A boundary integral equation method for Navier-Stokes equations—application to flow in annulus of eccentric cylinders

A novel Navier-Stokes solver based on the boundary integral equation method is presented. The solver can be used to obtain flow solutions in arbitrary 2D geometries with modest computational effort. The vorticity transport equation is modelled as a modified Helmholtz equation with the wave number dependent on the flow Reynolds number. The non-linear inertial terms partly manifest themselves as volume vorticity sources which are computed iteratively by tracking flow trajectories. The integral equation representations of the Helmholtz equation for vorticity and Poisson equation for streamfunction are solved directly for the unknown vorticity boundary conditions. Rapid computation of the flow and vorticity field in the volume at each iteration level is achieved by precomputing the influence coefficient matrices. The pressure field can be extracted from the converged streamfunction and vorticity fields. The solver is validated by considering flow in a converging channel (Hamel flow). The solver is then applied to flow in the annulus of eccentric cylinders. Results are presented for various Reynolds numbers and compared with the literature.     

Laminar flow and heat transfer over a two-dimensional triangular step

The velocity correction algorithm is used in the finite element method to solve forced convection problems between parallel plates with a triangular step, for Reynolds numbers up to 1000. Equal-order interpolation functions for velocity, pressure and temperature are used. The solutions show a smooth variation of pressure. The streamfunction, isotherms, isobars and velocity profiles are presented for a typical Reynolds number of 500. The skin friction and heat transfer results are presented for Reynolds numbers up to 1000.     

Finite difference and finite element methods for mhd channel flows

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.     

Calculation of oseen flows past a circular cylinder at low reynolds numbers

The velocity, pressure, vorticity and streamfunction are computed in the Oseen hydrodynamic field of an unbounded fluid past a circular cylinder in the Reynolds Number range going from 0.4 to 12. The boundary condition is satisfied by means of the method of least squares that determines suitable coefficients for Faxén series. Particular investigation is made of the wake region in which calculations are made of flow patterns, velocity and vorticity distributions. It is shown that, attached vortices arise at the rear of the cylinder at Reynolds Number Re=3.025. Calculated drag coefficients are in good agreement with known results of the works of several authors up to a Reynolds Number of 20.     

Finite difference solution of a Newtonian jet swell problem

A finite difference technique has been developed to study the Newtonian jet swell problem. The streamfunction and vorticity were used as dependent variables to describe the jet flow. The boundary-fitted co-ordinate transformation method was adopted to map the flow geometry into a rectangular domain. The standard finite difference method was then applied for solving the flow equations. The location of the jet free surface was updated by the kinematic boundary condition, and an adjustable parameter was included in the free-surface iteration. We could obtain numerical solutions for the Reynolds number as high as 100, and the differences between the present study and previous finite element simulations on the jet swell ratio are less than 5%.     

A comparative study of central and upwind difference schemes using the primitive variables

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

Finite element solution of the streamfunction-vorticity equations for incompressible two-dimensional flows

The streamfunction-vorticity equations for incompressible two-dimensional flows are uncoupled and solved in sequence by the finite element method. The vorticity at no-slip boundaries is evaluated in the framework of the streamfunction equation. The resulting scheme achieves convergence, even for very high values of the Reynolds number, without the traditional need for upwinding. The stability and accuracy of the approach are demonstrated by the solution of two well-known benchmark problems: flow in a lid-driven cavity at Re ? 10,000 and flow over a backward-facing step at Re = 800.     

Integral transform solution for the lid-driven cavity flow problem in streamfunction-only formulation

The basic ideas in the generalized integral transform technique are further advanced to allow for the hybrid numerical-analytical solution of the two-dimensional steady Navier-Stokes equations in streamfunction-only formulation. The classical lid-driven square cavity problem is selected for illustration of the approach. The corresponding biharmonic-type non-linear partial differential equation for the streamfunction is integral transformed in one of the co-ordinates and an infinite system of coupled non-linear ODEs for the transformed potential results in the other independent variable. Upon truncation to an appropriate finite order, the ODE system is numerically solved by well-established algorithms with automatic error control devices. The convergence behaviour of the eigenfunction expansion is demonstrated and reference results are provided for typical values of Reynolds number.     

Application of fourth‒order difference method to prediction of turbulence control in two‒dimensional channel with surface roughness

The fourth‒order finite difference method is combined with the vorticity–streamfunction formulation in generalized co‒ordinates. Direct numerical simulations are performed for channel flows with and without surface roughness at a Reynolds number of 10⁴. The present results are in good agreement with those of the pseudospectral method with respect to the flow in a smooth channel. It is shown that the present method predicts well the precise change in the flow with the channel length and roughness height. The turbulence is generally weakened by the roughness. Laminarization is also accomplished under the appropriate condition. © 1997 John Wiley & Sons, Ltd.     

A numerical study of heat and momentum transfer for tube bundles in crossflow

A numerical scheme is developed to predict the heat transfer and pressure drop coefficients in flow through rigid tube bundles. The scheme uses the Galerkin finite element technique. The conservation equations for laminar steady-state flow are cast in the form of streamfunction and vorticity equations. A Picard iteration method is used for the solution of the resulting system of non-linear algebraic equations. Results for the heat transfer and pressure drop coefficients are obtained for tube arrays of pitch ratios of 1·5 and 2·0. Very good agreement of the present results and experimental data obtained in the past is observed up to Reynolds numbers of 1000. It is also observed that the results of the present method show better agreement with the experimental data and that they are applicable for higher Reynolds numbers than results of other studies.     

Numerical solutions of 2‐D steady incompressible driven cavity flow at high Reynolds numbers

Numerical calculations of the 2‐D steady incompressible driven cavity flow are presented. The Navier–Stokes equations in streamfunction and vorticity formulation are solved numerically using a fine uniform grid mesh of 601 × 601. The steady driven cavity flow solutions are computed for Re ? 21 000 with a maximum absolute residuals of the governing equations that were less than 10^?10. A new quaternary vortex at the bottom left corner and a new tertiary vortex at the top left corner of the cavity are observed in the flow field as the Reynolds number increases. Detailed results are presented and comparisons are made with benchmark solutions found in the literature. Copyright © 2005 John Wiley & Sons, Ltd.     

Effects of Periodic Excitation on the Flow Around a D-Shaped Cylinder at Low Reynolds Numbers

The baseline and forced flow around a bluff body with semi-elliptical D-shape was investigated by solving the 2D Navier–Stokes equations at low Reynolds numbers. A D-shape rather than the canonic circular-cylinder was selected due to the fixed separation points in the latter, enabling to study a pure wake rather than boundary-layer control. The correlation between Strouhal and Reynolds numbers, the mean drag, the lift and drag oscillations vs. the Reynolds number and wake structure were investigated and compared to experimental and numerical data. Effects of open-loop forcing, resulting from the influence of zero-mass-flux actuators located at the fixed separation points, were studied at a Reynolds number of 150. Fluidic rather than body motion or volume forcing was selected due to applicability considerations. The motivation for the study was to quantify the changes in the flow field features, as captured by Proper Orthogonal Decomposition (POD) analysis, due to open-loop forcing, inside and outside the “lock-in” regime. This is done in order to evaluate the suitability of low-order-models based on POD modes of this changing flow field, for future feed-back flow control studies. The evolution of the natural and the excited vortices in the Kármán wake were also investigated. The formation and convection regions of the vortex evolution were documented. It was found that the forcing causes an earlier detachment of the vortices from the boundary-layers, but does not affect their circulation or convection speeds. The results of the POD analysis of the near-wake flow show that the influence of the bluff body shape (“D”-shaped versus circular cylinder) on the baseline POD wake modes is small. It was found that the eigenfunctions (mode-shapes) of the POD velocity modes are less sensitive to slot excitation than the vorticity modes. As a result of the open-loop excitation, two types of mode-shape-change were observed: a mode can be exchanged with a lower-energy mode or shifted to a low energy level. In the latter case, the most energetic mode becomes the “actuator” mode. The evolution of one-slot excitation on still fluid (“Synthetic jet”) was studied and compared to published data and to “actuator” modes with external flow present. Based on the current findings, it is hypothesized that the cross-flow velocity POD modes are suitable for feedback control of wake flow using periodic excitation, due to their low sensitivity to the excitation as compared to the streamwise velocity or vorticity modes.     

Analysis of incompressible massively separated viscous flows using unsteady Navier–Stokes equations

The unsteady incompressible Navier–Stokes equations are formulated in terms of vorticity and stream-function in generalized curvilinear orthogonal co-ordinates to facilitate analysis of flow configurations with general geometries. The numerical method developed solves the conservative form of the vorticity transport equation using the alternating direction implicit method, whereas the streamfunction equation is solved by direct block Gaussian elimination. The method is applied to a model problem of flow over a backstep in a doubly infinite channel, using clustered conformal co-ordinates. One-dimensional stretching functions, dependent on the Reynolds number and the asymptotic behaviour of the flow, are used to provide suitable grid distribution in the separation and reattachment regions, as well as in the inflow and outflow regions. The optimum grid distribution selected attempts to honour the multiple length scales of the separated flow model problem. The asymptotic behaviour of the finite differenced transport equation near infinity is examined and the numerical method is carefully developed so as to lead to spatially second-order-accurate wiggle-free solutions, i.e. with minimum dispersive error. Results have been obtained in the entire laminar range for the backstep channel and are in good agreement with the available experimental data for this flow problem, prior to the onset of three-dimensionality in the experiment.     

Heat transfer characteristics and Nusselt number correlation of turbulent pulsating pipe air flows

Heat transfer characteristics to turbulent pulsating pipe flows under a wide range of Reynolds number and pulsation frequency were experimentally investigated under uniform heat flux condition. Reynolds number was varied from 8462 to 48540 while the frequency of pulsation ranged from 1 to 29.5 Hz. The results showed that the relative mean Nusselt number is strongly affected by both pulsation frequency and Reynolds number. Enhancements in mean Nusselt number of up to 50% were obtained at medium pulsation frequency between 4.1 and 13.9 Hz for Reynolds number range of 8462 to 14581. An enhancement of up to 50% in mean Nusselt number was obtained at high pulsation frequency range between 13.9 and 29.5 Hz, specially as Reynolds number is close to 15000, while a reduction was observed at higher Reynolds number more than 21200. This reduction, at high Reynolds number, increased as pulsation frequency increased. Also, there was a reduction in mean Nusselt number of up to 20% that obtained at low pulsation frequency range between 1 and 4.1 Hz for Reynolds number range of 8462 to 48543. A significant reduction in mean Nusselt number of up to 40% was obtained at medium pulsation frequency between 4.1 and 13.9 Hz for Reynolds number range of 21208 to 48543. Empirical equations have been developed for the relative mean Nusselt number that related to Reynolds number and dimensionless frequency with about uncertainty of 10% rms.The support of both King Fahd University of Petroleum and Minerals and Cairo University for this research is acknowledged.     

Unsteady mixed convection heat transfer from tandem square cylinders in cross flow at low Reynolds numbers

A two-dimensional numerical study is carried out to understand the influence of cross buoyancy on the vortex shedding processes behind two equal isothermal square cylinders placed in a tandem arrangement at low Reynolds numbers. The spacing between the cylinders is fixed with five widths of the cylinder dimension. The flow is considered in an unbounded medium, however, fictitious confining boundaries are chosen to make the problem computationally feasible. Numerical calculations are performed by using a finite volume method based on the PISO algorithm in a collocated grid system. The range of Reynolds number is chosen to be 50–150. The flow is unsteady laminar and two-dimensional in this Reynolds number range. The mixed convection effect is studied for Richardson number range of 0–2 and the Prandtl number is chosen constant as 0.71. The effect of superimposed thermal buoyancy on flow and isotherm patterns are presented and discussed. The global flow and heat transfer quantities such as overall drag and lift coefficients, local and surface average Nusselt numbers and Strouhal number are calculated and discussed for various Reynolds and Richardson numbers.     

A Lagrangian vortex method for unbounded flows

A technique is presented for velocity calculations on the highly distorted node distributions typical of those found in Lagrangian vortex methods. The method solves the partial differential equation for streamfunction directly on the nodes, via a sparse, symmetric system of equations that can be solved using standard iterative solvers. When implemented in a triangulated vortex method, the technique gives computation times which scale as N^1.23, where N is the number of nodes. The computation scheme is derived for two‐dimensional problems and applied to the prediction of the evolution of perturbed multipolar vortices. Due to the numerical performance of the method, it has been possible to examine such evolution at higher and lower Reynolds numbers than have been considered in published numerical studies. Copyright © 2008 John Wiley & Sons, Ltd.     

Reynolds number effects on a turbulent boundary layer with separation,reattachment, and recovery

The present paper addresses experimental studies of Reynolds number effects on a turbulent boundary layer with separation, reattachment, and recovery. A momentum thickness Reynolds number varies from 1,100 to 20,100 with a wind tunnel enclosed in a pressure vessel by varying the air density and wind tunnel speed. A custom-built, high-resolution laser Doppler anemometer provides fully resolved turbulence measurements over the full Reynolds number range. The experiments show that the mean flow is at most a very weak function of Reynolds number while turbulence quantities strongly depend on Reynolds number. Roller vortices are generated in the separated shear layer caused by the Kelvin–Helmholtz instability. Empirical Reynolds number scalings for the mean velocity and Reynolds stresses are proposed for the upstream boundary layer, the separated region, and the recovery region. The inflectional instability plays a critical role in the scaling in the separated region. The near-wall flow recovers quickly downstream of reattachment even if the outer layer is far from an equilibrium state. As a result, a stress equilibrium layer where a flat-plate boundary layer scaling is valid develops in the recovery region and grows outward moving downstream.     

Double-diffusive convection in a cubical lid-driven cavity with opposing temperature and concentration gradients

A numerical study of three-dimensional incompressible viscous flow inside a cubical lid-driven cavity is presented. The flow is governed by two mechanisms: (1) the sliding of the upper surface of the cavity at a constant velocity and (2) the creation of an external gradient for temperature and solutal fields. Extensive numerical results of the three-dimensional flow field governed by the Navier-Stokes equations are obtained over a wide range of physical parameters, namely Reynolds number, Grashof number and the ratio of buoyancy forces. The preceding numerical results obtained have a good agreement with the available numerical results and the experimental observations. The deviation of the flow characteristics from its two-dimensional form is emphasized. The changes in main characteristics of the flow due to variation of Reynolds number are elaborated. The effective difference between the two-dimensional and three-dimensional results for average Nusselt number and Sherwood number at high Reynolds numbers along the heated wall is analyzed. It has been observed that the substantial transverse velocity that occurs at a higher range of Reynolds number disturbs the two-dimensional nature of the flow.     

The effect of Reynolds number on boundary layer turbulence

A new facility for studying high Reynolds number incompressible turbulent boundary layer flows has been constructed. It consists of a moderately sized wind tunnel, completely enclosed by a pressure vessel, which can raise the ambient air pressure in and around the wind tunnel to 8 atmospheres. This results in a Reynolds number range of about 20:1, while maintaining incompressible flow. Results are presented for the zero pressure gradient flat plate boundary layer over a momentum thickness Reynolds number range 1500–15?000. Scaling issues for high Reynolds number non-equilibrium boundary layers are discussed, with data comparing the three-dimensional turbulent boundary layer flow over a swept bump at Reynolds numbers of 3800 and 8600. It is found that successful prediction of these types of flows must include length scales which do not scale on Reynolds number, but are inherent to the geometry of the flow.