NUMERICAL SIMULATION OF FLOW PAST A SQUARE CYLINDER

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1 Proceedings of FEDSM99 3rd ASME/JSME Join Fluids Engineering Conference Jul 83, 999, San Francisco, California, USA FEDSM99-77 NUMERICAL SIMULATION OF FLOW PAST A SQUARE CYLINDER Ahmad Sohankar Thermo and Fluid Dnamics Chalmers Universi of Technolog S-4 96 Göeborg Sweden sohankar@fd.chalmers.se C. Norberg Division of Hea Transfer Lund Insiue of Technolog Bo 8, S Lund Sweden chris@ms.vok.lh.se L. Davidson Thermo and Fluid Dnamics Chalmers Universi of Technolog S-4 96 Göeborg Sweden lada@fd.chalmers.se ABSTRACT D and 3D unsead flow pas a rigid prism of a square cross-secion wih one side facing he oncoming flow is numericall simulaed for Renolds numbers beween o 5. An incompressible code is used emploing an implici fracional sep mehod finie-volume wih second-order accurac in space and ime. For D flow, i is found ha, for Re 3, he ime-mean flow paerns are no perfecl smmeric wih respec o he oncoming flow. In conras, such a non-smmeric mean flow paern is no observed in 3D simulaions. There is a marked and characerisic pulsaion in he force componens in he 3D flow for Re 3. These force pulsaions conain characerisic ime periods wih high and low levels of forces. I is found ha when he flow is wihin a low levels of forces he inensi of hreedimensional effecs becomes sronger, whereas he opposie is rue wihin a high level of forces. INTRODUCTION In he pas, he flow around bluff bodies, such as circular and recangular clinders, has been eensivel sudied owing o is relevance o echnical problems associaed wih energ conversion and srucural design. Over he las wen ears, a vas amoun of sudies has been conduced o increase he undersanding of differen ransiion processes of he flow pas a circular clinder eperimenall, numericall and heoreicall, for a review see Williamson (996). B conras, here are ver few similar sudies found on flow pas recangular clindrical srucures, e.g. he square clinder, a moderae Renolds numbers where a D/3D wake ransiion occurs. As an eample, a Renolds numbers less han abou 5 here is onl one single se of eperimens reporing he influence of Renolds number on he mean drag coefficien, see Okajima e al. (995). Recenl, wo ransiion processes were numericall invesigaed b he presen auhors. Firs, in Sohankar e al. (998), for a square clinder a various angles of incidence, he onse of vore shedding is invesigaed b using he Suar-Landau equaion and second, in Sohankar e al. (999) he ransiional 3D wake flow behind a square clinder is sudied. In he second work, i is found ha a ransiion from D o 3D shedding flow beween Re 5 and Re occurs and ha boh spanwise insabili modes, A and B, are presen in he wake ransiional process, similar o he flow around a circular clinder. However, seemingl in conras o a circular clinder, he ransiional flow around a square clinder ehibis a phenomenon of disinc low frequenc force pulsaions (Re 3) wih period abou 6 imes he shedding period. Recenl, force pulsaions wih a period of approimael imes he primar shedding period are also repored for a zero-hickness fla plae a Re 5 b Najjar & Balachandar (998). In Sohankar e al. (999), i is also shown ha he Srouhal number and he mean drag coefficien for he 3D simulaions are in general agreemen wih eising eperimens. The D resuls for mean drag are in reasonable agreemen wih eperimens, alhough oher quaniies and flow characerisics for Re are for he mos par in sharp conras wih available eperimenal daa and 3D resuls. The main objecives of he presen sud were o furher in- Coprigh 999 b ASME

2 vesigae 3D ransiional feaures in he flow pas a square clinder and o make addiional comparisons wih D resuls a moderae Renolds numbers, Re 5. z Compuaional Deails The flow is described in a Caresian coordinae ssem z, in which he -ais is aligned wih he inle flow direcion, he z-ais is parallel wih he clinder ais and he -ais is perpendicular o boh direcions, as shown in Fig. PSfrag. A fied wo-dimensional square clinder wih a side d is eposed o a consan free sream veloci, U. An incompressible flow wih consan fluid properies is assumed. The Renolds number is defined as Re U d ν. All geomerical lenghs are scaled wih d. Scaling wih d also applies o he Srouhal number, S f S d U, where f S is he shedding frequenc, and for all forces. Velociies are also scaled wih U, and phsical imes wih d U. In -direcion, he verical disance beween he upper and lower walls, H, defines he solid blockage of he confined flow (blockage parameer, β H). An incompressible finie volume code, which is based on a fracional sep echnique, is used and emplos a non-saggered grid arrangemen. The scheme is implici in ime, and a second order Crank-Nicolson scheme is used. All erms are discreized using he second-order cenral differencing scheme. The imemarching calculaions are sared wih he fluid a res, and a consan ime sep 5 is used. The calculaions are carried ou for differen resoluions, 69 5 (3D simulaion, A 6), 69 4 (3D simulaion, A ), (3D simulaion, fine resoluion for Re 5, A 6) and 69 (D simulaions). The following boundar condiions were used. A uniform flow was prescribed a he inle, which is locaed X u unis upsream of he clinder. A he oule, locaed X d unis downsream of he clinder, he convecive boundar condiion was used for all veloci componens. No-slip condiions were prescribed a he bod surfaces. Smmer condiions simulaing a fricionless wall were used a he upper and lower boundaries. A periodic boundar condiion was used in he spanwise direcion. The normal derivaive for he pressure was se o zero a all boundaries. In he presen sud, X u, X d and H were se o 8.5,.5 and 8, respecivel, see Fig.. Wake Flow Transiion Wihin he earl sages of he wake ransiion process (Re 5) a marked and characerisic pulsaion occurs in he force componens, see Sohankar e al. (999). Such a pulsaion is seen in Fig. for Re bu is no observed for Re 5. These force pulsaions conain characerisic ime periods wih high and low levels of forces which are referred o as HF and LF regions, respecivel. Associaed wih he presence Table. X u d X d Figure. Flow configuraion. Re LF HF R ω R ω R ω3 ω3 C P3 LF HF LF HF Summar of resuls aken from spanwise vorici conours a minimum lif insances, A 6, see e.g. Fig. 3. The raio of spanwise vorici ( ) and magniude vorici (ω= ω ω ), Rω, and pressure coefficien, C P, are chosen a poin where is ereme. Indees -3 in his able refer o developing vorices from he upper side (black one), he lower side (whie one) and deached vore (whie one) a he posiion of 4 5, respecivel. of force pulsaions, a ime-shifed posiive coupling is observed beween he insananeous shedding frequenc and he lif ampliude, wih shedding frequenc leading he ampliude, see Sohankar e al. (999). In he ime regions of high force levels, he flow is in an ordered sae wih relaivel small spanwise variaions, whereas he hree-dimensional effecs are srong in low force regions wih a seemingl chaoic spanwise flow srucure. The pulsaing forces, especiall he drag, are closel relaed o he acivi of he secondar vorices, see Sohankar e al. (999). Here we r o epand his maer furher b suding he vore srucure in HF and LF regions o find he main feaure of he flow in hese wo regions. Spanwise vorici conours in he mid-plane (z ) a he insance of minimum lif force for Re 5 are shown in Fig. 3. In his figure and oher figures Coprigh 999 b ASME A

3 4 Re A 3 PSfrag Re A 4 Re 5 A PSfrag Figure 3. Spanwise vorici conours in he plane z in he posiion of minimum lif force. (3D-simulaions, A 6, Re 5). Top: LF region ( 6); boom: HF region ( 34). Figure. Time hisor of spanwise-averaged lif and drag coefficiens (3D simulaions, A ). Top: Re (S 6, 4, 3, ), boom: Re 5 (S, 87, 48, 5). no shown here, i is observed ha he shear laers developing from he upsream edges of he clinder are no eacl similar in HF and LF regions. In he LF region, he free shear laers from he upper edge (black one in Fig. 3) are eended farher downsream in comparison o he case in he HF region, before rolling up o spanwise (von Kármán) vorices. In addiion, in he HF region, he free shear laers from he lower edge (whie one in Fig. 3) are rolled up closer o he clinder han is he case in he LF region. Thus, in he HF region he firs vore is shed in he wake a a posiion closer o he bod. This is also consisen wih he fac ha he frequenc of vore shedding is higher in he HF region han in he LF region. The higher frequenc means ha he process of spanwise vore shedding is faser. Similar findings were repored b Najjar & Balachandar (998) for flow pas a normal plae. The found ha he spanwise vorices in he HF region are more compac and roll up closer o he back side of he normal plae wih significanl shorer mean recirculaion region han is seen in he LF region. Sohankar e al. (999) repor ha he spanwise coupling of forces and he associaed near-wake flow componens were higher in HF regions han in LF regions. For eample, i is shown ha he spanwise variaions of he secional drag in he HF region are small in ransiion (Re and Re 5), whereas he variaions in he LF region are significanl higher. This means ha he degree of hree dimensionali is higher in he LF regions han in he HF regions. Furher invesigaion using differen componens of vorici also confirms his finding. Table provides he raio of spanwise o oal vorici a minimum lif insances R ω (= ω ω ), a he posiion of ereme in hree posiions in he wake region for Re and Re 5. These hree posiions of ereme are locaed on he core of he developing vorices from he upper side (black one), he lower side (whie one) and deached vore (whie one) a he posiion of 4 5, respecivel, e.g see Fig 3. Similar resuls o hose provided in Table, were also observed for he insan of maimum lif. From Table, i is seen ha he R ω has higher values (close o one) in HF regions han in LF regions for Re and Re 5. High values of R ω in he order of one mean ha he magniude of and ω are close or ha he effec of wo oher 3 Coprigh 999 b ASME

4 Γ z Γ z. PSfrag Γ z Γ z Figure 4. Spanwise circulaion (Γ z ) in plane z ogeher wih spanwise-averaged lif coefficien, spanwise vorici ( ) and vorici raio ( = ω ω ω ω ) vs. ime for Re, A 6. lef) LF region, righ) HF region. The circulaion is calculaed in he domain 6,. The and are aken from a poin in he free shear laer a he locaion of, 6 and z. componens, ω and ω, are negligible. Thus, he degree of wo dimensionali is higher in HF regions han in LF regions. I is imporan o menion ha he corresponding values of peak vorici, ω, and pressure coefficien, C P, a he cener of he firs deached vore (e.g see Fig. 3 a he posiion of 4 5) in LF regions are higher han in HF regions, see Table. This means ha more inense vorices (higher ω and C P in he core of vorices) are generaed in LF regions han in HF regions. In general, he level of spanwise vorici in HF and LF regions is approimael similar, e.g. see Fig. 4. As a crude measure of he inensi of he Kármán vorices, he spanwise circulaion, Γ z, was calculaed a mid-span (z ) in he region, 6. In referring o his figure, for Re, i can be noed ha, in LF and HF regions, he ime variaions in he spanwise circulaion, Γ z, are approimael sable wih regard o heir ampliude and var beween he limis of wih a frequenc corresponding o he lif signal. Such variaions are also seen in his figure for spanwise vorici,, a a seleced poin in he free shear laer (, 6, z ). In confirmaion of his finding, i was also found ha he spanwise circulaion raio, Γ zhf /Γ zlf, is.97 for he deached vore a he posiion of 4 5 in he insance shown in Fig. 3. On he oher hand, he srengh of he sreamwise vorici becomes sronger in he LF region. As is seen, he level and variaion of he vorici raio, (= ω ω ω ω ), a a seleced poin in he free shear laer (, 6, z ) are higher in he LF region han in he HF region. I was observed ha he inensi of he sreamwise vorici and he calculaed sreamwise circulaion, Γ, in differen sreamwise (z) planes were higher when he flow is in a LF region (hese plos are no shown here, see Sohankar (998)). Similar findings as for Re were also observed for Re 5. In general, referring o Figs. 3-4 and Table and as well as oher findings no repored here (see Sohankar (998)), i is concluded when he flow is in a LF region he inensi of hreedimensional effecs becomes sronger while he opposie is rue in a HF region. Comparison beween D and 3D resuls Sohankar e al. (999) repor ha he D resuls for mean drag are in reasonable agreemen wih eperimens, alhough oher quaniies for Re are for he mos par in sharp conras wih available eperimenal daa and he 3D resuls. Here, we r o make addiional comparisons b comparing D and 3D flow characerisics. Time averaged sreamlines of D simulaions for Re 3 and Re 5 and 3D simulaion for Re 5 (A ) are shown in Fig. 5. I was found ha he ime-mean flow paerns were perfecl smmeric for Re 3 (Fig. 5, upper), whereas he flow paerns were no perfecl smmeric wih respec o he oncoming flow for Re 4 and Re 5. This non-smmeric mean flow paern in D simulaions is shown in Fig. 5 (middle) for Re 5. Such a non-smmeric mean flow paern has also been repored b Breuer(998) for D flow around a circular clinder a Re 39. As is seen from Fig. 5 (lower) he 3Dsimulaed mean flow paern is smmeric for Re 5. As is observed in Fig.5, he mean flow paern for D and 3D simulaions a Re 5 is approimael similar on he fron, op and boom sides of he clinder, whereas here are man differences on he rear side where he wake flow develops. This difference is also clearl seen in Fig 6, which shows he pressure coefficien, C P, 4 Coprigh 999 b ASME

5 around he clinder. On he basis of his figure, he differences in C P for D and 3D simulaions are smaller on he sides of he bod hen on he rear. In spie of he differences in he values of C P on he rear side, he averaged value of C P for D simulaion on his side was he same as for 3D simulaion. This is he reason wh he drag coefficien for Re 5 is in good agreemen in D and 3D simulaions (D: 9, 3D: 87). When comparing averaged 3D resuls wih D resuls for RMS drag, RMS lif, RMS base pressure coefficien and Srouhal number, he differences were abou 63%, 6%, 4% and 7%, respecivel. For D flow, a non-zero mean lif was found for Re 3. For insance, he ime-mean lif coefficien for Re 4 PSfrag was equal o 4. There is also a significan difference beween D and 3D simulaions in pressure coefficien and sreamwise veloci a he cenerline, see Fig. 6. I is imporan o menion ha he D ime-averaged resuls did no change even when he averaging ime was increased up o 5 shedding periods. In Sohankar e al. (999) i is shown ha he lif and drag specra from D simulaions for Re 3 ehibi a scenario of perioddoublings while here are no signs of such period-doublings in he 3D simulaions, see Sohankar e al. (999). In connecion wih period-doublings for Re 4 5, i is observed ha here are wo and four peaks in each main shedding period in he ime hisor of he lif and drag forces, respecivel, see Sohankar PSfrag e al. (999). Corresponding o hese four peaks for drag, four vorices are shed during each period which lead o a sudden drop in Re 3 he ime hisor of he drag force beween each wo peaks. The D inensi and he formaion ime of hese vorices are differen, which leads o differen lif and drag force levels. For some reason one of hese era wo vorices which are creaed during one main shedding period occurs more frequenl on one side han on he oher. B muual ineracion opposiel-signed vorices amalgamae ogeher in he near wake wih he sronges pair of such a cluser of wo vorices having a preference o one of he side of he cenerline, see Sohankar e al. (999). ThisPSfrag causes an asmmeric behavior in D simulaions for Re 3. I should be emphasized ha he degree of asmmer could be dependen Re 3 of compuaional facors such as he compuaional domain size D and he spaial/emporal resoluion. Neverheless, i is quie clear from he 3D simulaions (Sohankar e al. 999) and laboraor eperimens (Okajima 98) ha he flow around a sufficienl Re 5 long clinder is inherenl hree-dimensional a hese Renolds D numbers. Thus, i is epeced ha using D simulaions of inherenl 3D flow leads o a non-accurae predicion of he flow srucures and global quaniies because he srucure of flow in he spanwise direcion is compleel eliminaed Re 3 D Re 5 D Re 5 3D Conclusions Numerical simulaions of unsead D and 3D flow around a square clinder a zero incidence are presened for Re 5. Figure 5. Time-averaged sream lines in D simulaions for Re 3 and Re 5 and a 3D simulaion for Re 5 (A ). 5 Coprigh 999 b ASME

6 C P Re 5 D 3D U B PSfragA D C C P Re 5 3D U D U. U C P Re 5 D D C P 3D C P.5. D A B A Figure 6. Comparison of D and 3D resuls for Re 5. lef) Time-averaged pressure coefficien around he clinder; righ) ime-averaged pressure coefficien and sreamwise veloci a he cenerline. For D flow, i is found ha, for Re 3, he ime-mean flow paerns are no perfecl smmeric wih respec o he oncoming flow. In conras, a smmeric mean flow paern is observed in 3D simulaions. I is believed ha he asmmeric behavior in D simulaions for Re 3 in some wa is relaed o he scenario of period-doublings. The phenomenon of perioddoubling is a dominan feaure in he D simulaions while here are no signs of such period doublings in he 3D-simulaions. Thus, i is epeced ha using D simulaions of inherenl 3D flow leads o a non-accurae predicion of flow srucures and global quaniies because he srucure of he flow in he spanwise direcion is compleel eliminaed. There is a marked and characerisic pulsaion in he force componens of hree dimensional flow for Re 3. These force pulsaions conain characerisic ime periods wih high (HF) and low (LF) levels of forces. The cenral pars of he von Kármán vorices a insans jus afer he have been deached from he clinder appears o be more inense in he LF region han in he HF region. The inensi in his cone is based on he magniude of he vorici vecor and he degree of pressure sucion. The level of non-spanwise vorici is higher when he flow is in a LF region, while he level of spanwise vorici in HF and LF regions is approimael similar. Thus, i is concluded ha he degree of wo dimensionali is higher in HF regions, while he opposie is rue in LF regions. REFERENCES Breuer, M. (998). Large edd simulaion of he subcriical flow pas a circular clinder: numerical and modeling aspecs. In. J. Num. Meh. Fluids 8, 8 3. Najjar, F. M. and S. Balachandar (998). Low-frequenc unseadiness in he wake of a normal fla plae. J. Fluid Mech. 37, 47. Okajima, A. (98). Srouhal numbers of recangular clinders. J. Fluid Mech. 3, Okajima, A. (995). Numerical analsis of he flow around an oscillaing clinder. In P. W. Bearman (Ed.), Proc. 6h In. Conf. Flow-Induced Vibraion, London, UK, April, pp. 7. Balkema, Roerdam. Sohankar, A. (998). Numerical Sud of Laminar, Transiional and Turbulen Flow Pas Recangular Clinders. Ph. D. hesis, Dep. of Thermo and Fluid Dnamics, Chalmers Universi of Technolog, Gohenburg. Sohankar, A., C. Norberg, and L. Davidson (998). Low- Renolds number flow around a square clinder a incidence: Sud of blockage, onse of vore shedding and oule boundar condiion. In. J. Num. Meh. Fluids 6, Sohankar, A., C. Norberg, and L. Davidson (999). Simulaion of hree-dimensional flow around a square clinder a moderae Renolds numbers. Phs. Fluids A, Williamson, C. H. K. (996). Vore dnamics in he clinder wake. Ann. Rev. Fluid Mech. 8, Coprigh 999 b ASME