NUMERICAL SIMULATION OF FLOW AND FORCED CONVECTION …

Suranaree J. Sci. Technol. 17(1):87-104

NUMERICAL SIMULATION OF FLOW AND FORCEDCONVECTION HEAT TRANSFER IN CROSSFLOW OFINCOMPRESSIBLE FLUID OVER TWO ROTATING CIRCULARCYLINDERS

Nikolay Pavlovich Moshkin*and Jakgrit SompongReceived: Aug 20, 2009; Revised: Oct 13, 2009; Accepted: Oct 20, 2009

Abstract

In this paper, the problem of laminar two dimensional heat transfer from two rotating circularcylinders in cross flow of incompressible fluid under isothermal boundary condition is investigated.The study is based on the numerical solution of the full conservation equations of mass, momentumand energy for Reynolds numbers (based on cylinder diameter and velocity of uniform stream) up to40 while Prandtl number ranges between 0.7 and 50. For the range of parameters considered, thestudy revealed that the rate of heat transfer decreases with the increase of speed of cylinders rotationfor the gap between cylinders more than one diameter. The increase of Prandtl number resulted inan appreciable increase in the average Nusselt number. The streamlines and isotherms are plottedfor a number of cases to show the details of the velocity and thermal fields.

Keywords: Heat transfer, flow over two rotating cylinders, incompressible fluid, finite differencemethod

Introduction

The flows of fluid and forced convection acrossa heated bluff body have been the subject ofconsiderable research interest because of theirrelevance to many engineering applications. Theflow past a cylinder is considered to be an idealbluff body by which to study the importantphenomena of heat and mass transfer. Forinstance, the knowledge of the hydrodynamicforces experienced by submerged cylindricalobjects such as off-shore pipelines is essentialfor the design of such structures. Furthermore,

because of changing process and climaticconditions, one also needs to determine the rateof heat transfer from such structures.

Heat transfer and fluid flow around a singlerotating cylinder has been studied by severalresearchers, see for example recent works of Badret al. (1989); Kang et al. (1999); Mahfouz andBard (1999a, 1999b); Stojkovi et al. (2002);Mittal and Kumar (2003); Gshwendtner (2004).

The heat transfer and flow around twostationary or rotating circular cylinders can be

Institute of science, School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, ThailandE-mail: [email protected]

* Corresponding author

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88 Numerical Simulation of Flow and Forced Convection Heat Transfer

considered as an elementary flow which ishelpful in understanding the flow patterns,heat transfer mechanism, and hydrodynamiccharacteristics around multiple bluff bodies inengineering practice.

Numerical computations for the flow overmultiple cylinders have intensified during thelast decade of the previous century. Chang andSong (1990) used a vorticity-stream functionmethod to compute the flow past a pair ofcylinders in side-by-side and tandem arrange-ments at Re = 100. Flow visualization andforce coefficients were shown to be in goodagreement with experiments. Mittal et al. (1997)used a finite element method to simulate threeconfigurations, side-by-side, tandem andstaggered arrangements of the cylinder pair atRe = 100 and 1000. Again, the results comparedwell with experiments. Recently, Kang (2003)investigated numerically the characteristics offlow around two side-by-side circular cylindersin the range of low Reynolds number definesas Re = U��D/v (where U��and v are the free-stream velocity and kinematic viscosity,respectively) over the range of 40 < Re < 160,respectively, and the normalize gap spacingg* < 5; he identified six kinds of wake patterns(g* = g / D, where g and D are the distancebetween two cylinders surfaces and the cylinderdiameter, respectively).

The problem of flow pass two rotatingcylinders in side-by-side arrangement has notbeen investigated widely. Papers of Sungnul andMoshkin (2006, 2008) are devoted to study theself-motion of two rotating circular cylinders.Only two recent studies (Yoon et al., 2007, 2009)have been found that investigate the flowaround two rotating circular cylinders in side-by-side arrangement in the range of α < 2for various gap spacing at Re = 100 (α is therotational speed at the cylinder surfacenormalized by the free-stream velocity). Onlyone paper of Joucaviel et al. (2008) has beenfound, which studies the thermal behavior ofan assembly of rotating cylinders aligned in across-flow.

As described above, the effect of rotationfor a single cylinder and of the gap spacingbetween two cylinders at rest in side-by-sidearrangement on the corresponding flow and

heat transfer has been studied by numerousresearchers. Only a few researchers havestudied the problem of flow pass two rotatingcylinders in a side-by-side arrangement.However, the heat transfer and fluid flow past apair of rotating circular cylinders in side-by-sidearrangement have not been addressed at all.This paper presents a numerical investigationof the characteristics of the two-dimensionalheat transfer and the laminar flow around tworotating circular cylinders in side-by-sidearrangements. In order to consider the combinedeffects of rotation and spacing between thetwo cylinders on the flow and heat transfer,numerical simulations are performed at avarious range of absolute rotational speeds( α < 2.5), for different gap spacing andReynolds number in the range Re < 40.Quantitative information about the flow andthe heat transfer variables such as the local andaverage Nusselt number, pressure and frictioncoefficients on the cylinder surfaces ishighlighted. The patterns of the flow andtemperature fields are analyzed for a wide rangeof parameters.

The mathematical formulation of theheat/mass transfer problem of a flow past tworotating circular cylinders is described in the nextsection. The problem is recast in terms of acylindrical bipolar coordinate system. Thefollowing sections present the details of thenumerical algorithm based on the projectionmethod to approximate the solution of themomentum equation, and the fractional stepstabilizing correction method to approximatethe solution of the energy equation. Thevalidation of the numerical algorithm was doneby comparing our computational results for largegap between the cylinders with availablenumerical and experimental data for flow andheat transfer over a single cylinder. The resultsof the various numerical experiments arereported and discussed in the final part ofour paper.

Mathematical Formulation of theProblem

Consider the flow of a viscous incompressiblefluid along the y-direction normal to the line

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89Suranaree J. Sci. Technol. Vol. 17 No. 1; January - March 2010

between the centers of two rotating circularcylinders with a constant velocity and tempera-ture at infinity, U�� , T�. The cylinders rotate abouttheir axes at angular velocities ωL and ωR ,assuming that a positive value correspondsto counter-clockwise rotation. A sketch ofthe flow geometry, coordinate system andnotations are shown in Figure 1.

Figure 1. Schematic representation of the flowand heat transfer over a pair ofrotating circular cylinders in side-by side arrangement

The present study is restricted to longcylinders and low Reynolds number, Re < 40.The flow across the cylinders is steady andtwo-dimensional. All flow variables arefunctions of the coordinates, x and y alone. Thethermodynamical properties of the fluid(density, specific heat capacity cp, thermalconductivity k) are assumed to be independentof temperature. Under these conditions, themomentum and energy equations are notcoupled.

For the study of heat transfer and fluidflow around two cylinders, the cylindrical bipo-lar coordinate system is the most suited. Theubiquitous definition of the cylindrical bipolarcoordinates (ξ , η , z ) is

(1)

where and ,a > 0 is the characteristic length in the cylindricalbipolar coordinates (so-called “focal distance”).The following identities show that curves of

constant ξ and η are circles in the xy -space

(2)

Figure 1 shows the two cylindersdescribed by η = ηR (with ηR > 0) and η = ηL

(withηL < 0), respectively. The cylinders’ radiirL and rR and the distances of their centers fromthe origin dL and dR are given by

(3)

The center-to-center distance between thecylinders is equal to d = dL + dR . If rL + rR andd are given, one can find a , ηL and ηR fromrelations (1)-(3) as follows

(4)

The fluid flow is governed by theconservation laws of momentum, mass andenergy. The nondimensional form of thegoverning equations in the cylindrical bipolarcoordinate system are

(5)

(6)

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(7)

(8)

where vξ and vη are the velocity components inξ and η directions, respectively, p is thepressure, t is the time, T is the fluid temperature,and h = a /(cosh η - cos ξ ). In the above equationsall quantities are rendered dimensionless, thevelocities by means of the free stream velocityU��, all lengths by means of the radius rR of

right cylinder, time by rR / U��, pressure by and temperature by (T - T�) (TR - T�). The twonondimensional parameters which appear in theabove equations are Reynolds number,

, and Prandtl number, where

μ is the viscosity of the fluid, ρ is the fluiddensity. On the cylinder surfaces the constanttemperature condition and the no-slip conditionfor the velocity vector are imposed. The constantstreamwise velocity and the uniform temperatureare specified at infinity.

Discretization of the GoverningEquations

Description of Grid

The computational domain is dividedinto a mesh by points ξ j = ( j - 0.5)Δξ andηi = ηL + (i - 0.5)Δη where Δξ = 2π/N, Δη =(ηR - ηL)/ M are spatial mesh sizes in both ξand η directions, respectively. A staggeredplacement of variables is used with velocitycomponents u = vη located on the vertical sidesof each cell and components v = vξ on thehorizontal sides of each cell. We used thefractional indexes to denote grid values ofvelocity components ui+1/2,j = u(ηi+1/2j ,ξ j ) andvi, j+1/2 = v(ηi ,ξ j+1/2) where ηi+1/2,= ηi + 0.5Δη ,ξ j+1/2 = ξ j + 0.5Δξ. The pressure p andtemperature T are represented at cell centers,pi, j = p(ηi ,ξ j), Ti, j = T(ηi ,ξ j). The upper indexn denotes values of variables at time tn = nτwhere τ is the step size in time.

Discretization of Navier-Stokes EquationsSimulating incompressible flows presents

a difficulty of satisfying the property of massconservation. The velocity field must satisfy theincompressibility constraint, which reflects theunability of pressure to do compression work.For developing numerical approximations to thisproblem, it is natural to exploit the techniquesof the fractional step projection method ofChorin (1968). The main idea of the fractionalstep projection method is the splitting of theviscosity effect from the incompressibility, whichare dealt with in two separate subsequent steps.

The time derivatives are representedby forward differences. In case of a steadysolution, time is considered as artificial(iterative) time. If the integer n represents thetime level, then the intermediate velocity fieldcan be calculated from

(9)

(10)

Here we used the following notation forconvective and diffusive terms

The second order central differencescheme has been used to discretize both theconvective and diffusive terms. The velocity

components and are computed for

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all faces of the cells except one, where thevelocity components are given by the boundarycondition. Figure 2 shows the location of gridpoints where velocity components are knownfrom the no-slip boundary condition or from theboundary condition at infinity. The boundarycondition at infinity is shifted on the boundaryof finite domains Ω1 and Ω2.

The explicit advanced tilde velocity maynot necessarily lead to a flow field with zero massdivergence in each cell. This is because at thisstage the pressure field not used. Pressure pn+1

and velocity components un+1 and vn+1 have tobe computed simultaneously in such a way thatno net mass flow takes place in or out of a cell.In such case, we make use of an iterativecorrection procedure in order to obtain adivergence free velocity field. First, the velocitycomponents are updated in the following form

(11)

(12)

where the index s is used to denote iterationnumber, s = 1, 2, 3,... . In the case s = 1, we assumethat pn+1,0 = pn. The point iterative pressureequation becomes

(13)

where is the unsatisfied divergence

at the ( , )thi j cell due to incorrect velocities

and .

The pressure (velocity) equations (11)-(13) are to be iterated until the continuityequation is satisfied to the prescribed accuracyand then the computation proceed to next timestep (artificial time)

(14)

where the discretized form of continuityequation is

(15)

The optimal value of the relaxationparameter β was found by trial and error.

The pressure advanced equation (13) canbe interpreted as Jacobi iterative method to

Figure 2. Sketch of computational grid and implementation of boundary condition

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solve the finite-difference analog of Poissonequation for pressure. Let iterative processconverges, i.e. vn+1= vn+1,s , un+1= un+1,s , andpn+1= pn+1,s . Substituting (11) and (12) into (15)

and requiring that , we get the

following finite-difference approximation ofPoisson equation for pressure,

(16)

Finite difference equation (16) is an algebraicsystem with respect to the unknown vector

. System ofequations (16) can be solved by the methodof false transient

(17)

where β is an iterative parameter. Equation(13) is a particular case of (17) where theforward finite difference is used to approximatethe left-hand-side in (17),

(18)

On the surface of the cylinders, the no-slip condition is applied, which is equivalent tosetting the tangential velocity at the boundaryto αi , i = L, R and the normal velocity to zero.

Where αi is defined as

, (ωi is the

angular velocity of the cylinders). Implementa-tion of no-slip and no-penetration boundaryconditions is straightforward. Because of thestaggered arrangement of the variables, we usedthe second order one-side finite differences to

approximate the derivatives and .

In a numerical simulation, it is impossible

to satisfy constant streamwise velocity as

. Usual practice involves theplacement of the conditions at a faraway(“artificial”) boundary, which is located at a largedistance from the body. In our computations,the far boundary coincides with lines ξ = constand η = const (recall that images of the infinityin computational domain are two points(ξ = 0, η = 0) and (ξ = 2π, η = 0) . To be moreexact, we choose far boundary as the boundaryof the following domain

where εη = +(kw +0.5)Δη , εξ = +(kd +0.5)Δξ , andkw , kd are integer numbers. In Figure 2 asketch of these domains Ω1 and Ω2 is shownby shadow regions.

At the nodes of the mesh which arelocated on the boundary of the regions Ω1 andΩ2 we assumed that the tangent component ofvelocity vector and pressure p = p�� are known.

Here we have utilized the idea that prescribingtangent component of velocity and pressuregives a well-posed problem for the Navier-Stokesequations (Antontsev et al. (1990), Moshkinand Yambangwai (2009)). The normal to theboundary component of the velocity vector iscomputed from the requirement of continuityequation for the cells contained in thisboundary. For example, for the case shown inFigure 2 the boundary of region Ω1 (/or Ω2)passes through points ‘N’, ‘S’, and ‘C’. Thetangential component of velocity and are given by boundary condition as well as thepressure at point ‘C’ is equal to pressure atinfinity. The component of velocity vector is computed from the requirement of zerodivergence for the cell centered at the point ‘C’

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Discretization of the Energy EquationThe momentum and energy equations are

not coupled. The energy equation is solvedseparately from the Navier-Stokes equations.When the steady solution of Navier-Stokesequations is computed, the iterative methodof stabilizing correction is used (Yanenko, 1971)to find the steady distribution of temperature.The structure of the scheme of stabilizingcorrection is the following:

• the first fractional step producesabsolute consistency with the energy equation,

• all succeeding fractional steps arecorrections and serve to improve the stability.

For the 2D case, the scheme possessesstrong stability and satisfies the property ofcomplete consistency. The requirement ofcomplete consistency guarantees convergenceof the unsteady solution to the steady solutionfor arbitrary time and space step size (Yaneko1971).

To describe the scheme of stabilizingcorrection, consider a two dimensionalconvection-diffusion equation in the form

(19)

where is the gradient

operator, and is the Laplace operator. To findthe steady solution of (19) we use the methodof false transient. Consider the related unsteadyproblem with “fiction” time

(20)

where represent the derivatives in onlyone space direction. For the solution of (20)

Douglas and Rachford (1956) proposed thefollowing scheme

(21)

(22)

Eliminating T k+1/2 we can rewrite system (21)and (22) in the following uniform scheme

(23)

It follows from this that scheme (21) and (22),and the equivalent scheme (23) approximateequation (20) with the same order of accuracyas the implicit scheme

(24)

Temperature references to nodes of maincomputational grid related with center of cells

of discrete domain

.On the boundary of the shadowed domains Ω1

and Ω2 the constant temperature of uniformstream T = T

� is prescribed. The constanttemperature at the cylinder surfaces is approxi-mated by the following

where T0,j and Tm+1,j are “ghost” points, whichintroduced for convenience in writing thecomputational code. For each fractional step asystem of linear algebraic equation withtridiagonal matrix is solved.

Computation of Heat Transfer and FlowCharacteristics

Two groups of characteristic quantitiesare of interest in the present study, one forcharacterizing the forces at the cylinder surface

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and the other for the heat transfer. If Fxi andFyi , i = L, R are the lift and drag forces on thecylinders; the lift and drag force coefficientsare defined by

(25)

and each consists of components due to thefriction forces and the pressure. Hence

(26)

where

Here are the unit vectors in x and y axes

directions, Σ is the cylinder perimeter, and isthe outward unit normal to surface vector. Toevaluate the integrals we used the middle point

rule, for example, in order to compute weused formula:

(27)

where i = L, R, that is, we computed thecoefficient on the left and right cylinder surfaces,respectively. Pressure on the cylinder surfacesis evaluated by extrapolation from interiorpoints.

The vorticity in the cylindrical bipolarcoordinate system is given by the following

formula

On the cylinder surfaces the tangent derivative

is equal zero . The normal derivative

is approximated by one-side second orderfinite-differences.

The important parameter of interest inheat transfer problems is the heat transfer rateper unit area from the cylinder wall to theambient fluid. The local Nusselt number in thecylindrical bipolar coordinate system based onthe diameter of the cylinder is

(28)

The average Nusselt number is calculated byaveraging the local Nusselt number over thesurface of the cylinder

(29)

The integral is approximated by the middlepoint rule.

Validations

The first task in any numerical work is tovalidate the codes ability to accuratelyreproduce published experimental andnumerical results. Unfortunately, there are nodata in the literature to verify the accuracy ofthe considered problem (flow over two rotatingcylinders). Since for large gap spacing betweenthe cylinders, the mutual influence of thecylinders on each other is negligible we canassume that the flow and heat transfer will besimilar to flow and heat transfer over a singlecylinder. The comparisons with flow and heat

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Figure 3. Validation of numerical algorithm (a)-pressure coefficient over cylinder surface,(b)-the vorticity distribution over cylinder surface, for the left cylinder at Re = 20, 40,and g = 14

transfer around a single cylinder can be viewedas a partial validation of the algorithm presented.Thus, the comparisons were carried out of thecharacteristics of flow and heat transfer asobtained from our numerical results (gap spacingg = 14, g = (d - rL - rR )/ DR ) with the publisheddata for a single cylinder. All computations wereperformed in a large domain in order to reducethe influence of the outer boundary. A sequenceof uniform grids is used. The wake behind asingle cylinder is steady in the flow regimeRe < 46 + 1. We assume that in the case of tworotating circular cylinders, the steady regimeexists at least for the same range of the Reynoldsnumbers. Due to symmetry with respect to they -axis, all our results are presented for the leftcylinder only.

The important characteristics of flow arethe drag coefficients, CD = CDf + CDp , the localand average Nusselt numbers. They are presentedin Table 1 for Re = 20, g = 14, and Pr = 0.7.

Table 1 shows convergence of our results on asequence of grids and good agreement withresults from other publications.

Representative results showing thevariation of the pressure coefficient CDp andvorticity ω on the surface of the left cylinder areplotted in Figure 3 for the two values of theReynolds number Re = 20 and 40 and for the gapspacing g = 14.

Comparison of our numerical simulationsis performed with the numerical results ofDennis and Chang (1970) for Re = 20 and 40,with the experimental results of Thom forRe = 36, 45 and Apelt for Re = 40 (Batchelor,2000). Our data come from simulations on thegrid with 81 � 81 nodes in the ξ - η plane. Theclosest distance from the cylinder surfaces tothe far boundary for this grid is 53 cylinderdiameters. The angle variable θ is zero at thefront stagnation point and increases in theclockwise direction on the left cylinder. The data

Table 1. Validations of the numerical algorithm. Effect of grid refinement upon CD, CDp, CDi

and average Nusselt number for Re = 20 and g = 14

Source CD, CDp, CDi (Pr = 0.7)

Present (21 � 21) 2.149 1.274 0.875 2.669

Present (41 � 41) 2.112 1.274 0.838 2.481

Present (81 � 81) 2.064 1.242 0.822 2.478

Soares (2005) et al. 2.035 1.193 0.842 2.430

St lberg et al. (2006) 2.052 1.229 0.823 -

Bharti et al. (2007) - - - 2.465

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from our simulations (∗ - sign for Re = 20 ando - sign for Re = 40) match the results from otherpublications.

To the authors’ knowledge, even for theflow past a single circular cylinder, there are onlya few published sets of data pertaining to thedrag and lift coefficients at Re < 40 andnonzero rotation. Table 2 shows comparisonsof our numerical results for Re = 20, g = 14 withthose numerically obtained by Badr et al. (1989);Ingham and Tang (1990); Sungnul and Moshkin(2006) and Chung (2006). The left cylinderrotates with constant angular velocity inclockwise direction and the right cylinderrotates with the same speed in counterclock-wise direction. The validation results inspireconfidence in the reliability and accuracy of the

numerical method used.

Results

Numerical computations were carried out forRe = 5 - 40, for a range of values of Prandtlnumber from 0.7 to 20, for range of rotationα = 0.1 + 2.5, and different gap spacing. In all cases we represent results obtained on a grid

with 81 � 81 nodes in the computational domain.

The influence of the gap spacing on the heattransfer rate in terms of the local Nusseltnumber Nu(θ) is shown by Figures 4(a) and 4(b)for Re = 20, α = 0 and Pr = 0.7 and 20. Thesefigures show that a significant influence of thedistance between cylinders on the local Nusseltnumber Nu(θ) is observed for g 5.

Table 2. Hydrodynamic parameters of flow over a rotating circular cylinder at Re = 20with g = 14

Contribution CD CL

α α α α α = 0.1 α α α α α = 1.0 α α α α α = 2.0 ααααα = 0.1 α α α α α = 1.0 α α α α α = 2.0

Present (21 ��21) 2.146 2.035 1.906 0.286 2.974 6.309

Present (41 ��41) 2.108 1.897 1.410 0.288 2.864 6.030

Present (81 ��81) 2.052 1.847 1.346 0.293 2.770 5.825

Sungnul & Moshkin (2006) 2.120 1.887 1.363 0.291 2.797 5.866

Badr et al. (1989) 1.990 2.000 — 0.276 2.740 —

Ingham et al.(1990) 1.995 1.925 1.627 0.254 2.617 5.719

Chung (2006) 2.043 1.888 1.361 0.258 2.629 5.507

Figure 4. Local Nusselt number for different gap spacing at Re = 20, α α α α α = 0, and (a)- Pr = 0.7,(b)- Pr = 20

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Another aspect that seems to be interestingis the decreasing Nu(θ) for withdecreasing g in the case of Pr = 0.7 (rate ofconvection and conduction almost equal). In thecase of Pr = 20 (convection is the dominantmechanism of heat transfer) Nu(θ) decreaseswith g decreasing for and

, and for the valuesof Nu(θ) increases with g decreasing. It is notsurprised that the average Nusselt number foreach individual cylinder increases with increasinggap spacing and tends to the average Nusseltnumber for a single cylinder. Figure 5 shows thevariation of the average Nusselt number withgap spacing between non-rotating cylinders atdifferent Prandtl numbers and fixed Reynoldsnumber, Re = 20.

As expected, the average Nusselt numberfor the cylinders increases with the Prandtlnumber.

Large Gap SpacingThe effect of steady rotation of the

cylinders on heat transfer is first studied forlarge gap spacing, g = 14. Figure 6 shows thevariation of the local Nusselt number Nu(θ) atRe = 20, Pr = 0.7 and Pr = 20, and for different

values of An increase in therotation of the cylinders leads to a displacementof the points of maximum and minimum Nusseltnumber in the direction of rotation. At the sametime, the maximum value of the local Nusseltnumber decreases with increase of rotationalspeed. The minimum value of the local Nusseltnumber slightly increases in step withincreasing rotation. For Pr = 20 and highrotational speed α = 2, the rate of heat transferaround the cylinder surface becomes almostuniform as shown in Figure 6(b). This behavioris quite expected, since due to the no-slipcondition the fluid layer adjacent to the cylinderenwraps the cylinder and rotates with almostthe same angular velocity. Increasing rotationalspeed to α = 2 creates a thick rotating layer(buffering layers) around the cylinder. The heattransfer through that layer is mostly due tothermal conduction.

Figure 7 shows streamline patterns andtemperature contours plot for large gapspacing g = 14, Reynolds number Re = 20 ,

Figure 5. Average Nusselt number for differentgap spacing at Re = 20, α α α α α = 0

Figure 6. Local Nusselt number variation on the surface of the circular cylinders at Re = 20,Pr = 20, Pr = 0.7 and 20, g = 14 for α α α α α = 0, 0.1, 0.5, 1.0, and 2.0

(a) (b)

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Pr = 0.7, 20, and α = 0.1, 1.0, 2.0. The rotationeffect substantially changes the flow pattern inthe vicinity of the cylinder. Without rotation,the flow field exhibits a symmetric pair ofstanding vortices behind the cylinder defininga closed recirculation region. With the increaseof the rotation rate, the flow becomesasymmetric: the vortex detaches from thecylinder and the stagnation point rotates in thedirection opposite to the direction of cylinderrotation departing from the surface of thecylinders (see first row in Figure 7).The minimumof the local Nusselt number is observed nearthis point. In downstream direction, the streamlinesare shifted clockwise and the same occurs forthe isotherms (see second and third rows in

Figure 7) resulting in the asymmetrical distributionof the local Nusselt number as shown inFigure 6. The oncoming fluid stream isaccelerated by the rotating cylinder on the“west’ side of an egg-shaped region of closedstreamlines. As a result, the convective heattransfer between the ambient fluid and the fluidwithin the egg-shaped region is increased at thepoint of maximum velocity. This observationexplains the deviation of the maximum of thelocal Nusselt number in the direction ofcylinder’s rotation. An increase in the Prandtlnumber increases the compactness of theisothermals toward the downstream direction(compare second and third rows of Figure 7).Owing to rotation a wake-shapes region of

Figure 7. Streamline patterns (first row), temperature contours over two circular cylinders atRe = 20, g = 14, α = 0.1, 1.0 amd 2.0 and for Pr = 0.7, (second row) and Pr = 20(third row)

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is otherms is shifted in the clockwise directionfor the left-hand side cylinder (see third row inFigure 7) and in the counterclockwise directionfor the right-hand side cylinder. This behaviorcan be explained by the increasing role ofconvection in the mechanism of heat transferwith increasing Pr number. We remind here thatwe represent results only for left-hand sidecylinder, which rotates in the clockwisedirection.

Figure 8(a) shows the dependence of theaverage Nusselt number on the rotational speedfor Re = 20 , g = 14 and Pr = 0.7, 1.0, 5.0, 10.0, and20.0. The average Nusselt number decreaseswith increasing α. In the case Pr = 1 the averageNusselt number drops down by 10% and in thecase of Pr = 20 drops down by almost 50%when α increases from 0 up to 2. This behaviorcan be explained by the fluid layers adjacent tothe rotating cylinders due to no-slip requirement.

Small Gap SpacingFor small gap spacing, g = 1, the effect of

rotational speed is shown in Figures 8-12, forRe = 10 and 20, Pr = 0.7 and 20.0, and αin the range 0 - 2.5. Representative plots of thestreamlines and isotherms for Re = 10 and 20,g = 1, Pr = 0.7, 10 and 20, and 0.5 < α < 20are shown in Figures 9-11. Due to the no-sliprequirement, there are regions of closedstreamlines near the cylinders for all values ofα. The size of these regions increases withincreasing rotational speed. At some α = α ∗ the

two regions merge together. The main streamflows around the fluid region, which surroundsboth cylinders and consists of two regions ofclosed streamlines. These regions acts as abuffer (blanket) isolating both cylinders fromthe main stream and causing a decrease in theoverall heat transfer rate from the cylindersurfaces. The values of α∗ depends on Reynoldsnumber. As can be seen from Figure 9 forRe = 10 and α = 0.5 main stream can not passthrough gap between cylinders, however forRe = 20 and α = 0.5 main stream is strong enoughto go through the gap between two cylinders. Afurther increase in angular velocity α > α ∗

is a reason of further increasing of regions sizein both x and y directions. The stagnation pointsare now located on the y _ axis, both upstreamand downstream, as illustrated in Figure 9.

The details of the steady thermal fieldsare presented in Figures 10 and 11 (for the casesof Re = 10 and 20, g = 1, Pr = 1, 10, 20, andα = 0.5,1.0,1.5, and 2.0) in the form of constanttemperature contours. Isotherm patterns dependon the Prandtl number. For Re = 20, Pr = 20 andα = 0.5 there are two almost separate wake-type isotherm patterns downstream of eachcylinders. One of the interesting features of thetemperature field is topological similarity ofisotherm patterns in cases of α = 1.5 andα = 2.0. For the cases of large Prandtl numbersPr = 40 and 20, the convection is dominantmechanism in heat transfer. The lines of constant

Figure 8. Average Nusselt number at Re = 20, (a)-g = 14 and (b)- g = 1, for different Prandtlnumbers

(a) (b)

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temperature in the regions of closed streamlinesfollow to the streamlines contours. Heat exchangefrom the cylinders to the main stream occurredthrough the boundary of the fluid region, whichenwraps both cylinders. Heat transfer inside thisregion is mostly due to conduction. By thisreason, the average Nusselt number almost doesnot change for α >1.5 as can be seen fromFigure 8(b). Comparison of isotherms in Figures10 and 11 shows that an increase in Prandtl

number makes wake-shape region oftemperature field more narrow downstreambehind the cylinder It is interesting to point outthat the rate of rotation does not significantlyaffect the size of the wake-shaped region.

Another interesting observation is thelocation of the saddle critical point in thetemperature field. This critical point is locatedon the line connecting the cylinder centers inthe cases of Re =10, α = 0.5, Pr = 0.7, and

Figure 9. Streamlines contours over two circular cylinders at Re = 10 left column andRe = 20 right column Pr = 1, g = 1, and α α α α α = 0.5, 1.0, 1.5, and 2.0

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Re = 20, α = 1.0, Pr = 0.7. For larger values ofα, α >1.5 the saddle point is shifted down (inthe negative direction of y-axis) by the fluidlayers which rotate in step with cylinders due tono-slip condition.

Figure 12 shows effect of rotation on heattransfer in terms of the local Nusselt numberdistribution. For Pr = 20 and α = 2.5 the localNusselt number is almost constant on the cylindersurfaces. As a result, the average Nusselt numberdrops as α increase. Dependence of the average

number on rotational speed is demonstrated

in Figure 8(b). Figure 8(b) shows that for

, decreases sufficiently fast,

whereas for the rate of decrease of is very low. This allows us to assume that heattransfer is almost insensitive to the speed ofrotation if .

Conclusions

A detailed numerical study of the two-dimen-sional heat-transfer problem and laminar flowaround two rotating circular cylinders in side-

Figure 10. Temperature contours over two circular cylinders at Re = 10, g = 1, Pr = 1, 10, 20,and α α α α α = 0.5, 1.0, 1.5, and 2.0

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by-side arrangement has been carried out. Anefficient finite difference algorithm has beendeveloped for the 2D Navier-Stokes equationin the cylindrical bi-polar coordinate system.Comparing the results for the case of large gapspacing with the available in the literatureexperimental and numerical data for the case ofa single cylinder shows a good agreement,

For the problem of heat-transfer and flowaround two rotating circular cylinders in side-by-side arrangement, the average Nusseltnumber decreases with increasing rotationalspeed.

For large Prandtl number (Pr = 20) andhigh rotational speed (α��2.0), the local Nusseltnumber distribution on the cylinder surfacesbecomes almost uniform. This is due to theno-slip condition, the increasing rotational speedcreates thick rotating layers of fluid (bufferlayers) around the cylinders.

In case of small gap spacing between thecylinders, there are two flow regimes. In the first,the main stream passes through the gap betweencylinders and in the second case, there is a fluidregion which surrounds both cylinders andwhich consists of two sub-regions of closed

Figure 11. Temperature contours of fluid flow over two circular cylinders at Re = 20, g = 1,Pr = 0.7, 10, 20 and α α α α α = 0.5, 1.0, 1.5 and 2.0

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streamlines.It has been observed that the temperature

contour pattern is similar to the streamlinecontour pattern. Owing to the rotation, thewake-shaped region of the isotherms is shiftedin direction of the cylinder rotation. An increasein the Prandtl number increases the compactnessof the isotherms toward the downstream.

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