bluuebunny/crossref_metadata_2025 huggingface dataset:
One‐, Two‐, and Three‐Dimensional Coordination Polymers Built from Large Mo36‐Polyoxometalate Anionic Units and Lanthanide Cations
Authors: Izarova Natalia V. , Sokolov Maxim N. , Samsonenko Denis G. , Rothenberger Alexander , Naumov Dmitrij Yu. , Fenske Dieter , Fedin Vladimir P.
Published: November 2005

The Mo36‐polyoxometalate [Mo36O108(NO)4(H2O)16]12– reacts with Ln3+ to give crystalline solids with 1D, 2D, and 3D polymeric frameworks, depending on the size of Ln3+ and the reaction conditions. Reaction of LaCl3 with (NH4)12[Mo36O108(NO)4(H2O)16] gives (H3O)2[{La(H2O)5}2{La(H2O)6}{La(H2O)5Cl}{Mo36(NO)4O108(H2O)16}]Cl·21H2O (1), inwhose structure the La3+ cations and the polyoxometalate anions are united in a 3D framework, whereas NdCl3 and SmCl3 give [{Ln(H2O)6}4{Ln(H2O)4}{Mo36(NO)4O108(H2O)16}]Cl3·nH2O [Ln = Nd, n = 24 (2); Ln = Sm, n = 21 (3)], which have 2D frameworks. Under different conditions, large Ln3+ ions (La–Nd) give the one‐dimensional coordination polymers [{Ln(H2O)6}2{Ln(H2O)7}2{Mo36(NO)4O108(H2O)16}]·nH2O [Ln = La, n = 29 (4); Ln = Ce, n = 41 (5), Ln = Pr; n = 40 (6), Ln = Nd, n = 36 (7)], whereas reaction with Gd(NO3)3 gives [{Gd(H2O)5}4{Mo36(NO)4O108(H2O)16}]·34H2O (8), which has a unique 2D layered structure. Smaller Ln3+ ions (Tb–Lu) give isostructural 1D chain coordination polymers (H3O)3[{Ln(H2O)6}2{Ln(H2O)4}{Mo36(NO)4O108(H2O)16}]·nH2O [Ln = Tb, n = 42 (9); Ln = Dy, n = 43 (10), Ln = Ho, n = 46 (11), Ln = Er, n = 50 (12); Ln = Yb, n = 56 (13); Ln = Lu, n = 31 (14)]. The first coordination polymer based on [Mo36O112(H2O)16]8–, namely (H3O)2[{Nd(H2O)5}2{Mo36O112(H2O)16}]·40H2O (15), was prepared from MoO42–, NH2OH, and Nd3+. It has a 2D layered structure. All these compounds are obtained in moderate to high yields (35–93 %), and were characterized by elemental analysis, IR spectroscopy, and single‐crystal X‐ray structural analysis. Direct coordination of Ln3+ to the oxygen atoms of terminal cis‐MoO22+ groups is responsible for the formation of these polymeric frameworks. The coordination numbers of Ln3+ change from 9 (La–Sm) to 8 (Gd–Lu). (© Wiley‐VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2005)
Structural Relationships in High‐Nuclearity Heterobimetallic Bismuth‐Oxo Clusters
Authors: Mehring Michael , Paalasmaa Sanna , Schürmann Markus
Published: November 2005

The novel heterobimetallic sodium‐bismuth‐oxo clusters [Bi2Na4O(OSiMe3)8] (1), [Bi10Na5O7(OH)6(OSiMe3)15]·1.5C7H8 (2·1.5C7H8), [Bi15Na3O18(OSiMe3)12]·C7H8 (3·C7H8) and [Bi14Na8O18(OSiMe3)14(THF)4]·C6H6 (4·C6H6) were prepared starting from BiCl3 and NaOSiMe3. Compound 1 crystallises in the trigonal space group R$\bar {3}$c with the lattice constants a = 12.8844(3) Å and c = 54.6565(3) Å, compound 2·1.5C7H8 crystallises in the triclinic space group P$\bar {1}$ with the lattice constants a = 15.0377(2) Å, b = 16.0373(2) Å, c = 27.8967(5) Å, α = 87.1321(6)°, β = 86.6530(7)° and γ = 63.6617(6)°, compound 3·C7H8 crystallises in the monoclinic space group C2/c with the lattice constants a = 54.311(11), b = 19.846(4), c = 22.885(5) Å and β = 112.32(3)°, and compound 4·C6H6 crystallises in the trigonal space group R$\bar {3}$ with the lattice constants a = 15.9786(4) Å and c = 46.8329(17) Å. The formation of M–O–M bonds results from both partial hydrolysis followed by condensation as well as from elimination of Me3SiOSiMe3 from M–OSiMe3 groups. The hexanuclear metal‐oxo silanolate 1 is more conveniently synthesised by the addition of NaOSiMe3 to a toluene solution of in situ‐prepared [Bi(OSiMe3)3]. The metal‐oxo(hydroxo) silanolates differ significantly in composition, but show similar building units. Thermal decomposition of the metal‐oxo silanolates in the solid state gave heterogeneous decomposition products containing bismuth silicates. (© Wiley‐VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2005)
Infinite Chains of Quasi‐Planar Hexameric Water Clusters Stabilized in a Metal‐Organic Framework Built from CoII and Pyrazine‐ 2,3,5,6‐tetracarboxylic Acid
Authors: Ghosh Sujit K. , Bharadwaj Parimal K.
Published: November 2005

Pyrazine‐2,3,5,6‐tetracarboxylic acid (ptcH4) reacts with Co(NO3)2·6H2O in 1:2 molar ratio in aqueous pyridine (py) at room temperature to form a coordination polymer with the empirical formula, {[Co2(ptc)·(py)2·4H2O]·4H2O}n, (1). The structure consists of 2D sheets of metal‐organic frameworks and infinite chains of cyclic water hexamers that join these sheets to form an overall 3D structure. Thermogravimetry, X‐ray powder diffraction and X‐ray structural analysis have been used in addition to infrared spectroscopy and elemental analysis to characterize 1. Crystal data for 1: monoclinic space group P21, a = 7.153(4), b = 17.003(2), c = 11.060(5) Å, β = 107.207(5)°, V = 1284.9(11) Å3, Z = 2, R1 = 0.0340, wR2 = 0.0875, S = 1.028. This structure demonstrates a new mode of association of water molecules. (© Wiley‐VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2005)
Octameric Water Clusters of Staircase Structure Present in a Metal‐Organic Framework Built from Helical Lanthanide Coordination Polymers
Authors: Ghosh Sujit K. , Bharadwaj Parimal K.
Published: November 2005

Two lanthanide‐carboxylate polymeric complexes containing nine‐coordinate lanthanide metal centers, {[Pr(ptc–3)·2H2O]·2H2O} (1), {[Nd(ptc–3)·2H2O]·2H2O} (2) have been prepared by hydrothermal synthesis of Pr(NO3)3·6H2O or Nd(NO3)3·6H2O with pyridine‐2,4,6‐tricarboxylic acid (ptcH3). These network structures are characterized by X‐ray crystallography, powder X‐ray diffraction, infrared and thermal gravimetric analyses. Both 1 and 2 form infinite single helical chains with large widths and pitches containing six metal ions per turn. The overall structure consists of two helices connected via discrete octameric water clusters. (© Wiley‐VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2005)
Evaluation of a coupled solution approach for fluid flow calculations in body‐fitted co‐ordinates
Authors: Karki K. C. , Mongia H. C.
Published: November 2005

An investigation has been conducted to study the performance of a coupled solution approach for solving the fluid flow equations in body‐fitted co‐ordinates using a covariant velocity formulation. The set of discrete equations is solved using a sparse matrix variant of LU decomposition. A series of test problems have been considered to study the effects of Reynolds number, mesh skewness and grid size on the convergence rate. The coupled solution approach has been compared with a state‐of‐the‐art sequential approach. The results indicate that the coupled technique is faster by a factor of about two for reasonable grids. For severely non‐orthogonal grids the performance of the coupled solution approach deteriorates considerably.
Taylor–least‐squares finite element for two‐dimensional advection‐dominated unsteady advection–diffusion problems
Authors: Park Nam‐Sik , Liggett James A.
Published: November 2005

A new finite element method, the Taylor–least‐squares, is proposed to approximate the advection‐dominated unsteady advection–diffusion equation. The new scheme is a direct generalization of the Taylor–Galerkin and least‐squares finite element methods. Higher‐order spatial derivatives in the new formulation necessitate higher‐degree polynomials. Hermite cubic shape functions are used. Extensive comparisons with other methods in one dimension proved that the new scheme is a step forward in modelling this difficult problem. The method offers straightforward generalizations to higher dimensions without losing the accuracy demonstrated in one dimension, i.e. the method preserves the important property of the Taylor–Galerkin scheme of being free of numerical crosswind diffusion. Several numerical experiments were made in two dimensions and excellent results were obtained from the representative experiments.
Fourier analysis of the Eulerian–Lagrangian least squares collocation method
Authors: Bentley L. R. , Aldama A. , Pinder G. F.
Published: November 2005

A Fourier analysis was performed in order to study the numerical characteristics of the effective Eulerian–Lagrangian least squares collocation (ELLESCO) method. As applied to the transport equation, ELLESCO requires a C1‐continuous trial space and has two degrees of freedom per node. Two coupled discrete equations are generated for a typical interior node for a one‐dimensional problem. Each degree of freedom is expanded separately in a Fourier series and is substituted into the discrete equations to form a homogeneous matrix equation. The required singularity of the system matrix leads to a ‘physical’ amplification factor that characterizes the numerical propagation of the initial conditions and a ‘computational’ one that can affect stability.

Unconditional stability for time‐stepping weights greater than or equal to 0‐5 is demonstrated. With advection only, ELLESCO accurately propagates spatial wavelengths down to 2Δx. As the dimensionless dispersion number becomes large, implicit formulations accurately propagate the phase, but the higher‐wave‐number components are underdamped. At large dispersion numbers, phase errors combined with underdamping cause oscillations in Crank–Nicolson solutions. These effects lead to limits on the temporal discretization when dispersion is present. Increases in the number of collocation points per element improve the spectral behaviour of ELLESCO.
GSMAC finite element method for unsteady incompressible Navier–Stokes equations at high reynolds numbers
Authors: Tanahashi T. , Okanaga H. , Saito T.
Published: November 2005

A new finite element technique is developed for predicting the velocity and the pressure in transient incompressible viscous fluid flows at high Reynolds numbers. The new method is based on the generalized and simplified marker‐and‐cell method (GSMAC) and has two characteristics: one is an application of the Bernoulli function and the implicit pressure solution algorithm to the explicit fractional time step method; the other is a high‐order flux calculation to prevent the pressure field from oscillating. Two examples, driven cavity flows at high Reynolds numbers and vortex shedding behind a circular cylinder, are presented. Satisfactory agreement with experiment is demonstrated.
On the quality of viscoelastic flow solutions: An adaptive refinement study of a Newtonian and a maxwell fluid
Authors: Rao Rekha R. , Finlayson Bruce A.
Published: November 2005

Little adaptive finite element work has been done in the area of viscoelastic flow in complex geometries. In this paper an adaptive finite element mesh refinement study is carried out on a Newtonian fluid and a Maxwell fluid in an axisymmetric 4:1 contraction. The error indicator used for the refinement is the local norm of the residual in an element. For the Newtonian fluid, steady improvement with refinement is seen, though this is not the case for the Maxwell fluid, which never achieves a solution of good quality.
Compressible viscous flow calculations using compatible finite element approximations
Authors: Bristeau M. O. , Glowinski R. , Dutto L. , Périaux J. , Rogé G.
Published: November 2005

We discuss in this paper the numerical simulation of compressible viscous flows by a combination of finite element methods for the space approximation, an implicit second‐order multistep scheme for the time discretization and GMRES iterative methods for solving the non‐linear problems encountered at each time step. Numerical results corresponding to flows around aerofoils and aerospace vehicles illustrate the possibilities of these methods.
Godunov–mixed methods for immiscible displacement
Authors: Dawson Clint N.
Published: November 2005

The immiscible displacement problem in reservoir engineering can be formulated as a system of partial differential equations which includes an elliptic pressure–velocity equation and a degenerate parabolic saturation equation. We apply a sequential numerical scheme to this problem where time splitting is used to solve the saturation equation. In this procedure one approximates advection by a higher‐order Godunov method and diffusion by a mixed finite element method. Numerical results for this scheme applied to gas–oil centrifuge experiments are given.
Finite difference and finite element methods for mhd channel flows
Authors: Ramos J. I. , Winowich N. S.
Published: November 2005

A Galerkin finite element method and two finite difference techniques of the control volume variety have been used to study magnetohydrodynamic channel flows as a function of the Reynolds number, interaction parameter, electrode length and wall conductivity. The finite element and finite difference formulations use unequally spaced grids to accurately resolve the flow field near the channel wall and electrode edges where steep flow gradients are expected. It is shown that the axial velocity profiles are distorted into M‐shapes by the applied electromagnetic field and that the distortion increases as the Reynolds number, interaction parameter and electrode length are increased. It is also shown that the finite element method predicts larger electromagnetic pinch effects at the electrode entrance and exit and larger pressure rises along the electrodes than the primitive‐variable and streamfunction–vorticity finite difference formulations. However, the primitive‐variable formulation predicts steeper axial velocity gradients at the channel walls and lower axial velocities at the channel centreline than the streamfunction–vorticity finite difference and the finite element methods. The differences between the results of the finite difference and finite element methods are attributed to the different grids used in the calculations and to the methods used to evaluate the pressure field. In particular, the computation of the velocity field from the streamfunction–vorticity formulation introduces computational noise, which is somewhat smoothed out when the pressure field is calculated by integrating the Navier–Stokes equations. It is also shown that the wall electric potential increases as the wall conductivity increases and that, at sufficiently high interaction parameters, recirculation zones may be created at the channel centreline, whereas the flow near the wall may show jet‐like characteristics.
A technique for analysing finite element methods for viscous incompressible flow
Authors: Stenberg Rolf
Published: November 2005

We give a self‐contained presentation of our macroelement technique for verifying the stability of finite element discretizations of the Navier–Stokes equations in the velocity–pressure formulation.
A test problem for outflow boundary conditions—flow over a backward‐facing step
Authors: Gartling David K.
Published: November 2005

A numerical solution for steady incompressible flow over a two‐dimensional backward‐facing step is developed using a Galerkin‐based finite element method. The Reynolds number for the simulations is 800. Computations are performed on an extended channel length to minimize the effect of the outflow boundary on the upstream recirculation zones. A thorough mesh refinement study is performed to validate the results. Extensive profile data at several channel locations are provided to allow future testing and evaluation of outflow boundary conditions.
A Chebyshev matrix method for the spatial modes of the Orr–Sommerfeld equation
Authors: Danabasoglu Gökhan , Biringen Sedat
Published: November 2005

The Chebyshev matrix collocation method is applied to obtain the spatial modes of the Orr‐Sommerfeld equation for Poiseuille flow and the Blasius boundary layer. The problem is linearized by the companion matrix technique. For semi‐infinite domains a mapping transformation is used. The method can be easily adapted to problems with boundary conditions requiring different transformations.
Laminar mixed convection in the entrance region of horizontal rectangular ducts
Authors: Nonino C. , Giudice S. Del
Published: November 2005

A generally applicable finite element procedure for the prediction of laminar mixed convection in horizontal straight ducts of arbitrary cross‐section is presented. The procedure, based on the parabolized simplification of the complete Navier‐Stokes equations and on the Boussinesq approximation of the buoyancy terms, is validated through comparisons of computed results with the available literature data for mixed convection in the entrance region of a rectangular duct of aspect ratio a=2. Uniform heating at different sides is considered as the thermal boundary condition, although the proposed formulation allows specification of most thermal boundary conditions of practical interest.
Characteristic‐based flux limiters of an essentially third‐order flux‐splitting method for hyperbolic conservation laws
Authors: Lin Herng , Chieng Ching‐Chang
Published: November 2005

A flux limiter based on characteristic variables is extended by a control volume flux formulation to approximate the convection term at the cell interface for an essentially third‐order‐accurate scheme. The basic algorithm uses implicit MUSCL‐type flux splitting and the approximate factorization method. It is applied to three test problems: (i) a one‐dimensional shock tube problem; (ii) a two‐dimensional problem of an oblique shock step with Mach numbers 3 and 10 and a shock angle of 59°;(iii) a two‐dimensional problem of transonic inviscid flow past an NACA0012 aerofoil with Mach number 0·8 at zero angle of attack. The computational results by the new flux limiter function are compared with the results of direct applications of the SMART algorithm, Leonard's SHARP algorithm, the third‐order Van Leer flux‐splitting method with a smooth limiter, Harten's second‐order unwind‐biased TVD scheme, Chakravarthy's third‐order MUSCL‐type TVD scheme and the exact solution. The comparison shows that the present method gives the most accurate and least oscillatory results with a rapid rate of convergence.
Three‐dimensional laminar flow and heat transfer in the entrance region of trapezoidal ducts
Authors: Farhanieh Bijan , Sunden Bengt
Published: November 2005

The laminar convective flow and heat transfer in a duct with a trapezoidal cross‐sectional area are studied numerically. The governing equations are solved numerically by a finite volume formulation in complex three‐dimensional geometries using co‐located variables and Cartesian velocity components. Details of the numerical method are presented. The accuracy of the method was also established by comparing the calculated results with the analytical and numerical results available in the open literature. The Nusselt numbers are obtained for the boundary condition of a uniform wall temperature whereas the friction factors are calculated for no‐slip conditions at the walls. The asymptotic values of the Nusselt numbers, friction factors. incremental pressure drops, axial velocity and momentum rate and kinetic energy correction factors approach the available fully developed values. Various geometrical dimensions of the cross‐section are considered.
Evaluation of the drag force by integrating the energy dissipation rate in stokes flow for 2D domains using the FEM
Authors: Ganesh Ram K.
Published: November 2005

The total drag force on the surface of a body, which is the sum of the form drag and the skin friction drag in a 2D domain, is numerically evaluated by integrating the energy dissipation rate in the whole domain for an incompressible Stokes fluid. The finite element method is used to calculate both the energy dissipation rate in the whole domain as well as the drag on the boundary of the body. The evaluation of the drag and the energy dissipation rate are post‐processing operations which are carried out after the velocity field and the pressure field for the flow over a particular profile have been obtained. The results obtained for the flow over three different but constant area profiles—a circle, an ellipse and a cross‐section of a prolate spheroid—with uniform inlet velocity are presented and it is shown that the total drag force times the velocity is equal to the total energy dissipation rate in the entire finite flow domain. Hence, by calculating the energy dissipation rate in the domain with unit velocity specified at the far‐field boundary enclosing the domain, the drag force on the boundary of the body can be obtained.
Calculating hydrodynamics problems for a liquid film flowing radially on a flat horizontal surface
Authors: Siwoń Bogdan , Ennaoui Mustapha
Published: November 2005

A mathematical hydrodynamic model of a thin liquid film flowing radially on a flat horizontal surface has been elaborated. The model consisted of continuity and momentum equations which were resolved by means of the integral method. In fact, several versions of the model were examined; they differed mainly in film velocity distribution. The predictions of the different versions were then compared with liquid film thicknesses obtained from experimental investigation. The best version was applied in further calculations.
A spectral collocation method for compressible, non‐similar boundary layers
Authors: Pruett C. David , Streett Craig L.
Published: November 2005

An efficient and highly accurate algorithm based on a spectral collocation method is developed for numerical solution of the compressible, two‐dimensional and axisymmetric boundary layer equations. The numerical method incorporates a fifth‐order, fully implicit marching scheme in the streamwise (timelike) dimension and a spectral collocation method based on Chebyshev polynomial expansions in the wall‐normal (spacelike) dimension. The discrete governing equations are cast in residual form and the residuals are minimized at each marching step by a preconditioned Richardson iteration scheme which fully couples energy, momentum and continuity equations. Preconditioning on the basis of the finite difference analogues of the governing equations results in a computationally efficient iteration with acceptable convergence properties. A practical application of the algorithm arises in the area of compressible linear stability theory, in the investigation of the effects of transverse curvature on the stability of flows over axisymmetric bodies. The spectral collocation algorithm is used to derive the non‐similar mean velocity and temperature profiles in the boundary layer of a ‘fuselage’ (cylinder) in a high‐speed (Mach 5) flow parallel to its axis. The stability of the flow is shown to be sensitive to the gradual streamwise evolution of the mean flow and it is concluded that the effects of transverse curvature on stability should not be ignored routinely.
Supercomputer simulation of liquid drop formation on a solid surface
Authors: Greenspan Donald
Published: November 2005

Using a molecular‐dynamics‐type approach, we show how to simulate the formation of a liquid drop on a solid surface. Application is made to the case in which the liquid is water and the solid is graphite. The dynamical equations are large systems of nonlinear, ordinary differential equations which must be solved numerically. CRAY X‐MP/24 simulations and related contact angle calculations are described and discussed.
Numerical analysis of time‐dependent Boussinesq models
Authors: van der Houwen P. J. , Mooiman J. , Wubs F. W.
Published: November 2005

In this paper we analyse numerical models for time‐dependent Boussinesq equations. These equations arise when so‐called Boussinesq terms are introduced into the shallow water equations. We use the Boussinesq terms proposed by Katapodes and Dingemans. These terms generalize the constant depth terms given by Broer. The shallow water equations are discretized by using fourth‐order finite difference formulae for the space derivatives and a fourth‐order explicit time integrator. The effect on the stability and accuracy of various discrete Boussinesq terms is investigated. Numerical experiments are presented in the case of a fourth‐order Runge‐Kutta time integrator.
The integrated singular basis function method for the stick‐slip and the die‐swell problems
Authors: Georgiou Georgios , Olson Lorraine , Schultz William
Published: November 2005

We further develop a new singular finite element method, the integrated singular basis function method (ISBFM), for the solution of Newtonian flow problems with stress singularities. The ISBFM is based on the direct subtraction of the leading local solution terms from the governing equations and boundary conditions of the original problem, followed by a double integration by parts applied to those integrals with singular contributions. The method is applied to the stick‐slip and the die‐swell problems and improves the accuracy of the numerical results in both cases. In the case of the die‐swell problem it considerably accelerates the convergence of the free surface profile with mesh refinement. The advantages and disadvantages of the ISBFM when compared to other singular methods are also discussed.
Application of a moving grid method to a class of 1D brine transport problems in porous media
Authors: Zegeling P. A. , Verwer J. G. , Van Eijkeren J. C. H.
Published: November 2005

The background of this paper is the study of transport of pollutants by groundwater flow when released from a repository in a rock salt formation. Flow in regions surrounding such formations may be strongly influenced by variations in salt concentrations, a factor requiring special attention in the development of realistic mathematical models for predicting transport of pollutants. Indispensable for this development are advanced numerical methods. The aim of this paper is to illustrate the application of such a method to a class of non‐linear brine transport problems in one space dimension. Our method is based on the method of lines for solving time‐dependent partial differential equations. The method is of the finite difference type, implicit and thus applicable to wide classes of (one‐space‐dimensional) partial differential equation systems. The main feature of the method, however, is that it can automatically move the spatial grid for evolving time and thus is able to refine the grid in regions with large, special transitions. The grid refinement has provea to be a very valuable facility in the numerical modelling of brine transport problems involving low and high salt concentrations. From the user's point of view an additional advantage of the moving grid method is that it can be implemented in advanced, user‐oriented method‐of‐lines software packages based on implicit stiff ODE solvers. In the brine transport application discussed here we have used the package SPRINT.
A finite volume Navier‐Stokes algorithm for adaptive grids
Authors: Kallinderis Yannis
Published: November 2005

A compact, finite volume, time‐marching scheme for the two‐dimensional Navier‐Stokes equations of viscous fluid flow is presented. The scheme is designed for unstructured (locally refined) quadrilateral meshes. An earlier inviscid equation (Euler) scheme is employed for the convective terms and the emphasis is on treatment of the viscous terms. An essential feature of the algorithm is that all necessary operations are restricted to within each cell, which is very important when dealing with unstructured grids. Numerical issues which have to be addressed when developing a Navier‐Stokes scheme are investigated. These issues are not limited to the particular Navier‐Stokes scheme developed in the present work but are general problems. Specifically, the extent of the numerical molecule, which is related to the compactness of the scheme and to its suitability for unstructured grids, is examined. An approach which considers suppression of odd‐even mode decoupling of the solution when designing a scheme is presented. In addition, accuracy issues related to grid stretching as well as boundary layer solution contamination due to artificial dissipation are addressed. Although the above issues are investigated with respect to the specific scheme presented, the conclusions are valid for an entire class of finite volume algorithms. The Navier‐Stokes solver is validated through test cases which involve comparisons with analytical, numerical and experimental results. The solver is coupled to an adaptive algorithm for high‐Reynolds‐number aerofoil flow computations.
Numerical modelling of cavities on axisymmetric bodies at zero and non‐zero angle of attack
Authors: Ingber Marc S. , Hailey Chris E.
Published: November 2005

The flow about submerged, fully cavitating axisymmetric bodies at both zero and non‐zero angle of attack is considered in this paper. A cavity closure model that relates the point of detachment, the angle that the separating streamline makes with the body and the cavity length is described. The direct boundary element method is used to solve the potential flow problem and to determine the cavity shape. A momentum integral boundary layer solver is included in the formulation so that shear stresses can be incorporated into the drag calculations. The numerical predictions based on the proposed closure model are compared with water tunnel measurements and photographs.
Invariant discretization of the incompressible Navier‐Stokes equations in boundary fitted co‐ordinates
Authors: Segal A. , Wesseling P. , Van Kan J. , Oosterlee C. W. , Kassels K.
Published: November 2005

The discretization of the incompressible Navier‐Stokes equation on boundary‐fitted curvilinear grids is considered. The discretization is based on a staggered grid arrangement and the Navier‐;Stokes equations in tensor formulation including Christoffel symbols. It is shown that discretization accuracy is much enhanced by choosing the velocity variables in a special way. The time‐dependent equations are solved by a pressure‐correction method in combination with a GMRES method. Special attention is paid to the discretization of several types of boundary conditions. It is shown that fairly non‐smooth grids may be used using our approach.
Large‐eddy simulation of turbulent flow and heat transfer in plane and rib‐roughened channels
Authors: Ciofalo M. , Collins M. W.
Published: November 2005

Large‐eddy simulation results are presented and discussed for turbulent flow and heat transfer in a plane channel with and without transverse square ribs on one of the walls. They were obtained with the finite‐difference code Harwell‐FLOW3D, Release 2, by using the PISOC pressure‐velocity coupling algorithm, central differencing in space, and Crank‐Nicolson time stepping. A simple Smagorinsky model, with van Driest damping near the walls, was implemented to model subgrid scale effects. Periodic boundary conditions were imposed in the streamwise and spanwise directions. The Reynolds number based on hydraulic diameter (twice the channel height) ranged from 10 000 to 40 000. Results are compared with experimental data, k‐ϵ predictions, and previous large‐eddy simulations.
Flow field modelling near a well with a conductive fracture
Authors: Li Y. C. , Huang N. C.
Published: November 2005

In the technology of oil recovery the oil production rate can be increased by generation of a vertical sand‐filld conductive fracture on the wall of the well. Oil diffuses through the conductive fracture to the well. In this paper the seepage flow and isothermal deformation fields in both the formation and fracture and the oil production rate at the well are studied by modelling the formation as an infinite poroelastic medium saturated with a one‐phase compressible fluid. The fracture is treated as a one‐dimensional poroelastic medium. Darcy flows are considered in both the formation and fracture. The plane strain condition is imposed. Our solution is obtained numerically by a finite element method based on a variational principle. The accuracy of the analysis is studied by comparison of the numerical solutions of some problems with their analytical solutions. Since we are dealing with the transient flow problem of an infinite region, an extrapolation technique is employed to find the finite element solution. The production rate of a well with the conductive fracture is compared with that of a well without the conductive fracture.
A finite element method for a three‐dimensional second‐order closure model of turbulent diffusion in a convective boundary layer
Authors: Pai Prasad , Tsang Tate T. H.
Published: November 2005

A three‐dimensional second‐order closure dispersion model is used to simulate the plume behaviour of a passive contaminant in a convective boundary layer. A time‐splitting finite element method together with a non‐linear filtering scheme is used to solve the three‐dimensional second‐order closure transport equations. The model results show good agreement with laboratory data for a ground level source.
Numerical simulation of laminar and turbulent flows around rectangular cylinders
Authors: Okajima Atsushi , Ueno Hisanori , Sakai Haruhisa
Published: November 2005

By a finite volume method, laminar flows around bluff bodies with a rectangular cross‐section of various width‐to‐height ratios from 0·2 to 10 and with a cross‐section of a round leading edge and a square trailing edge have been computed on body‐fitted curvilinear co‐ordinates at Reynolds numbers of (1, 4, 7) × 103. Turbulent flows have also been computed by a standard k‐ϵ turbulence model. Computed results are compared with experimental data at a Reynolds number of 103 and clearly show the effects of the shape of the bluff body on the aerodynamic characteristics. We can successfully simulate some interesting phenomena whereby the flow pattern changes critically when the side ratio B/H is about 2·1 and 6; that is, a fully separated flow, an alternately reattached one and a stationarily reattached one. The results also reveal interactions between the wake and separation bubbles. There are, however, significant discrepancies between the results from the k‐ϵ turbulence model and experiments.
Stepwise increase in the Strouhal number for flows around flat plates
Authors: Ozono Shigehira , Ohya Yuji , Nakamura Yasuharu , Nakayama Ryuzo
Published: November 2005

The unsteady viscous flow around flat plates with square leading and trailing edges is investigated by using a finite difference computation of the incompressible two‐dimensional Navier‐Stokes equations. The chord‐to‐thickness ratio of a plate ranges from d/h=3 to 9, with a Reynolds number based on the plate's thickness equal to 103. The numerical analysis confirms the finding obtained in our previous experiment that vortex shedding from flat plates with square leading and trailing edges is caused by the impinging shear layer instability. The Strouhal number based on the plate's chord increases stepwise with increasing d/h in accordance with the experiment. The numerical analysis also gives some crucial information on the complicated vortical flow occurring near the trailing edge. Finally, the mechanism of the impinging shear layer instability is discussed in the light of the experimental and numerical findings.
Turbulence modelling for deep ocean radionuclide disposal
Authors: Onishi Yasuo , Trent Donald S.
Published: November 2005

In this study the adequacy of the k‐ϵ turbulence model and the feasibility of the three‐dimensional hydrodynamic‐transport models TEMPEST and FLESCOT for deep ocean radionuclide disposal assessment were evaluated qualitatively. TEMPEST and FLESCOT were applied to a hypothetical, two‐dimensional, deep ocean case with and without stratifications. TEMPEST with the k‐ϵ model was applied to obtain quasi‐steady state eddy viscosity distributions. With calculated eddy viscosity distributions as part of the input, FLESCOT then calculated distributions of velocity, water temperature, sediment and the dissolved and sediment‐sorbed radionuclide, assuming that the radionuclide was disposed on the ocean bottom.

Results revealed that the computed eddy viscosity increased almost linearly with vertical distance near the ocean bottom, then rapidly decreased towards a molecular viscosity value when the vertical gradient of the velocity distribution became very small. The results also demonstrate the importance of the density gradient to suppress the turbulent kinetic energy production, resulting in reduced eddy viscosity, producing the maximum computed eddy viscosity of 0·2 Pa s, which compares well with the reported value of 0·07 Pa s in the deep ocean. Thus the k‐ϵ turbulence model appears to be qualitatively applicable to the deep ocean environment.
A practical implementation of spectral methods resistant to the generation of spurious eigenvalues
Authors: Lindsay K. A. , Ogden R. R.
Published: November 2005

This work describes a practical way of constructing a spectral representation of linear boundary value problems (BVPs) using a tau method. All BVPs are treated as first‐order systems, unlike most implementations which tend to view the problem in terms of a single high‐order differential equation. For most applications this formulation will adhere more closely to the natural derivation of the original equations from, for example, a series of conservation laws. The technique is exemplified for Chebyshev polynomials in a variety of real applications, although detailed results are provided for any polynomial basis.
Computation of flow of viscoelastic fluids by parameter differentiation
Authors: Ariel P. D.
Published: November 2005

A technique combining the features of parameter differentiation and finite differences is presented to compute the flow of viscoelastic fluids. Two flow problems are considered: (i) three‐dimensional flow near a stagnation point and (ii) axisymmetric flow due to stretching of a sheet. Both flows are characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. The exact numerical solutions are obtained using the technique described in the paper. Also, the first‐order perturbation solutions (in terms of the viscoelastic fluid parameter) are derived. A comparison of the results shows that the perturbation method is inadequate in predicting some of the vital characteristic features of the flows, which can possibly be revealed only by the exact numerical solution.
Turbulent flow through bifurcated nozzles
Authors: Hershey Donald E. , Thomas Brian G. , Najjar Fady M.
Published: November 2005

A finite‐element model has been used to study steady‐state turbulent flow through bifurcated submerged‐entry nozzles with oversized ports typical of those used in the continuous casting of steel. Both 2D and 3D simulations have been performed with the commercial code FIDAP, using the standard K–ϵ turbulence model. Predicted velocities from 3D simulations compare reasonably with experimental measurements using a hot‐wire anemometer conducted in a physical water model, where severe turbulent fluctuations are present. Results show that a 2D simulation can also capture the main flow characteristics of the jet existing the nozzle and requires two orders of magnitude less computer time than the 3D simulation. A model combining the nozzle and mould was set up to study the effect of the outlet boundary conditions of the nozzle on the jet characteristics. This modelling technique will assist in the design of submerged‐entry nozzles, especially as applied to enhance steel quality in the continuous casting process. Further, the model will provide appropriate inlet boundary conditions for a separate numerical model of the mould.
Spectral methods for the viscoelastic time‐dependent flow equations with applications to Taylor–Couette flow
Authors: Avgousti Marios , Liu Baichen , Beris Antony N.
Published: November 2005

The time evolution of finite amplitude axisymmetric perturbations (Taylor cells) to the purely azimuthal, viscoelastic, cylindrical Couette flow was numerically simulated. Two time integration numerical methods were developed, both based on a pseudospectral spatial approximation of the variables, efficiently implemented using fast Poisson solvers and optimal filtering routines. The first method, applicable for finite Re numbers, is based on a time‐splitting integration with the divergence‐free condition enforced through an influence matrix technique. The second one, is based on a semi‐implicit time integration of the constitutive equation with both the continuity and the momentum equations enforced as constraints. Stability results for an upper convected Maxwell fluid were obtained for the supercritical bifurcations, either steady or time‐periodic, developed after the onset of instabilities in the primary flow. At small elasticity values, ϵ ≡ De/Re, the time integration of finite amplitude disturbances confirms the stability of the single branch of steady Taylor cells. At intermediate ϵ values the rotating wave family of time‐periodic solutions developed at the onset of instability is stable, whereas the standing wave is found to be unstable. At high ϵ values, and in particular for the limit of creeping flow (ϵ = ∞), the present study shows that the rotating wave family is unstable and the standing (radial) wave is stable, in agreement with previous finite‐element investigations. It is thus shown that spectral techniques provide a robust and computationally efficient method for the simulation of complex, non‐linear, time‐dependent viscoelastic flows.
Segregated finite element algorithms for the numerical solution of large‐scale incompressible flow problems
Authors: Haroutunian Vahé , Engelman Michael S. , Hasbani Isaac
Published: November 2005

This paper presents results of an ongoing research program directed towards developing fast and efficient finite element solution algorithms for the simulation of large‐scale flow problems. Two main steps were taken towards achieving this goal. The first step was to employ segregated solution schemes as opposed to the fully coupled solution approach traditionally used in many finite element solution algorithms. The second step was to replace the direct Gaussian elimination linear equation solvers used in the first step with iterative solvers of the conjugate gradient and conjugate residual type. The three segregated solution algorithms developed in step one are first presented and their integrity and relative performance demonstrated by way of a few examples. Next, the four types of iterative solvers (i.e. two options for solving the symmetric pressure type equations and two options for solving the non‐symmetric advection–diffusion type equations resulting from the segregated algorithms) together with the two preconditioning strategies employed in our study are presented. Finally, using examples of practical relevance the paper documents the large gains which result in computational efficiency, over fully coupled solution algorithms, as each of the above two main steps are introduced. It is shown that these gains become increasingly more dramatic as the complexity and size of the problem is increased.
A new method for accelerating the rate of convergence of the simple‐like algorithm
Authors: Wen X. , Ingham D. B.
Published: November 2005

A pressure correction formula is proposed for the SIMPLE‐like algorithm in order to improve the rate of the convergence when solving laminar Navier–Stokes equations when there is rapidly varying pressure. Based on global mass conservation, a line average pressure correction is derived by integration of the momentum equation for approximate one‐dimensional flow. The use of this formula with the SIMPLE‐like algorithm can rapidly build up the pressure distribution in the region where the pressure undergoes a very large change, which normally causes the rate of convergence of the SIMPLE or the SIMPLEC schemes to be slow. In order to illustrate the technique, the performances of SIMPLE and of SIMPLEC with the average pressur correction are investigated for axisymmetric flow past and through a sampler. A comparison of these two techniques shows that the average pressure correction proposed in this paper significantly accelerates the rate of convergence.
Methods for extending high‐resolution schemes to non‐linear systems of hyperbolic conservation laws
Authors: Rider William J.
Published: November 2005

In extending high‐resolution methods from the scalar case to systems of equations there are a number of options available. These options include working with either conservative or primitive variables, characteristic decomposition, two‐step methods, or component‐wise extension. In this paper, several of these options are presented and compared in terms of economy and solution accuracy. The characteristic extension with either conservative or primitive variables produces excellent results with all the problems solved. Using primitive variables, the two‐step formulation produces high‐quality results in a more economical manner. This method can also be extended to multiple dimensions without resorting to dimensional splitting. Proper selection of limiters is also important in non‐characteristic extension to systems.
A multigrid procedure for three‐dimensional flows on non‐orthogonal collocated grids
Authors: Smith K. M. , Cope W. K. , Vanka S. P.
Published: November 2005

The development of a multigrid solution algorithm for the computation of three‐dimensional laminar fully‐elliptic incompressible flows is presented. The procedure utilizes a non‐orthogonal collocated arrangement of the primitive variables in generalized curvilinear co‐ordinates. The momentum and continuity equations are solved in a decoupled manner and a pressure‐correction equation is used to update the pressures such that the fluxes at the cell faces satisfy local mass continuity. The convergence of the numerical solution is accelerated by the use of a Full Approximation Storage (FAS) multigrid technique. Numerical computations of the laminar flow in a 90° strongly curved pipe are performed for several finite‐volume grids and Reynolds numbers to demonstrate the efficiency of the present numerical scheme. The rates of convergence, computational times, and multigrid performance indicators are reported for each case. Despite the additional computational overhead required in the restriction and prolongation phases of the multigrid cycling, the superior convergence of the present algorithm is shown to result in significantly reduced overall CPU times.
Accuracy and convergence of element‐by‐element iterative solvers for incompressible fluid flows using penalty finite element model
Authors: Reddy M. P. , Reifschneider L. G. , Reddy J. N. , Akay H. U.
Published: November 2005

The ability of two types of Conjugate Gradient like iterative solvers (GMRES and ORTHOMIN) to resolve large‐scale phenomena as a function of mesh density and convergence tolerance limit is investigated. The flow of an incompressible fluid inside a sudden expansion channel is analysed using three meshes of 400, 1600 and 6400 bilinear elements. The iterative solvers utilize the element‐by‐element data structure of the finite element technique to store and maintain the data at the element level. Both the mesh density and the penalty parameter are found to influence the choice of the convergence tolerance limit needed to obtain accurate results. An empirical relationship between the element size, the penalty parameter, and the convergence tolerance is presented. This relationship can be used to predict the proper choice of the convergence tolerance for a given penalty parameter and element size.