for the subsonic portion of the flow. For the supercritical portion of the flow, a characteristic marching procedure in the hodograph plane is used to solve for the supersonic flow. The results are then mapped back to the physical plane to determine the blade shape and the supercritical pressures. Examples of shock‐free compressor blade designs are presented. They show good agreement with the direct computation of the flow past the designed blade.
Computer‐extended series for a source/sink driven gas centrifuge
Authors: Berger M. H.
Published: August 2005

We have reformulated the general problem of internal flow in a modern, high speed gas centrifuge with sources and sinks in such a way as to obtain new, simple, rigorous closed form analytical solutions. Both symmetric and antisymmetric drives lead us to an ordinary differential equation in place of the usual inhomogeneous Onsager partial differential equation. Owing to the difficulties of exactly solving this sixth order, inhomogeneous, variable coefficient ordinary differential equation we appeal to the power of perturbation theory and techniques. Two extreme parameter regimes are identified, the so‐called semi‐long bowl approximation and a new short bowl approximation. Only the former class of problems is treated here. The long bowl solution for axial drive is the correct leading order term, just as for pure thermal drive. New O(1) results are derived for radial, drag and heat drives in two dimensions. Then regular asymptotic, even ordered power series expansions for the flow field are carried out on the computer to O(ε4) using MACSYMA. These approximations are valid for values of ε near unity. In the spirit of Van Dyke, one can carry out this expansion process, in theory, to apparently arbitrary order for arbitrary but finite decay length ratio. Curiously, the flows induced by axial and radial forces are proportional for asymptotically large source scale heights, x*. Corresponding isotope separation integral parameters will be given in a companion paper.
Flow simulation in channels with distorted geometry using a spectral code with co‐ordinate transformation
Authors: Fodemski T. R. , Voke P. R. , Collins M. W.
Published: August 2005

Turbulence is essentially four‐dimensional in character, and requires the corresponding treatment of the well‐known Navier‐Stokes equations. However, this has only been possible over the past twenty years and then by using the largest computers available. Interest is now turning from the initial, mainly smooth channel, simulations to geometries of eventual engineering significance. This paper reports a new code using the spectral methods of Orszag, but also incorporating a novel generalized co‐ordinate transformation approach.

Initial predictions for smooth channels agree well with published data. For distorted geometries, the initial velocity field has considerable influence on the success of the simulations. This is accommodated by gradual (step) changes towards the required distortion, so that the initial velocity field for the ‘new’ geometry is the final field from the previous step. Examples are given of different two‐dimensional channel geometries achieved, and these include the successful prediction of recirculating flows.
On the development of Taylor vortices in a vertical annulus with a heated rotating inner cylinder
Authors: Ball Kenneth S. , Farouk Bakhtier
Published: August 2005

A numerical study has been conducted to determine the heat transfer characteristics and flow patterns which develop around a rotating, heated vertical cylinder enclosed within a stationary concentric cylinder. A tall annulus (aspect ratio of 10) with fixed, adiabatic horizontal end‐plates and a radius ratio of 0·5 has been considered. Furthermore, the effect that the introduction of buoyancy forces by heating the inner cylinder has on the development of the Taylor vortex flow is examined. It is observed that the formation of the Taylor vortices is delayed until the rotational parameter σ = Gr/Re2 has a value below unity for any given Reynolds number Re which is above the critical value Recrit for the formation of Taylor vortices in an isothermal flow. Also, the Taylor cells first appear at the top of the annulus. As σ is gradually decreased below unity, bifurcations to other states are observed. The final structure of the secondary flow is noticeably distorted in the mixed‐convection mode, with the size of the Taylor cells varying greatly along the height of the annulus. This distortion diminishes as σ is further decreased, until the isothermal flow pattern is nearly recovered below σ = 0·01.
On pressure boundary conditions for the incompressible Navier‐Stokes equations
Authors: Gresho Philip M. , Sani Robert L.
Published: August 2005

The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence‐free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time‐varying divergence‐free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well‐defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.
Finite‐element analysis of fluid flow and heat transfer for staggered bundles of cylinders in cross flow
Authors: Dhaubhadel M. N. , Reddy J. N. , Telionis D. P.
Published: August 2005

The two‐dimensional Navier‐Stokes equations and the energy equation governing steady laminar incompressible flow are solved by a penalty finite‐element model for flow across finite depth, five‐row deep, staggered bundles of cylinders. Pitch to diameter ratios of 1·5 and 2·0 are considered for cylinders in equilateral triangular and square arrangements. Reynolds numbers studied range from 100 to 400, and a Prandtl number of 0·7 is used. Velocity vector fields, streamline patterns, vorticity, pressure and temperature contours, local and average Nusselt numbers, pressure and shear stress distributions around the cylinder walls and drag coefficients are presented. The results obtained agree well with available experimental and numerical data.
Computation of three‐dimensional transonic flows using two stream functions
Authors: Sherif A. , Hafez M.
Published: August 2005

A computational method for three‐dimensional flows is presented in terms of two stream functions, which may be considered as two components of a generalized vector potential. An iterative scheme is developed such that only a sequence of two‐dimensional‐like problems, for each function, is solved. The convergence of the iterative scheme is studied based on von Neumann linear analysis. For transonic flow calculation, numerical methods used for potential flows are readily applied, namely artificial density and Zebra relaxation. Results of transonic flow calculations around a wing are presented.
Numerical solution of a free surface problem by a boundary element method
Authors: Aitchison Joyce M. , Karageorghis Andreas
Published: August 2005

This paper describes a method for the numerical solution of a Riabouchinsky cavity flow. Application of a boundary element method leads to a system of non‐linear equations. The mild singularity appearing at the separation point is treated with the introduction of a curved boundary element, which satisfies the exact behaviour of the free boundary in that neighbourhood.
A method for generating irregular computational grids in multiply connected planar domains
Authors: Weatherill N. P.
Published: August 2005

A method for generating irregular triangular computational grids in two‐dimensional multiply connected domains is described. A set of points around each body is defined using a simple grid generation technique appropriate to the geometry of each body. The Voronoi regions associated with the resulting global point distribution are constructed from which the Delaunay triangulation of the set of points is thus obtained. The definition of Voronoi regions ensures that the triangulation produces triangles of reasonable aspect ratios given a grid point distribution. The approach readily accommodates local clustering of grid points to facilitate variable resolution of the domain. The technique is generally applicable and has been used with success in computing triangular grids in multiply connected planar domains. The suitability of such grids for flow calculations is demonstrated using a finite element method for solution of the inviscid transonic flow over two‐ dimensional high‐lift aerofoil configurations.
A new fluid‐structure interaction analysis based on higher‐order boundary elements
Authors: Hearn Grant E. , Donati Edilio
Published: August 2005

In strip theory analysis the vessel is represented by a series of 2D transverse sections. For 2D arbitrary‐shaped sections either floating in the free surface or totally submerged, a higher‐order boundary element analysis has been developed to permit determination of the associated radiation and diffraction velocity potentials.

In this paper the formulation of the cited interaction problems is reworked to reflect the new capability of permitting curved boundary elements to represent the geometry and a higher‐order functional behaviour of the unknown velocity potentials over that geometry. This is in direct contrast to the usual technique of using straight‐line geometric panels and invariant behaviour of the required potentials over these simple panels.

Applications to representative sections of floating ships and the fully submerged pontoon section of a semi‐submersible are presented. Within these applications the results of the standard Frank close‐fit technique, of linear panels and constant behaviour, are compared with different combinations of higher‐order representations of the geometry and the determined velocity potentials.

Conclusions regarding the advantages and limitations of the procedures developed are discussed.
A flux‐vector splitting method for steady Navier‐Stokes equations
Authors: Dick E.
Published: August 2005

The flux‐vector splitting method is applied to the convective part of the steady Navier‐Stokes equations for incompressible flow. By the use of partial upwind differences in the split first‐order part and central differences in the second‐order part, a set of discrete equations is obtained which can be solved by vector variants of classical relaxation schemes. It is shown that accurate results can be obtained on one of the GAMM backward‐facing step test problems.