AN EVALUATION OF THE BOUNCE-BACK BOUNDARY CONDITION FOR LATTICE BOLTZMANN SIMULATIONS

The bounce-back boundary condition for lattice Boltzmann simulations is evaluated for flow about an infinite periodic array of cylinders. The solution is compared with results from a more accurate boundary condition formulation for the lattice Boltemann method and with finite difference solutions. The bounce-back boundary condition is used to simulate boundaries of cylinders with both circular and octagonal cross-sections. The convergences of the velocity and total drag associated with this method are slightly sublinear with grid spacing. Error is also a function of relaxation time, increasing exponentially for large relaxation times. However, the accuracy does not exhibit a trend with Reynolds number between 0·1 and 100. The square lattice Boltzmann grid conforms to the octagonal cylinder but only approximates the circular cylinder, and the resulting error associated with the octagonal cylinder is half the error of the circular cylinder. The bounce-back boundary condition is shown to yield accurate lattice Boltzmann simulations with reduced computational requirements for computational grids of 170×170 or finer, a relaxation time less than 1·5 and any Reynolds number from 0·1 to 100. For this range of parameters the root mean square error in velocity and the relative error in drag coefficient are less than 1 per cent for the octagonal cylinder and 2 per cent for the circular cylinder. © 1997 John Wiley & Sons, Ltd.     

New analytical solutions to water wave radiation by vertical truncated cylinders through multi-term Galerkin method

New analytical solutions to water wave radiation by vertical truncated circular cylinders are developed based on linear potential flow theory. Two typical cylinder configurations of a surface-piercing cylinder and a submerged floating cylinder are considered. The multi-term Galerkin method is employed in the solution procedure, in which the fluid velocity on the interface between different regions is expanded into a set of basis function involving the Gegenbauer polynomials, and the cube-root singularity of fluid velocity at the side edges of the truncated cylinders is correctly modeled. The present solutions have the merits of very rapid convergence. The results with six-figure accuracy for added mass and radiation damping can be obtained using a few truncated numbers in the basis function for three motions (surge, heave and roll). The calculated results of the present solutions agree well with that by a higher-order boundary element method solution. Calculation examples are presented to investigate the influence of the motion frequency on the added mass and the radiation damping of the truncated cylinders with different geometric parameters. The present solutions can be used as a reliable benchmark for numerical solutions to water wave radiation by complicated structures.

     

The method of fundamental solutions for Stokes flow in a rectangular cavity with cylinders

Numerical solutions based on the method of fundamental solutions are discussed for Stokes flow inside a rectangular cavity in the presence of circular cylinders. The Stokeslets are used as the fundamental solutions to obtain the solution for the flow field by a linear combination of fundamental solutions. Flow results on the cellular structure of flow field resulting from the dynamics of cylinders and the horizontal walls of the cavity are reported for (i) one rotating cylinder in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions, (ii) two rotating cylinders kept apart in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions. The effect of aspect ratio of the rectangular cavity, direction of movement of the two parallel horizontal sides of the cavity and the diameter of the rotating cylinder on the flow structure are studied. The flow results obtained for the single cylinder case are in accordance with the results available in the literature. From the computational point of view, the present numerical procedure based on the method of fundamental solutions is efficient and simple to implement as compared to the mesh-dependent schemes, which needs complex mesh generation procedure for the multiply connected geometrical domains considered in this article.     

A two-dimensional model of the flow of ordered suspensions of rods

Analysis of creeping flow over an array of freely-rotating cylinders sandwiched between two sliding parallel plates is studied using a finite-difference and a least-squares numerical technique. The flow pattern was found to be very much influenced by the cylinder-to-cylinder spacing and by the gap width of the parallel plates. The shear stress on the cylinder surface and on the parallel plates was found to be a strong function of position. The viscosity of a suspension composed of an array of freely-rotating cylinders was deduced from the applied shear rate and the evaluated shear stress on the parallel plates. Experimental results confirm the numerical findings.     

Measurements of velocity fields in finite cylinder arrays with and without tip clearance

The velocity field in a finite cylinder array was investigated experimentally in a water towing tank and an acoustic Doppler velocimeter (ADV). The experimental system consisted of a staggered cylinder array having 14 rows to permit streamwise evolution of the flow. The boundaries were manipulated to create several global flow configurations. Three basic configurations were studied: a globally unidirectional flow, a flow with partial lateral blockage at the inlet and outlet planes, and a flow with the top boundary separated from the cylinders creating a tip clearance. The three components of the velocity vector were measured at various points within the model. Time-averaged results are presented for the different flow configurations. The results provide insight into the development of the flow field in cases of a finite array with complex geometry and boundary effects.     

Elastic and viscoelastic solutions to rotating functionally graded hollow and solid cylinders

Analytical solutions to rotating functionally graded hollow and solid long cylinders are developed. Young＇s modulus and material density of the cylinder are assumed to vary exponentially in the radial direction, and Poisson＇s ratio is assumed to be constant. A unified governing equation is derived from the equilibrium equations, compatibility equation, deformation theory of elasticity and the stress-strain relationship. The governing second-order differential equation is solved in terms of a hypergeometric function for the elastic deformation of rotating functionally graded cylinders. Dependence of stresses in the cylinder on the inhomogeneous parameters, geometry and boundary conditions is examined and discussed. The proposed solution is validated by comparing the results for rotating functionally graded hollow and solid cylinders with the results for rotating homogeneous isotropic cylinders. In addition, a viscoelastic solution to the rotating viscoelastic cylinder is presented, and dependence of stresses in hollow and solid cylinders on the time parameter is examined.     

The Slip Velocity at the Edge of a Porous Medium: Effects of Interior Resistance and Interface Curvature

Singularity methods are used to analyze creeping planar flow in the annulus between concentric cylinders, when a portion of the annulus is filled with an array of regularly spaced rods adjacent to the inner cylinder. The rods are evenly spaced on concentric circles, and the circles are spaced such that the array resembles a square lattice bent into a circle. The rods and inner cylinder are stationary, and steady rotation of the outer cylinder generates the flow. The quantity of interest is the slip velocity, the mean velocity at the interface between the array and the unfilled portion of the annulus. The primary part of the study concerns the influence of the interior rods on the interfacial velocity, and to this end the velocity is found as successive circles of rods are removed, starting with the circle closest to the inner cylinder. The calculations are carried out for solid volume fractions from 0.0001 to 0.1, and these show that the slip velocity is virtually unchanged as the interior circles of rods are removed, until only one circle remains and then the velocity is of order 10% larger than that for the full array. Hence the velocity at the edge of a sparse porous medium depends minimally on the hydrodynamic resistance of the obstacles in the interior. In the secondary part of the study, it is found that curvature of the interface does not influence the velocity there.     

Incompressible fluid flow past a circular cylinder with linear relation between the vorticity and the stream function

Incompressible fluid flow with a linear relationship between the vorticity and the stream function past a circular cylinder is studied.Vortical flows about profiles have been considered in several studies [1–15], but in all these studies with the exception of [15] a constant vorticity was assumed (in [15] an approximate solution is found of the problem of incompressible fluid flow about a Zhukovskii profile with parabolic distribution of the velocities in the approaching stream).A freestream velocity profile similar to that considered below occurs, for example, in a planar jet (laminar or turbulent), in the wake behind a bluff body, in the boundary layer along an infinite plane [4,13], in turbulent jet flows with reverse fluid currents [16]. A similar situation also arises in the flow past an array of cylinders with large spacing which is located in the wake of another array.The author wishes to thank V. E. Davidson for posing the problem and for guidance in its solution.     

Dual cylinder inviscid vortex sound of low Mach number

The unsteady motions of an inviscid vortex under the influence of a cylinder pair in the presence of a low Mach number mean flow and the corresponding sound generation are examined in the present study. The two cylinders are in close proximity. A semi-analytical approach using the conformal mapping together with the potential theory is adopted. The results show that the vortex will interact intensively with the cylinders under the right combinations of mean flow direction and initial vortex position. Such interactions result in a high rate of change of vortex propagation velocity, strong fluctuating forces on cylinder and strong sound radiations. However, it is found that much stronger acoustic energy radiation will result when the vortex approaches the cylinder pair from the bottom than from the top, unless the mean flow is nearly perpendicular to the horizontal cylinder pair axis. Stronger sound radiation is also observed for the identical cylinder cases in general, except the flow direction is close to some critical values.     

Flow of wormlike micelle solutions past a confined circular cylinder

In this paper, we present the results of an investigation into the flow of a series of viscoelastic wormlike micelle solutions past a confined circular cylinder. Although this benchmark flow has been studied in great detail for polymer solutions, this paper reports the first experiments to use a viscoelastic wormlike micelle solution as the test fluid. The flow kinematics, stability and pressure drop were examined for two different wormlike micelle solutions over a wide range of Deborah numbers and cylinder to channel aspect ratios. A combination of particle image velocimetry and pressure drop measurements were used to characterize the flow kinematics, while flow-induced birefringence measurements were used to measure the micelle deformation and alignment in the flow. The pressure drop was found to decrease initially due to the shear thinning of the test fluid before increasing at higher flow rates as elastic effects begin to dominate the flow. Above a critical Deborah number, an elastic instability was observed for just one of the test fluids studied, the other remained stable for all Deborah number tested. Flow-induced birefringence and velocimetry measurements showed that observed instability originates in the extensional flow in the wake of the cylinder and appears not as periodic counter-rotating vortices as has been observed in the flow of polymer solutions past circular cylinders, but as a chaotic rupture event in the wake of the cylinder that propagates axially along the cylinder. Reducing the cylinder to channel aspect ratio and the degree of shearing introduced by the channel walls had a weak impact on the stability of the flow. These measurements, when taken in conjunction with previous work on flow of wormlike micelle solutions through a periodic array of cylinders, definitively show that the instability can be attributed to a breakdown of the wormlike micelle solutions in the extensional flow in the wake of the cylinder.     

Heat transfer promotion with a cylinder array located near the wall

Heat transfer from a flat plate has been investigated when a cylinder array is located near the wall. Each cylinder in the cylinder array was positioned normal to the flow direction and parallel to the flat plate surface. Measurements of the heat transfer coefficient and the optimum value for the cylinder pitch and spacing between the cylinders and the flat plate surface were obtained. A comparison of the heat transfer mechanism in this flow system with that obtained previously for the case when a single cylinder is inserted in the boundary layer was made.     

A class of exact solutions of a nonlinear boundary-value problem for fluid flow in a rotating cylinder

The flow of an ideal incompressible weightless fluid that fills a rotating cylinder is investigated. The rotation axis of the cylinder is outside it and parallel to the cylinder generator, and the form of the cylinder section is determined in the process of solution of the problem. In the paper, a class of exact solutions of the problem is obtained in terms of elementary functions for different angular velocities of the cylinder. In these solutions, the flow field is formed by two straight vortex filaments parallel to the cylinder generator. The intensities of the vortex filaments are determined by the angular velocity . Investigations of ideal fluid flow in rotating vessels were begun already in the last century by Stokes and Zhukovskii [1]. The subject has been reviewed in monographs [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 1, pp. 71–75, January–February, 1984.     

Hysteresis phenomenon at interaction of shock waves generated by a cylinder array

This paper describes investigations of the interaction between bow shock waves generated by cylindrical bodies in a supersonic flow. Numerical simulations are performed using the inviscid Euler equations for cylinders whose axes are parallel to each other and normal to the flow direction. Mostly an infinite periodical cylinder array is considered, but the case of two cylinders is also briefly discussed. Three different regimes of the shock wave interaction, a regular interaction, a Mach interaction, and a choked flow, have been observed for the flow through the periodical cylinder array. In the case of the flow around two bodies, the choked flow is replaced by a regime with a collective bow shock. The transition between different flow regimes is studied by varying the inflow Mach number or the distance between the cylinders. A hysteretic behavior at the transition between the regular and Mach interactions has been observed. The transition is governed by the theoretical detachment and von Neumann criteria based on the local shock wave inclination at the interaction point.     

Natural convection in vertical concentric annuli under a radial magnetic field

Exact solutions for fully developed natural convection in open-ended vertical concentric annuli under a radial magnetic field are presented. Expressions for velocity field, temperature field, mass flow rate and skin-friction are given, under more general thermal boundary conditions. It is observed that both velocity as well as temperature of the fluid is more in case of isothermal condition compared with constant heat flux case when gap between cylinders is less or equal to radius of inner cylinder while reverse phenomena occur when the gap between cylinders is greater than radius of inner cylinder.     

Effect of induced magnetic field on natural convection in vertical concentric annuli

In the present paper,we have considered the steady fully developed laminar natural convective flow in open ended vertical concentric annuli in the presence of a radial magnetic field.The induced magnetic field produced by the motion of an electrically conducting fluid is taken into account.The transport equations concerned with the considered model are first recast in the non-dimensional form and then unified analytical solutions for the velocity,induced magnetic field and temperature field are obtained for the cases of isothermal and constant heat flux on the inner cylinder of concentric annuli.The effects of the various physical parameters appearing into the model are demonstrated through graphs and tables.It is found that the magnitude of maximum value of the fluid velocity as well as induced magnetic field is greater in the case of isothermal condition compared with the constant heat flux case when the gap between the cylinders is less or equal to 1.70 times the radius of inner cylinder,while reverse trend occurs when the gap between the cylinders is greater than 1.71 times the radius of inner cylinder.These fields are almost the same when the gap between the cylinders is equal to 1.71 times the radius of inner cylinder for both the cases.It is also found that as the Hartmann number increases,there is a flattening tendency for both the velocity and the induced magnetic field.The influence of the induced magnetic field is to increase the velocity profiles.     

Near-wake flow structure of elliptic cylinders close to a free surface: effect of cylinder aspect ratio

The flow fields behind elliptic cylinders adjacent to a free surface were investigated experimentally in a circulating water channel. A range of cylinder aspect ratios (AR=2, 3, 4) were considered, while the cross-sectional area of the elliptical cylinder was kept constant. The main objective of this study was to investigate the effect of cylinder aspect ratio and a free surface on the flow structure in the near-wake behind elliptic cylinders. For each elliptic cylinder, the flow structure was analyzed for various values of the submergence depth of the cylinder beneath the free surface. The flow fields were measured using a single-frame double-exposure PIV (Particle Image Velocimetry) system. For each experimental condition, 350 instantaneous velocity fields were obtained and ensemble-averaged to obtain the mean velocity field and spatial distribution of the mean vorticity statistics. The results show that near-wake can be classified into three typical flow patterns: formation of a Coanda flow, generation of substantial jet-like flow, and attachment of this jet flow to the free surface. The general flow structure observed behind the elliptic cylinders resembles the structure previously reported for a circular cylinder submerged near a free surface. However, the wake width and the angle of downward deflection of the shear layer developed from the lower surface of the elliptic cylinder differ from those observed for a circular cylinder. These trends are enhanced as cylinder aspect ratio is increased. In addition, the free surface distortion is also discussed in the paper.     

Unsteady flow of viscoelastic fluid between two cylinders using fractional Maxwell model

The unsteady flow of an incompressible fractional Maxwell fluid between two infinite coaxial cylinders is studied by means of integral transforms.The motion of the fluid is due to the inner cylinder that applies a time dependent torsional shear to the fluid.The exact solutions for velocity and shear stress are presented in series form in terms of some generalized functions.They can easily be particularized to give similar solutions for Maxwell and Newtonian fluids.Finally,the influence of pertinent parameters on the fluid motion,as well as a comparison between models,is highlighted by graphical illustrations.     

Numerical predictions of low Reynolds number flows over two tandem circular cylinders

Flows over two tandem cylinders were analysed using the newly developed collocated unstructured computational fluid dynamics (CUCFD) code, which is capable of handling complex geometries. A Reynolds number of 100, based on cylinder diameter, was used to ensure that the flow remained laminar. The validity of the code was tested through comparisons with benchmark solutions for flow in a lid‐friven cavity and flow around a single cylinder. For the tandem cylinder flow, also mesh convergence was demonstrated, to within a couple of percent for the RMS lift coefficient. The mean and fluctuating lift and drag coefficients were recorded for centre‐to‐centre cylinder spacings between 2 and 10 diameters. A critical cylinder spacing was found between 3.75 and 4 diameters. The fluctuating forces jumped appreciably at the critical spacing. It was found that there exists only one reattachment and one separation point on the downstream cylinder for spacings greater than the critical spacing. The mean and the fluctuating surface pressure distributions were compared as a function of the cylinder spacing. The mean and the fluctuating pressures were significantly different between the upstream and the downstream cylinders. These pressures also differed with the cylinder spacing. Copyright © 2004 John Wiley & Sons, Ltd.     

New exact solution of the problem of rotationally symmetric Couette-Poiseuille flow

An exact solution is obtained for the problem of steady-state viscous incompressible flow under a pressure difference in the gap between coaxial cylinders for the case where the inner cylinder rotates at a constant angular velocity. The solution differs from the classical Couette-Poiseuille result by the presence of radial mass transfer, which provides for interaction between the poloidal and azimuthal circulations. The flow rate is found to depend linearly on the angular velocity of rotation of the inner cylinder. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 71–77, September–October, 2007.     

A finite element formulation for global linear stability analysis of a nominally two‐dimensional base flow

A stabilized finite element method, to carry out the linear stability analysis of a two‐dimensional base flow to three‐dimensional perturbations that are periodic along span, is presented. The resulting equations for the time evolution of the disturbance requires a solution to the generalized eigenvalue problem. The analysis is global in nature and is also applicable to non‐parallel flows. Equal‐order‐interpolation functions for velocity and pressure are utilized. Stabilization terms are added to the Galerkin formulation to admit the use of equal‐order‐interpolation functions and to eliminate node‐to‐node oscillations that might arise in advection‐dominated flows. The proposed formulation is tested on two flow problems. First, the mode transitions in the circular Couette flow are investigated. Two scenarios are considered. In the first one, the outer cylinder is at rest, while the inner one spins. Two linearly unstable modes are identified. The primary mode is real and represents the axisymmetric Taylor vortices. The second mode is complex and consists of spiral vortices. For the counter‐rotating cylinders, the primary transition is via the appearance of spiral vortices. Excellent agreement with results from earlier studies is observed. The formulation is also utilized to investigate the parallel and oblique modes of vortex shedding past a cylinder for the Re = 100 flow. It is found that the flow is associated with a large number of unstable oblique shedding modes. The parallel mode of vortex shedding is a special case of this family of modes and is associated with the largest growth rate. Copyright © 2014 John Wiley & Sons, Ltd.