A STUDY ON THE RESPONSES OF FREE SHEAR LAYERS UNDER EXTERNAL EXCITATIONS

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1 ISTP-6, 5, PRAGE 6 TH INTERNATIONAL SYMPOSIM ON TRANSPORT PHENOMENA A STDY ON THE RESPONSES OF FREE SHEAR LAYERS NDER EXTERNAL EXCITATIONS Hsu, Chng-Chiang Dpartmnt of Airraft Enginring, Air For Institut of Thnology Kaohsiung, Taiwan 8, R. O. C. Kywords: vortx mthod, mixing layr, ontinuous priodi transvrs foring, puls b i Abstrat Vortx modl is usd to study th rsponss of ford mixing layrs. In th prsnt study, th mployd xtrnal xiting approahs inlud ontinuous priodi transvrs foring and puls foring. In th ontinuous xiting ass, th ontinuous sinusoidal transvrs prturbations ar imposd on th origin of mixing layr. Dtaild dynami vortial struturs, sptrums and flow proprtis ar throughout studid and ompard with xprimnts. In th sptrum of vloity flutuation, th dominat frqunis ar xitd on ( f ) and its first harmoni ( f ).This phnomnon is onsistnt with th rsults of Wisbrot and Wygnanski [6]. Diffrnt saturatd loations ar found at low and high amplitud foring ass. As foring amplitud is high, th saturation loation at foring frquny and its sond harmoni is at RX normalizd loation, X S = =.. But, in th low amplitud foring, th saturation loations RX ar about at normalizd loation, X S = =.85. As saturatd loation apparing at high amplitud foring, th spatial volutions of momntum thiknss ar ompard with xprimnts, and gt good oinidn. RX RX Intrstingly, as X s = and Y s = ar takn as non-dimnsional spial oordinat, similar bhavior ar obtaind among flow filds undr diffrnt foring frqunis. In th puls foring as, only on sinusoidal prturbation is posd on th origin of mixing layr. At th bginning, prturbation amplifis and onvts stram-wisly. Thn, prturbd wav slf-grows to b a vortx strutur and sprads th disturbans to nighboring vortx struturs. Puls prturbation transporting prosss undr onvtiv instabl mhanism ar shown larly in th prsnt study.. Introdution Th xistn and rol of larg-sal ohrnt struturs in fr shar flows dvlopmnt has bn onfirmd by an ovrwhlming numbr of xprimntal obsrvations [,, 3, 4]. Th dvloping flow in both axis-symmtri jt and two-dimnsional shar layrs is dominatd by larg-sal wavlik ohrnt struturs. Ths obsrvations indiat that suh a strutur starts as an instability wav on th shar layr, th amplitud of whih rahs a maximum and thn days gradually downstram. Ths struturs an also mrg with thir nighbors as th shar layr dvlops downstram. Mungal and Hollingsworth [5] analyzd th photograph of th turbulnt plum from th ground tst of a Titan IV rokt motor. In thir analysis, turbulnt motions of many sals an b obsrvd, from ddis and bulgs omparabl in siz to th width of th plum, to th smallst sals th amra an b rsolvd. It mans that th ohrnt struturs ar th intrinsi faturs of turbulnt flow. Globally, th ohrnt strutur an b viw as a larg-sal organizd vortial strutur within turbulnt shar layr. Th intration of vortx struturs plays a lading rol in th dvlopmnt of turbulnt fr shar layrs. Studying th volution of th vortiity fild is

2 Chng-Chiang Hsu th ky to undrstanding th flow fild. Th dominat mhanism of fr shar layr is th Klvin-Hlmhotz invisid instability. Fr shar flow originat from som kind of surfa upstram, b it a nozzl, a moving body, or a splittr plat. As w hav ralizd, th dvlopmnt of fr shar layrs ar snsitiv to th prturbation in th initial stag. Flow ontrol is on of th popular fluid dynamis rsarh topis. Studying th ffts of xitation of shar layr is important for undrstanding flow ontrol [6, 7]. Although, manipulating th dvlopmnt of fr shar layr by artifiial xitation has mad a grat progrss, how dos th xtrnal disturban transport in th flow fild? is still a problm. So, studis on th transportation of xitation in th flow fild ar worthwhil to b prformd and thy ar positiv for ralizing th transition pross. Thr ar two distint mods about disturban transporting in th flow fild. On is th onvtiv instability mod and th othr is th absolut instability mod [8]. For fr shar layrs, as R = <. 34, th + dominant instabl mhanism is onvtiv. Th xtrnal disturban amplifis and onvts downstram. Th flow systm bhavs as a nois amplifir. Th vortx mthod has bn widly usd to study various typs of vortial flow, suh as mixing layrs, wak, jt, osillating wing and sparatd flow of blund-body [9,, ]. In th simulation of fr shar layr, Inou [, 3] introdud a vortx modl to study th turbulnt mixing layr. In his simulation, th profils of th man, flutuation vloitis and th Rynolds strss of unford-mixing-layr show similar trnds. So, th haoti haratrs of point vortis volving in th fr shar layrs ould b analogy to th ral on. Inou [4] also usd his vortx modl to simulat th dvlopmnt of fr shar layrs undr bi-mod xitation and sussfully aught th paring and oalsn of ohrnt vortial struturs. Du to th suss of vortx simulation on intrations of vortial struturs in fr shar layrs, th vortx simulating thniqu is suitabl for rsarhing on th volution of flow fild whih is dominatd by vortial struturs. In th prsnt papr, a vortx modl will b introdud to study th spatial bhaviors of mixing layr, whih is undr transvrs priodi foring at diffrnt frqunis and amplituds. Th gnral bhaviors of ford-mixing-layr and th haoti proprtis of flow fild will b throughout studid. And, th volution of puls sin disturban on th initial pla is studid.. Vortx modl for turbulnt mixing layr As shown in Fig., th flow fild of th turbulnt mixing layr is dividd into two vloity strams by a splittr plat. Th vloity diffrn btwn th top and bottom of th splittr is dnotd by. If w modl th splittr plat by a vortx sht of strngth pr unit lngth, thn =, whr and ar th rsptiv flow vloitis abov and blow th splittr plat. Th onvtion spd,, of vortis in th - D turbulnt mixing layr is onstant and is givn by ( ) + =. Fig. : Vortx modl for mixing layr Th point of origin in th omputational oordinat systm is takn to b at th nd of th splittr plat. Aftr laving th origin, all

3 A Study on th Rsponss of Fr Shar Layrs undr Extrnal Exitations vortis ar assumd to mov undr th ombind influn of th potntial fild introdud by th individual vortis, and th onvtiv vloity. Thrfor, th ontrol paramtrs for th dvlopmnt of th mixing layr ar and th vloity ratio r =. In th ass undr xtrnal foring, ah nw disrt vortx that appars at th origin is subjtd to a singl mod transvrs displamnt Y f = Asin( π ft) t. A is th foring amplitud. f is th foring frquny, and t =.5. In ordr to mak lar omparisons btwn th prsnt alulations and th xprimnts [6], ths numrial xprimnts ar prformd with r =.6, and = 3.. On travling in th mixing layr, ah vortx attains a vloity V n = ( n, Vn ). In trms of th omplx potntial W (Z), th following rlationship an b writtn: dw n ivn = dz Z = X + iy i( ) W ( Z) = ln( Z ξ ) dξ π + N k= Z up iγj ln( Z Z j ) + Z π Th first and sond trms of th right-hand sid ar th omplx potntials introdud by th splittr plat and th othr fr vortis, rsptivly; n dnots th n-th point vortx, ( X, Y ) th vortx position, Γ th strngth of irulation of th jth fr point vortx, N th numbr of point vortis, and Z th position of th upstram dg of th splittr plat givn in th omplx form. Th sond ordr Rung- Kutta shm is adoptd to updat th vortis trajtory. j up 3. Rsults and Disussion 3-. singl mod foring Although th vortx mthod is popular usd in turbulnt shar flow study, th quantitativ omparison of th numrial and xprimntal rsults is inadquat. This is baus propr physial sals ar not usd. In this papr, vortx simulation is mployd to onstrut a lar pitur of turbulnt mixing layrs ford by a singl mod at a varity of foring frqunis and amplituds. Qualitativ and quantitativ omparisons ar mad btwn th alulatd rsults and th last xprimnts [6]. Sin th bhavior of a mixing layr ford by a singl mod is mainly inflund by th amplitud and frquny of th imposd vloity disturban, th sltion of th foring frquny is th first topi for disussion. In th prsnt study, w intnd to onstrut a univrsal bhavior of singl mod ford mixing layrs, so th imposd frquny is somwhat arbitrary. In ordr to obtain high-rsolution vortx trajtoris in th data rording rgion, rfrn foring frquny f is hosn to b.483 aftr arful invstigation into th alulatd rsults Amplituds ffts In ordr to undrstand th global bhavior of ford mixing layrs, th strak-lins of vortx struturs for diffrnt foring amplituds ar alulatd. Som of th rsults at T=4 ar shown in Fig. In th low foring amplitud ass, th vortx struturs bhav randomly, and no apparntly rgular vortis ar formd. In th high foring amplitud ass, rgular and lar vortial struturs ar sn in th flow fild. Th distan btwn two adjant vortis is almost qual to on wav lngth ( = ) and similar vortx bhaviors an b f obsrvd at th sam stram-wis loation, 3 whil foring amplitud (A) xds V. 8 3

4 Chng-Chiang Hsu Fig. : Th foring amplitud fft on straklins. For onfirming th ffts of foring amplitud on th mixing layr dvlopmnt, th frquny sptra and ( u,v ) tim trajtoris undr diffrnt foring lvl ar shown in Fig. 3. In Fig. 3, w ould find that th frquny sptra undr low foring lvl ar with no apparntly dominating frquny in th flow fild and th ( u,v ) tim trajtoris bhav haoti. As th foring amplitud inrass, th phas-lokd ( u,v ) tim trajtoris appars and th frquny sptra hav two dominatd frqunis f and f. It says that th ohrnt flow struturs of mixing-layr undr a singl mod transvrs foring ar mainly manipulatd by th foring frquny ant its doubl. This phnomnon is onsistnt with th rsults of Wisbrot and Wygnanski [6] Similar bhaviors of high lvl foring r shar layrs Fig.4 is th instantanous kinti nrgy (E(t)) fild for diffrnt foring frqunis at high lvl amplitud. Whr E( t) = [( ( t) )] + [ V ( t)] and f r =.483. Th ratios of foring transvrs displaing amplituds to foring wav lngths ar maintaind to b onstant and V t qual to. In th figur, w ould find that r th largr foring frquny f ompanis th smallr rgular vortial struturs in th flow fild. Although, th sizs of vortial struturs undr diffrnt foring frquny ar dfinit, th instantanous kinti nrgy filds ar with similar spatial volutions. If th lngth sal, p is mployd to normaliz th spatial R oordinats of flow fild. Th non-dimnsional rsults ar shown in fig. 5. Th instantanous kinti nrgy filds for diffrnt foring frquny at high lvl amplitud in fig.4 ar almost th sam. Not only vortial struturs ar with th sam non-dimnsional siz, but also loations of th spatial volution. So, is p R th only lngth sal of ohrnt struturs in th flow fild undr high lvl singl mod foring. Fig. 3: Frquny sptra and ( u,v ) tim trajtoris undr diffrnt foring lvl (a) A=.35V (b) A=.5V () A=V. Fig. 4: Th instantanous kinti nrgy fild for diffrnt foring frquny at high lvl amplitud. 4

5 A Study on th Rsponss of Fr Shar Layrs undr Extrnal Exitations Fig. 5: th non-dimnsional instantanous kinti nrgy fild for diffrnt foring frquny at high lvl amplitud. Dirt Fourir transfr thniqu is mployd to invstigat th omponnt nrgy volution of stram-wis flutuation u' and transvrs flutuation v' at spifi frqunis. Comparing th alulatd rsults, it is found that th omponnt nrgis of f and f ar muh largr than thos of th othr frqunis. Ths rsults agr with thos from th frquny sptra analysis. Thus, th dominant frqunis in th flow fild ar f and f. Wisbrot and Wygnanski hav obsrvd th sam rsult in thir xprimnts [6]. Th volutions of th intgrals of omponnt nrgis u' and v' at xitation frquny ( ) and its sond harmoni ( f ) ar shown in Fig. 6. In th figur, w ould find that th volutions of th intgrals of omponnt nrgis u' and v' at f and f ar with th sam trnds. And, th maximum valus for u' and v' at th two diffrnt frqunis ar almost apparing at th sam stram-wis loation. Th similar phnomna ar also obsrvd in xprimntal rsults [6]. So, th frquny rspons of f and f ar ondutd by th sam ohrnt strutur in th flow fild. f / u f / v f / u f / v f.5 A= V RX/ RX/ RX/ RX/ Fig. 6: Transvrs intgrals of (a) stram-wis vloity flutuation at f (b) stramwis vloity flutuation at () transvrs vloity flutuation at f (d) transvrs vloity flutuation at. Th stram-wis loation, that th intgrals of omponnt nrgis u ' and v' at f and f rah maximum valus, ould b sn as th position for vortx strutur omplt formation [7] and will b alld th saturatd loation in th followings. Th saturatd loations at various foring lvl ar shown in Fig. 7. If th foring amplitud inrass, th saturatd loation movs up-stram. Exitingly, as foring lvl blow som valu, th saturatd loation also boms invariant and is onsistnt with that from xprimnts at low foring lvl [7]. And, as foring lvl abov som valu, th saturatd loation boms invariant and is onsistnt with that from xprimnts at high foring lvl [5,6]. R X/ Hsiao's rsults[7] Wisbrot and Wygnanski's rsults[6] o prsnt rsults - - foring amplitud Fig. 7: Th saturatd loations at various foring lvl. f f 5

6 Chng-Chiang Hsu Fig. 8 dpits th stram-wis variations of momntum thiknss,θ, for ford mixing layr at high amplitud foring lvl. And, θ is dfind as = θ dy. Th momntum thiknss is a dfinition of th loal width of th mixing layr. Th volution of a singl mod ford mixing layr is govrnd by a onvtiv vortx strutur, so th transvrs haratristi lngth sal,, and th stram-wis lngth sal, R, ar hosn in th normalization. Th omparison of th numrial volutions of non-dimnsional momntum thiknss in th normalizd stramwis dirtion is also shown in fig. 6. It an b sn that th urvs ar narly oinidnt and losly agr with th xprimntal data [6]. Aording th spatial volution of momntum thiknss and th strak-lin, w an roughly lassify th volution pross of a singl mod ford mixing layr into th following thr stags: vortx formation, vortx onntration, and vortx brakdown to haoti small ddis. In th vortx formation pross, th momntum thiknss inrass linarly with th inrasing distan for th initiation, whil it drass nonlinarly with distan in th onntration pross. As th haoti motion apparing, th onntration pross nds and th mixing layr grows again and sprads with distan inras. θ /.5..5 Prsnt Exp. [6] A= V RX/ Fig 8: Stram-wis volutions of momntum thiknss for high amplitud xitd mixing layr. 3-. Puls disturban transporting in fr shar layrs In th puls foring as, only on sinusoidal prturbation is posd on th origin of mixing layr. Whn th flow fild rahs statistially stabl, w tak som tim T o as rfrn point. Th following transvrs disturban Y = Asin( π f T )[ u( T To ) u( T To T )] is imposd on ah nw disrt vortx that appars at th origin. Whr, u is unit stp funtion and T =. Th flow fild undr f puls sin disturban is alld xitd flow fild and its vloity fild is shown as ( ( T ), V ( T )). W tak th undisturbd flow p p fild as ( n ( T ), Vn ( T )). Thn, th disturban fild introdud by th puls foring is V ) = p,v ). ( p, p ( n p V n In th first, th flutuant kinti nrgy filds introdud by puls foring ar trad to study th volution of xtrnal disturban. Th flutuant kinti nrgy filds introdud by puls sin foring at various tim ar shown in fig. 9. At th bginning, prturbation amplifis and onvts stram-wisly. Thn, prturbd wav slf-grows to b a vortx strutur and sprads th disturbans to nighboring vortx struturs. Th introdud disturbans from th puls wav mak largr influn in th downstram than in th upstram. So, th vortial struturs in th flow fild not only sustain, but also amplify th xtrnal disturbans. Th volutions of xtrnal puls sin disturban ould b dividd into two stags:. flow fild rivs th puls flutuation and forms a wav-typ vortial strutur.. wav-typ vortial strutur grows to b onntratd round-typ and sprads disturban to surroundings. In aording to th transportation of puls disturban in fig. 9, w ould asily raliz that th sprading of introdud disturban is ontrolld by two mhanisms. Ths two mhanisms ar disturban flowing downstram by onvtion and th flutuant 6

7 A Study on th Rsponss of Fr Shar Layrs undr Extrnal Exitations vloitis drivd by th xitd vortial struturs. Th fft of onvtiv mhanism just maks th disturban to travl downstram-wis, but th sprading flutuations introdud by xitd vortial struturs xist in anywhr and bom smallr with largr away distan. Disturbd signals transporting to upstram is th omptition by th two mhanisms. But, disturbd signals transporting to down-stram is th oopration by th two ons. As th flowing ffts ar dominatd in th transportation of disturban, th instabl mhanism in th flow fild is onvtiv instability. Othrwis, th instabl mhanism is absolut instability. Th volutions of puls sin disturban in tmporal dirtion ar shown in fig. and typial onvtiv instabl wavpakt ould b obsrvd. Fig. 9: Th flutuant kinti nrgy filds introdud by puls sin foring at various tim. 4. Conluding rmarking Vortx simulation thniqu is usd to invstigat th dynami bhaviors of ford turbulnt mixing layrs. In th study of th dvlopmnt of singl-mod and puls ford mixing layrs undr diffrnt foring amplituds in th initiation, dtaild dynami vortial struturs and flow proprtis ar throughout invstigatd and ompard with xprimnts. In th prsnt study, th following onlusions ar obtaind: () As th foring amplitud inrasing, th haoti motions in th flow fild ar supprssd and th volution of vortx struturs will b phas-lokd by th imposd foring frquny. () ndr th singl-mod transvrs foring, th rsponsiv frqunis in th flow fild ar th foring frquny ( ) and its sond harmoni ( f ). This phnomnon is onsistnt with th xprimntal rsults [6]. (3) Th bhaviors of ford mixing layr in th low and high foring lvl ar with grat diffrn. In th ass of th low foring lvl, th saturation loation for th nrgis at foring frquny and its sond harmoni ar about at normalizd RX loation, X S = =.85. But for th high lvl foring ass, th saturation RX loation is about at X S = =.. f =5 =4 disturbd wavpakt (4) Th vortial struturs in fr shar layrs not only sustain, but also amplify th xtrnal disturbans. =3 = = X Fig. : Th volutions of puls sin disturban in tmporal dirtion. 7

8 Chng-Chiang Hsu Rfrns [] Browand, F. K. and Widman, P. D., Larg sals in th dvloping mixing layr, J. Fluid Mh., Vol. 74, pp. 7-44, 976. [] Brown, F. K. and Roshko, A., On dnsity ffts and larg strutur in turbulnt mixing layrs, J. Fluid Mh., Vol. 64, pp , 983. [3] Crow, S. C. and Champagn, F. H., Ordrly strutur in jt turbuln, J. Fluid Mh., Vol. 48, pp , 97. [4] Winant, C. D. and Browand, F. K., Vortx pairing th mhanism of turbulnt mixing-layr growth at modrat Rynolds numbr, J. Fluid Mh., Vol. 63, pp , 974. [5] Mungal, M. G. and Hollingsworth D., Organizd motion in a vry high Rynolds numbr jt, Phys. Fluids A, pp , 989. [6] Gad-l-Hak, Flow ontrol-th futur, Journal of Airraft, Vol. 38, No. 3, pp. 4-48,. [7] Grnblatt, D. and Wygnanski, I. J., Th ontrol of flow sparation by priodi xitation, Progrss in Arospa Sin, Vol. 36, pp ,. [8] Hurr P., Opn shar flow instabilitis, In prsptivs in Fluid Dynamis A Colltiv Introdution to Currnt Rsarh, d. By G. K. Bathlor, H. K. Moffatt, M. G. Worstr, Cambridg nivrsity prss,.. [9] Hou, T. Y. and Lowngrub, J., Convrgn of th point vortx mthod for 3-b Eulr- Equations, Commun. Pru Appl. Math., vol.43,99, pp [] Cottt, G. H. and Koumoutsakos, P. D., Vortx mthods-thory and pratis, Cambridg nivvsity prss,. [] Akbari, M H. and Pri,S. J., Simulation of dynamis stall for a NACA airfoil using a vortx Mthod, Journal of Fluids and Struturs, Vol. 7,3,pp [] Inou, O., Vortx simulation of turbulnt mixing layr, AIAA J., V ol. 3, pp [3] Inou, O. and Lonard, A., Vortx simulation of ford/unford mixing layrs, AIAA papr No , 987. [4] Inou, O., Doubl-frquny foring on spatial growing mixing layr, J. Fluid Mh., Vol. 34, pp , 99. [5] Wygnanski, I. And Ptrsn, R. A., Cohrnt motion in xitd fr shar flows,: AIAA J., Vol. 5, No., pp. -3, 987. [6] Wisbrot, I. And Wygnanski, I., On ohrnt struturs in a highly xitd mixing layr, J. Fluid Mh., Vol. 95, pp , 988. [7] Huang, J. M. and Hsiao, F. B., On th mod dvlopmnt in th dvloping rgion of a plan jt, Physis of Fluids, Vol., No. 7, PP ,