Hayashi, Tatsuo

Hayashi, Tsutomu

Waka, Ryoji

Yoshino, Fumio

Tottori University Research Result Repository

今すヽThe NuFneFiCal S01utiOn.of F10,w aFOund a RotatittgCircular Cylinder in a UnifOrm FlowbyFumiO YOsHINO与l TSutOmu HAYASHl・,Ryo,i VAKA・,TatsuO HAYASII・■ DepanmentばM Chanical EngheeringlRece?ed August名1984)The NavieF‐StOkes eq,ations were numericany solved On the fIOw aЮu d arotatil■gO■laF Cアlnder in a uniFoFm flow fOr various rotating spetts at theReyo91ds ntlrnber 6f 80,The aerOdylaanic forc“on the cylhdeF,St eam 14es andvorticity distFibutio囃arOulad l vere.deteFinhed,ヽ、IⅢ登彎「ヤ町!Fumio YOSHINO,Tsutomu HAYASHI,Ryoji WAKA and Tatsuo HAYASHI: The NumericalSolution of Flow around a Rotating Circular Cylinder in a Uniform Flow■. IntroductionThe history of the research on a rotating cy■inder is long. The flow a―round a rotating cylinder in a uniform flow has been analysed theoretica■■y byMoore:1)and experimented by Tanaka et a■:2)near the transition Reynolds number.Okajima et a■!3, made a research on a circu■ar and n e■■ipt c cylinder vi―brating circumferentia■■y in a uniform f■ow. From these researches, it is seenthat the f■ow pattern around a cy■inder changes drastica■ly wi h the rotatingspeed. For instancer Okajima et a■:a)fOund that vortex streets are formed evenat the Reynolds number of 40 when the ratio of surface speed of a cy■inder tove■ocity of a uniform f■ow is O.2. At this ■ow Reyno■d number′  the flow issteady and vortex streets are not formed when the cylinder is not rotating.These resu■ts′ however′ are those of ana■ys■8 0f f10w at a relative■y■ow ro―tating speed in a uniform flowr and did reveal neither the mechanism how sucha transition occurs nor the interre■ation betve n the rotation of a cy■inderand the aerodynam■c force on it.This motivated the authors to ca■cu■a e a more genera■cas   that is, af■ow around a rotating cェcu■ar cylinder in a uniform shear f■ow a■though on■ythe result of a cy■inder in a uniform flow is presented here. The finite dif―ference method was made use of to so■ve the Nav■er―Stokes equations at a rela―tive■y ■ow Reynolds number such as 80.aastC. and CdClp and CdpCls  nd CdsCp(η)and cT(η)(r′0)StrT and △Tu  and u∞      cV0Vζ and Vη(x,y)2. NomenC■ature: radius of the cェrcu■ar cy■inder: distance from the center of the cylinder to the stagnationpoint: lift and drag coeffic■ents′ respective■y. ressure components of C. and cd′ reSpectivelyt hear stress components of C. and Cd′ respective■y: coefficients of pressure and shear stress on the cy■indersurfacei cylindrica■ coordinates (Fig。■): the Strouha■ n mber (=2af/Uc): dimensとo less time and time stepr respective■y: velocity′ and that on x ax■s, at infinite■y far upstream′respective■y (Fig.■): rotationa■ speed ratio of the cylinder(=circumferentia■ speed of the cy■ind r/Uc): y component of ve■ocity at infinitely far upstream (Fig.1): ξ and n cOmponents of ve■ocity′ respective■y: rectangular coordinates (Fig.■)ξ(t′η )Reports Of the Faculty Of Engineering,TOttOri University,V01 15: dimension■ess ve―locity gradient(vorticity)。f theun■form shear f■Ow(=a/Uc Du∞/8y): vorticity: orthogona■ curv―■inear coOrdinates(Fig.■)V         = stream functiOnThe symbOls ― and～ On respective■tems above indicate time mean valueand amp■itude of periodic f■uctuationOf a related physica■ Or fie■d quantity,respectively.Fig.■ The coordinate systemsand nOmenc■ature3. Fundamenta■ Equ tiOns and BOundary cOnditionslt is assumed that the f■Ow is twO―dimensiOnal and incOmpressib■e.  Thecoordinate system as indicated in Fig.■ is used to give a magnified view par―ticularly of a cOmp■ic ted f■w near the cylinder.  The re■atiOns Of the co―ordinates (:′n)to (r,o)and (x′y)are as f。■■Ows,= = ■og(r/a)η=0x = aet cos ηy = aet sin η′ =ゞY/伯Uc)′甚Цuc/→′   h = h/afo■10ws,―譜器=竜(拳+辞)+鼎)(■),al:r:u:leti::l三ξ:lii Si=lξeをF71:き/i:」をicとを子/::ざPsi:ithttqIこ                   とthe stream functiOn to vOrticity are represented in (t,n)coordinates′  usingVorticityr stream functiOn, time′ pressure and sca■e factOr nondimensiOnalizedby a and uc  uch asζⅢ= ξ(a/uc)P = P/(pU色)The express■on  are ash2缶+器詣ヒー 詩(審(2)′(3)′Where Re  S the Reynolds number and the superscript tt tO show a dimens■On■essFumiO YOSHINO,Tsutomu HAYASHI,Ryoji VAKA and Tatsuo HAYASHI:The Nume?calSolutiOn of Flow arOund a Rotating Circular Cylinder in a Uniform F10wquantity is dropped for simp■icity hereafter.The no―s■ip condition on the surface of the cy■inder resu■ts in the fol―lowing boundary conditions,VE 器 =0 ′  Vη  ~亜 =V0where the rotational speed ratio vO is the ratio ofthe cy■inder to uc  The uniform shear f■ow infinit ■yexpressed as fo■■ows,(4),circumferen ia■ spe d offar from the cylinder is(5)′dimensェon■ess ve■ocity gradient∞   Uc(■ + ε普 ) V =0where ε is dimensionless vorticity and means aof the un■form shear flow.4. Method of ca■cu■ation                       ,4.l Finite difference equationsThe domain  (Fig.■) was divided into N equal parts in the n direCtiOn(subscript ぅ)starting from the trai■ing dge of the cy■inder and with  theinterva■ S (=2T/N)′ equa■ to that in the n direction′in th   t direction(subscript i)。  A forward difference was used for the time at each grid pointwhile a centra■ difference approximated by the average of the values at thetimes T and T+△T being used for (ξ′η) coordinates.  The finite differenceapproximations of Eqst(2)and (3)are as fol■ows,善(ξこ::T―II,ぅ)==手 猟・‐ 痢+・―・ ri+叱消‐+生痢‐~4Ei洵ド町+ ( `i+.′j tt ζi―■′ぅ + `i′う+■ tt ζi′う-1 ~ 4ξi,ぅ )T+△T}十十 き H(Vi+.,ぅ― Vi―与 j)T(ζttj+1-ζ与 ぅ―■)T十+(Vil′ぅ― Vi―馬 う)T+△T(ζ与 ぅ+ュー`ivう―■)T+△T}―― ((Vi′ぅ+1 - Vi′j-1)T(ζi+.′j ~ ξi―■′j )T 十+ (Vi′jtt. ― Virう―■)Tキ△T(`i+1,j ― ほュ_.′ぅ)T+△T}]Yp = (eζ - 1/eC)Sin ηY=Ψ +V(6)′υ堅::=V→ぅ+甍洋(V■.′ぅ+V■ij+Vュj+.+V主ちとュ_4Vュj+s2h全′ぅとI::T)(7),where αv is a re■axatiOn factor and m is number of iteratioAs.The boundary conditions on the surface of the cy■ind r are appr x■mated byΨ■,コ   0・ 商一缶椎ドЪj~叱河+吼消働―字(戦〆Reports Of the Facuity of Engineering,「Fottori U iversity, 「ヽol 15considering Eq。(4)(:' A■though there exists a problem such as determining aleve■ of the stream functiOn since the equatiOns in infinite doma■n are nu―merェca■y solved in finite doma■n′  he va■ue of V on the surface of the cyl…■nder is made equa■ tO zerO. This wi■l not ntroduce any large error since thedomain of calculation is taken wide enough as r∞>■ooa. Then Eq。(5)for theouter boundary of this domain results in the fol■Owing re■ation ,Two kinds of initial values were usedunder the boundary cOnditions (8)and (9)。around a stil■cy■inder in a unifOrm shear(3)′(9).in order to so■ve Eqs.(6)and (7)The invisc■d outionlⅢ)of the f■ow=■ow′ i.e.,V∞′う = et∞sin ηjζ∞′j=―C+ 号e2ξ∞(. _ cos 2nj )Y主′j=(eζ―ま)sinη+号{e2t(1_cos 2η)+著計―・〕`i′j=―e (10)was emp■oyed as the initial va■ue of the case of e■O and VO業0. In the case  ofe=o and vO＼o′ the resu■t of cOmputatiOn at vO was emp■oyed as the initia■va■ue fOr the computation of the next ■arger vO. Th  resu■t of computation wasconcluded to have converged when the residuals of iteration of respective Vand ξ at a time T became ■ess than ±12 and ±5名 of s3, respectively.4.2 Aerodynamic and stress cOefficientsThe so■utions V and ζ at a time T obtained by iterative computatiOn canbe used to calcu■ate the pressure and shear stress distributiOns on the sur―face of the cy■inder.  Rewriting the fundamental equatiOns by making use ofEqs,(4)on the surface yie■ds the relative pressure at a point of n to that atthe trai■ing edge of the cylinder(η=0)  a dimensiOnless form (divided bypU色/2), cp(η)―Cp(0)=竜お1(指)t=O dη            (・ ・ )・Fumio YOSHINO,Tsutomu HAYASHI,Ryoji WAKASolution of Flo帯arOund a Rotating Circular Cyhnderand Tatsuo HAYASHI:The Numericalin a Uniform FiowThe condition that Eq。(l■)shou■d give zero at n=2■is ot a■ways satisfieぎS'■n rea■ computation due to an error in the calculation.  Hence′ pressuresat η=±2T were averaged to give the value at η=0.The pressure (subscript p)and shear stress (subscript s)components ofthe ■ift and drag coefficients (c. and Cd′ respective■y)are obtained from thefol■Owing integration,The shear stress is easi■y obtained to giveCT価)=竜(艶―準)Clp   ―多 すCp(n)sinn dηC.s =  才ヂ CT(n)cosn dηCl = C■p + ClsThe definition of the Strouha■str   く計三C'   Cdp = ~を すCp(n)cosn dη'   Cds = ~タデ Ct(n)sinn dη′   Cd = Cdp + Cdsnumber str is as follows,(■2).(■3)where f is the frequency of the f■ctuating ■ ft。The computation was made using M-200 of Data Processing center of KyOtOUniversity.5. Resu■ts and DiscussionFigure 2 compares the relation ofthe drag coefficient to the Reynoとdsnumber of a still cy■inder in a uniformf■owァ 。btained by the present calcula―tion′ with the results of ca■cu■ationand experiment by the other research―ers.  The va■ues of the parameters usedfor the present computation are as fol―■ows: S=2T/40, △ =0.O■′αY=・84, r∞≒95aWhen Re=20 and S=2T/60r △T=0.o5, αΨ=■.60′ r∞≒l■■.3a, T≦■00.O When Re=80.The figure shows a good agreement of thepresent ca■culation with those by theothers(3',(5',16)which  imp■ies  that  thepresent ca■cul tion can we■■ be applied(■4)′Ex「℃rimental data◇:T?Цon t働Numerical solutions□i Oka)ma et a119)△:Taniura ct al(5,①:Present cal20   40 6080ReFig.2 Dependency of drag coefficienton the Reyno■ds number??Reports Of the Faculty of Engineering, TOttori University, Vol. 15tO the case of a rotating cylinder■owing computations were made withRe=80。Figures 3 and 4 shOw the time―dependent ■if and drag coeffic■ents′ re―spective■y, 。f the rotating cy■inder in a uniform f■。w′ in which the rota―tional speed ratio vO was changed from o t。 -5.5. That vOくo indiCates that theCy■inder rotates in the counter arrOwdirec iOn of Fig.1.  The figure shOwsthat va■ues Of c. and cd atta■n thOse of a statiOnary state for T>50, fromin u ifOrm and unifOrm shear f■ows.The fo■―the va■ues of the parameters given abOve atFig.5 Relation of lift and dragcoefficients tO rotationaユspeed ratiOwhich the time mean values cl and cd are the averiages Of cl and Cd for T>5(spectively. Figure 3 indicates that value of c. increases with  lvOl  andfOr T>50′re――  ―――マ ヤー  ヤ■ …ⅢⅢヤ…‐ЧOしo wとしよI  IYOI  ctllureaches a maximum of about 6。8 at vO=-4. The amp■itude of the periodic fluc―tuatiOn ■s tOO sma■■ to distinguish from the T ax■s when vO o thOugh c. surelyfluctuates pefioriCa■・y at this vO, ュn―Fig。3 VariatiOn Of ■iftwith timecreases with lvOI′ and becOmes ■丘rgerat vO=―と than ■o times that at vO=o。The fluctuatiOn′ howeverr dampens with:u:u::leiviingLiasihf!者 |とR!tigdtttS::e::r::neiきiSξ :::idモぇfξ:子nI!:: |とR!::rfどどie:Ob::::d :rq:1 :手: :d::i3ミshown in Fig。4,  the va■uc of cd tOOincreases with lvOI′ and the rate Ofincrease is particu■ar■y ■arge near v0Re=80 , c=o●:ε′ ●:ε`, c:εl.△:σど  △:εtrp  ム:σと,0.81』0.60.40.20U   50   80   10oTof drag cOefficient80   10oTcoefficient?。?、?．?、???、??Fig。4 variatiOnwith timeRe=80 ′ ε=oRe=80 , c=o_      VO=-5.5|と とlj°6FumiO YOSHINO,Tsutomu HAYASHI,Ryoji WAKA and Tatsuo HAYASHI:The NumericalSolution Of Flow around a Rotating Circular Cylinder in a Uniform Flow=-3. The amp■itude of the fluctuation of Cd has the same tendency as that ofC., but is smal■er. Fig!re 5 indicates Cl, Cd t°gether with he pressuFe andshear stress components of them against vO,  and confirms what was mentionedabove about c. and Cd. The figure a■so indicates that the pressure componentsC.p and cdp account′ broad■y speaking,  for 90名o  c. and cdP respectively′ atany vO.The stream ■ines and equ■―vorticity lines in the f■ow fie■d ar shown inFigs.6 and 7P respective■y. The figures (a)and (b)correspond to the cases of(VO=―■, T=96.0)and (VO=-4)′ respectively. The va■ue of C. is about minimumwhen T=96.O at VO=―■ (Figs。(a)). Figure 6 shows that the wake deflects down―wards of the figure aョlvOl increases.  This downward def■ection is consideredto be due to the fact that the tota■ vortici y in he wake becomes positive byaddition of pOsitive vortic■ty g nera ed by the rotation of the cylinder. Forexamp■e′ the positive vorticity near the lower surface of the cylinder is car―ried toward the upper sider in the rotational direction and into the wake byconvection and diffusion (Figs.7(a) and (b)).  The vorticity is in fact posi―tive anywhere in the wake at vO=-4.  This positive vorticity in the wake in―duces a downward ve■ocity which in turn def■ects the wake downwards and gen―(a)ρセ=8(I 1/。=-1,c=0、7｀=060(b)P=三80‰=-4,c‐0.T=1000Fig.6 Stream ■ines aroundrotating cy■inderthe Equエーvorticity ■ nes aroundthe rotating cy■der(a)Pr=801/a‐―l ●=O T=960(b〕 P,=8().1/0=-4,f=0.T ttilK 0Fig。7Reports Of the Facuity Of Elagineering,「rottori University, V01. 15erates an induced drag On the cy■indertO inCFeaSe Cd with ivOI (different inorigin from that of a finite span wing).:!aを。i:とd」どを啓 を:C::attSfi,ddととtと。 そRとincrease of induced drag. The increase°f Cd With lVOl  is observed in thecase of a cy■inder rOtating at a re■a―tive■y higher speed in a high Reynoldsnumber flow12, Figure 8 indicates thedistance tO the  upstream  stagnationpo■nt from the center Of the cy■inderobtained frOm stream ■ine distributiOn.The stagnatiOn poェnt is nOt On the sur―face Of the cy■inder when vO業o andmoves farther away from the cylinderas lvOl increases.  The angular posi―tion Of the point stays near O≒220。irrespective of vO as seen from Figs.6(a)and (b), which is cOnsidered tO bedue to cance■■ tion of thechange of O due to the increase of circu■ation withlVOl by that due to the increase Of intensity of dOwnwash.????Fig.8 Distance from the stagnationpoint tO the center Of thecy■inderReferences(■)Moore, D.W.′ 」. F■uid Mech., vOl.2,No.6(2)Tanaka′ H. and Nagano, s., Trans. 」apan6. conclusionsNumerica■ s。■utions fOr Re=80 were obtained on the f■ows around a circu―lar cy■inder with the rotationa■ speed ratiO vO of o～_5.5 in a unifOrm f■Ow(C=0). The fo■■Owing are conc■usions.(1)C. increases with lvOI′ reaC es a maximum (≒6.8)at vO=-4, and decreasesthereaftero  cd increases with lvOI・  Th  pressure compOnents c.p and Cdp ac―Cij争1年i°ie彗:i:s°毛x Ciaこと二u写狛。:ei;ii:よ子euと is nearュy zero for lvOI≧2.牝shows the same tendency as c. th°ugh the former is sma■■er than the latter.(3)The wake def■ects dOwnwards due to the rotatiOn (vOく 0)of the cy■inder′that is, due tO the induced veloc■ty in such a way that the def■ection is■arger when lvOl is larger.  The stagnation point moves away from the cy■inderaS IVOl increases. The angu■ar position of the point approaches, first,rapid■yO1220°′ but stays about there even if lvOl increases further.(■957-8), p.54■.Soc. Mech. Engrs, (in 」apanese),10  FtuniO YOSHINO,Tsutomu HAYASHI,Rycii WAKA and Tatsuo HAYASHI:The NumeFiCalSolution Of Flow―arotlnd a Rotating Circttlar Cylinder in a Uniform FlowVo■.38′ NO.310 (■972-6), p.■343(31, Okajimar A.′ et a■.P Trans. 」apan Soc. Мe h. Engrls。 (in Japanese)′Vo■.371 No.304 (■97■-12)r p.2300.(4)Frenkiel′ F.N. anS Temp■e, c.(ed.), App■id Mathematics and Mechanics,Vo■。8 (■964), pi■2P Acade4ic prees.(5)Tamura, H.r et a■.P Trans. Japan Soco Mech. Engrs, (in 」apanese),Voと。46′ No.494 (■980-4), p.555‐。(6)TrittOnT D`」.ュ J・F■uid Moch`P Vol.6, NO.4 (■954-■1)′ p.547.

The Numerical Solution of Flow around a Rotating Circular Cylinder in a Uniform Flow

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The Numerical Solution of Flow around a Rotating Circular Cylinder in a Uniform Flow

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