Flow patterns past two circular cylinders in proximity

Flow patterns past two circular cylinders in proximityKyongjun Lee,Kyung-Soo Yang *Department of Mechanical Engineering,Inha University,253Yonghyun-Dong,Nam-Gu,Incheon 402-751,Republic of Koreaa r t i c l e i n f o Article history:Received 18February 2008Received in revised form 6June 2008Accepted 24July 2008Available online 5August 2008Keywords:Circular cylinderImmersed boundary method Flow pattern Vortex shedding Wake interactiona b s t r a c tFlow patterns past two nearby circular cylinders of equal diameter immersed in the cross-flow at low Reynolds numbers (Re 6160),were numerically studied using an immersed boundary method.We con-sidered all possible arrangements of the two cylinders in terms of the distance between the two cylinders and the inclination angle of the line connecting the cylinder centers with respect to the direction of the main flow.Ten distinct flow patterns were identified in total based on vorticity contours and streamlines,which are Steady,Near-Steady,Base-Bleed,Biased-Base-Bleed,Shear-Layer-Reattachment,Induced-Sep-aration,Vortex-Impingement,Flip-Flopping,Modulated Periodic,and Synchronized-Vortex-Shedding.Collecting all the numerical results obtained,we propose a general flow-pattern diagram for each Re ,and a contour diagram on vortex-shedding frequency for each cylinder at Re =100.The perfect symmetry implied in the geometrical configuration allows one to use these diagrams to identify flow pattern and vortex-shedding frequencies in the presence of two circular cylinders of equal diameter arbitrarily posi-tioned in physical space with respect to the main-flow direction.Ó2008Elsevier Ltd.All rights reserved.1.IntroductionCross-flow past a group of cylinders is often found in practical engineering applications.For instance,air flow past a bundle of pipes in a chemical plant,coolant flow past tubes in a heat exchan-ger,sea-water flow past columns of a marine structure,and wind past an array of chimneys of a power plant,to name a few.Flow characteristics past each cylinder are affected by its neighbors via wake interaction,resulting in alteration of the overall flow pattern.Therefore,flow pattern past multiple cylinders heavily depends on their relative positions with respect to the main-flow direction.Consequently,vortex-shedding frequency of the individual cylin-der is accordingly determined,and serves as an important factor in generation of flow noise [1].Being motivated by this,many researchers have been involved in studying wake interaction be-tween two circular cylinders of equal diameter immersed in a cross-freestream as a basic wake-interaction model.Spatial arrangement of two cylinders can be classified into three catego-ries,namely,aligned with the direction of the main flow (in tan-dem),placed side-by-side,and placed in a staggered arrangement.When two circular cylinders of equal diameter are placed side-by-side,flow pattern varies with Reynolds number (Re )and the distance between the cylinders.Kang [2]identified six distinct flow patterns depending on Re and the distance between the cylinders.It was also found that when the surface-to-surface distance is long-er than 5D (here,D is the cylinder diameter),wake interaction van-ishes and flow past each cylinder behaves just like the flow past the single cylinder placed in the freestream [2].Williamson [3]classified the types of vortex shedding by using flow visualization.When Re =100and the surface-to-surface distance is longer than D ,the flow pattern past the two cylinders is periodic,and vortex shedding from each cylinder is either in phase or out of phase by 180°with the other one.However,when the surface-to-surface distance is shorter than D ,the flow pattern past the two cylinders is non-periodic and completely irregular [2,3].When the two cylinders are placed in tandem,there exists the critical distance between the two cylinders,below which vortex shedding of the upstream cylinder (hereafter,called ‘‘main cylin-der”,MC)does not occur;the value of the critical distance varies depending on Re [4–7].Mizushima and Suehiro [4]revealed that the flow pattern with L /D =2.0is different from that with L /D =4.0at Re =100mainly due to suppression of vortex shedding from MC.Here,L denotes the streamwise distance between the two cylinder centers (Fig.1(a)).They also predicted existence of bistable solutions in the range of 3.16L /D 63.5for Re =100.Sub-sequently,Tasaka et al.[5]performed experimental investigation to verify the prediction of Mizushima and Suehiro [4],and reported 4.66L/D 65.0as the range of L /D for existence of bistable solu-tions,being shifted from the numerical prediction of Mizushima and Suehiro [4].Tasaka et al.[5]attributed the discrepancy to the finite length of the cylinders in their experiments.Carmo and Meneghini [6]investigated the three-dimensional structures be-hind two circular cylinders when they are in tandem for the range of 160<Re <320.Sharman et al.[7]performed numerical predic-*Corresponding author.Tel.:+82328607322;fax:+puters &Fluids 38(2009)778–788Contents lists available at ScienceDirectComputers &Fluidsj o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /c om p flu idrevealing a sudden change in keyflow parameters between L/D= 3.75and4.0.Sumner et al.[8,9]experimentally studiedflow-induced forces andflow patterns for high-Re(8506Re61900)flows past two cir-cular cylinders in staggered arrangements;they identified nine dis-tinctiveflow patterns.Akbari and Price[10]identifiedfive distinct flow patterns at Re=800.Gu and Sun[11]and Sun et al.[12]also reportedflow patterns in turbulentflow regime.In spite of the numerous studies carried out so far,the case where the two cylinders are placed in a staggered position has not been systematically studied,especially in the laminarflow re-gime.In this investigation,flow patterns past two nearby circular cylinders of equal diameter immersed in the cross-flow at Re6160,based on the freestream velocity(U)and D,were numer-ically studied as a basic model for laminar wake interaction.An im-mersed boundary method[13]was employed for effective treatment of the cylinders on a Cartesian grid system.We consider all possible arrangements of the two circular cylinders in terms of the distance between them and the inclination angle of the line connecting their centers with respect to the mainflow direction. The current authors recently reported their numerical results on theflow-induced forces in the sameflow setting[14].As a fol-low-up study,the present paper classifies theflow patterns ob-served,and describes distinctive characteristics of eachflow pattern.Collecting all the results obtained,we propose aflow-pat-tern diagram(‘‘map”)for prediction of theflow pattern associated with a particular arrangement of the two cylinders as well as a Strouhal-number diagram for vortex shedding of each cylinder to provide an overall picture on the wake interaction.The perfect symmetry implied in the geometrical configuration allows one to use these diagrams to identifyflow pattern and vortex-shedding frequencies in the presence of two circular cylinders of equal diam-eter arbitrarily positioned in physical space with respect to the mainflow direction.2.Formulation and numerical methodologyThe current investigation requires a parametric study where numerous numerical simulations must be performed with various values of L and T.Here,T represents the vertical distance between the two cylinder centers(Fig.1(a)).This kind of parametric study demands considerable amount of computing resources;the com-puting efforts can be significantly reduced by employing an im-mersed boundary method which facilitates implementing the solid surfaces of the arbitrarily-positioned cylinders on a Cartesian o u jo x jÀq¼0ð1Þo u iþo u i u jj¼Ào piþ1o2u ij jþf i i¼1;2ð2Þwhere u i(or u,v),p,q and f i represent velocity component in x i(or x,y)direction,pressure,mass source/sink,and momentum forcing, respectively.All the physical variables except p are non-dimension-alized by U and D;pressure is non-dimensionalized by far-field pressure(P1)and the dynamic pressure.The governing equations were discretized using afinite-volume method in a nonuniform staggered Cartesian grid system.Spatial discretization is second-or-der accurate.A hybrid scheme is used for time advancement;non-linear terms are explicitly advanced by a third-order Runge–Kutta scheme,and the other terms are implicitly advanced by the Crank–Nicolson method.A fractional step method[15]was em-ployed to decouple the continuity and momentum equations.The Poisson equation resulted from the second stage of the fractional step method was solved by a multigrid method.For detailed description of the numerical method used in the current investiga-tion,see Yang and Ferziger[16].3.Boundary conditions and numerical parametersThe main cylinder isfixed at the origin of the coordinate system, and the downstream cylinder(hereafter,called‘‘surrounding cylin-der”,SC)is placed at various locations relative to MC,which are represented by dots in Fig.1(b).The total number of cases com-puted is208for Re=100,and58for Re=40,50,and160.In the case where the center distance(R,Fig.1(a))is minimum, L/D=0.25,T/D=1.0,and the inclination angle(/,Fig.1(a))is 75.96°.The entire computational domain was defined as À35D6x635D,andÀ50D6y650D.For each cylinder,32Â32 uniform grid cells in x-and y-directions,respectively,were allo-cated,and uniform grid cells of the same cell size as in the cylinder region were employed between the cylinders.In the other region of the domain,nonuniform grid cells were used.The numerical resolution was determined by a grid-refinement study to ensure grid-independency.In the case of L/D=1.5,T/D=0.5(Fig.2),for in-stance,the total number of grid cells was368Â208,while 480Â352grid cells were allocated in the most demanding case (L/D=5.0,T/D=5.0).No-slip condition was imposed on the cylinder surfaces;a Dirichlet boundary condition(u=U,v=0)was used on theK.Lee,K.-S.Yang/Computers&Fluids38(2009)778–7887794.Verification of the codeFor verifying our numerical methodology and resolution,the flow past a single circular cylinder at Re =100was computed and our results were compared with other authors’numerical [2,18]and experimental [19]results.The size of computational domain and the boundary conditions employed were as described in the previous section.The same number of uniform grid cells (32Â32)as in the two-cylinder cases was allocated for the cylin-der,and the total numerical resolution was 320Â192.Table 1pre-sents mean drag coefficient ðC d Þ,maximum magnitude of lift-coefficient fluctuation ðC 0l ;max Þ,and Strouhal number (St )of the vor-tex shedding in the literature,along with the present results.The present results are in excellent agreement with those of other authors,verifying that the numerical scheme and resolution used in the current simulation are adequate and reliable.5.Results and discussionIn the present investigation,we classified types of flow pattern based on streamlines and contours of spanwise vorticity (x z ).Spanwise vorticity is defined as 1o v Ào u,and contour lines cover À4.06x z 64.0with increment of 0.1in the following contour plots.Solid and dashed lines represent positive and negative vor-ticity,respectively.Four different values of Re (40,50,100,and 160)were considered.For Re =100and 160,the general trend can be summarized as follows.When the two cylinders are ‘‘very”close to each other,the vortex shedding occurs as if the two cylinders were just one object.When they stand ‘‘reasonably”apart,wake interaction takes place,resulting in various types of flow pattern depending on how the two cylinders are positioned.When they are ‘‘far”away from each other,wake interaction is weak and flow characteristics around each cylinder resemble those of a single cylinder at the same Re .In the current investigation,10distinct flow patterns were identified in total for Re 6160.To name the flow patterns,weadopted the terms in the previous studies [2,9],if applicable.Otherwise,we named the new flow patterns with our own terms.In the following section,detailed description of flow patterns is gi-ven for Re =100.The effect of Re on wake interaction is addressed in Section 5.2.5.1.Classification of flow pattern for Re =1005.1.1.Base bleed (BB)BB pattern is noticed when (L /D ,T/D )=(0.0,1.25),(0.25,1.0),(0.25,1.25),(0.5,1.0),(0.75,0.75).Even though the distance be-tween the two cylinders is quite short,the flow through the ‘‘gap”is significant owing to the large /.Fig.3presents snapshots of pressure contours together with streamlines through the gap at the four selected instants with equal time interval during one per-iod of vortex shedding (T p ).Fig.4shows the corresponding time histories of drag coefficient (C d )for MC and SC,respectively.In Fig.3(a),a low pressure region is seen behind MC,attracting the gap flow into this region and also causing a high drag force on MC (see A in Fig.4).The opposite is the case in Fig.3(c);pressure is low behind SC.As a result,the gap flow is diverted towards the low pressure region (Fig.3(c)),and SC experiences a high drag force (see B in Fig.4).In Fig.5,contours of x z are presented at the same instants as in Fig.3.The short distance between the cylinders pre-vents ‘‘individual”shedding;the vortices are shed in a fashion sim-ilar to that behind a single object.5.1.2.Biased base bleed (BBB)BBB pattern is found when (L /D ,T /D )=(1.0,0.25),(1.0,0.5).The gap between the cylinders is very narrow like the BB pattern,but /is small unlike the BB pattern.Fig.6shows snapshots of instanta-neous pressure contours and streamlines through the gap at the four selected instants with equal time interval during T p .The flow through the gap is always heading downwards due to the small /(Fig.6),while in the BB pattern,the direction of the gap flow is alternating up and down (Fig.3).In Fig.7,contours of x z are shown at the same instants as in Fig.6.Like in the BB pattern,vortices are shed as if the two cylinders were just one object.5.1.3.Shear layer reattachment (SLR)SLR pattern is observed approximately when 1.256L /D 64.0,T/D 60.75;in this range,/is less than 22°.Instantaneous stream-lines of a typical case of SLR pattern (L /D =3.0,T/D =0.5)are pre-sented in Fig.8.The shear layer formed above MC reattaches on SC.A lower portion of the shear layer is deflected by SC,and to-gether with the shear layer formed below MC,surrounds the recir-culation region behind MC.In Fig.9where instantaneous contours of x z are taken at four instants with equal time interval during T p ,it is seen that SC significantly interfere with the upper shear layer from MC.It is also noticed that vortex shedding from MC has been completely suppressed by SC,and vortex shedding from SC is somewhat delayed.This is one of the important features of SLR pat-tern.Sharman et al.[7]reported that in tandem arrangements of the two cylinders,vortex shedding from MC resumes at a certain ‘‘critical”distance between them which is in the range of 3.756L /D 64.0depending on Re .5.1.4.Induced separation (IS)IS pattern is noticed approximately when 1.06L /D 63.5,0.756T /D 61.75.In this region,/is relatively large compared to that of SLR pattern.Fig.10presents instantaneous contours of x z taken at the four instants with equal time interval during T p .TheTable 1Validation of numerical method:flow past a single cylinder at Re =100ReC d C 0l ;max St 780K.Lee,K.-S.Yang /Computers &Fluids 38(2009)778–788one vortex (C in Fig.10).However,negative vortices are mainly shed from SC (Fig.10).Fig.11shows a snapshot of streamlines for L /D =1.5,T /D =1.0.The gap flow actually induces separation on SC,enhancing vortex shedding from SC.Consequently,vortex shedding from SC is much more active in IS pattern than in SLR pattern.5.1.5.Vortex impingement (VI)VI pattern appears in the range of approximately L /D P 4.0,T /D 61.75,where the two cylinders are far apart (R /D P 4.0)and /is less than 24°.When the two cylinders are placed in tandem withK.Lee,K.-S.Yang /Computers &Fluids 38(2009)778–788781a short distance between them(L/D63.75for Re=100),SC sup-presses the vortex shedding of MC[7].With a larger distance,how-ever,SC fails to suppress the vortex shedding of MC,and vortices are shed from both cylinders.Vortices of positive and negative signs, shed from MC in an alternating manner,impinge upon SC.However, when the two cylinders are slightly‘‘off-tandem”(say,T/D P0.25),shedding with equal time interval when L/D=4.0,T/D=1.0.The negative vortex which developed in the upper shear layer of MC im-pinges upon SC(A in Fig.12(a)),and then split into two parts(A0and A00in Fig.12(b)).One of them(A00)is combined with the negative vor-tex shed from SC,and the other remains as a separate secondary vor-tex(B in Fig.12(c),and C in Fig.12(d)).782K.Lee,K.-S.Yang/Computers&Fluids38(2009)778–788where the distance between the two cylinders(1.346R/D63.20) is somewhat larger than those in BB or BBB patterns,and/P38°. In this pattern,both cylinders shed vortices,yielding two pairs of counter-rotating vortices.Unlike BB or BBB pattern,however,FF pattern is irregular in time,being consistent with Meneghini et al.[20]and Kang[2].Kang reported that in side-by-side arrange-ments of two cylinders,flow pattern is irregular when1.46T/D62.2.Fig.13shows temporal variations of drag and lift coefficients(C d,C l),respectively,for L/D=0.0,T/D=2.0;irregular force coeffi-cients reflect irregularflow behavior in FF pattern.5.1.7.Modulated periodic(MP)MP pattern occurs in the range of approximately0.56L/ D63.0,T/D P2.5where the two cylinders are far apart (2.696R/D65.59),and approximately/P45°.Combination of the large distance and the relatively high inclination angle yields weak interaction between the two wakes,resulting in slight mod-ulation in vortex shedding from each cylinder.It is noticed that the vortex-shedding frequency of MC is slightly higher than that of SC.K.Lee,K.-S.Yang/Computers&Fluids38(2009)778–788783Fig.14presents temporal variation of C l for L /D =1.5,T /D =3.0,revealing modulation of amplitude and frequency of the force coef-ficient.Instantaneous contours of x z at two distinct instants are illustrated in Fig.15.It is seen that two positive vortices shed from the cylinders are side-by-side in Fig.15(a).However,the opposite is the case at a later time (Fig.15(b)),indicating the difference be-tween the two frequencies of vortex shedding.MP pattern is sim-ilar to ‘VPE’(vortex pairing and enveloping)or ‘VPSE’(vortex pairing,splitting and enveloping)patterns in Sumner et al.[9]for 8506Re 61900.5.1.8.Synchronized vortex shedding (SVS)SVS pattern is noticed in the arrangements where the two cylinders is far apart (R /D P 2.5)and /P 20°.Due to the long dis-tance and high inclination angle,the interaction of the wakes is negligible,and vortices are shed just like the single cylinder in the freestream.Fig.16shows contours of x z at four instants of equal time interval during one period of vortex shedding for L /D =0.0,T /D =3.0.The cylinders shed vortices independently;interaction of wakes such as vortex merging is hardly seen.In this particular case,the vortices from the two cylinders are out of phase by 180°owing to the perfect geometrical symmetry with respect tothe center line between the two cylinders.However,in a staggered arrangement of the cylinders,the phase difference changes,and the wake interaction is still negligible in SVS pattern.5.2.Flow patterns for low-Re flowsThe following two flow patterns are uniquely observed in low-Re flows (Re =40and 50).784K.Lee,K.-S.Yang /Computers &Fluids 38(2009)778–7885.2.1.Steady(S)In this pattern,flow is steady in the entire domain.5.2.2.Near steady(NS)NS pattern is characterized by‘‘delayed”unsteadiness.In Fig.17,streamlines are presented at four instants of equal time interval during one period of vortex shedding for L/D=0.0, T/D=1.5at Re=50.In NS pattern,flow is steady around the two cylinders,but becomes unstable in the far downstream wake.It should be noted that NS pattern is also found at Re=40which is slightly below the critical Reynolds number(Re c%43)for the sin-gle-cylinder case.5.3.Flow-pattern diagramCollecting all the results obtained,we propose theflow-pattern diagrams on L–T-plane in Fig.18to provide an overall picture on theflow patterns past two nearby circular cylinders of equal diam-eter.All the patterns discussed earlier are indicated together with their‘‘schematic”boundaries for convenience.The cases actually computed are denoted by the dots.One can identify the effect of Re by comparing the diagrams.At a subcritical Reynolds number(Re=40,Fig.18(a)),theflow would be entirely steady,if there were only single cylinder present.at Re=50which is above Re c(Fig.18(b)).FF and VI patterns do not occur in the low-Reflows.For the high Reynolds numbers(Re=100 and160),S and NS patterns are not found(Fig.18(c)and(d)).As Re increases from Re=100to Re=160,FF and VI patterns occur in more arrangements as expected.5.4.Frequency of vortex sheddingFig.19presents contour diagrams of Strouhal number(St)of vortex shedding for MC(Fig.19(a))and SC(Fig.19(b)),respec-tively,for all possible arrangements of the two cylinders at Re=100.The Strouhal number was computed based on C l using fast Fourier transform(FFT),and normalized with that of the single cylinder(St s=0.165)immersed in the freestream at Re=100.When multiple dominant frequencies were identified (for example,in FF pattern),the most dominant one was reported in Fig.19.For the main cylinder(Fig.19(a)),the maximum St(=0.204)is identified for L/D=1.0,T/D=2.0.In the range of approximately T/D P2.0,the Strouhal number of MC is larger than St s,implying that SC induces more frequent vortex shedding from MC in this region.The opposite is the case when T/D62.0because SC suppresses vortex shedding from MC.It should be noted that in the range of approximately0.06L/D62.5,1.56T/D62.0,St ofK.Lee,K.-S.Yang/Computers&Fluids38(2009)778–788785786K.Lee,K.-S.Yang/Computers&Fluids38(2009)778–788other that the‘‘effective”characteristic length scale is larger than D.As a result,the‘‘effective”Reynolds number increases and St drops.Although the distance between the cylinders is identical (R/D=1.12)in both patterns,the inclination angle in BB pattern is larger than that in BBB pattern,yielding a bigger‘‘effective”char-acteristic length,thus smaller St.In the cases of SLR,IS,and VI patterns where/645°,the Strou-hal numbers of MC and SC are identical,but smaller than St s,indi-cating strong wake interaction due to small/.Nevertheless,the Strouhal number in VI pattern is larger than those in SLR and IS patterns,and more closer to St s,which implies that wake interac-tion is rather weak in VI pattern because the two cylinders are far apart in VI pattern compared to the other two patterns.In the cases of FF and MP patterns where/is large,the Strouhal number of MC is larger than that of SC as well as St s.One can con-jecture that in these patterns,vortex shedding from MC is more en-hanced by the presence of SC via wake interaction.However,as the two cylinders become far apart and/also increases,such as in SVS pattern,the Strouhal numbers of the two cylinders become identi-cal again,and approach St s.5.5.Extension of usage of the diagrams to other arrangementsEven though the diagrams(Figs.18and19)were constructed only for one quarter of all possible arrangements of MC and SC with MCfixed at the origin,one can still use them for the other arrangements using the symmetry of the geometrical configura-tion.The constant freestream velocity allows the symmetry of SC with respect to the x-axis(Fig.20(a)).For example,when SC is placed in the fourth quadrant,the diagrams are still effective as they are,as long as the vertical distance between SC and the x-axis is used for T.Furthermore,the symmetry with respect to the origin is valid as the two cylinders are identical(Fig.20(b)).Thus,as long as the upstream cylinder is designated as MC,and the downstream cylinder as SC according to our‘‘definitions”of MC and SC,and to-gether with the symmetry with respect to the x-axis if necessary, the diagrams can be used in any possible arrangement of the two cylinders.6.ConclusionIn this investigation,we considered cross-flow past two arbi-trarily-positioned nearby circular cylinders of equal diameter for Re6160,and classified itsflow pattern into10distinct ones based on streamlines and contours of spanwise vorticity.They are steady (S),near steady(NS),base bleed(BB),biased base bleed(BBB), shear layer reattachment(SLR),induced separation(IS),vortex impingement(VI),flipflopping(FF),modulated periodic(MP), and synchronized vortex shedding(SVS).Characteristics of each pattern were identified and described in detail.For high-Reflows(Re=100and160),when the two cylinders are very close to each other(such as in BB or BBB patterns),vortices are shed as if they were just one object;the Strouhal number of the vortex shedding is less than that of the single cylinder immersed in the freestream.Wake interaction becomes complicated depending on the distance between the two cylinders and the inclination an-gle of the line connecting the cylinder centers with respect to the mainflow.When the surrounding cylinder is located in the near-wake region of the main cylinder,the frequency of the vortex shed-ding is identical for both cylinders,and is less than that of the sin-gle cylinder.In FF and MP patterns where the inclination angle is large,the vortex-shedding frequency of the main cylinder is larger than that of the surrounding cylinder as well as that of the single cylinder,implying that the vortex shedding of the main cylinder is enhanced by the presence of the surrounding cylinder in the re-gion of high inclination angle.However,both frequencies approach that of the single cylinder as the two cylinders get far apart.It also turned out that presence of a neighboring cylinder can destabilize a subcriticalflow,and stabilize a slightly supercritical flow in the particular arrangements of the two cylinders.Collecting all the computed results,we propose theflow-pattern diagrams and the contour diagrams of Strouhal number of vortex shedding in thefirst quadrant of physical space with the main cylin-derfixed at the origin.Due to the symmetry implied in the geomet-K.Lee,K.-S.Yang/Computers&Fluids38(2009)778–788787sponding vortex-shedding frequencies sheds light on developing a wake-interaction model in engineeringflows.AcknowledgementThis work was supported by Inha University Research Grant. 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