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Large eddy simulations of flow interference between two unequal sizedsquare cylindersM.B. Shyam Kumar a;S. Vengadesan a

a Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, India

Online publication date: 16 April 2010

To cite this Article Kumar, M.B. Shyam andVengadesan, S.(2009) 'Large eddy simulations of flow interference betweentwo unequal sized square cylinders', International Journal of Computational Fluid Dynamics, 23: 10, 671 — 686To link to this Article: DOI: 10.1080/10618560903580013URL: http://dx.doi.org/10.1080/10618560903580013

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Large eddy simulations of flow interference between two unequal sized square cylinders

M.B. Shyam Kumar and S. Vengadesan*

Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600 036, India

(Received 20 March 2009; final version received 21 December 2009)

A uniform flow past two unequal sized square cylinders arranged in a side-by-side pattern and at a Reynolds numberof 50,000 has been investigated using large eddy simulation (LES) technique. The modelling of sub-grid scales ofturbulence is done using the Smagorinsky model. The effect of the transverse gap ratio (T/D) on the flowcharacteristics has been studied. Numerical simulations are carried out for five different transverse gap ratios (T/D),namely 1.120, 1.250, 1.375, 1.750 and 2.500. Results in terms of the aerodynamic forces, Strouhal number, meanbase pressure coefficient, streamlines, vorticity, surface pressure distribution, normal and shear stresses arepresented. A shift in the stagnation point for the small square cylinder from the centre of its front face towards itsgap side is seen at smaller T/D ratios. The presence of a jet-like flow seen in the gap side is more pronounced atT/D ¼ 1.12. A biased gap side flow towards the near wake of the small square cylinder is seen at smaller T/D ratios.No interference effect is seen at T/D ¼ 2.5. The flow behaviour is similar to that of the isolated square cylinder atthis gap ratio.

Keywords: large eddy simulation; tandem cylinders; square cylinder; wake; vortex shedding; turbulence; bluff bodies

1. Introduction

The phenomenon of flow around two bluff bodies intandem or in side-by-side arrangement is one of theinteresting fluid structure interaction problems inengineering. As defined by Zdravkovich (1987), theinterference brought about by the former arrangementis called wake interference and by the latter arrange-ment is called proximity interference. Cylinder-likestructures find many technological applications both inair and water flows.

The interference effect brought about by the bluffbodies placed side-by-side gives rise to a very complexflow pattern. The complexity lies in the interaction offour separated free shear layers, two Karman vortexformation and shedding processes and interactionsbetween two Karman vortex streets. The lift and dragforces acting on the two structures are quite differentfrom those acting on a single structure. The flowbehaviour and parameters like the Strouhal number(St), mean base pressure coefficient, coefficient of liftand drag strongly depend on many factors. Some ofthe parameters are the spacing between the centres ofthe two cylinders, blockage ratio, the approaching flowand the aspect ratio of the bluff bodies.

Wong et al. (1995) presented the experimentalresults for flow past two side-by-side square cylindersof unequal sizes at sub-critical flow regime. They foundthat the gap side shear layer of the large cylinder

reattaches to its inner surface for smaller separation.For larger separation, reattachment of the gap sideshear layer was found in the inner surface of the smallcylinder. It was shown that only skewed flow towardsthe small cylinder exists and no bistable flow wasobserved. As the separation increases, the skewness ofthe vortex streets towards the small cylinder decreases.Kolar et al. (1997) studied the unsteady flow aroundtwo identical square cylinders placed side-by-side usinglaser-Doppler velocimetry and ensemble averaging.They found a coupled vortex street with in-antiphasemode existing along with the flow being symmetricabout the centreline. In addition, they found a jet-like flow in the gap using the time averaged velocityfield.

Sumner et al. (1999) reported the dynamics of flowpast two and three circular cylinders of equal sizearranged side-by-side using flow visualisation, hot-filmanemometry and particle image velocimetry. Theyfound that the fluid dynamics of the side-by-sidearrangement of cylinders is quite insensitive for500 5Reynolds number (Re) 5 1,11,000. They alsofound that at small gap ratios, the higher momentumfluid through the gap increases the base pressure,reduces the drag of both cylinders and also increasesthe stream-wise extent of the vortex formation regionin case of two side-by-side cylinders. Sumner et al.(2000) investigated the flow around two equal sized

*Corresponding author. Email: [email protected]

International Journal of Computational Fluid Dynamics

Vol. 23, No. 10, December 2009, 671–686

ISSN 1061-8562 print/ISSN 1029-0257 online

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DOI: 10.1080/10618560903580013

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circular cylinders arranged in a staggered pattern usingflow visualisation and particle image velocimetry. Theyidentified nine different flow patterns and the processesof shear layer reattachment, induced separation, vortexpairing and synchronisation and vortex impingementwere clearly presented. Their study revealed that thevortex shedding frequencies are more properly asso-ciated with individual shear layers than with theindividual cylinders.

Kondo (2004) showed numerically that when thedistance between the two rectangular cylinders locatedin a uniform flow was short, a very complicated flowpattern arises. Kondo and Matsukuma (2005) numeri-cally obtained the phenomena of biased gap flow thatappeared when the two circular cylinders placed side-by-side are at a small separation distance. In addition,they found the flow pattern changing from the biasedgap flow to a coupled flow, when the two cylinders areat a larger spacing.

Agrawal et al. (2006) carried out low Re flowaround two identical square cylinders placed side-by-side using lattice Boltzmann method. They proved theexistence of both synchronised and flip-flop regimes. Inaddition, they found that the merging of wakes wasgradual in the synchronised regime and rapid in thecase of the flip-flop regime. They also observed theoccurrence of vortex shedding in both regimes to beeither in-phase or in-antiphase. Liu and Cui (2006)numerically studied three-dimensional flow over twoside-by-side circular cylinders at Re ¼ 200. Theyfound that the wake patterns depend not only on Reand gap spacing but also on the end-flow conditions.They found biasing of the gap flow intermittentlytowards one cylinder or the other.

Niu and Zhu (2006) presented numerical results forflow past two identical square cylinders in staggeredarrangement at Re ¼ 250. They examined the influ-ence of the cylinder spacing on the flow induced forcesand vortex shedding frequencies and the effect ofprimary wake interference on the secondary flowstructure. They reported that the momentum of thegap flow which was transferred into the near wake ofone cylinder not only suppresses the oscillations of thedrag and lift of that cylinder but also suppressesthe generation of the secondary organised structures inthe near wake of that cylinder. Several experimentaland numerical investigations have been made forsquare/circular cylinder in isolated condition andcircular cylinders in side-by-side arrangement. Tothe best of our knowledge, numerical studies oftwo unequal sized square cylinders in side-by-sidearrangement at high Re have not been reported inliterature.

In this article, our focus is to investigate numeri-cally the effect of transverse spacing (T/D) on the flow

characteristics past two square cylinders arranged in aside-by-side pattern. The two square cylinders areconsidered to be infinitely very long. The size of smallsquare cylinder (d) is half that of the large squarecylinder (D). The Re based on the large square cylinderdiameter (D) and the free stream velocity U? is 50,000.The transverse gap ratio (T/D) is defined as the ratiobetween the centre-to-centre distance (T) in the cross-stream-wise direction to the large cylinder diameter(D). The simulations are carried out for five differentT/D ratios, namely 1.120, 1.250, 1.375, 1.750 and2.500.

As the flow is unsteady and three-dimensional athigh Re, large eddy simulation (LES) technique isemployed for carrying out the study.

2. Mathematical formulation

Here, a uniform, viscous and incompressible fluidflow with constant fluid properties is consideredfor carrying out the simulations. The two squarecylinders of unequal size are placed in a side-by-sidepattern in such a way that the axis of the twocylinders is perpendicular to the direction of the freestream fluid flow. In LES, the contribution of large,energy-carrying eddies to momentum or energytransfer is computed exactly, and only the effect ofthe smallest scales is modelled. To separate the con-tribution of large scales from the small scales, LESmakes use of the filtering operation. Here, the griditself acts as a low-pass filter, which is the popularlyknown implicit filtering approach. Application ofa filtering operation to the continuity and Navier–Stokes equations results in the respective filteredgoverning equations. These equations in non-dimensional tensor form are given by Equations (1)and (2), respectively as,

@ uih i@xi

¼ 0 ð1Þ

@ uih i@tþ@ uih i uj

� �@xj

¼ � @ ph i@xiþ 1

Re

@2 uih i@xj@xj

þ @ti j@xj

ð2Þ

Here, hui and hpi represent the filtered velocity andpressure, respectively, tij ¼ 7huiuji þ huiihuji repre-sents the sub-grid turbulence stress and h i indicatesspatial filtering. The indices i and j ¼ 1, 2, 3 representthe three Cartesian coordinates (x, y, z) which in turnrepresent the stream-wise, span-wise and cross-stream-wise directions, respectively. All geometrical lengthsare normalised with D, velocities with U?, physicaltime with D/U? and the pressure with rU?

2.The turbulent fluctuations are accounted in the

filtered equations. In order to close the equations,

672 M.B. Shyam Kumar and S. Vengadesan


sub-grid turbulence stress modelling is required. Therole of the sub-grid-scale model should be to removeenergy from the resolved motion and dissipate it at theappropriate rate by the unresolved motion.

In this model, the sub-grid turbulence stress takesthe Boussinesq eddy viscosity form which is given by,

ti j ¼ � 23 ksdi j� �

þ 2uGSi j

� �. Here Sij ¼ 1

2@ uih i@xjþ @ ujh i

@xi

� �

is the rate of strain tensor of the filtered velocity, uG isthe sub-grid eddy-viscosity coefficient, ks is the sub-grid turbulence kinetic energy and dij is the Kroneckardelta. For uG and ks, we use the Smagorinsky (1963)sub-grid-scale model as shown in Equations (3)and (4).

uG ¼ 2 CsDð Þ2ffiffiffiffiffiffiffiffiffiffiSijSij

pð3Þ

ks ¼ uGð Þ2= CkDð Þ2 ð4Þ

In order to resolve the viscous sub-layer and toapply the no-slip boundary condition on the wall, thefirst grid point next to the wall in the normaldirection should be set close to the wall in terms ofnon-dimensional wall distance, which should be lessthan 20. This point becomes physically closer andcloser to the wall, as the Re increases. But it is foundfrom approximate calculation for the grid used in thisstudy that the first point was far away from thisregion. So, Van-Driest type near-wall damping foreddy viscosity is not used. Thus, Smagorinskyconstants – Cs and Ck are being set to the constantvalues of 0.13 and 0.094, respectively. D is the filterwidth (grid size) which is the characteristic lengthscale of the largest sub-grid-scale eddies and is taken

to be the geometrical average of the grid spacing inthe three directions, i.e.

D ¼ D x1 Dx2 D x3ð Þ1=3 ð5Þ

A typical computational domain along with theboundary conditions used for a transverse spacing of1.250 is shown in Figure 1. The flow is from left toright and the origin is positioned at the centre of thelarge square cylinder. The various boundary condi-tions employed for the simulations are as follows. Atthe inlet, a constant stream-wise velocity (u ¼ 1) isspecified with the other two velocities set to zero. Onthe surface of the two square cylinders, the standardboundary condition of no-slip (u ¼ v ¼ w ¼ 0) hasbeen employed. On the top and bottom boundaries,free-slip or symmetry boundary condition (@u/@y ¼@w/@y ¼ v ¼ 0) is used. At the exit, the convectiveboundary condition @ui/@t þ uc(@ui/@x) ¼ 0 is used.Here, u, v and w represent the velocity components inthe x, y and z directions, respectively, and uc is theconvective velocity.

The simulations for the T/D ratios 1.120,1.250, 1.375, 1.750 and 2.500 have been carried outusing structured grids of size 208 6 41 6 206,208 6 41 6 219, 208 6 41 6 226, 208 6 41 6 257and 208 6 41 6 298, respectively, each having ablockage ratio of about 8%. A typical grid generatedfor T/D ¼ 1.250 is shown in Figure 2a and a closerview of the grid near the two square cylinders is shownin Figure 2b. In all the cases, the first grid point is at adistance of 0.01 D from the surface of the cylinder andthe aspect ratio of the grid used is kept at 40. The totalnumber of grid points in x and y directions has been

Figure 1. Computational domain with boundary conditions for flow past side-by-side square cylinders at Re ¼ 50,000.

International Journal of Computational Fluid Dynamics 673


maintained constant for all T/D ratios. As the cylindersare moved apart in the z direction, the grid points inthis direction only are varied. In the stream-wise andcross-stream-wise directions they are stretched gradu-ally going away from the cylinders, on the surface ofwhich grids of non-uniform spacing are generated.The grid points are uniformly distributed in the span-wise direction.

3. Numerical details

A numerical code employing finite difference schemewith staggered grid arrangement is used to discretisethe governing equations. The velocity components aredefined at the midpoint of the cell face to which theyare normal and the pressure at the cell centre. Theviscous and sub-grid stress terms are discretised usingsecond-order accurate central differencing scheme.The convective terms are discretised by third-orderupwind biased scheme. The time integration is doneby using a second-order accurate explicit Adams–

Bashforth scheme in two stages. In stage one, thevelocity components are found using the previousvelocity and pressure values at all cells. Thesevelocities, however, do not satisfy the divergencefree condition for incompressible flow. In order toensure the mass conservation, adjustments must bemade in stage two. This is achieved by using thehighly simplified marker and cell (HSMAC) algo-rithm developed by Hirt and Cook (1972). The non-dimensional time step (dt) to be used, in order to takecare of the stability criteria is determined from theCourant–Friedrichs-Lewy condition. A time step of0.0005 was employed for all cases. In this study, aconvergence limit of 0.0015 was used in all thesimulations.

The basic validation of our computational code hasalready been carried out for laminar and turbulentflow over an isolated square cylinder. These can befound in Lankadasu and Vengadesan (2008a, 2008b,2009a) for laminar flows and in Nakayama andVengadesan (2002), Lankadasu and Vengadesan(2009b) for turbulent flows. In addition, for the sakeof the present study, we have carried out a separatevalidation study. We considered two cases: (i) laminarflow past two equal sized square cylinders at Re ¼ 250in staggered arrangement and (ii) turbulent flow pastisolated cylinder at Re ¼ 21,400. The results arecompared respectively with Niu and Zhu (2006) andLyn et al. (1995). The results in terms of the bulkparameters are shown in Table 1. It should be notedthat the work of Niu and Zhu (2006) is a numericalstudy. The discrepancy found could be due to thedifference in the numerical schemes such as meshsize, discretisation, etc. used. Figure 3a shows theinstantaneous span-wise vorticity, which are found tobe almost in good agreement. The solid lines in thepresent instantaneous span-wise vorticity plot denotenegative vorticity and dashed lines the positivevorticity, with a vorticity increment of 0.2. The flowpattern consists of the primary vortex pairing, splittingand enveloping pattern similar to those identified bySumner et al. (2000). The mean stream-wise velocityplotted along the geometric centreline in comparisonwith Lyn et al. (1995) is shown in Figure 3b. They arealso found to be in good agreement.

4. Results and discussions

The simulations are started with the fluid at rest andthen allowed to progress for sufficient time till theflow gets stabilised. The non-dimensional timecorresponding to this condition is around 120.Next, the time averaging was performed for over20 vortex shedding cycles, which corresponds toapproximately the non-dimensional time of 240 for

Figure 2. A grid of size 208 6 41 6 219 generated for T/D ¼ 1.250 and Re ¼ 50,000.



Table 1. Comparison of bulk parameters at Re ¼ 250 for flow past two equal sized square cylinders in staggered arrangement.

H/D ¼ 0.5 CLU CLD CDU CDD CrmsLU Crms

LD CrmsDU Crms

DD StU StD

Present simulation 70.425 70.066 1.839 2.135 0.198 0.751 0.157 0.291 0.045 0.091Niu and Zhu (2006) 70.475 70.076 1.834 2.061 0.177 0.631 0.136 0.292 0.081 0.081

Subscripts U and D represent upstream and downstream square cylinder, respectively.

Table 2. Comparison of bulk parameters at Re ¼ 50,000 for flow past two unequal sized square cylinders in side-by-sidearrangement.

T/D CLI CLII CrmsLI Crms

LII CDI CDII CrmsDI Crms

D II StI StII CpbI CpbII

1.120 0.106 70.004 0.085 0.156 1.739 2.090 0.051 0.125 0.092 0.275 71.042 71.5361.250 0.341 70.095 0.163 0.299 1.693 2.204 0.087 0.134 0.015 0.290 70.977 71.6251.375 0.224 70.229 0.146 0.209 1.706 2.161 0.068 0.107 0.015 0.015 70.998 71.5701.750 0.092 0.155 0.074 0.602 1.699 2.770 0.052 0.532 0.031 0.397 70.998 72.2072.500 0.398 70.285 1.587 1.259 2.665 2.734 0.354 0.495 0.137 0.290 72.136 72.260

Subscripts I and II represent large and small square cylinder, respectively, and overbars represent the mean qualities.

Figure 3. (a) Instantaneous span-wise vorticity at t ¼ 82 for flow past two equal sized square cylinders in staggeredarrangement; (oz min, oz max, Doz) ¼ (71, 1, 0.2). (b) Mean stream-wise velocity plotted along the geometric centreline for flowpast in an isolated square cylinder at Re ¼ 21,400.



all gap ratios. The bulk parameters, namely themean and root mean square (RMS) values of lift anddrag coefficients, St and the mean base pressure

coefficient for all T/D ratios are presented in Table 2.The suffix I and II for the above parametersrepresent the large and small square cylinders,

Figure 4. Time history of the lift, draft and moment signals for all T/D ratios. (a) Lift signal; (b) Drag signal; (c) Momentsignal.



respectively. The instantaneous lift, drag and mo-ment coefficients are determined using the followingexpressions.

CL ¼ 2FL= rAU2

1

ð6Þ

CD ¼ 2FD= rAU21

� �ð7Þ

CM ¼ 2 YFD � XFLð Þ= rDAU21

� �ð8Þ

Here, FL and FD represent the lift and drag forces,A the frontal area of the corresponding cylinders andX,Y represent the distances from the centre of gravityup to the surface of the cylinder in x–z plane. Thevariation of CL and CD from their mean values is

represented by the RMS values. It is found usingffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1 ðCLðiÞ � CLÞ2=n

q. Here, n is the total number of

discrete points over which time averaging has beencarried out. CL and CL represent the instantaneous andthe mean lift coefficients, respectively. The time historyof the lift (Figure 4a), drag (Figure 4b) and moment(Figure 4c) coefficients for all T/D ratios shows highunsteadiness due to the turbulent nature of the flow.The small square cylinder has the largest momentcoefficient for all the gap ratios. Also, both thecylinders have positive moment coefficients. It is clearthat the lift force acting on the large square cylinder ismore in comparison with that of the small squarecylinder for all T/D ratios except for T/D ¼ 1.75.

Figure 4. (Continued).



According to Shao and Zhang (2008), in case oftwo equal sized circular cylinders in side-by-sidearrangement, the cylinder to which the gap side

flow deflects has a higher drag coefficient. Theshear layer separated from the large cylinder in thegap side is biased towards the small cylinder and

Figure 4. (Continued).



transfers more momentum through the small-scaleinstabilities. This alters the near wake of the smallcylinder which results in a higher drag force. Thesmaller amplitude of oscillations seen in the liftsignals for smaller T/D ratios (1.12, 1.25, 1.375 and1.75) could be due to the interference effect. Thelarger amplitude of oscillations seen in the lift anddrag signals for T/D ¼ 2.5 is similar to that for theisolated square cylinder flow. This confirms that theeffect of interference is almost negligible for this gapratio.

The vortex shedding process is primarily charac-terised by the Strouhal number, St ¼ (fD)/U?, where fis the vortex shedding frequency. These are identifiedbased on the dominating frequency in Figure 5, whichis obtained by performing the Fast Fourier Transform(FFT) of the time variation of lift signals. The smallcylinder has a larger St in comparison with the largecylinder for all T/D ratios except for T/D ¼ 1.375. Inthis case, both the cylinders have identical values of St.The multiple peaks seen for smaller T/D ratios is dueto the interaction of the inner shear layers separated

Figure 5. Fast Fourier Transform of the lift signals for all T/D ratios.



from both cylinders. For T/D ¼ 2.5, we can see only asingle dominating peak which resembles that for anisolated cylinder.

The mean streamline plot for all T/D ratios isshown in Figure 6. The presence of a wider wakebehind the large cylinder and a narrow wake behindthe small cylinder is clearly seen. For T/D ¼ 1.12, 1.25and 1.375, the flow from the gap side consists of acentral jet surrounded by two shear layers separatedfrom the sharp corner of the two cylinders. The jet-likenature of the gap side flow is mainly due to theacceleration of the flow to maintain a constant flowrate. This jet-like flow in the gap side is similar to thebase bleed flow pattern present in side-by-side circularcylinders [Sumner et al. (1999, 2000)] at smallerspacing. This causes the stream-wise lengthening ofthe near wake region of both cylinders. Because of this,

the recirculation length increases when compared withthat of the isolated square cylinder.

The gap side flow is biased more towards the nearwake of the small cylinder for smaller T/D ratios. ForT/D ¼ 1.12, the reattachment of the separated shearlayer from the gap side leading corner of the largesquare cylinder to its inner surface is seen. ForT/D ¼ 1.75, the reattachment of the separated shearlayer from the gap side leading corner of the smallcylinder to its inner surface is also seen.

As T/D ratio increases, one can notice the decreasein the wake width for both the cylinders, in thetransverse direction. However, in comparison with thelarge cylinder, the decrease in wake width is not seenclearly for the small cylinder. Because of the inter-ference effect, the wake of both the cylinders isunsymmetrical in nature for T/D ¼ 1.12, 1.25, 1.375

Figure 6. Mean streamlines for all T/D ratios.



and 1.75. For T/D ¼ 2.5, the wake behind the largeand small cylinder is in resemblance with that of theisolated square cylinder. This indicates that theinterference effect is absent for this gap ratio.

The mean stream-wise velocity plotted at x/D ¼ 1.0 location is shown in Figure 7. It attains thefree stream velocity on the outside of the large squarecylinder at z/D ¼ 71.5 for all T/D ratios, while on theoutside of the small cylinder it attains free-streamvelocity at around z/D ¼ 2.0 for T/D ¼ 1.12, 1.25 and1.375 gap ratios. In case of T/D ¼ 1.75 and 2.5 thefree-stream velocity is attained at z/D ¼ 2.5 and 3.5,respectively. An additional shift in the distance overand above the displacement of the small cylinder isseen. This is due to the flattening of the parabolicvelocity profile which indicates less interference effect.In addition, the jet-like flow in the gap side almostvanishes for larger T/D ratios. It is more pronounced

for the smallest T/D ratio. The maximum velocitydefect point at z/D ¼ 1.5 near the small cylinder is at adistance of 0.38 from its centre for T/D ¼ 1.120. Thisdistance is 0.45, 0.43, 0.25 and 0.1 for T/D ¼ 1.25,1.375, 1.75 and 2.5, respectively. This decrease in thedistance as T/D increases is very similar to what isreported by Wong et al. (1995). The plot also shows theexistence of weaker mean velocity gradient on thesmall cylinder side and a stronger gradient on the largecylinder side, for all T/D ratios.

The mean base pressure coefficient, Cpb (Table 2) isdefined as the pressure at the centre point of the rearsurface of the cylinder. It is less for the small cylinderin comparison with that for the large cylinder. Fromthe table, it is very clear that the small square cylinderhas a lower Cpb, higher vortex shedding frequency andlarger drag values. Moreover, from the mean stream-lines (Figure 6), one can see a narrow wake behind thesmall cylinder. These results confirm with the findingsof Roshko (1955).

The mean stream-wise velocity plotted along thegeometric centreline for T/D ¼ 1.25 is shown inFigure 8a for both large and small square cylinders.In the upstream side of both cylinders, the velocity ispositive but decreases steadily as it approaches thecylinder. In the downstream side of the large cylinder,the velocity profile first decreases due to backflow andthen starts to increase. The separation of flow over thecylinder causes a pressure drop across its surface andleads to a pressure drag which results in a loss ofmomentum of the fluid in the wake. This is the reasonfor the time averaged stream-wise velocity at any pointin the near wake, to be smaller than that in the freestream.

Figure 7. Mean stream-wise velocity along x/D ¼ 1.0 forall T/D ratios (Two vertical thick solid lines represent rearside of the cylinders).

Figure 8. Mean stream-wise velocity along the geometric centreline. (a) T/D ¼ 1.250; (b) T/D ¼ 2.500.



After this, the recovery of velocity takes place dueto the entrainment of the free-stream fluid. Therecovery rate is directly related to the wake width.The narrower the wake, the faster is the velocity

recovery and vice versa. In case of the small cylinder,velocity profile just downstream of the cylinderdecreases due to loss of momentum of the fluid inthe wake. A sudden increase in the velocity profile is

Figure 9. Time averaged pressure coefficient over the cylinder surface for all T/D ratios.



seen after this due to more momentum transferredfrom the gap side into the near wake of the smallcylinder. After this, again a drop in velocity profile isseen as the fluid now moves through the wake of thelarge square cylinder. The velocity profile again starts

to increase as normal recovery due to entrainment ofthe free-stream fluid taking place. The above discus-sions are true for all T/D ratios except for T/D ¼ 2.5(Figure 8b). In this case, the cylinders behave likeisolated cylinders. The wake centreline coincides

Figure 10. (a) Mean span-wise vorticity for all T/D ratios, (oy min, oy max, Doy) ¼ (71, 1, 0.2); (b) Zoomed view of the meanspan-wise vorticity, (oy min, oy max, Doy) ¼ (74, 4, 0.3). Available in colour online.



with the geometric centreline and the velocityrecovery is similar to that observed in the isolatedcylinder case.

The surface pressure distribution around the largeand small square cylinders for all T/D ratios is shownin Figure 9. In the front portion of both the cylinders,the pressure is maximum at one point which is due tothe occurrence of the stagnation point. For the largecylinder, this point occurs exactly at the centre of thefront face of the cylinder. No shift in stagnation pointfor the large square cylinder is seen for all the T/Dratios. In case of the small cylinder, it is shiftedtowards its gap side from the centre of the front facefor T/D ¼ 1.12, 1.25 and 1.375. This is because of thelow pressure prevailing in its gap side. For T/D ¼ 1.75and 2.5, the stagnation point starts to move back tothe centre of the front face of the small cylinder.Similar inferences can be drawn by observing the meanstreamline plots (Figure 6) as well.

Because of the acceleration of flow, the pressureis lower in the gap side of both the cylinders forT/D ¼ 1.12, 1.25 and 1.375. Because of a very lowpressure in the gap side when compared to outside,both cylinders are under attraction for T/D ¼ 1.12,1.25 and 1.375. For T/D ¼ 1.75 and 2.5, there is anegligible difference in pressure between the flow ingap side and outside of both the cylinders. Similarresults are reported by Wong et al. (1995). ForT/D ¼ 1.12, another peak is seen on the gap side ofthe large cylinder which is due to the reattachment ofthe separated shear layer from the gap side leadingcorner. For T/D ¼ 1.75, the peak is found on thesmall cylinder side. No additional peak is found forT/D ¼ 1.25, 1.375 and 2.5. In the rear side, thepressure is small due to the presence of wake.

The mean span-wise vorticity plot for all the T/Dratios is shown in Figure 10a. The merging of shearlayers separated from the small cylinder and that fromthe top corner of the large cylinder, in the near wake ofthe small cylinder is seen for T/D ¼ 1.12, 1.25 and1.375. This leads to a very strong interference effect forthese gap ratios. Also, at these gap ratios the flowpattern resembles that of a single bluff body. The effectof interference is less felt in case of T/D ¼ 1.75. Theinterference effect is absent for T/D ¼ 2.5 as there is nomerging taking place between the shear layers sepa-rated from the two cylinders. The zoomed view of thevorticity plot near the two cylinders is shown inFigure 10b for few gap ratios. These show the flowstructures in the vicinity of the gap side, details of flowseparation, wake formation and deflections veryclearly.

The time averaged normal and shear stress profilesat x/D ¼ 1.0 location is shown for all T/D ratios inFigures 11 and 12, respectively. For all gap ratios, theFigure 10. (Continued).



normal stress is high in the regions of peak vorticity.This is similar to that reported by Lyn et al. (1995).It is low at places of zero vorticity. This place isgenerally along the gapside or geometric centreline ofthe cylinder and at the intersection of shear layersseparated from the gap side of the two cylinders.The shear stress profile for T/D ¼ 2.5 exhibits fourextrema. Two maximum in the shear layers separatedfrom the bottom corners of the cylinders, while twomaximum of opposite sign in the shear layersseparated from the top corners of both the cylinders.However, the magnitudes are same in the above twocases. It is minimum in the recirculation regions. Theprofile also shows the stress being zero along the gap

side or geometric centreline of both the cylinders.The shear stress profile at this gap ratio is verysimilar to that of an isolated square cylinder. Atsmaller T/D ratios, we see only a single maximum atthe place where merging of shear layers in the gapside takes place. This nature is due to the gap sideflow being deflected towards the near wake of thesmall cylinder.

5. Conclusions

The effect of gap ratio on the flow characteristics pasttwo unequal sized square cylinders in side-by-sidearrangement has been carried out at Re ¼ 50,000using LES. The coefficient of lift is high for the largecylinder in all gap ratios except for T/D ¼ 1.75. Thecoefficient of drag is high for the small cylinder in allthe gap ratios. The vortex shedding is found to be morein small cylinder when compared to that of the largecylinder, for all gap ratios. The mean base pressurecoefficient for the small cylinder is low when comparedto that of the large cylinder for all gap ratios. Atsmaller gap ratios, the presence of a jet-like flow ismore pronounced in the gap side. The reattachment ofthe shear layer separated from the gap side leadingcorner of the large cylinder to its inner surface is seenonly for T/D ¼ 1.12. For T/D ¼ 1.75, the reattach-ment of the separated shear layer from the gap sideleading corner of the small cylinder to its inner surfaceis seen. Because of the presence of a very low pressurein the gap side than that in the outside, the twocylinders are under attraction mode for T/D ¼ 1.12,1.25 and 1.375. There is only a negligible difference inpressure between the gap side and outside of the twocylinders in case of T/D ¼ 2.5. From the mean span-wise vorticity plots, it is clear that the interferenceeffect is present for T/D ¼ 1.12, 1.25 and 1.375 ratios.Moreover, the gap side flow is biased towards the nearwake of the small cylinder for the above gap ratios.The interference effect is less felt in case of T/D ¼ 1.75and absent in case of T/D ¼ 2.5. The flow behaviour atT/D ¼ 2.5 is very similar to that of the flow past twoisolated square cylinder.

Acknowledgements

The authors gratefully acknowledge the financial supportextended by the Council of Scientific and Industrial Research(CSIR, Scheme number 22/(379)/05/EMR–II), India. Theyalso acknowledge Mr. Y. Sudhakar, Junior Research Fellowfor extending his technical support.

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Figure 11. Normal stress along x/D ¼ 1.0 for all T/Dratios.

Figure 12. Shear stress along x/D ¼ 1.0 for all T/D ratios.



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