f"..- .. -.---~-
NI9C!H Tf? IOS-3
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
REPORT 1053
INVESTIGATION OF TURBULENT FL TWO-DIMENSIONAL CHANNEL
By JOHN LAUFER
REPRODUCED BY NATIONAL TECHNICAL INFORMATION SERVICE u. S. DEPARTMENT OF COMMERCE SPRINGFiElD, VA. 22161 ~r-Y~~------~ - .. I ~I
, . .
I REPORT 1053 I I
INVESTIGATION OF TURBULENT FLOW IN A TWO-DIMENSIONAL CHANNEL
By JOR LA FER
California In titute of Technology
",. . ';...... /' " --
• . ational Advisory COlnmittee for eronautlcs
H Ntdqnarters, 1724 F treft N Il'., Washingto n 25, n. ('.
Created by act of Congrcs approved M arch 3, 1915, fOL" the sup ervision and direction of th e cientific study of the problem ot flight (U. . Code, title 50, ec. 151) . Its membership wa increa ed from 12 to 15 by act approved M arch 2, 1929 , and to 17 by act approved :" 1ay 25 , 194 The members are appointed by the Presiden t, and erve a snch without compensat,ion .
J ERo~m C. H 'SAK 1, R, Sc. D., Ma achu~etts fnstitute of T eC hnology, Chainllan
ALEX AN DER \ VI" I' ~ I O l m, 'c. D., 'ecretary,. mithsonian I nstitution, Vice Chairman
DI~TL8V W . B RONK, PH. D., President, John H opkins Univer H ON. D o ALD W. NYROP, Chairman, C ivil A ronautic Board. s ity, Do ALD L . P UT'I', Major General, United States Ail' Force, J OR ' H. CASSADY, Vice Admiral, nited tate Navy, D eputy Acting D eputy Chief of taff (Development). Chief of Naval Operations. ARTHUR E. RAYMO n, . D ., Vice President, Engineering, EDWARD U. CONDON , PH . D., Director, 1 ational Bureau of D ougla Aircraft Co., Inc. Standards. FRANCIS W. REICHELDERFER, Sc. D., hief, United taies llON. THOMAS W. . D AVIS, Assistant ecretary of Commerce. Weather Bureau. J AME S H . D OOL IT'L' I"E, 'c. D ., Vi ce Pre ident, 'hE'll Oil o. R. ;-1. H AZEN, B. S., Director of Engineeri ng, Alii on Division, GORDO T P. AVILLE, Major General, nit,ed tates Air Force, General i\ l otors Corp. Deputy Chi f of taff- D evelopment. WILLI AM Ll'l"l'LEWOOD , i\I. E ., Vice President, Engineering, HON. W AL'l'ER G. WHI'nIAN, Chairman, Research and D evelop American Airline , Inc. ment Board. D epartmen t of D efense. THEODORE C. L ONNQUE '1', R eal' Admiral, United States Navy, THEODORE P. WRrG m " Sc. D ., Vice Pre irl e nt [or Re earc h, D eputy and A s i tant Chief of t he Bureau of Aeronautics. Cornell Un i versi ty.
H UG f! L. DRYD EN, PH, D., Dil'ector J OHN F . VI 'l'ORY, LL. D ., Executive Secretary
JOH N W. CROWLEY, JR., B. ., Associate Director fOT Research E. H. CHAMBERLI N, Exe~utive Office)'
H ENRY J. E. R EID, D. Eng., Director, Langley Aeronautical Laboratory, Langley Field, Va. MCTH J. D EFRANCE, B. S., Director, Ames Aeronautical Laboratory, Moffett Field, Calif. EDWARD R. S HAHP, C. D ., Director, Lewis Flight Propul ion Laboratory, Cle veland Airport, C leveland, Ohio
TECHNICAL COMMITTEES
AEIWDYN A ~ II C OPERA'rlNG PRORLEMS P OWE H PLAWl';; I'O R AIRCRAWl' INDUSTRY CONSULTI G AmCRAJi''I' CONSTRUCTION
Coordination oj Research Needs oj Military and Civil Aviation Prepamtion oj Research Programs Allocation oj Problems Prevention oj Duplication Consideration oj Inventions
LANGT"E Y AERONAUT ICAL L ARO HATORY , A~m , AERO ' AU1'fCAI. I"A llORA'T'O lt Y, Ll'; WIS }CLI GWI' ]~ROI'U r"s ION LARORA'1'ORY, Langley Field \·a. '\ lo fT E'tt Firkl, Calif. Clon ' la.n d Airport, Clevrland, Ohio Conduct, under unified control, Jor all agencies, aJ sc ientifi c: l'es arch on the Jundamental pl'oblems oj flight
OFnCE OF AERONAU'l'ICAL I N'l'ELLIGENCE, \Vas hington, D. C. Collection, classifi cation, comp-ilat'ion, Ilnd dissemination oj scientific and !echnical inJormation on aeronaut,'cs II , l ,
REPORT 1053
INVESTIGATION OF TURBULENT FLOW IN A TWO-DIMENSIONAL CHANNELl By J ohn L aufer
SUMMARY at thi. point extends to about 5000 cycles per 'econd it i8 evident that the highfr quencie are nearly i .~otropic in agreement with Ll detailed up/oration of the field of mean aneZ fluctuating Kolmogoroif's hypothesis. quantitie8 in a two··dimensional turbulent channel flow i presented. The mealiurem nt' wer repeated at three Reynolds INTRODUCTIO numbers, 12,300, 30,800, an(l 61,600, based on the ha~f width In rocent. year a con icl('ra bIe sLep forward wa made in of the channel and Ihe mal'imum mean velocity. .11 channel of Lho Lhoo ry of LUl'hul('nc(' . Kolmogorofl' (reference 1), 5··inch width (wri 12: 1 aspect ratio wall usedfor the investigation. H oisen berg (refer('nce 2), and On agel' (rci'('rence a) ob J fean··s peed (Lnd arial :fiucluation measurements were made tained in depend('nLly an ('nergy- p('ctrum jaw that hold in well withi II the lam i !lar .';ublaY fI' . The semitheo re6cal predic rather resLricLed Lypes of Lurbu1('nl flow. Thi progr(' s tions concerning th e ('.rtent oJ the laminar sublayer were con O'ave a n('w imp('tu Lo both th('oretical and experimental finned . The distn'bution oJ the velocity fluctuation s in the inve tigation. T ho exp('rimenLal worker may follow Lwo direction of mean flow u' 8how~ that the influence of the viscosity e.rtendsJartherfrom the wall than indicated by the mean velocity pr incipal method of approach Lo Lhe problem. FirsL, he may e Labli 11 flow fi eld which aLisfy sufficiently the profile, the region oJ illfluenC? being appro.J.'imately foul' limes assumptions of Lhe now theory, namcIy, that tho flow is as wide. i otl'opic and of high Reynolds number 0 lhat the influence Fluctuation!; perpendicular to th flow in the lateral and of vi '0 ity i a minimum and Lho erred of the turbulonce vert~cal direction' I,' and w', respectively, and the corr lation producing mechanism is mall. Uncler th(' e ondition he u'v' coefficient k= ,= = were al'o measu/·ecl. The turbulent may m easure quanti ties such a cOlTelaLion function , ales, -V U ' 2 V'2 and micro cale that arc defined exactly in the flow field shearing stress was computed by thre d~fferent methods. The and may compare his results wiLh tho e predicted by Lhe 1'e!:;ult8 show &ati.~ractory agreement for the two lowel' Reynolds t h ~o r y. T he main cli1fiC'ully wiLh Lhis meLhod i to predict number!;. I n the case oj tlu highest Neynolds number, hOWNlfl', 110\\' closely one has Lo approximate in the actual flow the th e totaLsliearing "tn·.· T obtained from the fluctuation measure ('ondition assum('d in lbe theory. In other words, Lhe mellt:,; was appro.rimately 20 peI'cent [OWN than that COm7)uted en itivi ly of lho Lheory to deviation from true i. otropic (1'01/1 the mean-velocily and mean-preS8ure measurements. ill eondi lion i noL known ane! in ca 0 the mea lU'('mc'nt do dimensionles; meanfluctuatillrJ quantitie,' Wei' found to decrea.· nol agr(,e with Lho lheory one does noL know whether to with increa;;ing Reynolds number. JIeasurement::; of tIL scale' allribule Lh e eli agreement to faulty a umptions i.n the oj turbulence Lv and Lz and micro. cale8 of turbulence Ay and Az Lh eOlY Or Lo th(' incompleLe isotropi.c condition of Lhe fl ow. across the channel ar presented and their 'variation with Ij ul'thermor(', Lh e imposition of Lbe concution or isoLropy Lo Reynolds number i.' (li.~cu8sed . ['sing a new technique, caLue.<; fl uctuating field ' may be a very Lrong restriction and the for the miCl'O.'cale Ax were obtained; a new methodJor estimating tudy of uch fidd may noL yield thc complet e picture of the the scale Lz is also [Jiven. LU1'bulence mechanism. TIL energy balance in the turbulent .flow field was calculated The econd method of approach i to e. Lah li h a simple .from the mpa.surcd quantilies. From thi. ca lculation it is noni otl'opic turhulenL flo\\' fidd ",ith wcll-clcfined boundary possible to Oil' ({ dl'scriptive pic/w'l' of turbnlent .. nel'{j!/ di.thl.,·ioll ('ondition and, in the light of Lh(' ('xi Ling theori(,s, (0 try in tlu center portioll rd tlu cflannel cross 8I'CtiOIl. Lo obtain information 011 (he m echanism of (,llergy tran fer POI' thl' flow correspondill{j to a Reynolds number of 30, 00 from the low freq ueL1 cie of the energy speclt'um to the high the energy spectl'wn oJ the u'~jluctuatio1l8 at variou points one. rrhe prcseuL llve tigaLion of a LlIl'bul nL channel flow across the channel, inciurhllg one in the laminar ublayer, was ha tbi pmpo e in mind in iL long-mnge program. obtained. The difficulLie or tht m('Lhod are immediately r ealized . it stalion y/d=O"L wh('}'(, y i.' Lateral distance and II i:-; ha(f Bccau e of Lh(' nonisotropie nature of the flow, chamcL ristic width oJ th e chalillel, th e contribution to the twblJ1 el/t shear quan iltlc uch a scale and micro ('ule arc 110 longe]' well -tress }rom Nlriou.<; frequency band' was measured and it was defined and it j' v(,ry po ible LhaL in lhe fir l stages of the found thai thl' contribution correspondino to Fequencie. abol'e inve liO'aLion c('l'tain quanliti('s will be m('a mod thaL later 1500 cycles pel' econd is negligible. Since the 'pectru'In of U '2 will prove Lo b(' trivial. On tbe oLhcr hand, Lhe main advanLago of a fu lly developed channel flow i Lhe fact that, 1 tiupcrscti('s i\ AC.\ 1.':"l 21Zi, " ln ve3ti~i:ltion of l'ul'bulcllL F'Io\\" in a Two-Dimensional Channel" by John Laufer, 1950. in contra L with the flow behind grid, flow condition are 1 f , t \
2 l'tEPORT l053- r J ATIONAL ADVISORY COMMITTEE FOR AERO AUTr C steacly; n o decay of m ean or flu ctuaLing guanliLie exi ts dor!' and Kueth ' (references 6 and 7), a nd R cich a rdt (/'cJer in th e di1'eC'lion o[ the flo\\" . on equenLi y , the turbulent. cnces a nd 9). A few unpublished measuremcnts have be n en ergy goes Lhroug h nil of it sLages of lra n fo rma ti on acr os ma de /'cc('nlly at Lhe P oly lcchnic InstiLute of Brooklyn. the ch annel section- Lurbulent-en ergy produclion from the D onch ' and N ikm'a cl se' mea urem ent w C' 1'e con cerned m ean flow , en ergy diffusion, and turbulen L and laminar only ' 'lith the m ean-velocily di tl'ibution and are llru oC dissipation- and one may t ud y th ose Lran format ion in n oL too much inter cst for a comparison with thc p1'C'scnt detail. Sl't o[ In l'aS UI ·eln en Ls. W a t tendod mca m ed Lhe intensity It has becom e d ear thaL th e phenomen ological Lbeo ri es of or thc flll eL ua Ling-velociL y component ancl then ded ueee! turbulence, uch as th e mi);ing-l engLh Lheorie , have 10sL the ('oHclation coeffi cient from Lh e m can-pressure mea m e most of their imp or Lan ce. These theorie , developed in the m ents. TllO te hnigue lor m easuring Lhe axial comp on enL late twentie and early thirti , wer e aimed sp ecifically at of Lh e velocity fJ uetuation wa well-developed at Lh e time, an eval uation of Lhe m ean-velocity clistribu tion in tUJ'bulen t but the cros componen L was only tentaLively mea ured. flm". The exi ting exp erimen tal evidence hows clearly th at The mo L complete et of m eaS llrem en ts is d ue Lo th e work the m ean-v elo ity distribution is very in sensitive to the of R eicb ard L, who mea Ltfed velocity flucL uation in the e en tial as umpLiolls in troduced into th e ph enomenological direction of the flow and normal Lo t he wall a well as Lhe theories. In fact, pUl'ely dinlen ionai arg um enLs generally LUJ'bulen L shear direcLly. R eichardt founel very good agrcc uffice to give Lhe hape of th e m ean veloeiLy profile wiLh m cnt b elwcen LllO sh earing Lrc determined in Lh e e Lwo ufficien t accuracy. For th e further developmenL of an ways; his pap l' com es cIo cst Lo Lb e pJ'escn t in ve LigaLion undersLanding of Lurbulen ce, detailed m ea urem enLs oJ th e and his )'e ulL will be used for comparison. T11 R cynold field of fluctuating r ather than of mean velocities ar e n eces number in R eichard t' m casUl'cmenL s w as 000, whicll i , sary. The progr am of e:".']J el'llnenLal r esear ch of whi ch th i lower than the mnge covercd by Lh e present m ea uremcJ}i s. work is one p ar t i ba cd on Lhi rea orung. lL is qu i1 A criLicism wh ich can b c made of R eichardt's invcstigation apparen t and n atural Lhat the am e conclusions ha" e been is IlLS u e of a LU11l1cl of onl)T 1:4 a pccL mtio. Thc Lwo drawn by worker in Lhe tmbulentficld el ewhe1' c and th at, ciimen ional ch araeler of thc Row is Lhu omcwh at cloubtful. in O'en ral, the main empba is of cu rl'ent experimcnLal inve - , iVattendor['s xperimcn l were macl c in a ch annel o[ ory Ligation i Lh e e"-1Jlol'aLion of th e field of the Ii ucL uaLing large a pect ratio (l : 1) and ar e Lhu free o[ this critici In. velocity componen ts. In a preliminary invesLigation a channel 1 inch wide and The present eL oJ experimen t deal wiLh flow in a Lwo 60 inches high was hosen . The m eaS Ul'em nt , h o,,' ('. CJ' , dimensional ch annel, that is, with press ure flow b etween have sh own th at in this case Lh e cale o[ Lurb ulence i 0 Lwo fl at walls. Ohannel How oC til i lype is Lhc simplesl mall th at greaL care musL be taken to co rrecL th c li ol Lype of 1mbulent fl ow n car solid b oul1cl al'ic whi ch can b c wire readings [01' lhe dIecL of wire len o'lh . ).[casul'cn1 en L produced experimen tally. The simpler oueLLc type fl ow of Lh e micro cales we1'C, in facL, impossiblc in Lh c ] -inch requires that one wall m ove with con Lan t velocit)·, a CO D cha nnel. The micros alc WC 1'e abo uL 0.1 c nlimet l' n, nd dilion which is difficul t to r eali ze experim entally. It can thu sm allcr lhan lli e length of tbe wire, The cO l'J'ections be approxima te 1 b y Lbe fl ow between con c ntl'ic cylindcr , ror Lh e ca e of m ea uremen Ls of v' and w' wer e ab out 30 Lo buL complieations clue eenLrifugal forces a ri c h e1'c. The pe1'cen l. in ce Lh e m ethod of cOlTccL ion becomes vcr)' inac simple geometry of a Lwo-climen iona l cbannel all ows an curate ror uch brge raLio of wir len g h to m icroscale, th integration of lhe R eynold equa tions, and th c tllrbulen t m ea urcm cn Ls wer e repealed in a 5-inch ch annel w ith a shcaring sLrcss can then be r elatcd direc Ll,Y to the shcarin g 12: 1 a pcel r atio. This mlio was still large enough to in m'c stress on Lbe ud'aee whi cb, in t llrn , ('an bc dclcl'mincr/ rrom lwo-dimcll sional fl ow a nd lbc lcng th co rl'cc tioll s W('1'(' grcatl y Lhe m ean-prcs lire gradient 0 1' the slope of m ean vcloc ity rcduced. profile aL tbe wall. This invesLigatio n was conduct ed aL lhe G uggenheim The r elation beLween th e appar ent slre s sand Lbe wall Aerona utical L aboraLor y, California In tiLuL e oJ T ch hearing tre can b e u e 1 to aclva n lage in two fa hions. n ology, uncler Lh e pon orship a nd with Lb e financial a si L It is h ere possible to obtain th c magni Lud e of tb e cou cla ti on ance of Lbe KaLional Advisory Oommittee for A eronautic. coe ffi cienL re ponsible for the appa rcn t sh ear by m ea LlJ'in g Tlte a Ulh or wish es to acknowlcdge Lhe CO l) tanL advice only Lhe inLen itie of lhe turbulen t flu ctuations. The and help in both experimcn Lal method and in terpretation LLu'bulenL bcaring Ll'es can al 0 h e m ea m ed c1ircclly by o[ 1' C ulL of D r. H . , V. Licpmanll during this inve tigaLion. means or the hoL-wire an em ometer. A compari on with the H e also wi h es Lo Lhank D r. O. B. Millikan fol' hi conLin shear di tl'ibution ob Lained from m ean-pressure-gradient or uou inter e t in this research. Tbe cooperaLion of Dr. mean-velocit)T-profile m ea m em enL er ves then a a very F. E . :;"[a rble a nd )'Ir. F . K. Chuan is much appreciatcd. useful ch eck on Lhe underlying a Llmp tions, on Lh e one hanel , o and pecificalJ y as a ch eck on Lbe r eliabili t~T and accurae.\' of YMBOLS th e direct mca LLr em ents, on Lh e oth er. ::-1 ea urements of channel fl ow h ave previously b ecn made .r lo ngitudinal coordinate m elire 'Lio n of flow ; by D on h (refer ence 4) , Ni kul'ad e (reference 5), WatLen- .1' = a con e ponel to chan nel exit ( ------.------~------l ! IN' 'ES'J'IGA.'l'IO OF 'fURB LEI 'J' FLOW IN A TWO-DIMEI SIONAL H ANNEL 3 y lalrl'ul coordin l1 Lr; y = 0 C01'1'r ponels to cha nnel scales of turbulcnce wall z vt'l'tical cool'c1inak (Lx= l '" RxdX, Lv= .ra'" RvdY, Lz=l '" RzelZ) d half width of cbannel (2.5 in.) X, } ', Z di taocc (in th r X-, y-, and z-direcLio n, J'C pec o LiJi cl(ll r "S of lami nar u blayer Lively) beLwcrn points at which con'daLion U mean velocity aL any point in channel flu cLuation are measured
Co maximum value of mr, n velocity (J ratio of quarr of comprllsaLrcl and Ullcompen U' insLanLanrou valu e of n loc iLy ilu cL uations in "atrcl flu cLuations di.rrcLion of m ean How J: M time con Lan expre ing thermal lng of hoL v', W' in lanLaneOllS valu r of velociLy flu ctuaLions \\,11'e normal Lo m ean flow in direcLions y and z, n frrquency, cyclc prJ' second _ rc pectively F;;;-;(n) fraction of LW'bulent cnergy u't associaLed \\'itll il' root-mrau- quarr valur of veloci ty flu ctuation band widLh dn in direction of m ean fl ow x F;r;r (n) fracLion of turbulen t s l1 r:1r u'v' a ocittLcd witll -, ,..." V ,W r ooL-mran- quarr valu rs o f vdocity fluctua band wid Lh dn Lio ns 1l000n1a l Lo mcan fl ow in direction, yand rr eli ipatio ll of L1Il:bulen t energy z, ]"rspecLively t time p prr me aL any poinL in cba nnel voltage flLlcLuntioll p air <1r11 ity T Lolal siwaring s Lrc s A ALYTICAL CONSlDERATIO S ::; hearill g tn's aL wall fric Lio n vel oci ty (.,Jr 0/ p ) EQ ATIO OF MOTION FOR T WO-DI MENSIONAL CK A EL FLOW absolu Le viscosi ty of air The m ean- and flucLuating-velocity component are II kinrmaLic vi cosity d noted by Vi an I 11;', 1'e pecLivcl)· ; CarLe ian roorc1inaLe H Rl'ynolcl " numhrr bas(' c1 o n ll a1£ width of cha ll a re de icrnated by .r- I ; and Plk denole the components of nl'1 a nd maximum lnrall vdocity Lhe s Lre s Len or which includes hoLh thr vi CO LI S a ncl ap k ('01Tc la lion ('o rfficienL H ' ponsibk for apparr nL parrilL (Rc),tlolds) str sscs. The R c)-nolcl quntion a nd t he conlinuiLy eq uaLion for Leady fl o\\' can Lhus be wriLleo, hear ( ~_) u,t V ' 2 using Cartesian Lensor notalion, condaLion codficirnt as funcLion of X, Y, and OV i OPtk Z, n' pt'ctivdy pUk-=- (la) Oh Ol' k mino cak of turhuien('('
I ~ = ((H~' ) 2J U'2 =2. (OU )2, u':=2. (OU' ) 2) (lb) \V ox A/ 2 oy Az- 2 OZ
where
.For channel flow between two parallel flat walls, du - , -, 'I11=U(Y) - p- pu12 J.I. --pU V 0 ely elu -,- , J.I. - -pU V -p-pv12 0 U2=O PH'= ely
U3= O 0 0 -p-pw12
J 4 REPORT l053- NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS where ,J' , y , and z replace Xi, and u', v' , and Wi replace u;' for (b) From lhe slope of Lhe m ean velocity profile near Lhe convenience in wri t ing equations (Ia) and (l b) in impler wall form , Equation ( I b) i.s automaLicall,\' satisfied and eq uation T - JJ. (dU) (1 a) becomes: 0- ely v-a
(c) From a direct m ea UJ'em ent of u'v' O=- 0 ( - p - pU12 ) + - 0 (dJJ. - u - pU- , V ') o x oy dy 1 -,-, cLU) d (2) T o= p u V - IJ. -{- --I ( Gy y-G O=- 0 ( - p - p12v ) + - 0 ( JJ. --elU p U- '-V ,) 1 oy o x dy J If y/el i not too small,
In fully c1 eyeloped turbulent flow Lhe variaL ion of m ean du« - , -, values of lbe fluctuating quantitie with :r should be zero ; IJ. -cyZ- p U V lhat is, lhe fl ow pattern is independent of the treamwise a nd then direction, H ence, equa tions (2) b eco~l1 e : - , -, d T o= P'IJ V d 1 op d 2u y - --= p- - --dH p o x dy2 dy The Lechnicall,V simple t \Va,v to determine T o, and Lhus in or 1 op 1 dT fa ct tl lC co mple Le shear dis lribution , i a m easuremenl of (3 a) Po x=p ely op/ox. This is the melhod applied in most inve liO'ation , T o deLermine Uw lope of lhe vd oeit\, profile neal' lhe wall , 1 o p OV'2 the profile It a to be kn own lip to points vc r." close lo t he Po y =- oy (3 b) wall , Lhat i , atlca t Lo dislances y/d= 1O -2. This in general ]'equi rcs the li se of the hot-wire anemo111eler and ve ry D iffe]'enti,Lt ing equation (3 b) with r espect to J' gives prec ise m eas urem enLs. 1 02p T he thil'cI m ethocl l'l'Cluires a direc t mea uremenl of th P2)X oy= O; con equently op/o x is ind ependent of y ;mc! cO ITc1alion betwcen the axial a ncl lat('1'a l velocil ,\' flu ct uations. eq uation (3a ) i immediatel." integrable, T h us The tecbniq ue of lhis Lype of mcas urcn1('nL is known .and was fir L applied by R eichardt and b ,\' Skramslad a nd , in y o p du - - -= V - -u'v' +Constant omewhat diA'erenL form , in recent investiga Lions at the pox dy I'\ational Bureau of S Landards, Pol,\' technic In tilute of In Lh e cen leI' of the eha nncl t he shear van is iJ es; hence if t he Bl'ookl\Tn , tHHI California Ins titute of T cc itn olog,\'. channel has the " 'id Lh 2d In Lhe presenl inve ligation all LI11'ee melhods have been applicd a nd lhe rcsulls compa red , with the exception of th d op Constant=- - fl ow al the lo\\'cs l R en101ds number. III thi case the pox pre s ll re O' l'adient is extl'emcl,\' small (approx. O.CO :3 mm of a nd henc(' alcohol/em) a nd rca onabl,\' accurale mea urCl11c nts were not I o p ( d) du - ,- , possible, This compariso n of t il e three methods has the -- y - = v - -u V p o x ely adva nLage tha L il give a good incii calion of t he absol ute aCC lll'ac,\' of m easul'emenls of lhe flu duali ng qU y -d o p JI du u'v' (4) ENERGY EQUATION pU/ ox=U/ dy - U/ 'Nri Ling equation (1 a) in the fo rm In terms of the hearing stress at the wall , To y -d JI elu u'v' (6) - pU 2 U 2 dy - (5) 0 ~= 0 u; and multiplyinO' it by U i , Lh e fo llowing relaLion obtained: It is evident from equations (4) and (5) that T o can be determined in three diff eren t way : 1 OU ,U;Uk (a) From the m ean-press Lll'e gradient 2" p O J!. ( 1) Transforming the last term b,\' partial integralion the energy T o=- l :;; ux equation of the m ean fl ow becomes ---~--~ I I VES'l'LGA.TION OF T RBULEN'f FLOW I A TWO-DIMEN 10 AL e RA NEL 5 see ",!leth('1' experimrn tal l'(' lilt confirm th is relation. Taking (7) a an example 1'e ult from l he m('asurements at R = 30, 00 and y/d=0.5 (where q3/D ",, 11 X 103 and /lq2/ ;\2"" 1.7 X l03) The velocity p('rlurbation can now be introduced : q= 52 ('(' n timc-ter p el' second ;\v=0.5 eenLimetel' Lh el'efore inee the veloeitie arc inde pcl1cknt of the ' oo l'din~lte .r a nd ~ t he following ('quation is oblain('d afler averaging: Thi ratio eems Lo bc' faid.\' con LnnL for difj'l'l'l' IlL ReYllolds numbe r and across the channel cross eclion. In view of Lhl' above dimcn ional con id(,l'ations, llll' -,-,du+ du'O'+ l d '( 12+ '2 W'2)=-U op_ 0 0'])+ p'U 0 PIt I -2 p -, 0 it 0 -(yz- l?! cy ox oy ene l'gy cq uaLion ma." be \\TiLten t 1 ( '2+ '2 + '2)_ /lU (Fu+l 2 ') /l ,d 2 U 0 W /l (OAut') (oAu/) (8) cy ~ (y u .rj U Xj ~Iakjn g USC' of equation (:~a) lhi implifie to The eq uation expresses l he facl that the energy produced b.v Lhe turbulent heal' forces a L a certain poinL i parlly W' 2) +v'p] + /l (dU)ely 2- e1 ifj'used and parll." eli ipaled. This equation has b('en u cd in e timating t hc enero-.\' difl'u ion across the channel, ineC' from the measured quantitie the produC'tion te rm and dissipation can be calculat d. This form was obtained by Von K a rm an (l'cI'c' r r llc' c 10) whilr EQUIPME T AND PROCEDURE discus ing it non isot l'Op'ic fl ow in terms o f thr s tati tica l WIND TU EL theory of lurbulrnc('. In order to er the 1' (' Iat iv(' ol'd (' l's Thl' inve Ligation wa C'a rried o ut in tbe wind tunnel of magn itud('s of the differenL term in lhe equatio n, Y on hown in fio-ure ]. The turbulcnC'e lew l is controllrd by a K arman expres e Lhem in t('l'm of a ingle w locil.,' honeycomb and scamIt- preci ion creell', followed by an ' 2 and a d13raclc'ri tic lrno-th ]) COJT(, ponel q=,'U'2+ Z,'2+W appl'oximatcI.\' 29: 1 ('on tracti n. Thc S('l'l'c ns Ita vc 1 _ ing 'to th(' width of t he chann('l. ... mc(' meshc per inch and a ",ire diam('ter of 0.01 in('h. The ho ne.\'comb con i t of paper mailing tube. , 6 inehe long and 1 inch in diameter. and dU = O (~~) = O (q) ely D [) ;;-232<~-fijf~=i-~~ 5 " channel ." , . _ ~ the above equation may be "'riLLen u. Access d oors .. ~" 'Honeycomb (Tubes 6 " long and I" diam.) Wind dlrectlon. ,, ~ox sec:: tion... Counfervanes.. ., where lh (' codIleients bave t he c: hanwlel'i t ics of correlation ..-___--,_-::.:::--;TlT, Conical diffuser.:.. /<./ function and;\ is the microscale of Lurbulence. For a hig h ------f-, R eynolds number flow gD/v»l, it follow that /l q2/D2«q3/D. : Thus the econd and third term on the right ide of th e equation may be n eglected. ince the p ertinent quantitie FIGt:KE I.-Diagram of two,dimeru;ional tunnel. h ave been mea ured during the pre ent \\'ork, the order of magnitude of the e term can be directly eyaluated and the Th over-all length of the channel i 23 feet. At the en omi ion of the term i found Lo b j u tified. Th aboYe lrance eetion it i 3 inche wide and ha an a p ct ratio of equation t hll conlains on1.\' lerms of the form g3/ D ancl 20: L. In a cli tance of 7 fe t it expand to a width of 5 inehe /lg2/;\2; thrse trl'm hould be of t he ame o rder of magnitudr and t he a peet ratio i I' clucccl to 12: 1. The " 'all of the exit ann therefor q;\2/ /I]) = 0(1 ). It is of con iderable interest to por Lion of the channC'l (about 6 ft) nre made of %-inch- r ---.... - 6 REPORT l053- NATIOI AL ADVI ORY COMMITTEE FOR AERO AUTIC thick plywood wi th a X-inch birch in ide (;Over. They were peciall.'" treated in order Lo acquire a m ooth fini h and were reinfo)"eed in order to avoid warping (Acr. 2); in pite or thi a few per("ent of ,,"idth vUl-laL.ioJ) c.'\:.isLI'.rl VWl'HE a. Traversing mec hanism. The lIe ro r eading of the travt'rsing ll1 eehani m (y = ) was ("a refull.'" found usin g th e followi ng method: The hoL-wire \Va placed dose Lo the 'mil (approx. 0.025 cm aW3)") ; the Ii tance be[\\"een the wire a nci its image in th e poli h ed wall wa m ea ured by means of an o("ular micrometer. The po ition o f the \\"ire i , of co urs(' , oll('-half of the observed eli tance and cOl Llc! be cletermin('ci willt an accurac.y of ± 0.0005 en ti mC'l er. In ord er to obLain the pressure di LribuLion along the middle of the ("hann el a 6-inch-long Lhin-walled t ube %-incb in ciiameter wa u ('(I wiL h a s ma ll s LaLi c-p re ure hole drilled ("lose to iLs end. The tube ,,"as free Lo slide in a uppo rL aL tlw entrance of tht' channel so that Lhe position of the pre. SlLl" e hole relative Lo Lhe (" haunel exit ("ould b e cha nged. The Cube wa kept sLraighl and uncier ten ion b)" m ean of weicrh L . HOT"WJRE EQUIPME T All vcloeit.\" m ea mement were made wi th h OL -wire ane monleLer ~ . The rreq uellc.,' I"e pOll e of tb e amplifier-compen ator unit or Lbe h ot-wire equipment, u ing a 0.00024-in('b wire aL standa rd operaLing condiLion , is fl at from approxi maLel.," 2 to ] 0,000 cyele pc]" second. The compensation of lhe h ol-wire fo r thermal lag i ac compli bed b.," a capac i L)~ network. The mncre or Lime con LanL was chosen from 0 to 1 milli ee nd co rre poneling ,·',r.nn: 2.-ExiL or channel. Lo the characterisL ics of 0.00005- Lo 0.00024-inch wire at Lhe o perating conditions emplo.,~ ed in general aL GALCIT. No The tunnel is operated by a 62-hor epower staLionary attempt wa made Lo exLend Lhe range of compen aL ion Lo natural-gas engine--normally operatincr at a fraction of it larger value of JJ, Illce III this case Lb noise I vel oon raLing- which drives two eigh t-bladed fan. The speed is becomes appreciable. remotel.v controlled by mean of a sm all electric motor whiell The con'ecL seLting of Lhe compensaLin cr lmiL \Va fOllllcl drives the throtLle through a gear and lead- cr ew system. usin g LIte quare-wave metbod dc cribed by Kova znay (rcf The experiment were calTied out at peed of 3, 7.5, a nd e1"e11('e 11) . The time consLa nL s of the wire u ed for t urb u 15 m eter per second. lence mea m em ents fell between 0.1 and 0.9 milli econd. The output readill"s were taken with a lher-mocoulle and TRAV ERSI NG MECHANISM preci ion poLentiometer. Figure 3 ho,," the type of traver ing mechanism u ed Mean-speed measurements.- 1 1/2-mil (0.0005-in.) plat during the experim ent. imply of a rnicromeLer inum ,,"ire of 1-centimeter length ,,"a u eel to mea Ufe the cr w on which the hot-wire upport is fa tened. The up mean peeel. The mea uremenLs were made b .'~ t11 on Lant port can rotate freely in a plane perpendicular to the air Dow ; re istance method. This m eLhod ha the advantage of thus, the hot-wire may be a Ijusted exactly parallel to th e keeping the wir Lemperature onstant throucrh Lhe v locity wall. fi eld. ------~------INVE TIGATIOI OF 'l'URB LE T FLOW I I A TWO-DlMEr 810 AL CHAI EL 7 Turbulence measurements.- l<'or tl1(' ill\Testigatioll 01' t ur of lbc u ' -flu clua Lio ns. From lhese co unts "xma~ T be cal bulent fluctua tions, 0.00024-iuch Wolla ton wire was used . culated directly by assmning a n ormal a nd independent T he wire wa oIL-soldered to Lh e tips of fine sewing n eedles di trib ll Lion for both u' a ncl oul /ox: aHer the silver coa Ling bad been etcbed ofL The two lateral components v' and w' and th e eorrela bon eo effi ci n t , , : .., = [0X A verage number of zero of u' per second k .J u ~ were measured using the X-wire tecbnique. The u'Z V ' 2 It i known Lba t Lhe eli tribution of u' i cl os cl~' a Gau sian m ethod wa essenLiall.\T lh e same a le cribed in delail in one even in n oni olropic Lurbulence ( ee , for in Lan ce, refer previou reports from Lhis labora tor.L (See, for exam ple, ence ] 5) ; h owever , for the case of AU' lox a mall. deviation reference 12.) from the normal d istribu lion was foun el (reference 13). F or T]JC X-m eLer for h ear meaSUrl',111 ent h ad angle b etween Lhe preliminary m ea ur menL of "xr eported pre entl.\', no tLe wire of approximalely 90°. The angles of th e v' ,w' correcLions were applied a )' eL for lbis eHe cL. Figure 4 met er w ere of the order of 30°. The.wire length of th e u' sbows an oscillograph trace of the u'-O ucL ua tion in the mid meler was ab out 1'.5 millimeter and lhat of th e v' -meLer clle of Lhe ch a nll el at R = 30, 00. The trace represenLs an wa 3 millimeters. interval of approxima tely 1/20 econd. The parallel-wire L chnique us d reccntly aL th e P ol.\"tecb uic InstituLe of Brooklyn wa also Lried in order Lo obtain the lateral component of th e veloeit:v fl uctuation and the sh ear close lo tbe wall. The parallel-wi re ins trument j uperio r to Lh e X-me Ler since it allow an exploration of Lhe turbulent fi eld up to eli Lances of a few tho ll sa nd th of an inch from lhe wall. H owever, i t wa found im possible to obLain reliable values wilh thi in trumen t. T h e corre - tion due to U' -flll ctuations and to lLn eqll al h ealing or th e wire were pronounced anel not ea ily acco unlable. The mcihod ,vas therefore Leml ora ril.\" ahandoncd after CO Jl si(J erable lime and labor had been spen t 0 ]] iL. ·Measurements of correlations between two points.- T ill' F Ir.l'I'E 4.-Typie,,1 oscillogralll of u'"f1uctuatioJl at R =30. 00 a nd !1/d= 1.0. Ilorizo nlul lill c correla tion function b etween value of u' at points along correspond to u'(t) = 0; interval is a pprox ima tely 1/20 second . the y- and z-axi , that i , Measurement of L T.- Th e following . imple procedure ~ \I'a applied Lo obtain a rough es tima te of calc of Lu rbulencc R = u' (O)u' CY) lI co rre poneling to co rrelation beLween points along the J' u ' ( 0 ) 2~u ' ( YF axi : D enoLe by Fu'2(n) the fraction of Lurbulen t j nLel1 s i t ~· wh icb i con l ributed b .\' freqllencie betwoe n 11 and n+ dn; u'(O)u'(Z ) tha Lis} u '(n? rln = u '2F u'2(n) dn were m ea ured u ing t be taudard techniqlle (reference 12) . and Lhu T he cale of turbulence Lv and Lz and t he microscales Au ( 00 F ,,'2(n) dn= 1 and "za t difreren L points across Lh e channel w ere obtained .1 0 from the e m easurements. Con icier no\\' an uncom pen aLe d h ot-wire. If Lh e Lime Measurements of "x.- A method sugge Lee! by T ownsend con tan t of thi LUlcompen atecl \\'i1'e JJ, the responso (reference 13) wa appliecl Lo m ea ure ing a n electronic "x. will be differenLia lion eircui t,. th e amplifier output ignal wa dif ferentia ted and "xco mputed from the Jollo\\"in o- relalion: [n' (n? lu ncomp. where OU')2= 2 (OU' )2= 2 U' 2 ( at u ox U Al The total in ten iLy for the uncompen aLed wire will then b e The error in volved in Lh e approximalion ~t-7U 00:1' can b e o- i, ' en. b.\' 3 e tima ted and it may b e een Lha t lhe above relalion llOl ds wiLh rea onable accuracy oyer th e large cen ter portion oJ th e channel. A n e\\" techniq ue fo r the m ea Ufe l11 en l of "xh a a1 0 been , T h is method was suggested by Dr. II. W. Liepmann. applied . The m eth od is de cribecl in deLail in r eference ] 4 3 Formulas of t his genera l type ha\'c becn proposed by Kampe de Fcriet and by Frcnkie I for determining the sp etrum of t urbulence fro m uncompensated hot"wire rncasur ments by and consist in (;ounting tbe zero of an 0 cillograph trace \'My in g .lI (reference ]U). 981431-52-2 L~ ____~ ( -,--~~.,....-~ . - ~ --~--r- - --- I I I REPORT I 053- NA'l' IONAL ADVISORY COMMITTEE FOR AERONAUTICS I Thus, if Lhe raLio of t he r e ~ po n e of Lhe ame wire eompen band ated ancl ullcompen ated i denoted by (1 , Lime Lhe valu e cOlTespo nci ing Lo zero frequency. I The hot-wire ignal wa f.,d into a Hewlett-Packa rd wave (9) analyzC'l', Lhe ouLput circuiL of which wa somewhat a.ltcred in orde l' Lo be f),ble to feed the outpu t ignal into a l"Oot In order Lo e timaLc Lx, F;;;z (n ) i a urn d to have Lhe simple m ean-square voltm eter. The averaging hal'acteri tic of form t he root-mean-square m eter \I 'ere checked and, [or a narrow band-\l'id th ignal, were found to give Lb e arne mean values 2 L IUO F u,2(n)=- x L 2 as l ho e calculated from lheJ'm oeollple reading . -rr . +? x 1 n- U 2 'iiVilh this method tb e enercr.\, o[ the 7~' 2 -fluctuation was 0 meas ured at c\i fl' erenL point aero s th e ch annel. At yld= O.4 4 w hich was given by Dry dell (refer ellce J 7) and cOlTespo nd the peelru m of the Lurbolen t shear u'v' was al 'o obtained. to an exponential correIa iOIl curvc. \. n X-t.\·pe meter wa used , both wire making an .. ngle of 2 2 Then 45° with the free-s l ream di rection. If e l a ncl e2 arc th e mcan- quare voltao-e flu ctuation «('orfected [or Lime lag) aero s L\H' L\VO wires i L may be s hown (reference 12) tha t Or with If el and e2 are feci inLo Lhe wave analyzer t ben the above relation h olds for all band widths dn; t hat i , a= NI Uol Lx tile/) PRELIMINARY MEAS REMENT IN A I-I CH CHA EL III th(' inilia l ·tug<'s of tile lurblllen t-c1 lu nn l'l-flow inve ti- ' H{'ll (' e gatioll a two-d im {' nsional (' hannel of 1-in('IJ wielt h was \I eel . .\l cas uJ'ell1eJlt · of m ean [Uld flu ctuating velocities and of tb correlaL ion coeffi cient k were completed. The calc Lv and Lz were also mea, urccl at the channel c nL e!' an I. w re found Thu by m ea uring the ratio of t he mean quares of the to be about 0.2 Lo 0.:3 centimeler. Thi small- calc t mbulence fluctuating veloci ties wi t it a nd wi thou t compensalion, Lx can ('x islin o- in t he cha nnel impo ed a d efinite limitation on ,the he estimatl'd. flccurae.\· of lIw f1ucluating meas uremenL . 'Wire-l ngth The procedure was chl'c kec\ ill lhe rIow Iw hind <1 }~ -i n c h correelions a,s high as 30 percen t had to be applied to lli e grid where tht' latpral calc \\'n mea tired a nd where l)(' c<1use mea ul"emenLs 0(" I" ancl w'. Fu rt hermore, no micro 'ales of i otropy Lhe relation Lx= 2Ly s hould hold fairly lI'ell , ('ouIe! be meas ured accurately; tllll one of the obj ective of The l"e su It \\'en': Lite inHstigaLioll , tile caleul a ti on of lhc energy eli sipation across lhc cha nnel, could nOl be obLain ed. .\everthcle , re ult bo\\' good con i lency ; the three independenl mea - .!if= 7 X IO- l econd uremenL of the turbulent hear indicale ati factor.\· acr ree m en t. { '0= L5 X 103 'e nLirnetel"' pC'r C' 'o nd It is of intet"(' L lo presen t t he e p reliminary meaSLll'emenL , tbcrcfore Lite Reynold number of whidl wa 12,200, and to compar e Lr= 1 .24 centimeters tbem with tho e obtained in Lbe 5-inch (' hannel. Fio-ure 5 hows lhe eli , tribution of a ll the meas ured quanlitie in Lhe Direct mea ucemellt raVe Lv= 0.55:3 cenLimetcr. lIC' ll ce l-inch channel and itIllay be dire lIy com.pared wiLh fjo-Ul'e 16 Lz= 1.11 'C' IlLimeters \\'bicb bow LbaL this m eL hod of wbic h hows t he quantilies meas ured in Lhe 5-inch channel aL estimating the value of Lx is ati factory. nearly Lhe ame R eynold number. Measurement of turbulent-energy spectrum,- In ol'der to AJ though the slope of the mean velocities at the wall arc 2 be able to m ea ure accurately the energy eli t ribu tion over a almo t the same in each ca e (for t be I-in. channel Tol pC0 = \ ide frequency bane!. it wa found \leees ary to usc extremely 0.0019; for the 5·· in. channel TolpC}=O.OOl ), the velocity thin wires with mall timp co n tnnt a nd t llLl to incrca e t il e rnLios ul e o in t he 5·· in ch cha lllw i ecm Lo be lower acro ignal-lo-Iloi l' ratio. W cre " of 0.00005-i nch diam eter, op- mos t of t he chan ll el. This ind icates t haL l1. more i l1l en e erating \\'illl time con tallL of approximately 2 X lO -~ 4 This III 'L hod appears to hun! b('c ll first, used by Dr. Slanley Corn;in in a turbulent jet "eCo ll c/ , \\'e)"(' 1I ('(I. In Lhi . way i L lI'a po ible Lo d pU'ct (prh 'ute communiC' ..llion ) . I l_ ~ ______~J ------~~ I I VES'I'IGA'I'ION OF 'l' RB LENT FLOW II\' J\ TWO-DIMEN 10 AL e RA NEL 9 IA 28 I I I I u'v' 0 --2 measured Uo 1.2 24 ____ u 'v ' calculat ed from velocity distribution 2 U0 W r-- , _ r_ 20 r-- u 11 u 'v ' L--- Vo flo ,.- Uo ~ ~ - Ie ~ _ L. ~ .8 ~- 1 ii' x/O V-~ Uo tI' 'ii' x / O I---t-----~-" U ~ •• ' Vo o .6 t- " Z - ~ :--- iii' x/O U o r( iii' ~ .. ' Vo ------~ .4 r------~ K 8 v' .. ,' ~ ~ Vo ---~ .2 K 4 ~ o o .1 .2 _3 .4 .5 .6 .7 .8 .9~ 1.0 V/ d ....·we RE 5.-Dislribulion or mN °ured QunnLities acroSS two-dimensional chatmel or I-inch width. R= 12 ,200. . - du 1 1 . I . I heigh L in the chanllrl: At position 6 il1che fl'om th L1I1'bulrn t··(' n(']'gy production T rly \'[1.((" P a('(' ] n L 1(' 5 .. ]n Cl b tLom , 6 in (' he h om th (' top, a nd at the middlt,. Agl'ee (' hatltwl. Ind('('d, t he tlll'hulen L-vrloC'j Ly fluC'Luil tion fill ( '0, mel) L I1 moll g the tlt]'ee ('l of mea un'mc' ll t confirmed the v' C o, and 'U/ / Co ((' peciall y fi' / C o) illcl'('fl. e at a fa t(''f' I'at(' Lwo-diml' tl iOll ality. A JlIl'thel' cll('ck was made on po ibl t' Loward I,h (' wall in the wie/e]' channel, although their \Talu e end eHecl thaI, mighl itlrtu(,llee the l'('suIL. The length of at the chal1nel ('(' nlr1' check V ('l'Y well in the two ca ('. The the cllanncl was extend('d by 6 in ches ilnd Lhe mean-velocity shea]' di ~ tl'ibuL o n T/pC/ i almo t t. he 8me rOl' th(' t.wo mea u)'emenl wen' )'('peated aL .r = - inehe '. :\0 chang(' ehann('[s; it fo llows, therefo'f'e, from th(' relaLion in the profile wa notic('d. Figure 6 bow the mean .. vclocity eli tl'ibution al tlll'e(' R ('ynold number , H = 61,600 , R = :30 00 , and I? = 12 ,:300. The c!i Ll'ihu Lion follol\' Yon KI1rm lln 's loga'l'iLhmic la.w VCl'y well , (' x('epl, of C'OllI'SC , Ileal' Lh a t J.: mu L have /), hi gh!']' ab olu te vallI(' in Lh(' J-ill ('h the wall ancl 1t Lh(' ('(' nt(' 1' of tll (' ('hall 11('[ (fig_ 7). 1'11<' channel inc(' uoth fi' j Coalldv' / [ 'o a'f'(, mall (, 'f' thel'e _ Ind('e cl vil lu e of C, wpr(' obtainNI from til c ye[oeiLy gradi('n L at th e maximum valu (' of k in Lhi ('a (' is - 0.6, lI'hile in th(' the wall. 5-- in ch clawl1l'1 it i - 0. 5_ A a matter of intel'(' Li t ('ould be m('ntioned that if tlt e 1.0 --~ basi of ('om pari on is no t R eynold numbC'l' but maximum .8 meall peed (for the I-in . chanllel [ '0= 15 m j ec) thcnn O' lIl'c 5 how a distribution 0) fi' , 75' , a nd W' YC'ry imib1' Lo Lhat or fi gw'e I , the ab olu t(' \'ailies bcing om('\\-ha 10ll'er ill JL Yo th 5-inch channel. .4 RE ULT A D DISCU ION .2 MEA -VELOCITY OI STHIBUTION A cu,rciul study of the two-dimen"ional na ture of Lhe 0 .I .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 challJwl flow Wil fir L made. ':'lean-veloei Ly mea U1'em(,11 t y/d \\' ('re can ie(/ ou t at approximately .r = - 2 incit e at d iA' (']'en t FIGt'HE 6. -:\lran-\-clocil)' distribution. ~re35urcd point. close to wall arc not shown . • " 10 EEPOE T 1053-1 A'rIONAL j DVISORY COMMITTEE FOR AERONAUT[ S 30 m cau value, SiJl ce th e m ean c urren t of Lhe wire varies wi th lh e fo urlh root of the veloeiLy, 25 ~ ~ ~ I I 4 - [ ( U)2 (VI)Z (W') 2J I 20 AO ~ -/u= "\ U 1 +u + 1; + u ~~"'" U lA r ~ U 15 t----f- ..Pd wh n' u is til t' v(' lo(' ity to "'hi eh the hoL-w in' r0s pond a nd l?'l' R V (; 12,200 ( I -in. channel) u is lh lrue m et n velociLy. Neal" Lll e wall ( ~)" < < 1 /0 W 1-1- o 12,300 (5-in. channel) V ~~ ],'IGnn; i .-Logari t. hmic L' C' pn'Sl'lIla tion of 11l(' ~l. n -\'(.' l oe il y dislribulion . )'1 l'a'Sul'el11l'n ls were made w iLb sp ec ia l car e c10 e lo Lbe wall , Y('10(' ities a l a large' nu m ber of poi nls were ]'e('orded wilhin lhe laminal' sub1 a TN ill 01'(\ (' 1' Lo establi h \\'itlt 111ce t be average oC t he odd t. erm is ze ro lhe erie con r ca o nabl(' accur acy Uw ~ h ape of lhe velociLY profil e aL Lhe verge rapidly, In lhe region in qur lion , lhe veloc iLy wall (fig, 8) , The thic\;:n ('ss of tlw lam ina r sublayer (tbe f111 clll fl.l ions oblain d wcr very high indeed (u l /u > O,30), poin l ,,'h('1' c lhe w ]ocily eli tribu Llo n cl evi ale ~ from Lhe Their a bsolu Le value , however, a rc probably even larger, loglll'ithmic Ill\\,) Wll. j'oun d Lo he 0 ""' :~OIl / I "T (fi g. 7). sin ce the nonlinearily effecL of Lhe large fl uctual.ion n the h l-wi re 1'e pon e Lill fu rther amplifi ed by Lh ver.\' low mean velD ilic i nol ta\;: n in lo accounL becau e the \Vire .8 lenglh correcLion are negleclecL The m ean-veloci Ly co r l'cclion LhCl'cfore given by th e above relation is too m all to gi IT e lhe req uil'ec\ negfl.tive eurvaLure of Lhe profile, .61---I--t-.y<--l- T nllULE C E LEVELS F igures 9 lo 14 represenL the 1'e ull of measuremrnL of l l1l e three component u , v' , a ncl Wi of Lhe lurbulent-velo i Ly f1 uduaLion dis Lri buLion in th e ch annel for lhe tlJl'ee R eyno} I. numbers, .18 q .16 .I I \ I .14 I ¢ \ R o .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .I I .12 o 12,300 - y/cI 0 30,800 0 61 ,600 .10 ~\ 1\ u' In figur(' iL can be ('en that for lhe ca e of lhe lowesl IT \ ~ ~ R('ynolcls number a fe" " poinl n ear lhe wall indicale v ry .D8 lO ll' velocilies. ince lhe poin t aL w]lich y/d= O h a b een .06 ~ b:' r---. '--0: c\eLe rminNI with greaL accuracy andinee the equa Lions of ~ moLion of t he channel now require a n ega Li ve curvalure foJ' .04 ~ [;:--t:F- lhe velocit,l- profil e, iL wa concluded that the h oL-wire ~ indicaLed loo low ve10cil i neal' Lh e waU for Lbe 10wesL .02 Reynold number. (The cl ash ed curve for thi ca e in fi g. show Lh e in lerpolaLed vcloeiLy di Lribu Lion for O< lI /d < O.O l .) o .I .2 .J .4 .5 .6 ,7 .8 .9 1.0 1./ II can be shown tha l very high 10cal- veloeiL.v flu cL ua li ons lJ/cI ('au e the hol-wire to read velocities 10ll" el' lhan lhe true FIGU Il E 9.-Vclocity flu ctuations u' relative to local spceds, I L______.---1L-- ______J j - -~~------~ r IXYES1'IGA'l'ION OF TURBULENT FLOW IN A 'l'WO-DIMENSIO AL CHAN EL 11 .45 .06 I IR o 12,300 .40 1/\ .04 Plr.l'n~ 10. Velocity fluctuations u' relntive to local spc ds n~nr wa ll. The velocity fluclualions 11,' relative Lo local speed .14 increase.' very rapidly 11('a[' lhe \I'all a i shown in fi gure.' 9 , , '. and 10, ~lea ul'emenls Yery do e lo the wall indicate Lhat .12 •, \ :U' /u reaell e a maximum within lhe laminar bow1dary la,\'er , , (y 7) and it lend toward a con lanL value at th wall .10 1\ , T/ II ""' ] , which i independent of the Re.\·nolds number. Tilis point ...... R I~ \ _ _.-8. 000 (reference 9) \\'ill be eli U ed in cl eLailunder "R eynolclsKumbel'Efl'ecL." .OB '---J __ __ J ii' T he ab olule value of the dislribuLion of 11,' how lhe same o ~ r 12 HEPOR1' J 053- rATIO -AL >\.D YISORY OM lITTEE FOR ~ERON A. TIC .5 0' tained de('l'ease slowly w it h in Tea ing R e.nlolds numbcr a nd SCALE A 0 MJ CROSCA LE MEAS RE LENTS th us show a defi nite R e.\'l101c1 s numbcr dependen cc. Exi Lin g I'C ults inciieate, h owe \'c L' , t hat paramctCl's othcl' than R e.\'ll F o)' a furt hcr unclcl' tand in g of t ilc stl'll cL ul'C of t hc t ur old numbcr also influence the value of k. The foll owi ng bulcnt field , co]']'clations of u'-fhl ctuaLioll s aL two differcnL table hows magnit udes of th e maximum correla ti o n coeffi poinLs \\' e1' e ('arried ouL. Sill ce Lh c field i no isotropic, cienl a o bta in ed b y diA'el'cn L invc tigatol' w orking with t hc calc a nd mic)'oscale m casu rem en Ls wc)' e 1' ep eatcd both ch anncl 0 f ' -a ri Oll s wid (h a ncl wi th va L' ious R c.\·nold s in t he y- a nd z-dircction foJ' difl'e1'ent values of y/d. number Hz-correlation,- Fig u1'c 20 s how typical R z-correlaLio n ChaJln cl width dis LribuLioll aL cilfferent valucs o f y/d co]']'c ponding to Experimcnts b,\' - (crn ) R " .. or R eichardt (rcfCl'cnce 0) _ _ _ _ 3-1. 6 , 000 - O. -+ 5 H = :30 , 00. Fo1' la rge1' val u c of z, inaccmacie i n t h c Laufel' ( pl' e~c IlL papcl') _ _ 2.5 ( I in .) 12, 200 - .63 m t'a urem cn L did not permiL t hc cx plor aLion of Lhe ncgativc Laufcr (prc'cnL papcr) __ J2. 7 (,- ill.) ]2,300 - . 50 rcgion of thc cOJ'l'cl ation di I'ibution. From th e mea Ul' cl WaUcndol'f (rcfcl'cncc 7)_ _ 5. 0 15, 500 - .52 dis Lribution of Hz at foul' s tations acros Lhe channel a nd L aufer (prcscnL paper) ___ 12.7 (.5 in.) 30, 00 - . 45 1'01' difl'el'en L valucs of H, th t' valucs of L z a nd Az wer e cal Laufcr (pl'cscnL papcr) __ _ 12. 7 (5 in.) 6 1, 600 - . -lO cli laLc ci (fi g. 2 1 a nd 22) . In t hc c fi g urcs Lz i ccn t be vVaLtcnclorf obtained his value of lc from his m casurcmcnt d(,(,rcasing u niformly towa]'d thc wall , \\'hilc Az ]'cachcs a of the quantiti c 'U' /Uo, v' /Uo, and u'v'/U/ . dcfini te m aximum at abou t y/cl = O.7 , nd t hen d ec]'ea. es his preliminary m ea Ul' cment o f v' a rc iocol'l'c('(; (ir cir \\'iLh d ccr easing yjcL. magnitude i la rger than Lhat of 'U' aL all value o f y acro H y· correlation.- Di t ribuLion of th c Ny-col'l'elaLio n at the cha nnel con trar y Lo r esult obtaincd b.\- R cicil a l'd l a ncl a given value of y/d we]'e obtain ed by fixing Lh e tationar y b.\- the pl'csco L writer. 'YauendOl'f him clf points o ut Lir e hoL-wire at th e givcll valuc of y/d and LJ'aversing wiLh t he moving hot-wirc a way hom Lhc fi xed one Loward the channel I inaccW'acy of th e value of v' in hi paper. Bccause o f th e ('c nLcr. Vahl es of L y a nci Ay wcr c t hen ca lculatcd from t he e la r O'e va l ~ l cs of v' hi compuLed co,lfj'cla ti o n cocfficicnt a rc di l rilnltion ' of R y. III t hc ]'co'io n 1.0 > y/d> 0.1 th c grad too 100L Tirc value lc "'flx=-0.52 lis tcd ill the above ta bl e iPll ls of Lhp various q ua ll titics arc li g h t; no s ig llifi ca n L WI1S o btaincd by using ' YaLtcndol'f's valucs u'o'/ ( '/= 0. 01 asymnwL ry ill Lhe fUll ctioll Ny is t hcrefo rc exp cctcd. :' Lea and 'U / Uo= 0.055 and t ir c valuc v' /Uu= 0.03 5 (!J /d= 0.5) m cm cn ts of the Lop parL of Lh c H y-co1'1'elatio n distribll Lio n obLained by lIl c a uLh or [01' b olh t hc I-inch (2.5-('m ) and ( 1.0 > Hv> 0. ) by Pl'a ndLl a nd RcichardL (r efer cnce 19) 5-inch (12 .7-cm) cha nnel at R = 12,2CO a nd H= 12,:300) jll tify Lhi belief fa irly well. re pectiYcl.,-. B y comparillg fi gUl'e 20 "'iLh figu rc 2:3 , Lhc sLrong e H' c t The variatio n of lcincii cu tes lhat for th e amc R cyn old of Lbc exi ting Il o ni otropy can immediaLely be een. AI number Lh e eO l'relation cocffi cient te nel s lo increa e with UlOu g b boLh Ny an d Hz corrcspond Lo Von Karma n' g illcl'easin O' maximum m ea n v 10(' it.\-(i. e. , in a cha nnel of fu 11 ('tion (rdcl'c nct' 10) t hcy cxhil it diffcl'ent behaviors decreasing ,,·idth). across tb e channcl. The H z"('o l'l'elation fun ction fall mo]'e Figmes 16 to] sh ow lhc distribu t ion of all th e m easured a nd more l'apidly Lo zer o with decrea ing yjct. Travcl' ing quantitic including t he sh ear d islribution . For Lir c ca c of f]'om yjd= 1. 0 Lo y jd= 0.70 Lll e dis LribliLion of R y i found to lh two highe r Reynold s numher , lhrce indepcndcnt bchavc simila rl y ; h owever , t he da hed curve mca ul'cd at meaSUl'cme nts wCl'e made for Lh e de tcrminatio n of t he sli ear y jd= O.4 (f ig. 2:3 ) indicatc;:, co n id erably int 1' casccl COl'J' el a dis tribution b.\- m e thod indicatcd in "Equatio n of :' Io tion lion for largcr vahlcs of }-. ] L j tln l 'ccn Lhat a round for Two-Dimc' ns ional Ch annel F low." F iO' ure ] 9 i ndicalc y jd= 0.7 Lh('l'(' i a dcfulite cl ec]'ca e in cncrgy conten L of th e t hal consistcn l l'c ull a rc obtained fo r th e value of T wheth er rIli ctuations having low fr qucncics. Further onsideration calc ulated from lh e prc s ure grad icn l along th e ch annel 01' of thi fact is C)'iven laLc)' in Lh e di Cll sion of t hc encr"'y from th e vclociL.\· O'racl ic nL aL Lhe wall. T he l urbulcnt-shear l>itl ancc in Lhe (' ha nnel. distributio ns obtained from h ol-wirc meaS Ul'ement and t hose calculn Led from fi g ure J 7 h ow sa t is factol'Y agrcemcn L. ~ rl'his rcsu l t. was accepted arter the absolute \"alucs were Ctll'cfully chccked; the Il'-wire response was compared wiLb u'-tlucWBtions in an isotropic (j Id and Lbe two-dim nsional However, for tb e rI ow at the hiO'11 e t R e.nlolds number, the characLer of Lhe flow was checked. I l .~----~-j \ \ INVESTIGATION OF T URB LEN T FLOW II TWO-DIMENSION AI., CHANNEL 13 1.4 28 I I I U'V' 0 --2 meosu,..,ed Vo 1.2 24 ------~ c olculat ed from velocity distribution 2 f\ ... ;, U0 ," P 0 1.0 20 u u Vo ~ V o - If f\ ~ -- .8 ~~ ~ I ii' xlO --- Vo p'- ~ i)' xlO -;;v _ " ...... Vo ~- .6 // - U 2 - ~ I 0 - , iV' x lO - ~~ ~ Vo .... ~-- ~ ------~ r--- ~ r-- i-----.. 8 .4 r ...... ii' ,," ...... - U w' , O ---~ ~ / Uo ~ --- / I ~ .2 ~ 4 / ~~ 1 1 1 o .1 .3 .4 .5 .6 .7 .8 .9~ 1.0 o .2 V/ d ~'J(tV l t£ iH.- Dislribulioll or IIlca n and fluctuating quulltities. 11.= 12,:mO. 1.4 28 I I I u 'v' 0 --2 measured Uo 1.2 24 ---- u'v ' calculated from velocity distrjbution U/ 1.0 20 u ~ Vo ~ u Vo - If .8 ~ ii' xlO V --- T------Vo ~U02 v' x/o V o ~ I---- .6 ~ -,-, ,""'''''' iii ' x/O uv " u- , Vo - ~ -- 1 0 ~". r------r-- "------~ .4 r-- - t--- I--- 8 /" -, , - I "7- ~ ,' .ML -- r--::: :::::::--- Vo Vo ~ ~ ~ ~ -- .2 ---"'" 4 ------~"'" ~ o o .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 V/ d FI<;cHE l •. - ilistribution of mc:\n and flu ctuating quantities. R=30,~IO. L J 1 14 REPORT 1053- ATIONAL ADVI SORY COMMITTEE FOR AERONA TICS 1.4 28 I I I ~ . 0 - -2 measured Vo 1.2 24 _ _ __ u 'v' calculated from velocity distribution 2 Uo 1. 0 20 , tJ f.------, U Vo Uo - If ~ .8 I 6 ,/' -- ii' x/ O UO ---- ~ v' x/a - - V 2 Vo o .6 K I 2 w' xlO -"' - Vo - u' ---- - I------...... 4 ---- ~ 8 I------, ~ ~ W _' v' .. ' -- ~ t::-- Vo Vo - ~ - -- 4 .2 ------~ ~ o o .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 !lI d 1" IGl'R E l .-.Distribution of mean alld flu ctuating quantities. R=6I,600. oo .06 v P'" The value of Lv= io RudY (fi g. 21 ) arc not so reliabl-,: as ..sv-~ 0 0 ...cr--- f-.--;; tho e of Lz in ce By could no t be m easured at uffici nLly ::- large valu es of Y because of tbe limitation of the traversing ....-"( )'..ifP" (a) mechanism u cd. o - The di tribution of Av acro s Lhe channel is similar to Lbat n n .06 of At (fig. 22) . From yjd= ].O Lo y jcl ~ 0.7 both Av and Az jn ~ ..a--- ...0-- crease almosL proportionally Lo u' , indicating that the Lur f-O" 2 Q., 1~ ·04 ~ bulenL-energy di ipation Wex: u'2jA i approximately CO L1- .8 of (J' js very do e to unity and, ince Lx i proporLional to _ 1_ , (J' should be knO\nl within a n accura 'y of 1 perce nt to .6 (J' - l fl" .4 cr i\'e consistent result for Lx. UnforLunaLely measurements of Lx ca nnot be made with this accuracy, particularly when .2 till' /i'-fl uctuations arc of rallH' 1' 10'\' frequencies a i the ea e 1' 01' R = 12 ,:300. It should be mentioned that the accuracy 0 50 of the determination of Lx is mol' limited b)' the inaccurately m easured value of (J' than b)- the fact t bat an approximate FIr.n'E 20.-R-correlation. R =30. 00. At y/d= J.O , L.=J. 65 centimeters, X.=0. 70 cenl i- 111 tcr; at. yJd=O.7, Lz= L. 55 centimeters, )..,:.:::0. 1 centimeter. pectrum funcLion F,• .z(n) is u cd in eq ua.tio n (9). I J I VESTIGATION OF '1'URBULEI '1' FLOW II A TWO-DIME SIO AL e RA EL 15 2.0 improved in Lrumentation it wa possible Lo red uce experi mental scaLter by a considerable amount. The accuracy of § ~ l- tbe pre eDt measurements i believed Lo be within ± ] 0 o L ~f ~ z percent with the exception of value cO l'1'espondinO' to low (a.) 0---- o Ly o frequencies (n .8 --;;;- I EO .6 6.0 U T I ¢ . .4 - -() 0 Zero c ounting I--- ~ -~ ¢ --< 2 - -- r'l Spectrum - 00 Differentiotlon 0 !.O 4.0 ~ -I-- ¢ 0 Lx EO.8 :;-' § o L U.6 V- z ),.4 b--- ·.f o Ly / ~ ¢ 2 -- 2.0 wrr 0 I 0- >-- !.O 7 0---- .8 ~ 1--J.-- r-- (c) 6.6 R - ~ o ----.4 .6 .8 1.0 .. .4 0 12,20.0 (S-in. channeO 30,80.0 (S-in. chonnel) y/d --< 0 2 0 61,600 (5-in. channel) (11) R =12,300. (b) R=30,800. 0 .I 2 .3 .4 .S .6 .7 .8 .9 1.0 !.I (c) R=61,600. y/d J<'lGURE 21.- cale of tu.rbulcnce distribution. FIG URE 22. ~ ::\licl'oscalc mcaSUI'('ml:nts. ).'he distribution of Ax obtained by the differentiation 1.0 method shows the arne behavior a that of A1/ and Az (fi g. 22) . .8 l\: LL has Lhe characteristic maximum at y /d """ 0.7. ..\ 0 mea 1I1'e \\~ ment wore made for R = 12 ,300 becau e of the inaccuracy .6 ~ ~"II d ue to the large noi e-to- ignal ratio . ,~~ vld -o. 4 A a matter of interest the re ult of orne mea m ement of \\ ...... ~ ..... -. I . -~ .··yld -/.O Ax obtained with the zero-counting method. are compared -- 1- _ ,(.. . ~ . '\' &0.5 ~ t:-:::; :""------.. _!ll d -O./ .2 \ with those described above. (See fi g. 22 .) in e the validity \ \ .. I f-'- __ !I.1 d oO.7~ .. ---- of the as LlInptions involved in thi method i not clarified \\ ,, '\' ''~ 56 ~ t-- yet, these mea lll'ement hould be regard ed a preliminfln·. o -- - PECTRUM MEAS REME ors .5 1.0 /.5 2.0 2.5 3.0 3.5 4.0 45 1'; cm The spectra of the velo city fluctuation U '2 have been ob :Flr.l-RE 23.-R.·corrclation. R=30, 00. At vld=l.O, L , =1.3 cenLi mcLCrs, X,=0.50 ocnli· tained at various values of y/d acro the channel. With m tcr; at y/d=0.7, £,=1.2 ce ntimetcrs, X,=O.56 centimcLcr. I I l. f 16 REPORT 1053- A'l'JONAL ADVISORY COMMr r TEE FOR AERONAU'l'I C TABLE I.- MEASURED VALUES OF THE SPECTR 1\ [ OF u'z F~(n) Pre- (sec) quency (cps) ~= O L 20 0 r--i---t-t~,--+--~~++--~--+-+-W ~ = I. O %= 0.7 %= 0.4 x Jo-' 30 ------_.-- .--.----.-- "" ------.------rn i 6 !)il 10-.1 10-' {B IS 40 56 1 X LO-' 585 X 10-' X 664 X 60 1 50 -- .------.------. 519 fil 349 'If>4 ------. ·136 70 ------.--- -" ------.-----. e34n5 SO 273 ZII 252 30S 309 {254 00 " " ------.------272 100 IS6 141 LS2.5 22fi 222 { 157 125 ------.------.------150 9J.(; 140 C Og l .'iO 95. 4 77. 3 114 6 . ., 1i .1 ----_.-. ------.-. ------{ 74 .9 .';2.6 9i.2 G3. 6 200 68. 2 56. S { G3.6 . 6 :€ :n. H 64.9 I ~ 2 f----Mrl---t-t-t---t--+-t--KI ~ +___+__l_1__J._ 250 2:3. I 45. S { 3 .0 { :l3.9 2:3. .'; 32.2 44. 8 23.8 ~ ' 1 \ 300 34. Ii 2.5. 6 34.7 51. 3 21. 5 400 19. 8 15. 0 19. Ii 25.2 . 03 ('0~/ r---~~r-~~--4_--+_4_++~-+---+-+~ .;00 J . 9 8.09 12.25 15.5 3. 74 .8t--t----lH -+++--+--H-+-+-*,,------\--\--l---l--I (lOO i . .~ i 5. 8(; i . 6 12 . .'; I. 83 p 700 S.27 4.0 5.14 . 73 I. 06 . 6r--+-f+-\ -H-+--l---+-+++\\--+-~----l----1 00 3. 31 2. 67 3. 53 I;' 07 . 53 1 900 2. 2\l 1. 77 2.51 4. 04 . 352 .4 +-+-t-+--l--+-I-++-+-l--I-\ -I---l---l 1000 1. 50 I. OS I. 76 2.99 . 1 5 J-t--\-\ .764 . 6n I. 01 1.80 . 000 1200 1.100 . 295 . 2.17 . 379 . i87 . 016 1750 . 14 7 . 16 . 22 1 . 422 . 008 .2 f--~\ t-+-H--+---+-4++----\-\+----I----l---I----j 2000 . 0832 . 0.184 . Ion . 224 . 0039 2.'>00 . 0192 . 0186 . 0487 . 0870 . 00102 3000 . 006 2 . 006f18 . 0141 .031:1 . 000354 r.00~./r--t-~b-+-~-_+-_+-4-++_-~-+_~~ :\.;00 . 03196 . 002i2 . 00 192 . 00121 . 000133 .08 4000 . 0008.1.1 . 00074 .0021 . 0043.1 . 0000664 \:;00 ------. 000608 . 00157 . 0029 ------.06r---r-~~-++---~-4~_++_--~~,-~ f>OOO . 000170 0 . 000.12 . 00117 ------1\ '>500 . ooofi2 ------. 04 r---t-~~~~--4_--+_4_++--~~~-+-H . 000252 . 0003J:3 6000 ------~ ------7000 ------. 000109 ------.021--+---+-H---t~ +--+--l-++-I-H \'-+-+---l-- correlation coefficien t is a F ourier transform of the spectrum !OOO~ . O~O~-~--~-L~/~O~O--L---L~~/O~O~O~~--L~-IOLJ,OOO fun ction it. fo ll ow (ll a t (1000) (10, 000) n, cp s Flfll "HE 2·1.- F' rcqucncy spec trum of U'2a t chann el c nLer. Seal" valu es wiLhin parentheses corres po nd to Curve on left. Tbus Ax can be co mputed from Lhe dis tribu tions of 50 n2F ;" z( n) . Thc calculated \"a lue check within a few per ~'" ce nt wiL h dircct meas urements (fi g. 22) . 40 Ir .. !lI d R~~ Y NO L O S NUMBER E F FECT Irr-<>-- o 1.0 '~ ,]0 i ~ 0 .7 Fig ure 7 shows the di tribu tion of mean vel cities plotted ItP lJ) ~ \1 .4 6 .1 in the form of " fr iction velocity" u/U, again t " friction I'€-" ;>--."~ ~ ~ ~ 0 .005 dis tance parameter" y U, /v. In this fo rm th e profil e are 16 '~ ~ indepcnd ent of the R e ~-n o ld s n umber Uod/v and fo ll ow Von \ ~..... f'-...... K arman's 10gar iLllmi c law: '"~ '---- ~ ~ k::- >- r-:--t-- ~ >- ~ - o 500 1000 1500 2000 2500 3000 n, cps F,,"; ,'RE Zo. - Spectrum distributions across cbannel. R =30, 00. The constan ts A and Bare 6.9 and 5.5, 1'e p ctively. Comparing the pre ent m ea LlTemen ts with previous one it For a comparison of the energy spectra at diff eren t poin ts is to be noted that beca use of the very mooth wall urfaces m the channel, figure 25 show the dist rib ut ion of n2Y;;;2 ( 1I ) Lls E' d , the value Ii i la rger tha n t hat of ot lI er in vestigator . in tead of that of F u'2( 1I ) as a Junct io n of the freq uellc.\". \la king all o\\-al1 CC for this eA'pc t, ])l ea Ui"emen t a re III T his method of presentation emphasizes cncrgics corl'espond raid.\" good agreement wit h lhose of D 6neh (reference 4) ing to the higher fl'equen i e hut gi\'es a clearer over-all and R eichar It (refe rence 9) . i\ikura cl se's channcl res ults com parison of the differen t set of mea urement . The (refe rence 5) cl iffeI' from all othcr exi ting experimen ts. It i. cbaracteri t ic maximum of the eli tribution of Ax immediately pertinen t to m ention that for the R e ~' no ld s number range in becomes ap paren t in this fi gure. From th e fact that the q uestion (R< l OO ,OOO ) the lope of the logar ithm ic mean- -----~j I TVESTIGATION OF 'l'URBULE T FLOW I A 'l'WO-DIMENSIO TAL CHAN EL 17 velo ity di tribuLion (the consLant 1) both in pi pes and sis Lent varia ti on wiL b velocity. Furthermore, measurements channel i larger Lhan Lha L eS Lablished by N ikuraclse's pipe in a I-inch 'hannel give a val ue fo)' L , five times lower and a m ea m em ents (A=5 .75) . ratio for L y om ewhat larger Lh an Lh e values obtained in The R eynold number has a definiLe in Au ence on th e the present investigation . ,velocity fluctua ti ons also. Along IDO t 01' th e cr oss section T he varia tion of A depends, of co ur e, on tbe velociL)7 and where the influence of the vi co ity is negligible the fluetua channel width. T he values of A decrease with incr ea ing Lions 11,', v', and W' decr ea e slowly with increasing R eynolds velocity; h owever , tbe varia Lion of A wilh channel width is number . less Lhan LhaL of L. Ver.,Tcl ose to tbe wall, accord ing to Pra ndtl' hypothesi , all vcloc itie of Lhe forID VelociL.v/UT musL be a fun cti on of the FULL Y D EVELOP ED C H AR ACTER OF TURBUL E CE friction-distance parameter only . I t has alreacl .,' been The fl ow in th e channel i called full)' develope J if tbe pointed ouL th a t tbe m ea n velocily obeys this similariL." variation of th e m ean values of the velocit)- and th e m ean law. Fig ure 26 sh ows Lh e dis Lribulion of 11,' / UT a a fu nction squares of th e velocit)T flu clua tion with ,r a re very m all. 3.0 That Lh e m ea n velociL.,- profi le docs n oL val'.'- down tream i evid en t from th pre sure-g radien t m easurem ent (fi g. 19). 0 '2 25 The gradient in J.. of '/1 was m easurecl on Lh e axis of the o 'b c 0 0 cbannel. It was fOUll I that u'" wa indeed decr ea ing wiLh 20 Q 1: . The grad ienL, however, wa very small a compared A 0 00 ii' A A A Q;) 0 0 <> . h 1 oJ? Ur 0 B 0 Wit --: /.S " 0. p 0 A OJ 0 0 0. 't A " "",, 0.01 .!. 0]) 12200 (I-in. channel) ox <> 0 1'1 0 0 /2.300 (S-in. channel) 0 0 .5 0 30.800 (S-,n. channel) H ence for all practical pur po (" 011' 2/0.1' can be neglec ted , A 61,600 (S-h channel) A o I I I I I K 0 measurements have be(' 11 made ('o l]('(' rn iI JO' OV' I" /o,!' , o 2 4 6 10 20 40 60 100 200 400600 1000 since the ('alter in lhl' value would ('OVe r a n." efl'e(' l. y Ur/v H owever, th ere i li llIe douh t tha t OU'V' /OJ.' i of th e same t J1~ I C; tORE 26. - LogariLhmic representat.ion of ve locity fl uctuations 'u , order a ou,t/o.t and th at the use of equations (3a) al) I (:3 b) therefore ju tifi ed h ere. of y UT / II for the differ ent fl ow co nditions inves tigated. The following remarks can be made with r efC'l'enee to this fig ure: EN E RGY BALANC E I N ~' L UC T UA TI G FIELD (1) F or values of yUT/ II < lOO , YHlues of 11,' / UT co rre ponel ing 1,0 the variol! cases indicate similar h eh avior. T h ey The energ.'- equation for a t\\"o-cli.m en ional channel has reach a m aximum at abou t yUT/ II""" 17. the forl11 given b~ ' equation ( ) and i valid throughout th e (2) It i believed t bat the sim ila rit." is arL ua ll.,- m or e ero eelion of tb e channel with the exception of a m all compleL than indicated in figllJ'c 26. B ecause tll(' micro reo- ion ncar tbe wall. The lerm T c1lu on the left sid e o r equa- scale is not kno\\-n in r eo-ions vel) ' cIo e to the wall no leng th ey corrections co uld be a pplied . These correction would of lioll ( ) co n 'e pond to the en(' rg," produced by the sllearing co urse b e apprec ia bl.r higher for tir e Lwo high-velocity fl ows t ress(' a ncl il ean be obtained di rectl.,' from the measllre (R =61,600, 2d=5 in . a nd R = ) 2,200, 2d= 1 in .) and wo uld m ents o /' T a nd from til e m ean " d oc iL." profil e. T Ile second th erefore b ring lbe various di tribulions of 11,' / UT d o er to . h· I, f term 0 11 tl1 e ng t J.L (0O11J'j t' ) (oOnt'Jj ) expre ses tue am ounL 0 gether in the r egion in que tion . (3) The efr ect of vi co ity i m or e pronoun ce I on Lbe energy lhaL i b iug el i ipa lecl beau e of Lhe breaking flucLuaLing q uantitie tha n on the m ean veloci t.,-. down of the lar o-e r edd ie Lo malleI' ones. T he term m ay (4) T aylor poinLed oul in 1932 (s upporting his a rg um ent be wriLLen expliciLl.,' by F age and T ownend' ultramicroscope m ea urem n t ) lhat 11,' /u and w' /u approach a finite value at the wall (reference 18). I t follow from th imilariLy law that this value hould be an ab olute constant independen t of 'the R eynold number. Fig Ul'e 10 indicates this to b e true, the con tant being (11,' /u)v=o"""O. l . It i of con iderable int r t Lo di cus the variation f lbe T be probl em i lo expres these fu nclion in terms of calc an d microscale wilh R eynolds number. F or fl OK be easil ." measurable q ua nlll ie In Lhe case or isot ropie hincl o-rids \\-he)'e th e tu)'bulence is i otl'opic lhe scale i in t urbulence T aylor olne! tJ) (' probl em by introd uc in g the dependen t of the m ean velocil.,- and depends on tlle m h m1 1'0 calc of turbulence A a nd oblained 1'0 1' the di ipa tion ize of th e grid. imilar b ehavior wa fo und for Lh e channel fl o,,". Figure 21 h ows the di lrib utions of L y a nd L z fo r (J 0) difl' eren t velocitie . The e distr ibutions ind icate no co n- ( T- 18 REPORT 1053- '\T101 AL DVI ORY OMMITTEE FOR AERONAUTIC IL is aLtempLe 1 hcre Lo carryover hi analy is LO the case of Lhe channel. T aylor obta ined a s imilar di lribution of of channel fiO\L T h e following a umpLions have to be the dissipated energy acros Lhe chanllel (reference 20); made in thi connection: h owever , hi numerical magnitude are too high inee he (a) The gradient of tbe velocily fluctualion i mall in ubsLituted in Lbe i otropic relalion, in quatiol1 (10) , Lh all three direction. With L11 excep Lion of the feaioD n car values of Ay, w hich t urn o uL to b e Ie than those of Ax the wall Lhis is ju tifiable from the m ea urements. and Az. (b) ince o nl .\~ correlation of u' have b een mca ured, it From the known distributions of the e n e' J -g,\~ production is assume I lila t the correlatio n of w' as functions of ~ a ncl cli ipation the diffusion of e n erg ~' i ea il~ ' calcul ated (Von IGl rm ~1, n ' J-funetion) have lhe same curval urc al the from equation () . It houlcl he pointed ouL that hee-a u e origin (Y= .% = O) as tbe co rrelation o f u' has as a funcl ion of the approximation involved in Lhe calculation of the of .\; tha l i , dissipa tion a nd clifru ion term , Lbe cone-hi ion derived from lhem arc m ore or Ie of a qualiLative nature. It i 02Bz:') =(0 2R;:') = _~ cell from fi a ure 27 that at y/d "", 0. 7 Lhe din'u io n term j ( oZ 0 aX 0 Ax ze ro. It wa aJ 0 poinL ed 01l t ('adier LhaL in thi region FurLhermore thf' corrclaLioll s of 'V' as fU ll c lion of 'X a nd ~ tile' /(y-co rrclaLiolls how a CO il iderable cba nae in ha p a nd the co rrf'1alions of w' as functio ns of X (Von IGum~Ul ' s inclieaLing a shift in ene' rgy from Lhe lower to hig her g-functions) have the a me curvature at lhe origin as Lhe frequencie of Lhe velocit y rlu cL uations. The e Lwo facL correlation of 1/' ha a a fundion of .%; that is, ugge·t lhat Lllf' ellergy difrusioll i associate'cl rna inl.\' with the low frequencies of thf' flu ctuations. T he eqll aLiolt ('XPI'(' 'in g tbe balance of Lhe tbree forms of nerg,v fumi he tbe followi.ng picL ure of Lhe' Lurbule nL n \ fi eld in tbe cbannel where viscous dis ipation i till neg l~ ina ll~' , ligible: T \\,o plane pa ing through poinL where Lhe dif Iu ion of en rg,\' yanishes divide the channel fl o,,' into L1n'ee PHrL S. The n1!dclle region, l he width of w hich i of Lite order "vVith thesf' u'sumptiolls Ihf' dl'l'ivativcs of the Huctuat iolls or L r, re(,eives 11l 0st of iLs energ,\' b.\' cl ifru . ive arLion and this could be cx prf'ssf'cl in Lerms of the m eil s ul'CCl "ailles of Ar, eIH'rg,y is dissipa Lc'd bere a L 11 ('011 ta n l rate. 1 n Lhe two ou t- ide region ,1,11 th ree energ,\' [('I'Il1S increase rapidl.\', the pro Ay , and Az: duction Lerm being Lhe dominant one, ancl Lheir inleraction is more involved. T his picture of th e' flow field i only of a de criptive nature. T h(' purpose' or furL her inve tigations should h to obLain At Lbe middle of the cli an nc1 l V turned ouL to be 2.62 erg information OJI l he devclopme'nt and m echani m of uch per cubic centimeter per econd for R = 30 , 0. U in g the an energ,y halance. i otropic rela lion LO C AL LY ISOTROPI C C H ARACTER OF TURB LE T C H AN EL I"LOW T he conee pL of locally iso lropic lurbu1en e obtain ed by K olmogorofI: req uire t he mailer eddie in turbulent flow to approach i olrop.\-. , mall pr e'd di es arc Lhe' ones with the \'alu f' 2. ,4.:3, a nd 2.24 erg pel' cubic centimeter pel' leng th dimension l small e-o mpa red with Llw calc of turbu s cond were obtained depe'nding upon whether Ax, Ay, or Az lellce L. T h e' smalksL chill'ilcLeri tic length in turbulenL was u cd. (For t he value of u't the algebrai· m ean of the m otion j K olm ogoro Cl" 11 clcfin d as q uares of the flu ctuations wa u cd .) Figure 27 show the cI istribu tion of 11 ' in the ce n tel' region 40 where E is h e total dissipated energy. learly, to approach ", 30 '\ locally isotropic ondition it i nece ary that U) E: (» ~ \\1\ ,- 'Production ~ B' 2 0 , ~ , , It follows from this h ypothesi that be~T ond uffici ntly ;;:. hig h freqllen cie (of th e order of n:\') no correlati n Di ~ 11,' / 11, 10 ~ I l ~ exi t b eL\\'een the compon ent of the veloei t.\, fluctuation . r QJ DiSSiP o tio n~ 1 ~ tS Fiaul'C 2 h o\\- the mea LU'ed pecLrum of 'l/'2 a c mparcd "'" ~ I 0 I ---- with the peeLrum of 1/'/.' at y/d = O.4. It j en that Diffusion.---- the hear p cLrum tend Lo z 1'0 at a frequency of about I I .I .2 .3 .4 .5 .6 . 7 .8 .9 1.0 ] 500 cycles per econd whil F";;;2 (n) till ha a detecLable Ij/ d val ll at 5000 cycle per ccon 1. Thi rc ulL verille K ol FIGU RE 27 .-Turbulcm-cncrgy balance in center region of channel. R =30, 00. mogoroff' a umpLion. It h owd be m entioned hat the ------~ ------~------~------~ INVESTIGATIO r OF 'l'URBULEN'l' FLOW IN A 'l'WO-DIMENSIO AL CHAr NEL 19 35 tli lriblltioll as mCH , lIred I1 rrt' ar(' ill good agrt'emell t with }b general [JwoJ'elical ('xlwc:tations. 30 ~I ea m ement of lht' 1mbulent fielel show that the bol-wire ,. te hnique i well en ough developed to give consistent 1'e ult ~ 25 fA 1\ for the intensities an correlaLion functions. (f) 1 ---<: 2 b \ o n F u.z{n) Dt'lailed m ra Ul·t'm enl oJ thr vt'loeity flu ctuations in the 2O \ 2 dil't'clion or the flow u' w('re carried out well within the 'Th '\ o n F u·v·(nj ~ laminar s llblayt'l'. JL was foull I lhat Ul(' imilal'iLy law for I:: /5 \ \) ?i'/ UT , wlJCl'e U T i hic Li on velociLy, llOld fairly well in the § \ \ vicinity 01 Lhe wall and as a con equencc lhe magnitude of ~ /0 ?i' /u, wbere U is local mean velocity, approaches an ab oluLe I~ con tant at the wall (approx. 0.18), the value being inde ~ 5 \0'Q, ~ I:: ~ pendent of the R eynolls number. The magnitudes of tbe ~ -----. '- velocity fluctuation normal to thefiow in the laLeral and 0 v verLical directions Z;' and ·il)' arc nearly lhe ame in tIl(' middle region of the ('hannel, 'w' inc]'t'asing more rapidly -50 /000 2000 3000 4000 5000 Loward Lh e ' all. l n. cps The meas ured microscales of lurbulence Ax, Ay, and Az con I<'lI auE 2R-Compari;o;on of Sp(\Clra o f U'2 tlnc] iL'II' at.. !lld=OA. sistently show a m aximum at y/d ""'0. 7, where y j lateral distance and d is bal( width of the channel; tbey incr easc ab olute value of F "'v,(n) are not 0 aCC LU'ate as tho e of proporLionally wiLh ?i', indicating a constant raLe of energy F';;;2(n) since they represent the small difference between dissipation IT ' in the cent('l' portion of Lhe channel since Lwo large value of hoL-wire ignal. If' cx: U'"/A2. J Lito be noted tba t evell though local isotropy was The s ale of Lurbulence L y and L . in the center region are hown to ex i l in lhe ch annel flow, Lhe value of Lbe variol! in JependenL of R eynold number and depend only on tbe vorticity terms (U//A/) differ appreciably, indicating Lhat th e channel widtb. The micros ales, however, show a depend ri lative magnitudes of lhe e term do not constitute a ency on tbe R eynolds number. se ll itive Le t for lhe exi Lence of local i otropy. Tili is From the cal ulated magnitudes of Lbe nergy produced (' vici enL from the comparison of tb distribution n2F,,'2(n ) by Lhe sh earing sLrt's es and of th e di sipated energy a and n2F;;-;;(n ) (fi g. 2 ) . It is apparent from the fi O'Ul'e cit' cl'iptive picture of Lh t' n er O' ~T cliffu ion in th e cenLt'r region of the channel i oblained. thaL a large parL of the conlribution to 1'" n2Fu' 2( n) dn The p t'clrum m ea Lll'ements of Lbe u'"-fIuctlla.t.ions tend lo indicaLe LhaL F;;;2 (n) b ehaves as n-7 over a large band in comes from a frequen cy well b elow] 500 cycles per econel. lhe high-freqLlency region (where Fu'2(n) is fraction of Local i olropy, accordinO' to the sh ear spectrLlm , exists onl y turbulenL energy associaLed 'with bane! w idth dn and n i above a frequency of 1500 cycle p er second. frequency). From the comparison of Lhe spectra of U'2 and CO CLUm G REMARKS of the turbulent h ear the exis tence of local isoLropy in Lhe hannel How is verified. The m ea lIremen ts presented h er co nfll'lTl the genrral eoncrptions eo ncrrning the m ean velocity profile in a t lIrhll G U(1G J ~ N H mM Aj·aw A 'rICA I ~ L ABOR.\TORY, lenL cbanne1. The extent of Lhe laminar ubI a)'el' , the C.\ LIFORNIA IN '1'ITU'J'lD OF T ECH NOLOGY, velocil)T profile in tbe ublay 1' , and Lhe over-all velocily P A. ADI ~ .\ , ALIF., Septem ber 1, 1.94.9. ~~1------~ __ ~----- 20 REPORT 10 53- r t\']'I ON t\L t\DVISORY COMMI'l'TEE FOR AERON I\. TI REFERENC ES I I. I( ol'f\szna.,·, L.: Calibration and :'Ir ('a~ m e mrn L ill T urbulence He s<'arch b.,' thr llol- \\' ire :'I e thod . XACA 11 30, J 9H. I. -I-':ollllol!;ororr. _\ . : T il(' I.o('al Slrue(u r(' of 'J' 1IJ'blllcn('(' in fn co ll1 - 1'1\ [ p r ('~~ ibl c \ - i~{ ' o ll " Fluid fo r \ '(' IT Larl!;(' l l(' .' · nold ~' ;\u lll b c r~ . 12. I.ippmallll, II ans \\'ol fgang, and Laufcr, .Jo hn : In " csl il!;a t ions of Comp. rc nd. , acado sci. 1.' HSfi, " 0 1. 30, 11 0. -J, Iccb. 10, In'! I, pp. 1'rr(' Turbulrll l :'Il ixing. ..'\.\ ' 1\ T:\ .12-7, 1\)47. 30 I 30.5. 13. TO\\,ll:- U S , GOVERNMENT PRINTING OFFICE: 1952 l