REVISIÓN REVISTA MEXICANA DE FíSICA 4. (1) 1-7

Vortex packets and the structure 01' wall turbulence

RJ. Adrian1 ami S. Balaehandar1

Del'artment nfTheoreticnl am/ Al'l'fied Mec/ul1Iics, Universily of Illinois ot Ur/Jat/a-Champaign2/6,6/80/ T,,{bOl Lab, Urbal/a, /L

e.mllil: 1 r-adrian@uiuc.edu; 1s-/Jalll@uiuc.edu

Recihido el 9 dc septiembre de 1999; aceprado el 24 de septiembre de 1999

FEBRERO 2000

Experimental evidence in low lo moderate Reynolds llumber wal! tlows shows thal hairpin vortiees (including asymmelric inelincd vorticcs)OCl:urin groups that propagate as a wholc wilh relatively slow dispcrsion. Thesc groups. or "packets". grow upwards from the buffer layerlo atxmt onc.half of Ihe thickness of the boundary layer. Direcr numerical simulalions of the growlh of a single hairpin eddy in a c1eanbackground now show how thesc packets may he formed in Ihc ncar wal! (Iow Reynolds numocr) region by a viscous autogeneralionIllechanism Ihat is simi lar in many regards to lhe mechanism proposcd by Smith {.(a/. (Phi/O.f. TnlTl.f. R. SoCo LondofJ, Ser. A 336 (1991) 131].Thc orgnnization of hnirpin eddics inlo packets nnd the internctiolls 01'those packets is an imponanl fe3lure of wal! turhulencc that providesa ne\.','paradigm by which many scemingly unconnected aspccls 01" wal! turbulcnce can be explained. These indude lhe inordinately largeamnunl of slreamwisc kinetic cnergy that resides in ver)' long ~Ireamwise wavelcngths. the occurrence of multiple Q2 evenls pcr turbulentbllrst. Ihe formation of new strcamwise vorticity. and the characterislic angles of inclinalion of fmnls. The nutogeneration process may nlsoexplain the fonnalion of long quasi-slreamwise vortices in the buffer layer nnd the associatcd low-spced streaks.

K"Yl1'ords: Vortex packets; low Reynolds number

Evidencia experimental en la región de baja a moderada del número de Reynolds en nujos de pared muestra que vórtices (incluyendo vórticesasimétricos inclinados) ocurren en grupos que se propagan como un rodo con dispersión relativnmenle baja. Estos grupos, o "paquetes",crecen a partir de la c"pn basal n cerca de la mitnd del espesor de la capa limítrofe. Simulaciones numéricas direclas del crecimiento deun solo remolino en un nujo circundante limpio mueslrnn como estos pnqueles pueden formarse en la región cercana a una pared (bajomímero de Reynolds) por un mecanismo de autogenemción viscosa que es similar en muchos aspectos al mecanismo propuesto por Smilh el

al. fPhi/os. Tram. R. SOCo Lo"do", Ser. A 336 (1991) 131). La organización de remolinos en paquctes y In interacción de esos pnquetes esun hecho importante de turbulencia de pnredes que proveé un lluevo paradigma por el cunl muchos aspectos nparenlemente no coneCIa.dosde turbulencia de paredes pueden ser explicados. Éstos incluyen la gran canlidad no ordinaria de energía cinética de torrentes que reside enlongitudes de onda de IOrrentes grandes, la ocurrencia dt.:múltiples eventos Q2 por ocurrencia de lurhulencia.la formación de nueva vorlicidadtorrencial. y los ángulos característicos de inclinación de frentes. El proceso de :lutogencrnción puede explicar también la formación degrandes vórtices quasi-torrenciales en la capa basal y las líneas asoci:ldas de baja velocidad.

lJeSCrll'10res: Paquetes de vórtices; bajo número de Reynolds

rAes, 47.32.-y; 47.27.-i

1. Introduction

llairpill shaped vortices are Ihought by Ill<lny lo he a centralfeature of turhulcnl wall laycrs. An idcalized hairpin vorlex(onSiSls al' a pair uf eounter.rotating quasi-streamwise vor-tices that are lillel! upwards along lhe downstrcam dircctionand iJ hairpin head that connccls lO the quasi-slreamwise vor.tices al their downstream cnds. as shown in Fig. l. I-Iair-pin vorlices ohscrvcd in expcrimcnts and compulalions sel-d(lm possess perfcct spanwisc syml11ctry; more oncn Ihey areasymlllclric. left- 01" right-handcd cane-Iikc vortices. which(onsist of a hcad. a ncck and one dominant quasi-strcalllwiseIcg 11. 2J. Nevcrtheless, a piclurc of the turhulent wall layeras a distribution 01" hairpin vortices provides a rcasonablc mo-del for many of the llow fcatures {hat have heen ohserved anddO(lIll1Cnled in the past (c.f Ref. 4 for example).

In the streamwisc wall nonnal :ry plane. the velocily sig-naturc of a hairpin vortex is (haraClerized hy a circular vortex(ore and a slrong olltward pumping of low mOlllenlum fluid.

L _low Speed Streak

FIGURE l. Schemalic 01'a h:lirpin vortex and associated flow fieldpropcrties lhnr form the signaturc of a hairpin on the xy planc.

This hairpin vortex signalure is relatively inscnsitivc lo Ihedegree of aSYllllllelry 01" lhe hairpin. Recenl PIV measurc-Illents [.1-Gj in a lurhulellt houndary Jayer Qver a range ofReynoltls numhers clearly show numcrous hairpin vortex sig-natures wilhin the bOllllllary layer. providing strong evidenccthat Ihe turbulcnl \Vall layer at modcratc Reynolds numhcrsis thickly populnlcd with hairpin vortices.

In lile cxpcrimcntallllcasurcmcnts ciled ahove Ihe hairpinvorticcs wcrc oftcn obscrved lo occur one hchind the olhcr as

2 RJ AIlRIAN ANO S. RALACHANOAR

a Irain in lhe strcall1wisc dircction fonning a cohcrcnt groupor packct nf hairpin vorticcs. Although lhe shapcs and si/es01' Ihe packct varicd, lhe lendenc)' lo fOfl11 group 01'hairpinswas ohscrvcu al aH insl<lnccs. Thc strcamwise coherence 01'Ihe I1car-wall hairpin VOrliccs pcrsistcd cven al highcr Rcy-nolds l1ulllhcrs. Thc hairrin vorticcs willlín a packet \VeTe oh-scrvctl 10 work coopcralivcly passing low-spccd fluid fromIhe downstrcam mus! vortcx lO ils upstrcam ncighbor ami so011ovcr several hairpin v<mices 10 forrn a low-spccd slrcak oflcn,glh signiticanliy longcr timo a single hairpin vorlCX. As a•.csult, transport propcrtics, such as Rcynolds strcsscs, nI' Ihepackcl could signiflcantly cxceed a simple sum 01' lhe contri.hUlinll frolll each individual hairpin within lhe packet. ThusIhe arrangclllcnt nI' hairpin vortices into packets with definitcdistribution 01' sizc, age amI spatial separation has a po(entia-lIy large elfcet 011the overall rnOlllentum and he:lt transportfrolll the \vall. For instance, signifkant drag reduetion can heanlicipaled hy disturhing lhe streamwisc alignmenl of h:i.ir-pins within:l packel.

The pn:sent paper will focus on the following issues: a) Ahricf revie\l,' 01' all experimental cvidellce supporting the exi~-tenee nI' hairpins: b) a rcview of computational reslIlIs on the:llIlOgcllcralion 01' new hairrin vorticcs: e) explanalion on thehasis 01' lhc hairpin packel paradigm 01' experimental ohser.valiolls thar eannol he explained hy single hairpin Illodcls; d)

(he illlplieations of hailpins oeeurring as paekels. insteau ofhcing randolllly scattered lhroughout lhe houndary layer.

Hairpin \'ortices have a long hislury. Theodorscn 171 wasthe tirsl lo suggcst the importance 01' hairpin-Iypc vorticcs inturhulenl wall layers. His original proposal consistcd of hor-scshoe vortices with ol1lega-shapcd head and ncck regioll lhateXII:nded spanwise to form spanwisc nJrtex legs.

The visualizalion expcriments of Hcad and Bandyopadh-yay [.~[ infcrred Ihat slrelched vorlex loops (or hairrin vOrli-ces) inclilled al ahmH .IJo are a major componenl 01' the (ur-hu1enl waJl layer. There are two major pieccs 01' cxperimen-tal c\'idcl1ce showing that hairpins oceur in groups. First. [heside \'ie,," slllokc now visualizations 01' Head and Bandyop-hadh)'a)' [~Irevcaleu large scale slruelures, inclincd at a cha-raclcrislic :lIlgle 01' ahout ~WO, lhat lIl:lrked rhe otiler edge 01'Ihe turhlllenl houndary laycr. They infcrred that Ihis slruclu-re \Vas a group of individual hairpin vorticcs. cach stretchingfmm lhe \\'alllo the oUler cdgc orlhe houndary Jayer. Fig. 2a.(See ,,1st) lilc model 01' Bandyopadhyay 19]) The angle nI' cn-velopc orille group was arrroximately 18°. Second, hascd nnexpl.'rimenlal ohservations of hydrogcn huhhle pattcrn in theIlcar-,,"all rcgion 01' a lo\\' Reynolds nlllllhcr turhlllelll houn-dary layer. Smi!h (101 reponed that Ihe rcsults are conSislen!wilh al Icast three hairpin vortices I'orming wilh aliglllllcntalong Ihe strcanl\'.'ise direction (near-wall rcgion 01' Fig. 2h).Suhscquelll work hy Smith and coworkers uemonslraled Iheronnalioll of sequences 01' hairpins hy a slationary hemisphe-ril.al hUlllp 011the wall [11. 12) anu hy impulsive injection oflluid [131 in a laminar boundary layer. More recent direclnu-11lcric;¡J silllulations hy ZhOll. Adrian, al1li Balaehandar 11.11and Zholl. Adrian. Balachandar, ano Kendall [15] on the evo.

la)

Ih)

FI{;URE 2. Hairpin padet moocb. (a) Head ano Bandyopadh-Y<lY[81: (b) Hicrarc.::hy 01' packcts a:-.obscrved in Rcf. 6.

lution of a single initial hairpin vorlcx in a unidirectional me-an lurhulent ehannel l10w have identificu (he mcchanistic dc-lails hchind the ilutogcneration of sccondary hairpin vonicesin Ihe ncar-wall region Icading lo (he forrnalion 01' a hair-pin paeket. A significant outCOIllC of thcse simulations is Iha!new hairpins are forllled bOlh on lhe upstream and downslre-am sities of lhe initial hairpin resulting in a hall'~diamond ortcnt-like hairpin packc!. This shapc for thc hairpin packet isnlllsislenl wilh recenl experimental measureIllcnts tú] se he-malically rcprescnled in Fig. 2b. Thc eompulations also showthal the aulogeneration proccss is rohust and occurs more rc-adi!y in the casc uf an asymmctric initial hairpin. cvcntuallyforllling a slaggcrcd array of one.sided hairpin vorlices. Thisis in accordanee \l,.'ith Ihe predominantly ollc-sided hairpinsnOlcd by GUC/CIIIlCCand Choi jI] and Rohinsoll [21.

2. Experimental eviclence ",r packets

The velocity signaturc 01' a h:lirpin vortex in a spanwise-wallnormal y= planc passing lhrough the quasi-strcamwisc vor-tex legs is characterizcd hy a pair 01' eounter rotating vorticespumping fluid away fmm rhe \\'all. On the olher hand. in astreanl\vise-wall normal .l',f) planc rhe hairpin vorlex is cha-ractcrized hy a) ti slrong oulward pUlllping 01"low morncntumIluid 011 the in-board side 01' the quasi-slrcamwise vorticcs.This quadranl-(wo (Q2) now encounters the high-speeJ frcc-slrcam alld forms a shear layer which is inclincd 450 lo thl~horizontal; and h) eJosed/spiraling slrc3mlines corrcsponuinglo Ihe circular vortex core nI" Ihe hairpin head in a frame nI'rcferellce travcling downstream with (he hairpin vorlcx.

The schemalicshowninFig.lclearly illuslrates Ihe typi-cal hairpin vorlcx sigllalurc in Ihl' .17.11 planc. provided it passeshClween lhe quasi~slreall1\\'ise Icgs. Howcver. it musl he ernp-hasi/cd lhat lhis hairpin signalure is rclalively insensitive tolhe degrcc of as)'mllletry 01' lhe hairpin. The hairpin vortex

Nn'. Mn. rú. -t6 ( I ) <20(0) I~7

VORTEX PACKETS ANll TIIE STRUCTURE OF WALL TURBULENCE 3

FIGURE 3. PIV mcasurcrncnts of instantancous vclocity lIclds in(he .l'!J plane 01"a turbulcnt boundary layer. (a) Rf'8 ;;; 1015: (h)RI'¡1;;; 770¿;.

signalurc will (!len allow rcasonably accurate itlcntilicatiollof hairpin vorliccs from quantitativc IllCasurclllcnt nI" lhe vc-iOl:ily licltl within lhe lurbulcnt wal! layef. Jt should he cmp-hasizcd Ihal lhe Q2 vcctors cxhihit a maxilllul11 sOl11cwhcrc!lelow lhe vortcx hcad and that jt is charactcristic of Ihe COIll.hincd induction associatcd with Ihe proximity of Ihe vortcxhcad ano Icgs. This peak in Ihe Q2 vclocity providcs a cIcarevidence for lhe existence 01' a ¡hree-dimensional vortcx, asa (\l/o-dimensional vortex, such as a vortcx Iinc, is in generalinenrahlc 01' gencrnting such a local velocity maxilllulTl.

Figure 3a shows a velocity vector plot in Ihe streamwisc-wnll normal .,.u planc obtained from high resolution PIV lIIC-asurement of a zero prcssurc gradicnt houndary laycr wilhH('n = 1015 IG]. A conslant convection velocity of Uf' =O.DI Uoo (Uro is lhc frcc strearn velocity) has heell sllhtraclcdfmm Ihc streamwise velocity in order to bring out the packetof hairpins, whose heads are clearly itlentiflcJ in the figure.The hairpins \\'ithin the packel are ohserved lo extcnd ffOmthe wall up lo U/;; ~ 0.5 to O.G, where 6 ddllles Ihe edgcof thc houndary layer. Frorn lhe conveclion velocily il can beinferred that lhe hairpin packet propagalcs aloog lhe stream.wise direclion al 0.0 to 0.9 Uoc' Thc PIV measurcments cm'era wide slreamwise range of up to 3.06 and over this extendedsfrcamwise rangc Tomkins [6] observed differenl geomelricsllapes ror the hairpin packet inclllding: lIninlerrupled strc-amwise growth, sawtooth shapc, half-Jialllond or lcnl shape¡¡nd constanl hcight. Fig. 3a might fit the descriplion 01' a pac-kel nI" constanl height. In Ihe case a ramp-Iike envclope rOl"thl.":hairpin packet, as in uninterrupted growth, sawtooth andhalf-dialllond shape, the mean angle 01' the ramp is ohservedlo oe aonut 15°, which is consistenl with earlier ohservationsof Head and 13andyopadhyay [8). Furthcnnore, over lhe Rey-nolds numher (Reo) rangc form 1000 lo 7705, the packel isohserved to conlain from abom el hairpins to as rnany as ninehairpins.

The coopcrative aClion 01' the slreamwise aligned hairpinswithin lhe packet can he clearly ohser"ed in Fig. 3a as Ihelone 01' strong negative vclocity thal Hes belo\\' Ihe hairpinvorlex heads. This ,-olle of low mOIllcntum also exisls at hig-her Reynolds nurnhcrs. as sho\\'1l in Fig. 3h foe Re= = 770;:;hut is reslricted closcr lo lhe wall in terrns of OUler units. Athigher Reynolds nlllllbcr lTlllltiple zones nI' almost unifonnrnornentulll can he obscrvcd 14. ;,). Thc interface hetw('en anytwo adjacent lones is lllarked hy a sequencc oí' vortex cores,which contrihulcs lo Ihl.":ncar uniforlll velocity jump acrossthelll. These vOrlex corcs can he idcntitled as the heads ofl1carly streanl\\'ise aligned hairpin vorlices and thus Ihe inter-face hctween tlle dillerenl unifonn momcnlum zones can heinterpreled as the enliclop nf a hicrarchical hairpin packet.

The uniform rnOlllellllllll 10l1e ohserved close to Ihe wal!in Figs. 3a and 3h cxtends over more than one lhousand vis-cous wall unils along Ihe streamwise direclion. \Vhile il re-sembles the low speed strcak lhal occurs in the huffer layer. itis a ne\V and quile distincl phenolllenon as il occurs well aho-ve Ihe huffer layer and e.'l((ends rast the loganihmic layer [5].Howcver the 10\V 1ll0lllelll1l1ll lones are associated wilh nc-ar wall streaks, since lhe hairpin vortices anJ hairpin paCkl.":lSare hclicvcd lo grow out of the near-wall streaks, at kast inlhe tirst zOlle closest lo Ihe wall. The packets are also nol lheconvcnlional hulges, hUI Ihe largest packets in Ihe hierarchyof hairpin packels mighl cause Ihe hulgcs. The organizalioll01' the hulges appears 10 he less coherent Ihan the packcts,sllggesting that the packcls inleract in a complex way.

Thc freqllent occurrcnce nf vorlex packets in Ihe cxpcrimentsrequires cxplanalion. The kernel exreriments of Acarla!" amiSmilh [11. 121 l'oll(}\wd lhe process of continuous general ion01' a train al' hairpin vorlices hehind a hemispherical hllmp ina laminar houndary layer. SlIhseqllcnt experiments hy lIai-dari anJ Smilh Il:~] (,'onsidered Ihe more relevant casI.": 01asinglc hairpin ,"orlex gl.":nl.":ralcdhy an impulsive injection offluid into the houndary Iayer ami Ihe suhsequent general ionof secondar)' hairpins to fOl"ln a packet. This process (Jf ¡ni-lial hairpin formal ion l'rolll fluid injection and its suhsequentevolulion and I"onnation 01"addilional vortical structures \Vasstudied nurnerically hy Singcr and Joslin [16J. Based on Iheexperimenlal ohservalions and inviscid computations. Smilhet al. [1 7J ollered a conceplual inviscid modcl for Ihe me-chanislll hy which Ilew hairpins can he naturally gel1cralcdout 01"a single hairpin. Whilc Ihe ahove cited investigalionsaH pertain lo Ihe fOl"lnalion 01" ti hairpin packet in a laminarhoul1dary layer /lo\\', lhe experimental evidencc [5. G, S, 10]suggesls Ihat once a hairpin is I"ormed hy a localized, lowrnOlllenlUIll agcncy !leal' Ihe \\'all, il can, under a range of cir-('Ulllstan(TS, pro,,:cL'd lo gcncralC a scqucncc 01"new vorliceslúrming a cohcrcnt packct of hairpin vortices. A fundamen-lally similar general lllechanisl11 lIlay occur in lurhulenl llo\\'as wel!. hui if so. il l1eeds lo he understood in that conlext.

3. Generatioll 01'hairpin \'ortex packets

3

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yló04

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R('I'. Mex. ¡:ú. 4f) (1) (2000) 1-7

4 R.l ADRIAN AND S. BALACHANDAR

Thc numcrical simulations of lhe growth 01' a singlehairpin vorlcx in lhe background al' a low Rcynolds num-he!" unidircclional mean lurbulent channcl now 114,151 of-fer additional insight iolo lhe mechanisms, which could cx-plain lhe formution of new hairpin vorticcs, and their spa-tial arrangcmcnt ¡nto packets. Thesc studics diffcr fram lhework describcd ahoye in that lhe ¡nirial tlcld was a viscous,hairpin vorlcx-Iikc slructurc tha! was extracted from the fu-11 two-raint turbulcnt corrclation tensor of a ReT :: 180channel now direc! nurncrical simulation of Kim, Muserand Moio [18] hy lhe proccss of stochastic cstimation (c1Rcf. 19), rathcr Ihan created hy external forcing. By appro-priatcly choosing lhc event vector in the stochastic eslil1lationproccss. the slructllre 01' lhe initial vorlcx can be varied over awide rangc in a systel1lalic l1lanner to represenl form an idc-alized symmetric hairpin to a more realislic asyml1lelric orone.sided hairpin.

Zhou el al. [14,15] considered both symmclric and asym-mctric Q2 cvent vcctors given by a second quadranl velo-city [u = nJ1 - (j'l'ltm• V = 0')1 -/J2vm, W = /3(u;n +V~Jl/2] speciflcd at a single paint within the channel. Here(11m, vllJ is chosen to maximize the producl llmVm weightedby the probabililY density 01' its occurrence ¡(um, Vm) andthereby miJximize lhe contrihution to mean Reynolds shearstress. The factor O: is a scaling I~lctor. which determines thevortica! strength ol' lhe initial struclure rclative to the vorlicityof lhe mean Ilow. The factor 13 is the asymmetry para meter;tJ = ° corresponds lo a symmetric event and results in aninitia! symlllctric hairpin. For a rcpresentative Q2 evcnl, Ihestochastic estimation proccss guarantees thal the initial fieldpossesscd lhe correct Icngth scales, shape, and vorlicity dis-trihution 01' a typical Q2 struclurc. Thus we helieve that theinitial hairpin structure is more representalive of the hairpinvorticcs ohscrved in real turbulence.

Further. lhe background flow in which lhe hairpin vor-lex is cmheddcd is chosen to he a unidirectional llow obtai-ned 1'1'0111 the mean streamwise velocity profile of ReT = 180channcl now dirccl numcrical simulation [18], rather than aPoiseuille Ilow. This difference at f1rsl glance mighl seem nolso imporlanl; however, in the case of the lurhulent mean Ilowprotlle the mean shear is predominanlly contained close tolhe chanllel walls wilhin approximately 25% 01' the chanllelhall" heighl (y+ < 45), whereas in the case ol' aplane Poi-seuil1e now the mean shear extends over the enlire channel.As a rcsult the peak shear in the case ol' lhe mean turbulentproflle is abolll factor fOllf grealer than that of the Poiseui-He Ilow. A delicate halance hetween lhe self-induced vclocitylhat tentls to cur! up lhe vorlex and lhe inlluence 01' the meanshear, which tcnds lo stretch lhe hairpin vortex, governs lhecvolution of lhe hairpin vortex. Thus differenccs in the meanbackground Ilow will have a strong innuence on the dynamics01"the initial hairpin vortex and lhe formation of the hairpinpackel. In particular. the impacl on lhe spatial and temporalscalcs 01"the resulting struclure is Iikely to be strong.

In a real turhulent boundary layer the formalion and evo-lutioll oí" lhe hairpin packet occurs in the presence of other

hairpin packels, vortical dehris, ouler layer perturhations andso on. These disturbanccs are in deed collectively responsi.b!e for the mean turbulent profile, bul of course in a time.averaged sense. Thc rational for using the unidirectional me-an turbulenl profile as the background now is to account forthe inl1uence 01"lhe olher turhulent structures al least in a sla-tistical sense, ami at the S<lme lime maintain the hairpin evo-lution simple ami cOlllrolled so as lo he ahle to follow il inclose detail withoul any clutler from other vortical slructures.

The simulations were performed at a Reynolds number01"Re, = 180 in a hox 01"streamwise (x) wall normal (y)and spanwise (z) size 41r. 2 ami 4n /3 respectively. The iso-surface of the imaginary parl of the eigenvalue of lhe velocilygradient lensor [15] is used to visualize vortices in lhe presenlstudy.

Time evolulion of bolh symmelric and asymmetric inilialstructures were followed in delail with a direcl Ilumerical si-mulalioll. In hoth cases lhe qUJsi-streamwise vortices quickly!in away fmm the boundary due lo mutual induction and thelift-up is lhe slrnngcst at the downslream end. Simultaneou-sly a shear layer forms where the Q2 velocily encounters fhemean now. Spanwise vorticity associated with lhis shear layerquick!y rolls-up and forms a cOlllpact spanwise vortex IOC3-ted jusI ahoye the downstream end of lhe quasi-strealllwisevorticcs. By ¡+ ~ 25 the rol!ed-up spanwisc vortex viscou-sly connecls with the lirted quasi-streamwise vortices lO forma hairpin structure. The geolllelry of this vortex resemblesin appearallce Ihe hairpin vortices ohserved in many experi-menls. Fig. 4a shows the hairpin-like vortex al t+ ~ 27 thatresulted from an asyrnmetric inilia! struclure wilh y;t = 30,(l = 2 and (3 = 0.5. Apart from the hairpin vortex. markedby the head and an asymmetric pair of quasi-streamwise vor-lices on the upstream side 01"the hairpin head. a pair of vorti-cal tongues can be seen on the downstrcam side of the head.These vortical longues will later develop info new downstre-am hairpin vorlices lhat will form parl of the hairpin packet.

The head orthe primary hairpin vortex continues 10 lin up¡nto a near vertical orientation and also grows wider evolvinginlo the characteristic !l-shape. Simuhaneously lhe quasi.slrearnwise legs slretch and form a kink as a resu!t of mutualinduction and interaclion wilh the head. A shear layer forrnsahove the kink and quickly intensifies. Suhsequently the ups-trearn portion 01"the quasi-strearnwise !egs detach from lheprimary hairpin at the kink and merge wilh the rolled up she-al' layer to form a sccondary hairpin which remains distinctI"mm lhe primary hairpin.

This process of new hairpin generation conlinues alongsimilar lines bolh upstream ami downstrcam 01"lhe primaryhairpin resulting in a hairpin packet. which is shown inrigs. 4c ami 4d al ¡+ ~ 144. The envelope oí' lhe packet ofhairpin vortices has a tenl Iikc appearancc with an approxima-te angle 01" l(JO upstrcam and 7° downslream oí' the primaryvortex. The velocity signature of the computed hairpin packet(jualitalivcly compares wilh lhe PIV measurernents similar toIhose shown in Fig. 3. However discrepancies exist since thecomputed hairpins are scparatcd along the streamwise direc-

Re\!. Mex. ¡':ú. 46 (I) (2000) 1-7

VORTEX PACKETS ANO THE STRUCTURE OF WALL TURBULENCE s

(a)

One must appeal to a coherent packet of streamwisc aligncdhairpin vorticcs in oreJer to consistently cxplanation othcr ex-perimelllal ami computational obscrvations of the pas!. Bc-low wc lisl all lhe importanl features that can be explaincdon lhe hasis that the houndmy layer is made up 01' hairpinpackels.

5. Hairpin ¡Jacket paradi~m

4. Sin~le hairpin paradi~m

Bere we will consider signifkant experimental and compula-tional obscrvations Ihal are consistent with lhe picture 01' lhelurhulent houndary layer populated with a forest 01' symme-trie and canc-lype hairpins.

l. The often ohserved quasi-streamwisc vorlices, hairpinshape and horscshoc (omega) shaped vortices. onc-sided caBc lypc vorticcs are aH part 01' lhe sume entityat variolls stages 01' lhcir evolulion and with diffcrcnldcgrce 01' asyrnmctry.

2. The range 01' hairpin vortcx ungle frum 15°_75° (wilh45° hcing more typical) is consistent with the mnge 01'angles observecl from the quasi-strcamwise vortices tothe hairpin heacl. FurlhermOfc. the tiIt or angle of lhehairpin head is a slrong function of its location; the he-ad takes a near verlical orienlation in the ouler regions01' tlle houndary layer, while near the wall it takcs amore conventional 45° angle.

3. The experimcntally ohscrved inclined internal shcar la-yers can he explaincd on lhe basis of lhe Q2 nuid pum-ped hy lhe hairpin encountering lhe free slrearn llow.

(ion aoout 200-LiSO viscolls wall units. whilc in Ihe cxperi-I11cntsIhe dislancc bctwcen hairpins is ooscfvcd lo be around100-150 viscous units.

Thc cffcet 01'asymmctry on lhe cvolulion of hairpin vor-tices is also invcstigatcd by systcmatically ¡ncrcasing lhe (3paramctcr fmm O. Il is ohscrvcd thal lhe proccss by whichnew hairpins are autogcncratcJ rcmaios qualitatively Ihe Sil-

me as in lhe symmctric casco Thus lhe formation 01' a co-hefenl hairpin packet is a robusl rncchanisrn active undcr awide range of conditions. Bowcver, the asymrnetric hairpinsare one-sided and the packcl cnnsisls 01' a streamwise train 01'alternating leh and right handed one-sidcd hairpins that arestaggered along the span. Both the symmelric and asymme-trie cases display a threshold behavior; hairpins only abo vea ccrtain tllrcshold inilial amplitude rcsult in lhe autogenera-lion 01' ne\V hairpins and fonnalion 01' a hairpin packet. Thethreshold amplilUde is observed lo decreasc with the dcgreenI' asymmelry suggesling Ihal asyrnrnelrie hairpins are morelikely lo aulogcnerate and fmm hairpin packets lhan the idc-:Ilizco sYllllllclrie (mes. This rcsult is in IOlal agreernent withlhe experimental ami computational observation 01' predomi-nuntly one-sided hairpins.

(e)

J60 ~o no

(b)

o

(d)

~-~ISO. I

. ~'---... ..__c~_~__Jo

FIGURE 4. Hairpin packet that cvolvcs from an asymmctric ¡nilialdisturhance; (a) Pcrspectivc view, t+ = 27; (h) Pcrspcctive vicw.f+ = 36; (e) Pcrspcclive view, f+ = 144; (d) Tor vicw, t+ = 144:(prom Zhou el al. (15».

ReI'. Mex. rú. ~(,(1) (2000) 1-7

RJ. ADRIAN AND S_ BALACHANDAR

5.1. l..ow~speedstreaks

l. Thc cooperalivc Q2 pUlllping 01' the near-wall lluidby the streamwise aligned hairpin vonices cxplain lhever)' long (more than 1000 waH unils) low-speed stre-aks.

2. The spanwise staggering nI' the one-sided hairpins inthe aSYlllllletrie case explain the onen observed span-\\'ise jogging of the low-speed steaks.

3. Reccnl Il1casuremenls by Meinharl and AJrian [.1] in-tlicalc that the 10w-spceJ streaks are not Iimited to thebuffer-Iayer. bULextcnd into lhe log-Iayer as wel!. SLre-alll\vise alignco hairpin vonices Lhat extend inlD thelog-Iayer. Lhrough thcir cOlllbincd indueed velocity canexplain lhe long log-Iayer Inw-specd streaks.

4. The packcI paradigrrl also explains Lhe long tai! ohser-\'ed in Lhe ll-corre1<uion (20,21].

5.2. Burst proccss

l. There is similarity 10 the POO results 01' Sirovich [22],in whieh Ihe 1110stimportant moJes are the strearnwiseindependent moJes that hug c10se Lo lhe walls. The-se modcs correspond lo the streamwise aligned quasi-sLrcamwise Icgs of the hairpins amI Lhe resulting longlow-specd strcaks. We interpret the propagating Illodesthat trigger the onsel of hursl-like aClivily to he lile pro-jections of lhe hairpin heads.

2. The streall1wise arrangeIllent 01' the hairpin vortices isin agreelllcnt with lhe measurements of [23-251, \'iherethe near-wall nurst process was onserved lo he typiea-lIy maJe IIp of multiple Q2 e\/ents. In this sense lhehairpin packet conslilutes a hurst.

3. Thc auj.H:cncy 01"hairpins within lhe packel reslIlts instrong internal shear layers \\'here the inducell úown-Ilo\\" (Q.I) "rom the upstream vortex heal! meels theImv-specd lIpllow (Q2) inducell hy the downstreamvortex. This Q2/Q4 slagnalion point Ilow provides eon-vineing evidence for lhe VITA signaturc, unen lIscd toidentify hurst process.

4. The cOllventional view of the hurst process as wavyoscillatioll 01' the low-speed streaks ano Q2 eruptionat lhe ellu also fits inlo lhe framework of hairpin pac-keL. \Vhile the low-speed (Q2) pumping 01' lhe llowns-lream hairpins are sOJ1lewhat rnitigaled hy (he corres-ponding Q-l induced vclocity 01' Iheir upSlreí.lm llcigh-hors. the low-speed pumping 01' Lhe chronologicallylasl upslrCalll-1ll0st hairpin rcmains unopposed and infacl is rcinforeed hy its uuwnstrcam neighhors.

5.J. Structurc ami s)'mmetr.y

l. Preference ror cane.shaped one-sitletl hairpins in theexperimcntal 111 ami computational reslllts 121 can hee.\plaincll on the hasis that lhe autogeneralion of new

hairpins is much more rohust and readily occurringwhen they are asymmelrie.

2. Often ohserved ¡ni ine alignment of lhe streamwise VOf-lices is tlue to the quasi-streamwise legs nI' lhe stream-wise aligned hairpins \vilhin lhe packet.

3. In lhe evenl 01"strong asymmelry, only Lhe altcrnali-ve qUJsi-strealllwise legs are significanl-the packet ismalle up of allcrnating 5equence of righl and len han-deJ canes. Ir one confines atlenlion to only {he nearwaH region (y+ < 60) lhe hehavior of the packet isvery similar to that proposcd hy [2G).

SA. Hieran.hy of packets

l. Recl.:nt P1V IllCaSUrelllents IGJ over a rangc of Rey-nolds 11lll11hersshow velocily signature thal is consis-tenL wilh a piclllre 01' lhe turblllent bOllndary layer madeup of hicrarchy of hairpin packets Ihat lravel downstre-am cohercntly.

2. \Ve helieve that the hydrogen huhhle measurernenlsnf 1101 and the smoke visualizations of Head andI3adyopadhyay 18) extraclthe near-wall and outermostgroups 01"hairpins. rcspectively. This suggesls the pre-sence ofhairpin packcls righl from Ihe near-wall regionto lhe oulermost reaches of Lhe houndary layer.

3. Thc (Ju(er scalc slructures such as the hacks and hu)-ges are consisLent wiLh a largc-scale hairpin packel thalextends inlo IhL' outer layer.

4. Thc scaling 01"ditTercnt quantiLies on inner, ouler antlmixcd scales is compatihle with lhe notion nf a hierar-chy nI"hairpin packcts.

5. The smallesl near-wall hairpin packet providcs adeqlla.te cxplanation ror lhe bulTer-layer nehavior. The !les.ting 01' smaller packets within larger OBes, which arewilhin even largerones, and 50 on may explain the log-layer.

6. Tlle hicrarchy 01"packcts can explain Rohinson's 00-scrvation lhat the huffer-region contains predominan-11)' quasi-strealllwisc yorticcs. the log-region containshOlh quasi-strealllwise vorlices and hairpin heads andthe wakl.:-region contains Illoslly hairpin heads.

(,. implicatinns nI' packet paradigm

Implicatiolls nf lhe hairpin packet paradigm for predic.tion. conlrol and 1ll0Jcling 01" lurhlllenl boundary layer are\'cry many. The superposilion 01' lhe induced lIelds of the~trcalll\Vise-aligned hairpins resull in a slrong Q2 \/cloeilydul.: (o the "solcnoid elfce!". This cooperative aetion Icads lo apotcntially very largc inCfease in mornenlulTl and he al trans-pOrl I"rom Lhe wall. For example, lhe total Reynolds stressof N incohcrelltly dislrihutcd (spatially uncorrclaled) hairpinvortices is silllply N Limes the individual conLrihution arisingfrom the Q2 pUlllping 01' cach 01' the hairpin, On the other

RI'I'. M('x. F/I"- 4(,(1)(2000) 1-7

VORTEX PACKETS AND THE STRUCTURE OF WALL TURBULENCE 7

extreme, ir lhe N hairpins aTe pcrfcctly ovcrlaid (perfect co-rrclation) lhe rcsulting total Reynolds stress is N'l times lheindividual contrihution. Thc Reynolds stress contribution 01'a real hairpin rackc! is likcly lo scalc somcwhere in bctwccnand dcpenJ critically 011 lhe strcngth and spatial scalcs 01' lhehairpin packct.

Thc ahoye ohscrvalion suggesls tha! houndary layer 1110.

deis that are hased on a hierarchy of randol1l distribution ofhairpins as in Pcrry el al. [3] have many of lhe ingredients nc-ccssary rOl" an accuratc dcscription uf lhe turbulent houndaryIayer, cxccpt for lhe coherence of hairpins within a packct. Inti similar manncr lhe phcnomcnological semi-Markov moJelorlhe turhulent houndary layer [2i] does not fully accounL rorlhe spalial coherence within the hairpin packet. By incorpo.raling this important feature lhe prcdietive capability of theselllodcls can he signifkantly advanced.

Hairpin packets also provide exciting possibilities ror thecontrol 01' hounJary ¡ayer turhulcnec and henec Jrag mana-gClllenl. Qne simple conceptual approach to drag reductionis lo Jclay the formation 01' secondary ami suhscqucnt vorti-ces and therehy increase lhe strcamwisc spaeing hctwecn the

1. YG. Gllczenncc and W.c. Choi. in Proc. ZOrtlll P. Zaric Me-lI/orilJ/ 11l/l'r/l(ltiOlu¡fSemillar 0/1 Neor Wafl Turbll/ellce. [Hoy/988. cditcJ hy SJ. Kline and N.H. Afgan. (Hemisphcre. NcwYork. 19H9) p. 420.

2. S.K. Rohinson.Awl. Re\'. Fluid Mech. 23(1991)601.

3. A.E. Pcrry. S. Henhcst. and M.S. Chongo 1. Fluid Mech. 165(19H6) 163.

4. C.D. Meinhart ami RJ. Adrian. Phys. Fluids 7 (1996) 694.

5. C.D. Meinhart and R.l Adrian. (manuscript in prcparation.199H).

G. C.D. Tomkins. M.S. Thesis. Univcrsity 01' IlIinois. Urbmw. Illi-flois. 1998.

7. T Thcodorsen, in Proc. Seco"d Mid~vestem Con! (~(Fluid AJe-c1ul/lics, (Ohio State Universiry, Columbus, Ohio. 1(52) p. 1.

8. ~1.R. Head and P. Bandyopadhyay. 1. Fluid Mi'ch. 107 (19R1)297.

9. P. Bandyopadhyay. Phys. FllIids 23 (1980) 2326.

10. c.R. Smith. "A synthcsizcd modcl 01' the near-wall hehaviorin turhulent boundary layers", in Proc. Eighth S.\'I1II'.on Tur.hlllence edited by G.K. Pattcrson anJ lK. Zakin, (Universityof !\lissollri-Rolla. Dcpt. 01' Chem. Engng .• Rolla. Missouri.19H4)

11. M.S. Acarlar anJ C.R. Smith, 1. Fll/id Mech. 175 (!987a) l.

12. 7\l.S. AcarIar and c.R. Smilh. 1. Fluid Mech. 175 (19X7h) 43.

13. AH. Haidari. and eR. Smith. J. Fll/id Meeli. 277 (1994) 135.

14. J. Zhou, RJ. Aurian. and S. Balaehandar, Phys. FIlIids8 (1996)

2HK

hairpins within lhe packet. The resulting reduced cooperationamong the hairpins \ViII signifkantly cut down the Reynoldsshear stress ami the Illomentum transfer. Rccent results bySirovich and coworkers [221 indicmc that reasonablc drag re-duction can he achieveJ hy ranJom distribution 01' roughncsselernents.

They hypolhesized Ihat the roughness elements disturhthe propagating moJes and therehy control the bursting pro-cess. An allcrnative, hut eertainly related. viewpoint will belhat Ihe roughness elelllents advcrsely affeet the hairpin ge-neration process and miligale lhe internal eoherenee withinpacket~.

Experimenlal and complltational evidence [11-16] sug-gests th,H Illany types of low Illomcntum events al lhe wa-11can crcate a sllceession 01' hairpins. This along with [22]can he taken to suggest thm by appropriate surfaee excita-tion, for examplc hy momenllllll addition to overeorne a locallow 1ll0menlUlll cvcnt thc hairpin formation proecss can becontrollcd to reduce drago Sueh exciting possibilities must hepurslled in the fUlure.

15. J. ZhOll. R.J. Adrian. S. Balachanoar. amI TM. Kendall. 1. FluidMcch. (1998) (sublllitteo).

1G. B.A. Singer and R.D. Joslin. Phys. Fluids 6 (1994) 3724.

lío C.R. Smilh, lD.A. Walker, A.H. lIaidari. and U. Sobrun. Phi-los. T,.(//!.\". R. Soco LOI/(/Oll,Ser. A 336 (1991) 131.

18. l Kim. P. Moin. amI R.D. Moser, J. Fluid Mech. 177 (1987)133.

19. RJ. Adrian. "Stochastic ESlimation 01' the Structurc of Turhu-lent Ficlos". in Eddy S/fl/ewre Idelllijicariofl. edile by lP. Bon-Ilet. (Springer. Bcrlin. 1996) p. 145.

20. A.A. Townscnd. TI/{' .\.trl/Cll/reo( /urlmlellt sl1('arjloll'. (Cam-bridge University Prcss. 1976).

21. 1lL. GranL 1. Fluid Mec"- 4 (1958) 149.

22. L. Sirovich. in Sd( Sus/al"i"g Mecha/lislIIs 01 Wall Tlfrbu/ell-CC. edited by R.L. Panton, (Compulalionnl Mechanics Publica-tions. Southnmpton. UK) p. 333.

23. D.G. Bogard amI W.G. Tiedcrman. 1. FllIid Mecll 162 (1986)3H9.

24. TS. Luchik ami W.G. Ticderman. J. Fluid Mech. 174 (1987)529.

25. F. Tarou, ex!,- Flllitl.\.19 (1995) 112.

26. J. ]cong and F. Ilussain. 1. Fluid Mech. 285 (1995) 69.

27. lC.S. rvlcng. in Se~( SIIS((Iillillg Mec/ulfIisms 01 \Vall Tllrbl/len.('('. cdited by R.L. Panlon. (Computational Mechanics Puhlica-lion ••.Southampton. UK) p. 201.

Rn'. Mee tú. 4(, (\) (20<Xl) 1-7