• 1
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
REPORT 1040
SPECTRA AND DIFFUSION IN A ROUND TURBULENT JET
By STANLEY CORRSIN and MAHINDER S. UB EROI
1951
RfPRODUCED BY NATIONAL TECHNICAL INFORMATION SERVICE US DEPARTMEN T OF COMMERCE .. SP RINGF iElD. VA. 22 161 r------.
REPORT 1040
SPECTRA AND DIFFUSION IN A ROUND TURBULENT JET
By STANLEY CORR IN and MAHINDER S. UBEROI
The Johns Hopkin: niversity
I
I I \ \
REPRODUC EO BY NATIONAL TECHN ICAL INFORMATION SERVICE u.s. DEPART MENT Of COMMERCE r SPRI NGfiElD . VA . 22161 I L ational dvisory Conlmittee for Aeronautic
Headquarters, 1724 F treet TT'., 'Washington 25, D. C. roated by act of Congre approved March 3, 1915, for the upervl ion and direction of the cientific tudy of the problems of fligh t (U. . Code, title 50, ec. 15 1). It member hip wa increa ed from 12 to 15 by act approved March 2, 1929, and to 17 by act approved May 25, 194 The member are appointed by the Pre ident, and serve a uch without compensation.
J EHOME C. Hu. SAKE H, Sc. D., 1\Jas achu tts Institute of Technology, Chairman
LEXANDEU WETMOHE, Sc. D ., Secretary, mith!;lonian In titution, 1fice Chairman
DETLEV W. BRONK, PH. D., Pre ident, John Hopkin Diver- HON. DO NALD 'vV. Nyuop, Chairman, Civil Aeronautic Board. sity. Do ALD L. P UTT, lIajor General, nited tates Air Force, J OHN I-I. ASSADY, Vice Admiral, United 'tates 1 avy, D eputy Acting Deputy hief of Staff (Development). Chief of Naval Operation RTHUU E. RAY 10 'D , Sc. D ., Vice President, Engineering, EDWARD U. CONDON, PH. D., Director, National Bureau of Douglas Aircraft Co., Inc. tandard . FUANCIS VlT. REI UELDE RFEU, . D., Chief, United tates HON. THOMA W. S. DAVI , A istant Secretary of Commerce. VlTe ather Bureau. JAMES II. DOOLI1."l'LE, c. D., Vice P re ident, hell Oil o. R. 1\1. HAZEN, B. S., Director of Engineering, Alii on Division, G OUDON P. SA VILLE, Major G nerai, United tat S Air Force, General Motor Corp. D eputy Chief of taff-Dev 1 pment. WILLIAM L ITTLEWOOD , 1\1. E., Vice Presid ent, Engineering, HON. WALTEU G. WIIl'l'MAN, Chairman, R esear ch and Develop American Airlines, Inc. ment Board, D epar t ment of D efen e. THEODOUE . LON ' QUE '1', Rear Admiral, United tates Javy, THEODORE P . WUIGfI'l', C. D., Vice Pre ident for R esearch, D eputy and A i tant Chief of the Bureau of Aeronautics. Cornell University.
HUGH L. DRYDE " PH. D., Director JOH F. VIC'l'OHY, LL. D., Executive ecretary
JOH . W. HOWLEY, JR., B. ., Associate Director for Research E. H. H A~ I BERLIN , Executive Officer
I'll" UY J. E. RI~ lD, D. Eng., Director, Langley Aeronautical Laboratory, Langley Field, Va.
ll'I'H J . DEli'UA 'CE, B. ., Director, Ames Aeronautical Laboratory, Moffett Field, alif.
EDWAUD R . HARP, C. D., Director , Lewis Flight Propul ion Laboratory, Cleveland Airport, Cleveland, Ohio
TECHNICAL COMMITTEE
AERODYN AAIJC OPEUA'l'IN G PROBLEMS POWER P LAN'l'S FOR ArnCRA FT TND STRY 0 SULTI G ATRCRAFT ONSTRUCTIO '
Coordination oj Reseal'e lL Needs oj l1Jilitary and ivil Aviation Prepamtion of Hesea1'ch Progmms A llocation of P roblems Prevention oj Duplication Consideration of Inventions
LANGLEY AEUONAU1'ICA L LABORATORY, AAIES AERO.NA 'rICA !J LABORATOUY, LEWI FLIGHT PROJ'UL ION LABOUATORY, Langley Field, Va. Moffett Field, alif. Cleveland Airport, Cleveland, Ohio Conduct, under unified control, for aU agencies, of scientific resem'ch on the fundamental problems of flight
OFFICE OF AEuo AunCAL I 1'ELLIGENCE, Washington, D. Collection, classification, compilation, and dissemination of scientific and technical information on aeronautics II
l ------/ REPORT 1040
SPECTRA AND DIFFUSION IN A ROUND TURBULENT JET 1
By TANLEY CORRSI N and 1JAHINDER . UBEROI
SUMMARY (a) An aLtempt to e Labli h Lhe pre ence or absence of local . In a round tubulent j et at room tempemture, measurement isoLropy, (b) a compari on of velocity- and temperature oj the hear correlation coefficient as a junction oj jrequency £luctuation field when the over-all boundary condiLion on (through band-pa s jilters) has given a mther direct verijication mean velocity and LemperaLul"e arc effeeLively the allie, and oj Kolmogoroif'slocal-isotropy hypothesis. (c) a tudy of the diffusion of heat from a local (line) ouree On e -~imensional power spectm oj velocity and tempemture in Lhe turbulent flow. fluctuatwns, mpa ured in unheated and heated jets , re pectively, . The only specific experimental verification of local isotropy have been contrasted. Under the same condition the two Cor III a turbulent hear flow to date wa bv Town end in tbe responding tran vel' e correlation j1tnction have b ~en measured plane wake behind a circular rod (refere~ce 3 and 4). H e and compal'ed. found Lhat the skewnc and 11attening factor of the prob Finally, measurements have been made oj the mean thermal ability density of au/at in the hear flow ar very closely wakes behind local (lin e) heat sources in the unheated turbulent equal to tho e in the (eff ecLively i otropic) turbulence far jet, and the order oj magnitude oj the temperature fluctuation behind a grid. incc difl'crenLiaLion empha ize the ilia-her has been determined. frequencie , hi mea urement how , in e ence, thaI, b the value of certain sLaLi Lical quantities related 1,0 th e smaller INTRODUCTION eddies in a shear flow are Lhe arne as the value for the smaller eddie in isotropic turbulence. H e al 0 found th e microscale At tbe present time there apparently exi ts no stati tical of u in Lhe tream direcLion Lo be nearly Lime Lhe micro- Lheory of Lurbulcnt heal' flow. On ignificant theoreLical -/2 cal of v in that direction, a relation which i exactly true consideraLion has been proposed: The hypoLhe i of local for i otr-opic turbulence. i otropy, originated by Kolmogoroff (reference 1 and 2). ntil recently, only mean-velociLy and mean-temperaLw·e Kolmogoroff ha ugge ted that the fine tructure in turbu eli tribution were mea ured 1,0 provide a compari on of the lent heal' now may be i otropic; recent experiment of tran fer rates of momentum an l of heat in Lmbulent hear Town end (ref rences 3 and 4) in a turbulent wake seem to flow wiLh over-all heal, tran fer. The previou reporL in this verify Lhi hypothesis. roun i-jeL inve Ligation (1' f l' nee 10) included a beginning on In view of Lhi situation, th variou experimental re- Lhe problem of diJ."ect comparison of the velocity and tempera earch~ in lh~ field follow two oour es: Fir t, they attempL Lure 11ueLuation a well a om mea urements of velociLy 1,0 ve~'ify or dIsprove local isotropy; seconel , they try in all temperaLw·e correIa lion. The flu cLuation in a warm conceIvable ,~ay to make mea urement that may hed light Lmbulent wake have b en ludied by TO"~1 end (reference upon the ba IC naLure of the turbulen hear llow 0 that the ll). foundation for a ucce ful theory can be laid. ' p to Lhe pre ent Lime , however, th re appeal' till Lo be no During recenL year iL bas become videnL that mea Ul" - , ucce ful hypothe j Lo account for lll(' well-known fact thaI, ment of th(' inten ity of turbulence alone cannot provide heat (and othcr calar quantilie , like material) i diffused suffi ciCI~t information about tb e taLi tical and dynamical more rapidly than momenLwn in a tmbulenL 11ow. Thus, propertlC of the flow field. Such quanLiLi a correlation more detailed tudy of Lhe f1u cLuaLion eerned in order . specLra, probability densities, th various term in the tUl"bu~ The fn." t real analy i on lhe liITusive proper y (for scalar lenL kinetic-energy balance, and so forLh may be expected 1,0 quan Litic ) of a homogeneous tmbulent field was Taylor' reveal many e enLial feaLme of the problem. . well-known work " DiHu ion by onLinuou i\Iovcmcnt " The Lypc of turbulent shear fiow that .ha.v come lU1cler (referen e 12). . do e exp('rimenLal Cl·uLiny are Lhe boundary layer (referenc(, The mean thermal wake behind a line OUl·ce of heal, in a 5 and 6), Lbe planc channel (reference 7), the plane wake flowing i oLropic tmbulence ha been carefully mea UT d by (references 3 and 4), the plane ingle free-mi.;'{ing region chubau I' (refer('nce 13) in the region clo e to the mce, and (referenc(, ), and the round jet (reference 9 and 10). Th by immon (mca memcn L l' ported in reference 14) over an pre ent work is a continuation of tha I, reported in referen ce extended range. Taylor' theory of diffusion by conLinuou 9 and ]0. movement i cLirectly applicabl to the diffu ion from a line The general objecLive of thi inve tigation have been to omce of heat in a homogen ou tmbulent field, and he ha learn omeLhing more about the £l ow in a fully developed round turbulenL jet and about Lhe heat tran fer in uch a I Supersedes :\'ACA T:\, 2124, " peclrums and DiITusioll in a Round T urbulent Jet" by flow. 1'he work reported here ha fallen into three phase : Stan l ~y Con·sin and IVfahinder s. Ubcroi, 1950. 1
I L 2 I {I~ POR'I' lO-lO- XATlONAL AD\'U:iO RY COt- l iVIiTTEE FOR AERONAU'I'l :-;
made a generalization Lo pC'rmit application of th e m eth od in (J maz maximum m ean tempC'rature clift rence at a a dC'C<1ying iso lropic tmbulC']1 cC' (rdC'rC'llce ]4) . c ro section
Tn a, lu)'lmlenl h ear Jl ow, Ih C' only P1lhli sh C'cl ll'wasUJ'C'J11t'n L 00 maximum mC'an trmpe rature in jet at orifi C'C' of l1 \(, lh rrmaJ wake of a local SO lll'('(' C'('Tn Lo h C' lhos C' o f {} instn nta,l1('oll. tC' mp e r aturr flu ct u ation SkJ'aJ11 Inci and Schubau C'J' ill a lurhulC'nl houncla , )'.\~ In.\·cr .z (fJ = O- O) ThC' e w C'rC' rC'porlC' I b.,~ lhC' C'x perirnen I ('1' (rde]'ence 15) and fJ' = fJ Z b!' Dr!' cl C'n (refC')'ence ] 6). The LC' J11p(')' alure distr iblltion tim C' aero lhe \\~ a ke in h ear Jl ow is dcc icl cell~T kew. On th e oLhel' e voltagC' flu ctuation hanel, in lhe i otropie turl)ulen ce, it i 10 all inten t and pm CY. , f3 r ll s itivity of a diagonal hoL-wi rC' to 11 a nd po e a Gall ian ClUTe. Of coW" e, in a h omogencou fidd to I', r C' peetiwly this nU've (measured close enough Lo lhe source so lh aL Ilw B u. hear correlation coeffi cie nt (uv/u'v') Lagrangian correlation cocfficiC'nt i still e ff ect i ve l !~ unit~T) i s 1/" ", l'n ins tanta nC'o us contribution of u(t) and vet), impl.\T Ih C' probability den i l \ ~ of lh.C' laleral yelo ci t !~ flucLua r rspC'c tively , ill a nalTO\\' freq Lleo cy band Lion. This l'C'lation ma)~ b e rouO'hl.,' Lrue for sh ear fl ow a of n ominal frequency n ; in a Fouri r S Tie well, in which case th e contrastin g r esult mC'ntioned abovC' el i e u ion, t he nth harmoni c of p eriodic would J11 0a n LbaL Lh e pl'o bahiliL.\' clcn s il.'~ of vet) is kew in 11(t) and v(t), respectively sh ear flow, but Gau jan in decaying iso tropic 1m bulenco. u,/ = U/ X 0 mea moment l1 ad been made of th e fluctuations in thesC' thC'rma l wake , ancl it wa fC'li thaL some information on Vn'= V; the nalurC' of thC'se m igh t hrlp fmlhC'l' a gen C'l'aJ unckr ,,fl ur shC'al' e l'l' C' lation coctrwiC' n t for a narrow band Landing of th e difl'u iv e proce . of frequencies (u" Vn/un 'vn'); th i function referred to a th C' (( hC'a r- orrC' lation This illwsligalion was conduelC'd at the Acron91ILi cs D epartment of The John H opkin niversity under lhe sp etl'um" or , b rie fl y, the" lwar pec trurn" spon orship and with Lh e finanC'ial a is tancc of th e KaLional phase angle
Advi or~~ Committee for Aeronautics. The auth ors would Il. F "L ' tl F" in Fou ri er eric an alysis, nF"=O,,z/ i!; a/, like to acknowledge man.\~ s limuJa LinO' con vcrsations wi th Dr. L. 8 . G. Kova zna)-, and to Lh ank Dr. F. H , Clauser for nFv= bn2/i!;b n2, whC're an a nd b" a r C' Fourier his h Ipful critici m. D onation of the h ot-j e t unit by Dr. sC'riC's co ffi eient B. :.\[illi klln, D irector of the Guggenheim Aeronautical 71 cyelie fr q Llency in gen ral ; in particular, Lab raLor .'~ at Lhe alifornia In LituLe of TeclU1olooy , i magnitu Ie of radial coordinate in t hr€'e• gl'eatl.\~ appreciated. :'\1r. Philip Lehowilz h elped to sct lip dim n ional frequency p ace much of Lhe laboralory equipment. cycli c IrC'qu en cy or one-cl imensional , p ec ka of u(t) and fJ(t ) SYMBOLS wave-lllffi1ber magnitudC' fo r t hr ee ~dim e n ional p ctra (27rn!U) d diame ter of orifi ce (1 in.) wavC'-number maO'ni tude f or oll e-clim C' o ional x axial distan ce from orin cC' pC'ctra (27r1l JU) T raclilll di stance fl' m jC't axi t1 rrferencC' constant with dim C' n ion s of Wllve llxial compon ent of mC'all Yclocity C numi C' r V radial com pon ent of m ean vC'loeity onC'-ciimC' n ional p ower sp C'cL rlllll of n (t ) m W~ tan gential compon en t of m ean Yeloc i ty t<' I'IllS of \\'aVC' numbC'r maximum U at a C1'O . ection (on llx i ) Uma:r onC'- ciinw ns iona l powC' r prciruill of net) In II axial componC'nt of III tan tan C'O Li yeloc ity tC'l'm or frC'quC'ncy fl lletliation thrC'C'-dimC'n ional power I C'druill of velocity v radial component or to tan taneO Li vel city fl uc tllation fiuc tuation one-dimensional po\\~ e r Sp C'C trUn1 0 r fJ (t) J n w tangen tial componen t of instan taoeou veloc tC'l'ms o f \\'a\'e number ity flu ctuation ,= one-ciim C' nsion al power spC'ct runl of fJ (t) 1Il u' ='Y U2 tC'rm of frC'quency Z v' = V G(k) thrl'C' -ciim C' osiooal power spectrum of tem o instan tan eou tC'mper ature difl' renee (m ea - perature fl uc tuation tran YC' r e corrdation function of m ea ured uJ'ed abo ve room temperatuTe as 7'ejerellce) R V u o m ean tempera turC' cli[rC' rC' ncC' (m C'as urr ci abol'e syrnm C't ri cally about jet axi (V(uz!UY) rOom temperatlll'e as 1'eje7'ellce) longi tu cl ina1 correlation f une tion or U t rn.nsy('J' C' eorrclation function of 1') m rasurC' d
' Wicghordt bas recenlly measured the di tTu sion or h at rrolll a local source in a turbulenl symm ctri cally about .i et axi (tJ\fJ 2(J2) boundary layer (rererence Ii) , but his sourcc wa s flu sh with the so lid surracc and thus he was longitudinal cal C' of u-£luctuations studying a dinerenl prohlem, that i<. onc more dircctl y rclot.cd to micromclcorolog ica l conditions. I ngitllclioal seal of fJ -fluctuation
I , I l ~~j PECTRA Am DIFF IO T IN A ROUND TURB LENT JET 3
latentl calc of ~l-flu ctuat i o n s jet total head in lhe nmge from 3.5 lo 5.0 lnche of water. lateml scale of t?-Iluctuu tion In free tmbulent flow there i no deLectable efl'ect of jet longitudinal micro calc of u -fiuetuation R eynolds number ove ], a much wideI' range of R e.mold lateral microscale of u-rtuctuations number th an till . lateral micl'oscale of t?-Illletuation vVh cn the jet wa run unhealed, Lh ere was a li ght Ll'mper longitudinal micro calc of t?-fluctuations atme rise Llu'oughLh' hlowe]' and duel. In th e meaSID'e distance downstrenID from local h eat Ollree m enL of th ermal wake b{'hil1d a local h eat o Ul'CC , corre('lion in x-direction for thi ambienL-tempera t lIJ'e field wa ncee ar.'". For all lateral distance, perpendicular to source line, hot run , the orifice air trmperatuTe \\Tas ,-el'.\- C'lose to 200 0 from local heat ource C, about 1750 above room temperalme. dimen ionIc t mperatul'e ratio (7J{8,,,ax) Three different " loeal heat, olll'ces" " 'ere llsed: (a ) A LraighL diametri 'ally Lnmg \\'ire of 0.00 -in ch standard deviation of m ean-tempera tlU'e dis- ~ichrome tribution in wake I ehind local heat ource (b ) A 2-ineh-diametel' Xidu'ome ring T pul e pacing (c) A 4-illch-diametpJ' 1\idnome ring It pulse heighL Because of the extremel.\' high Lurbulel1('e levl'l PJlco unted j pulse " rjdth in a free jet, a mea ura ble thermal wake could onl.\' be ob tained b.\T using source t emperature in the range from :300 0 EQUIPMENT to 700 0 Thi unc/ouht{'dl.\- led to omc local buo.\'anc.\' AEHODY . AMIC EQ l PMENT eft'ecL , buL, even " ' ith thi order of temperature , tIl(' thermal The l-ilH'h h ot-j et unit i h OWJ1 ehcnhL ticall.\' in fi.gure ]. wake wa barel.,' detectable 1 inch down tream. The centrifugal blower is drive]) by a }6-hor epower direet The R eynolds number of (l1.('se heat-source " 'ires were ClIJTeJlt motor. lJ eaL is added llu'ough two double bank about a follow: of eo il of 1\0. 16 Nicro'orne wire. A. eun be cen in the For straight wU'e on axis, sketch , a good parl of the h ealed air i directed around the 150, ha ed on air temperature outside of the jel-air pipe in order to maintain a fl aL initial 22, ba '('(L on wire LempC'ratll1'e temperature Figure 2 i, a photograph of the unit as se t up previousl.\' wire I und no aver agc momentum ddeeL co uld be detecLed (rcierence , 0) ; th e ]Jrt'sen t al'ru ngemeJlt is essentially the with n f1 atlened lOlaJ-h eacl tube, ('ven as do e as I ~ inch same. clown tream. All turbuleJlce m ea w'emenls were made wi lh an initial Centrifu gal blower - • Screen, JD' mesh<, , -Cenlrifugal blower :' Scr een, 3D' mesh Diffuser-' -Grid, Ys-inch - diam. rods L-71320 on !1; -in. t/;.'s FIG lIRE I.-Schematic diagram or I,inch hot-jeL uniL FIGURE 2.-Tbo jet unit. l, 4 REPORT l040- NATIONAL ADVISORY COMMITTEE FOR AERONAUTIC j MEA U1UNG EQUIPMENT either with a millivoltmeter or by tbe av rage de£l. ction rate The mea uring in trument u ed were: ToLal-h ad tube, of a fllL'(meter. Chromel-Alumcl thermocouple, and hot-,yire anemometer The various spectra reportecl here w re mea m ed with a (a1 0 u ed a 1'e istance thermometer). mo li.fi. ed General Radio Type 760- A So und Analyzer The hot-wire were nominally 0.000635-centimeter plat (refe rence 19). The change in output tage (fig. 6) were inum, about l.5 millin1eter in length , eLehed from Wolla ton made to eliminate the direet-eLLlTent component and to wire. The etched platinum \\-a soft- oldered to the tip obtain linear inslea 1 of logarithmic 1'0 pon e. As modified, of mall Leel n edJe upports. A di cu sion of heat 10 the sound analyzer had rather unde iJ'able frequ eney from a wire at various ambient temperatmes is aiven in re ponse eharaeLe1'i tics, particularly a day-to-day hift in appendL,( A. relative amplification of the higher-frequency range . The The basic hot-wire-anemomeLry equipment \Va pmchased fl' equency-ro ponse calibration in fi gm'e 7 i plotted in Lerms from Mr. Carl L . Thiele of Altauena, California. One of of voltage squar d, since thi was tbe quantity ultimately the two identical heaLing circuits is hown in figme 3. measm ed. The amplifier, with re i tance-capaeitance compen ation The frequency pa band for this analyzer i far from lhe network, i giv en in figure 4. The uncompen ated gain i OptUnlill1 rectangular hape. However , the slopes of the two con tant to within ± 2 percent over a frequency range from id es are su£ficien tly steep that no apprcciable errol' i 3 to 12,000 cycle pel' econd (fig. 5). With the wire and attributable to noninfinite lopes with the spectra meas operating conditions used (tin1e constants on the order of ured in thi inve tigation. Fiaure i an experimentally 1 milli econd), the over-all compen a ted respon e wa goo d determined band hape. There \Va fair imilari ty of band over the am range. COlT ct etting of the comp nation shape over Lhc entire frequ ency range. For computational network was determined by uperin1po ina a square wave purposes, an equivalen t re 'lanaular pass band was dcfiJl cd as upon the ho t-wire bridge (reference 1 ). Unfortunately, in a indicated in the figure. free tmbulent hear £low the ambien di turbance i so areat The in LrumenL is a type recording constant-percen t band (becau e of the extremely high turbulen e levels) that wi dth, mea lIl'ing Lhe product of power spectrum times calibration cannot be made in the flow to be tudied. frequency. This ha obviou. advantage in th e high The vacuum-thermo ouple signaJ output wa measured frequency range where there is so li ttle tW'bulen t energy. Caltbralion signal circu it S" Sh r------~ ~------_, ?II Switch position Healinq cir cuit I ond (2) o I 24 volts (direct currenl) - o ?II + - 225 volts p.. OX I 02 PI,! (P2.1) (direct curren I),! ob 03 25n 01j2 04 x 2 05 10 steps. ° 1j2 ob 06 lOOn each °Z 1 ° Y 2 o':: z 07 RI,I (R2./) 02 02 Bridqe circuit 1 2 oYz 08 100000n . 10 steps I,OOOn. IO steps oS" oSh 09 D 3 0 4 0/0 10 steps, ~ 00--<> 01I Ion each Ps RI.2 (R22) Amplt'fier selector slVltch SlA.ftC!7 10 steps, posHion I n each °lfl 0I RI,3 (R2.3) 02 1 02 °1j2 03 4 2 04 °A p 05 oA." oA, 00 oT' oT- 07 oS, 9 08 <>--<'I c>----<> 09 P., Brass 10-32 binding posts 100000n o Potentiometer selector switch l~rBP4 Square-wave generator inp ut circuit FIGURE 3.-Con rol circ ui ts. "\ ( \ r P ECTRA AND DIFFU ION IN A ROUND TURB LE TT JET 5 1603 6J 5 1603 6J 5 1603 1603 6J5 135 volt0 ~~ + 2501i 250k 1M 125 k t To qalvanometer or p otentiomefer Ii ... ~ '. Thermocouple 250k - /.5 - 1.5 volts volts ,n (1L-___J-r ____L-f_---'pP ilot liqht .rC Al soon, 10 s teps; 6. 3 volfs (direct cu rren t) volto qe r atio, 2j1 p er s tep Storage battery Az 2000n , 10 steps of 200n e ac h C1 1000pJ1.f air c onden ser C2 10, 000fll4 decade condenser; 1000flfli per s tep ® Bross 10-32 binding p osts FIGU RE I. - flot-wire-anemometor a mplifier. B~ B+ 1. 0 -....., .P-' I t I I lOOk \1.8 \ O.Bpl () I <\ lr) I I I C\J .6 \' I o ,\ 560k .C 0. 4 ~ .C I , ~. 2 - .- I r-- ;.... I I !I (a) ~O 0 10 I 10) < 10 3 10 4 Fr equency, cps F lG t'ln: 5. ~ \Illplifi c r frequency res pon se wit.hout compensation. 100H Pos ibly the chid disadvantage of the actual band shape i O.8fll lh e extr mdy harp peak, causin g a great cleal of fluctua tion 27H in the output signal, and making any imple meter-rea ding Inpu t I 0. 025f1 f 4 7k t chillque vi ltually impos ible in the lo\\"- and meclium fr quency ranO'e. Oonsequently an inleO'!'atino- teclmique 560k \Va devi cd, making u e of Lhe negligible restoring-torque Phone haracteri tics of a nsiti eRe earch o. :Ou;aneter. The p lug integrating technique u eel is shown schemaLicall y in fi gure 9. Actually a bank of vacuum Lhc1'mocouples \Va us d, and the (b) re istance bown are ju L Lyp ical valu e. The ignal put ouL (al Original omput circui t. by tll thermocouple i a highly fluctuatin g direct C LlIT nt. (b) "[odified output circuit. 'rhe bucking circuit wa nece itate 1 by th e followin o FIGliHE 6. ;\Iod ificution of General Radio Typ e 760-A ' ound Analyzer. combination of req uil'emen t : (a) F or th lowest frequencies rea onablc con i LeDcy ncgJjgible in the rang of very large deflection. H ence it wa co uld be obtain d only by integraLing OVel" periods a long a de iTable to keep total deflection to a minimum. 3 minutes. TIm , mo t of the average direct-curren I, component of the (b) Appreciable tatic bearing friction in the iluxmeLer thermocouple io-nal wa bucked out, and the constant buck demanded more or Ie s continuol! motion of the needle. ing currenli wa read on a preci ion microammelier. The ( ) The 1'e Loring torq ue of the f1.lL'illl e tel' IS no longer f1.uxm tel' needle Iluetuated more or Ie about the zero- 6 REPOR'l' l040-NATIONAL ADVI ORY COMMITTEE FOR AERO "AUTr c 40 with a con tinuol! Jy adjustable band-pa s filter, as described II j undcl' " Eq uipmen l." I I 0' TEM P.ER ATURE S J'ECTR UM // /1) ~ / ~ ~ The power s pectrum of the temperature fluctuations ill the heated jet was mea ured by usi.ng the hot-wi re effectively -~ # ":.:-- as a simple resistance thermometcr (reference 20) . T he ampliJied voltage signal was analyzed exactly as in the 2 102 Z 5 103 mea ul'cment of velocity spectra. Frequency, cps FIGURE 7.- Response of sound analyzer with modified output circuit. S H EAR- CO UHE LATION SPECTR M HoI' tb e sheal'-co LT elation pectl'um, the quantity to be £ ffective bond width. measul'ed is the correlation coeffi cient beLween a narroW ~ /.0 1- - frequeney band of u-fiu etuation and the same narrow fre I quency band of v-Auctuations, at tbe amc point in the flo " 1 n o .8 I o 40 field. 1 o 120 T il e method, fo r any particular nominal frequency, was 1 o 300 1 b. 4000 to pas the variou volla·e signals (el' ez, el+eZ, el-ez) from 1 I an X -type s11 eal'- (01' 1" -) meter Lh l'ough the band-pass 1 1 filter after amplification . By appropriate combination of I the mean-square value of Lhese rour signa.ls (identical with .2 toLal-shea r meaSUl'emell t), Lhel'e resulLs q.o .95 FIGURE .-Filtration characteristics of SQ UlJ d analyzer with modified output circui t.. A r(' :l ,,·here Lue subs0rlpL n indicates Lhe n anow band of nomina.l of rectangle equals area UIlder CLLrve. frequen cy It cyeles pel' second. \. justification for the validiLy of th is proeedm e is ob tainable by considering the t\\·o velo city-flu 'luation componen ts a.s periodi function. Of co urse, thi is no t a real proof. If it symmetrical X -meter IS a umed, the two m tan taneous voltage iglla.ls are el = au+ {3 V} (1 ) Sound ,-onalyzer F:luxmeter ez= au-{3v '\" looon Direct current F or perlodi0 flu ctuations, '--_-" ~/ternotir7(; ' -Vocuum thermocouple current FIGURE g.-Integrator and bucking circuit. (2) defl ection point dming the time of integration, and it reading a t the end of thi time ordinarily gave a mall correction on the re ult. III this 'imple 0a 'e, the correlation cocfficient fo r any specLra.J In the highest-frequency range, overload considerations on line is m erely the sound analyzer limited the signal drastically, an 1 only a "il",=cos (cf> ,,- f n) (3) part of the thermocouple direct cmrent was bucked. For the determination of average wake temperatme IC eq uations (2) arc substituted into equation (1), it i behind the local h eat onrces, the thermocouple voltage was easily shown LhiLt m easured with a L eeds & Northrup type K - 2 poten tiometer. o cillogram were taken from a blue 0 cillo cope tube by (4) mean of a General R adio T ype 651- AE cam era, u ing fa t film. P ROCEDURES in co mplete analogy to the conventional meLhod for measUl' ing B u. with an X -metel" . The algebraic details are "'iven VELOCI TY SPECTRUM in appendix B . The power (or energy) spectrum of the longitudinal IiVhen tbe meter is not perfectly ymmetrical ( al~a 2 ; velocity fluctuation at a point in the unheated jet was {3, ~ (3z ), the formal proce se, both algebraic and experi measmed by conventional hot-wire-anemometry technique, m ental, becom e exces ively involved. Consequently , in PECTHA AND DIFFU ION I A ROUND T RBULENT JET 7 actllul pructice tJ w efl' (,(,t oIunavoicbLbk unsynuncLl'Y in the hypotlw sis of 10cHI isotropy i, ('(' n to I e \'erifi ed in a wry X -me(('r was essentially llullin<'d by taking double sets of lirect wn y. The yalue of tllt' dir('ctly )lwas uJ'ecl totul-sl1l'ar rcadings at etwh frequency, ro tating Lhe insLr ument ]80 0 corrc1ation coeffIcient R".=uv/u'v' is indicated in the fi gure. about the axi of fl ow di rec ti on between ets. VELOCITY AND TEMPEHATURE S P ECTRA VELOCITY CORm, LA T10N F NCTION The one-dimensional power spec-trum F,(k,) of t he longi tudina.l velocity flu ctuation 1l(t) \Va m('a, ur('d a,t two radial The double cO lTela ti on Rv between longitudinal velocity po itions in the unheated j('t at j"/d= 20. Figur(' J l giv(' t he flu ctuation at pair of points 011 opposiLe ides of thc jet two spectra, ne measured on the axi and one mea ured at axis \Va measurecl only in the unheaLed jet. H n ee Lhe about the lllax-imulll- heal' location. P lotted against wave standarcl hot-wirc-anemomet ry Leclmiq ue wa u cd. The two hot-wires were always equicli LanL from the axi (on a number (k1 =27r11, /U), the two sp('ctra ar c identical within diametcr), so t ha t they w(' re under iden t iea I operating the experimental eatt('r. The solid line drawn a approxi ('oneli t ions. mation Lo the points is made up a follow: (n) 1"or O< k, < 1.25 , it i Von Karman's emi('mp irical TEMPERATURE CORR ELATION FUNCT ION rorlllula (reference 21) : In th(' hoI j('t, the wir('s trav('r ing symmetricall y as [or (5) R v were operated ilS simple ,.esi La n ce tll ermometers, 0 thaL F.(Z:) ~ [ l~(i:);J ". the double temperatur(' cOlTe lal ion Sv could be determine 1 eli,.(' elly. (b) For k l > 1.25, a nonana.l ytical CUl"Ye bas been faired in MEAN TEMPEHATUHES BEHIND LOCAL SO H CE I To c1C'termin e mean temperature behin 1 a local o Ul"c e, ~ I traver ('S were made with a Chromel-Alumcl thermoco uple, 0 who e voltage was mea Ul"ed with a Leed & Torthrllp type 10 ""- Radial position K 2 potentiometer. D On jet axis ( r = 0) TEMPEHAT R E FLUCTUATIONS BEHIND LO CAL o u n CES o In maximum - shear region (r : 4 em) I 1'0 determine tcmperatme flu ctuation behind Joc-al ouree , the fine platinum wire wa operated at mall CUl" rents, .0 that it worked essen tially a a re I ta nce thermom -1 '\ 10 f--- etC'r. -\- f== ~ 1-- -I-I- EXPERIME TAL RE LTS 0. - ~ :'Iean-veloc-ity and mean-temperature distribu ion for b -- yurious orifi c(' tempera -w'e are pre en ted in ref rence ] O. .~ SHEAH-CORRELATION S )'ECTR M \ 2 I~ The peetrum of hear correlation coeffici en t n R",= 10- 11" V,,/71,/V,Z' was mea llre 1 in the lin heated j t at xfd= 20 at - a radial tation ("orre ponding to maxirnllm shea r at thi s (TO eetion . Figure 10 hows quite definitely that ,J?lI.(n) :g 0 is it fun ction decrcasin g monotonically to zer . Thu , th(' C(' 3 ~ \ 7r-f\ I ! .6 rn I i \ .'5 -- \ \ ------1\:~~a~:"''':O::':~~~~lo,,-~o::~~~~!!.:,= .4 4 I ~ \1,\ -+- t , I I- \ ~ \ .1 ~ I I , 10- 5 t-r- -1 0 1 t , /0 10 10 10 ~ 1 2 Wave number, k l : 2 fml/U, em- 11" cps FJr.l"llf: Il. On~-dim('n'iontllllOWl'r spectra of u(t) m~tlsured in I-inch unheau'd jN alf/ll=20. FU1I'RE 10. "3rin,ion Of shear corr~ l !lt ion co~mcient ,R•• wilh frequency. Round turbulent Computcd scales: £,=:1.:1 centimeter. for both stations, },,=O.31 cenlillll'tcr for 001/1 jet. r/d=20; U=270 ~enlimcters per second. stations. 971526-52-2 HEPOHT I O~ O - ~;\'r LO:,\AL ADYISOHY C'O:\ I i\ IITTEE TOH AEHOJ:\;\UTI CS Thl' Y OIl Karm,l n exprc's. io n wns [I sed prim,uily to TRANS VERSE C" ORRELATJO ' F UNCTJO 'S simplify t ilt' p l'O bl eJ11 of extrnpoln t ion to k l= O llild to s hortell TIl(' doubl(' correlntio l1 func tion Hy = U,u2/u l'n2', meas ured the w ork of tran formation to t hree-dimensional spectra . ymmetrically about t he axis at ,r/d= 20 in thr unheated jet, (See " Veloci ty a nd T empernture prrtra" uncl er " Analy i, is plotted in fi gurr l5. Of co urse, sin ce t he wire ar c in of Rrsult, .") id r n tica.l fl ow ('o nd i tion ,ul ' = 1L2' = u' (say) , and R/f=U,U2{U:,. Thr ('o lTrsponding onr -diJ11el1 siona l powr r spectra of The doublr ('ol'l'('lation func tion S y= 1)11} 2/tJ, '1}/ in t hr t<'m])rrature fluctuations CI (k,) II' C'J'e m ea l1rrd in the heated heated jet at J-jd = 20 i pl otted in rio'ure ] G. incr t hi s II-ns jet COo= 170 0 C) at .l'/d= 20. One pectrum i on the jet axis; al 0 mrasul"r cl symmrt ri cl1 l1 y, S y= 1},1}2{J2 , whe!"r 1}, ' = 1} 2' = 1} ' onr i at about the radial tation of maximum heat transfer (say) . (and s hear) . From figurr 12, it can b e seen tbat they diA'cl' Clearly, the range of m easurable temperatur r correlation noti('eably ill t it r high-frequrnc:y range, but a re e entia ll y exceed the rangr of l11ra urable Yclocity correlation by a n idell Li a l in t hr Io\\,- and moderate-frequency ran ges. The amount greater t hn,n ('an br attributed imply to t he Juct ('urvrs used to appr oximate the exp rimen tal p oin ts a rr a t hat the hot jet is widrr t hn n t he un heated jrt (referenC'r ] 0). foll ows: On the axis- t be Von ]\:.urman form ula is used oYC' r MEAN T HE RMAL·.WAKR ~ BE:HI ' o _LOC AlZlJ EAT 'oun C ES thr enti re range. At t he maximum-heat-transfer poin ' I\ 'pi cal radial di slribulions o( average lcmperaturc behind (a) F or 0<)1'1 < 1.0, the Von K a rm a n formula, is u eel a traigh l (diametrical) wire, a 2-inch-cli amrler rin D", and a (1) ) F or Ir, > 1.0, a nonanalytical C' urye ha been faired lJ1 4-il) C' h-cliamelcr r in D" iLL 1,/d= 20 in lh e unhealed jrL nre Figure 1;3 a nd ] 4 contrast YCloc ity prctra with trmpera- hown ill figures 17 , 1 , and 19, l'especlivcl.I-. The point in ture prctra at corresponding radinl station . Lh ese figure nre ]l ot c/irrc: t experimental poinl , but mr['r1.1- r-o- -_ I I II 1'0,« o Fi (k ) , spectrum of u ( t) in 10 e ° ° /I III j ~ unheated jet II 1 I 1 Radial posdion ° (~ o Gj (k j ) , spectrum of 11(t) in 0 On jet axis ( r • 0) . heated jet , o I In mi::,ximu m - shear region (r : 4.8 ern) 1\ 1 ~ j \ :t>, \ !-;. \\ f'. -- \ '\ 0.'\ 2 1\ \ 6\1\ r'Q - ~ J--1-1- f-- \ --r--' ~ ° o 6 1\ 3 r\: 0 0\ 3 \\ \ r \ h\ \ ° ° \ IQ_ 0 ? \ ° \ ~\ \ \ /0- 4 \ - 1\ 1-- - 1\ - l- , /0 e /0 _ /0 2 Wove number, leI = 2 :n:n / U, cm-1 1 - 100 10 1 i"I GI'R E 12. - 0nc-dimensional power spectra of ~(t) 111 ~s u r~d in I·i nch heutcd j~1 at x 1d=20. Wove number, k j = 2 :n:n l/[i, cm-' Computcd sctlle : (O)=2. 1S ('c mimNcr!' for both staliol1s. i\ .I' =-=fr. I F' (;I'H ~ la. - One·dimensional speclra 01 II(/) and ~(t) on axis of I-inch Jet atxld=20. 'PJ!;(;'I' HA ,\ ND DIFl<' ~10~ 1i':' A HOUND TUHBULE ' T ,JE'r ene Lo ilHlicate the Jaircd l'esulL for d ifl'el'cnL cli stam'os 1.0 clown Lream, All these have been ('olTeeted for the pre viously menLioned mall ambient-temperaLure field in lhe unheated jct. There \nlS rather large scatlc]' (illu strated only in fig, 20) due to the small temperature diffcrences mea ured and to the eXLremely largc ckgree of flu etuation pre elll. The 4-inch ring j s li ghll ,\~ outside of the [lI 11~ ' tur .6 bulent jet core (reference 9); con equently the 1'e ults for thi ea e arc noL of dircct interesL in a tucl ~- of fully dcveloped R y turbulence. The figure how that each of the th ermal .4 wake pos (' ses imilariLy well within the accuracy of mea uremenl. All of the thennal \I'ake spread linearl y in th e mcn.sUl' eri .2 range (fig. 21 , 22, and 23r From Taylor's thc01',\' f li(I'u- ion b,\' c ntinuou movement , this impl~ ' indic aLes Lhnt, for the maximum dowl1 tream station tucliecl, th e Lagran gian ('orrelaLion coefficient of t,h c v-flucLuation ha still not a deparLed Hpprceiably from uniLy, For a traighl-linc ource at thc jet axis, because of CO I1- -.2 0 2 3 tlr; m . F((;I'U E \.; , Srmllll'tric l n tlUWt'I'SC corr('i{llion of 11, m t"ll" U1 l'd uhout :,l\is ~I t .r/d=2Q·ill I· inch oF; T(kj'), 'sk~J~~Lm or' u (1) , ~ ~~ unheated jet CIt r = 4 cm I I I I I I IIII 1 1 10 0 ~ o GJ k l ) , spectrum of 11 (l ) I~_ heated jet o f r = 4,8 cm .0 j \, ' q \ b': .2 ~, o ~------~~------~ 2 \ ~------~ .1 I 2 3 ~ ------:!5 .lr, in. Fl t1I ' II f': IU. SY Illflletric trans\'PI'S(' correlation of fJ, IIH.'Hs urcd UL.l/d=20 ill l· illch heated j et, S'=~IU2ry', ~ 3 I~\ !O ~ .t \ .8 (in) o 1/2 \ .79r '\ o 3/4 \i J ' J .6 o 1 \ "\ Jl .51 {)max 4 r- .3 .2 a <> . I <> 0 0 4 3 2 2 3 4 - 102. FI GURE 17.- im ilarily of temperalure bt'hind line source of heat on jot ",is a l x/ L J ]0 R8POHT l04 0- l\ATLO.'JAL ADVLORY COMMITTE8 FOR AERONAUTIC.' \ 1.0 - .25 \ ..if at t =O 0< t J-f t. ) 10 \ Q;- ~ \ "0 I (in) .20 \ .8 "\ -8 ~ Ok \ "- 0 % "- 0 .15 "- -j6 ~ .6 O '};6 ·t ...... ~ ~ .fL ~ ...... /0 0 -l4 ~ 8, ,, , 0 .4 ---- IQ;[ .05 2 .2 _0 0 25 0 .25 .50 .I:J 1.00 oo~ ~ce~:~ {, in. °4~~~3~~-2~-~--0~-~/ --J2---~3~~ru4 FIGf"J": 21.- ' 1)I'ead of hClil from a Jine SO UI"{ 'C ill CC lltcr of ullheated jci. x/d=20. . I - " t/Hf, +Sz) l",=l" at which 0=2 Omaz. F lG! ' UE lS.-Simiiarity of temperature distribution behind 2-inch-diam ter-ring source of - 1- o heat ill unheated jet at x/(/=20. fl, f,=f at which O=ijlm ... \ . ')5~ \ \ if ~- t \ \ .2U "\ .... ~ .... 1.0 ~ "' "' + 'r .9 ~ o ( ..... In,· U .8 (in) ------.7 o 9/32 .05- o 1/2 .6 026/32 ol ~. 5 -.25 0 .25 . '50 .75 Omax t, 1/ • . 4 FIGUl»; 22.-SI read of heat from 2.i neh·diamcter.ring source ill unheated jet. x/d=20 . .3 . - 1 - f,. f=f al \\'llI eh 0=2 Om ... .2 Jet center . / "(,0 0 .5 \ 10 0 \ ij 01 t J 4 3 2 o 2 3 4 \ t/H J1+J ) , Z .4 \ o .... FI GURE 19.-Similaril)' of temperature behind 4-iLleh-diametcr-rillgsouree of heat in unheated .C: f . - 1- "- jet at r/d=20. flo f'=f a t wilieh O=;fm". "- "' '- I ',..... y,~;lJI ...... '-'. ~ .... rC"\,j·'------0 --- .I 10 , o .25 .:.;0 .75 8 FIGL'UE 23.- Spread of hea l from 4·inch·diamcter.riug source ill unhealed jet. xld=20. ---.. . - 1 - ~ fl, f,= f at which 0= 2 Om.,. Cl V III ~6 servaLlon of heaL, i.L fo llows immedi aLely hom si.milarity and ~ l. li neal' spread th aL Lb e ma.xim.um temperature at aero s :-!:: ~ ection in the wake (J max mu L decrease hyperbolically with 1.4 ...Q. inc)' fi in g down tl'eum dislance. L et 7i/O max= if> (7/ ), whore I 0 0 0 ( '" eel t=Con Luut (6) .J 0 0 0 1.2 .8 .4 0 .4 .8 1.2 whoro Lhe mean-velocity ch ange arE' neglecLed. Thon, t , em FIG! ' " E 2O.-Experimentlll scaLler; tempera lure behind lin e So urce of heat on jet ax is. ~= 1.2; centimeters; r/d=20. (7) SP.8 'THA .\ :-10 Dl[i'FVSlO.'\ IN A RO "D TURB LENT JET 11 Lbe sam e a III un [i tw:bed Dow. On tll(' oth er lliUld , til e (( llU'bulent" th('rmaJ ,,'ak(' do e to the source mu t 1)(' simply a Yery narrow laminar th ermal " 'alee which i fluctuat ino- in dir('clion it vet) fluctuatcs. i11('e all of lhe fluid out The SaJlle is true of the anllulnJ' wake m tbe range of ~ so side of this U11 leady lamina r wake j of con Lan L tempera small that o«r. tur and the temperatur e flu c'luation can o nl~T be po ilive, in glc traverse were al 0 m ade behin 1 a Lraight-line h eat th(' general char uclt'1' of the oscillogram of ~ (t) in fio-Ul'e 26 souree for lWO oLiI <.'r ca es: I under Landable. '1'1)(' (' 1'cco1'd we1" takt'll about ~I i1l ch (a) " Ti lh the line source set perpendicular to r at a l'adiu down lream from th e t:raight-1ine beat OlIl'ce, and about of 1 in ch , a temperature travel' e wa made in th r-dil'ecLion, %6in eh oIT lhe wake axis. All of the 0 cillogram w (,1'e made at ~ =}~ inch (fig. 24) with in ujficient c01npell sat?'on for lh (' hol-wi1' th('rmal lag, (b) With th e li m source eL on a di clm eLralline, a tem in order to llppn'ss thc (hio- l1 fJ'eqllCl1('Y ) noise and Lhu perature traverse was made perpendicular to l' at a radius of permil the basic form of ~(t ) to stand out. From lhese two 1 inch; ~ =}~ inch (fig. 25 ) oscilloo·ra.m of 11 (t) and ('\l(' 'k mca u)'em(,l1 l of the turhu lence levels £01' th e Lwo ca e , it appears that the so urce wirc TEMPERAT U nJ~ FLUCT ATJONS BEHI ND LOCAL HEAT SOUJl CE ha m ade no apprec iable ch ange in the turbulence. As can be anticipateel, the temperatu1'e fl u ctuation dose .\I eu ur('l1]ents of the intcil ity or l hc temperature flu clua behind a local h eat Ource arc quite cl ifl'el'ent in nature from lions acros n eelion at ~ = O.4 illC'b are o-ivcn in figw'e 27 . the w locily fluctuation at the ame point or from the tem perature and ve]o('it.'- flu rLuation in a turbulent JIow with over-ali heat lran fer. ill('e a uiLable ource produce no 2 00 cps f""-,...., ,,r- ,, ,,, '\l' aeldi tional l urbulellce ,3 the vcloci ty flueluat ion houlcl h e v v ...., v V v 1.0 r u (t ) undIsturbed -j" i 0 ~~------~ ~~ .l Flow .8 perpendiculor Line 10 poqe source u (t) _ .6 ." ------.' ~- --."- " ' ~ ~---~ ------'. ---.. -- 8 -- (J~a x .4 .2 Jet cenlc r FIGUIlE 26.-0seillograms of velocity and tcnIJ lI'rul urc fluctuations. 0 .4 .3 .2 .I 0 .I .2 .3 .4 J. in. 2 .0 0 1'1 o .4 .2 .I o .I .2 .3 .4 j ,in. FIGl' RE 2" . - 'l' e mp e ra tur~ behind lin ' source of h at in unheated jet. x/d='20; ~= l '2 inch. Lin SOUI'C(' S t nil a dinm tral lin(' . 'l'ravC'rsc made perpendicular to r at a ra dius or OL-__L- __~ __~ __~ __~ __J- __~ __-L __ -L __ ~ { inch. .2 0 ./6 ./2 .08 .04 .04 3 An ideal source would also produce no a"cragc momcntum dt' fecl. lIowe,cr, !1 men· tloned previously,the momentum wake of the 10t", 1 source werr relatively so small as to be Fl r- I'HE 27. Trmpernture fl uctuations bch ind line source of heat in ce nt r of unheated jet. L-o mplct rly undrtcrtahl r as close as 14 inch downstrea m. Ild='20; ~=O.4 inch. 12 REP ORT 1 O.J.O - XAT lO_ -AL ADYlSORY OM}'JlTTEE FOB A8BOi\A GTlCl-i In the ame vicin it y the values f u' le and v'lu ar c on th e m easul'cd \'alu e of 0.44, especially ince ther e i no r ca on to ordel' of 20 perccD t. 1'11 extr em ely b igh values of 1')' ro ar c suppo e tba t the peel rum of Lhe v-fluctuation is identical not urpri ing ince tb e mean tempcratm e cl ifl'eren e is d ue wi th tll e spec tJ'um of the 1t-flu ctua Lions, t'xcc pL in Lhe range only to the pre ence of the fl uctuation. Since the imple of local iso tro py . re i lance-Lh el'm meter th eory i ba. cd upon the a ump t ion VELOCIT Y AND T1, MJ'E RAT URE S PECTHA of small fluctuation com pared wi th ab olu te temp eratul'e (r efel'<'D ce 20), it is ex,,!)eetecl tll a t th e e meaSLU'eme n ts are 'lhe one-climen ional peelra of velocity a nd temperature about as accurate as tbe m eaS ll l'em ent of m uch 1011' ('1' 1')' 10 flu ctua tioll s, a plo tted in fi gure J J a nd J 2, respectively , a l' e in the hot jet (rcf('t'ence 10). a l' a-normali ze d ; thaL is, they arc defin eci llch that A ALY818 OF RES LT8 (1 0) S H E AR-CORRELATIO ' S P ECTR UM A very convenien t cbeck upon tbe h a1'- pectJ'um measu]'e m ents can be gotten by the sim ple expecuen L of com p ut ing (11 ) tb e lotal (01' " net") turbulen t h eal' ol'l'clatioll c effic ienL (wh ich wa a1 0 elirectly measured) from this spectrum a nd H owcver , the origin al m ea Ul' Cm cllt · w cr e made 0 11 an tb e turbulent-energy spectrum . AO'ain, an eleffirn lal'Y ab 'o lulf - IJaLue ba is, a tbaL the total flu ctuation level u' /U FOLLl'ier serie treatment se rve to ju tify (not proY(') the n, ncl 1')' /e could be used a ch ecks on tbe spectl'a. The pectra inLuitive idea that Lh e toLal eO lTelatio n coeffi cient is B UD as measured were FI*(nl) and ('I *(nl), defined such that, imply a weighted a\'eraO'e of th nRuD (i. e., th e co me of id eally , the ph ase angles), weigh ted imply by th e pr oduct of th e square rool of th e energy pectra of 1t and v. The calcu la (12) tion is given in a ppendix B , an I yield, the J' elation '" (01*( IlI )d711 = 1') 2 (13) R UD = ~ (" F u"F .) 1 /2" R uD ( ) .J o ,,=1 IntcO"rat ion of fi \ * as illcii catetl III eq uation (12) yielded Unfortunately, 25 cycles pel' second is lll e lo\\'el' limit of t he following: the mea uJ' ed frequency range, so tba L som e extrapolation On Lb e axis (7'=0), mu L be made to lower frequencies ,,-hi ·h con tain much 01' tbe tmbulent energy. Since no theor etical bn is yet J ( ( '" )1 /2 exist to guide thi s extrapolatio n (like th e Von K :1rrn:1n fo r L' J /\*dll l = 0.2 mula in the c11. e of en el'gy pectrum), O' uesses had to bc (u' ([= 0.22, d il'ectly measul'ed) be made a to Lh m aximum a nd minimum of r ea onable looking extrapolaLion CU l'\" " These arc ploLted in figurr 2 , III tbc mnx imum-slH' J ( ( '" )l /t 8 .J oG1 *dll l = 0 .2 1 Computed: Buu (1')' /e= 0. 18, dir cLly mea med) Extrapolation CD 0.52 @ .50 @ .46 In the maximum- hear region (/' = 4. 'm), Directly measured .44 o (1')'/0= 0.36, d il'ecLly mea. ur cd ) On Lite jet axis, where con ventional m all-per turbation hot-wirc th eory may till be moderately accumte, the agl'e - o .4 .8 1.2 1.6 20 24 28 3.2 3.6 ment i atisfaetory . Wave number; k -271'1Z 1/D. cm-1 l I t hould be n oted in pas ing Lhat th e lire tJy m ea ured FIGI'RE 2,~. - S h ear pccll'um and power spectrum in I-inch unhealed j t atxfd=20 in region of maximum shear. value of 1')' /8 a rc a pprecia bl y higher than those l'epol'ted in JI ,'PECTRA AND DIFFUSIOX I I A ROUND l ' RBULE IT JET 13 rrfereHcr ]0. K o rxplanaLion for (his diH'erence is apparrn( . Again , an nnalogou approach to tem perature fluctuations The longit udinal scale of u -flu ctuations obtfl inable gives appro)"'1matcly from th e power pcctl'um : (19) Unfort una, tely , the high-frequen cy r ange of Lh e G'/s ar e too uncertain Lo p ermit r ca on able extrapolation an d th e whieh follow from t11 r funclamentfll Fou ri er (I' R)1 S;o n r.H u sc of equ ation (19), al though it can be seen from figure 13 (i on , that Ax>lx. However , equation (18) ha been used to com (J 5) pute Lhe long itu d inal v elocity m i l'oscale. F igure 29 is a plot of the integrand of equation (18) . The in tegration and appropriate computation give ill (he limit kl O. An analogous treatment of a trmprra Lure-flu ctuation ft'l et Ax= 0.44 cen timeter len ds LO an ic!c>nt ieal expression fo)' th E' longit udina.l , calc of -J-fl Llc tlJa( ions: and the am e value [01' bo th r adial position. Since th e turbulence on the axis of such a jet e m s Lo (1 6) b e rather i o tropic (the e_-pe l'imenLal eviden ce is th at uv=O anci u' ""V'), t he la teral micro calc Within the aceU!'acy of m easurem ent, the long itllclinal / scale of u(t) was found to b e the same on the jet, axis and in 2 JI2 A= [ - Ry" (O) (20) (h e maximum-sht'a!' I'rgion : L x=3 .6 eentimC'lr l's is of the or cie l' of AxI 2; th aL i , A"" 0.31 centimeter Th same was true of the longitudinal tt'mperalllre seak s in the ho t jr l: \Vi th the a, umption of i otropic turbulence on th e jet Ax = 2.4 centimeter. axi , it is po, ible to compu Le the three-dimensional p ower p ee tn lm F (k ) from the one-dimensiona,l pectl'um PI (k l ). In all the e com p uta tion s the m easUl' ed s pectra were H ei enberg (refer ence 23 ) ha given the inverse trans r xtrapola ted ( 0 ZN O wave number with a p a rabola, as fil ' t formation s uggested by Dry dcn (refercnce 22). ApPl'Oximately the F1(lcl)=l ( '" (P-k 2) dk same numerical values are obtained with th Von K a rman P (~l J (2 1) 4.J kl k- formula iIlu (m trcl in th e plot Led curve. In order Lo compa re L x in the unh E'aLed jet witb Ax in tbe and th e de ired P(}I\) i readily found to be h eaLed jet, L x rna. be multiplied by 1.1 5, the jeL-width ratio F (k )=2k [k lFt (kl )-F/ (kl)] (22) at l'/d=20 lor these two iniLial temperatures (ref ren ce 10) . l l The "eorrecLed " L x is then 4.1 5 c ntime tr r . ~ Llu'ee-climel1 ional pectrum ('omputed in thi way 1 gIven lL mus t be reeaj]rd tbaL the Fourier tran forma ( ion rela in figul' 30. tion brt wren Lim r p r C'lr um n.n 1 pa e cO lTela ti n w ul 1 be The corresponding pectral tran formation for th tm.'ee exactly Lnlr only if tll r tllrbulr nt flu ctuaLions at a point dim en sional isotropic. fluctuation field of a calar quan tity were due to pUl'(' rectilinear t ran ja t ion (by - ) of a fixed flu ctuation pa ttei'll. F or the f1' cC jet flow, th e extrem ely 5 high turbuknce levels m a ke s uell a t ran forma tion very uneerta in. Therefore, equ ation (14) an d (16) (and the I' e- 4 ulting cales) can only b e considered as cru de a pproxima tions. The longitud inal microscale, 3 (17) may a,lso be computed approximatciy 1'rom th one-climen ional p ower pect rum, by I -, (1 ) o 10 20 30 40 50 60 70 80 90 100 110 120 k (Prime signify cliA'eren t iation ",he'n applied to Ol'l'clation FIG1'RE 29.-Yiscous di ipulion and microscale). on axis of I-inch jet. xld=20: and sp ectra.) ~2= f o'" k,'PICk,) dk,; X=O.31 centimeter. 14 REPOR'l' 1 040- TA'l'IONAL ADVISORY COMMI'l'TEE FOR AERONA TIC dirrel' l'llt in shape. 'The con lrast JS sho\\,t in fi gure 31. learly, even Lho ugh Lh(' net areas ar c idenLi. cal, th ere is nonzero tempemtu I' e correlation over appreciably gr eater eli tance . A rough approximation to the microscal e of tmbulen ce can b e gotLen by gue ing at th e osclllating parabola for Ll1'= 0. In this particular case "guessing" is more appropriate t han. "fitting," since th e job i entil'ely extrapolatory in nature. Figure 32 (a) shows th e ver tex region of By with th e parabola that correspond t o 2.0 2.4 2.8 32 36 (27) em-' v,,"' ,,;: :10. O(' ncr,, ) naturc of three-dimension,)) speclra on jet ax is. 1'/<1 = 20 'fhi valu e' i in smpl'isingly good agreement wi th the 0. 31 (tcmperaLure, for example) ar c simpl)' (r efer en ce 24 ) centimeter obtained fr om th e' power pectrum on th e axi . In fact, th e agreem ent must b e r egarded as fortuitou , since Q (k ) - ~ ~ '" Q(le) n (23 ) the' difl'er ence is appreciably less than th e exp rimcntal I 1 - 2 k T ( • 1 uncer tainty. and II th e temperatm e-flu ctuation field is again considered (24) analogou ly , th e transverse microscale of temperature With Lhe as wnpLion of isotropic temperature flucLuaL ions flu ctuation (fig. 32 (b») is on Lh e jeL a;\i s, equalion (24) h as been used lo complll Q(k). 1.0 The LhJ' ee-climensional velocity and lemperatme spectra on th e jeL axis ar c ver.\' n earl)T lh e sam e- provided Lhat Lh e isotropy a mnp tion is r eason abl\' good. The curve in figure 30, which h ows th e gen eral na LLIl'e of boLh F and G, is simply Lhe Lra!). formation of the Von K arman a pproxima Li ons to F, and GI . lL must be recall d, how vcr, th a t F, is in Lhe Ll11he aLe d jet, while GI is in th e h eated jet. Alth ough tll · l'esulL of refer ences 9 and 10 indicate no e senbal ch ange in th e de tailed d)'namics of jet turbulence as a r e ult of moderate incr ease in jel temperature, ther e is an a ppreciable increase in jet width at a given x/d. A mentioned earli er , the width r atio beLween 80 = ]75° an d 80 = 0° i about 1.15 at x/d= 20. O' ~------~~------~~=------ THANSVEH SE C O]~ RELATION FUNCTIONS ___LI _ --1- The in te o'n11 area unclCl' th e vcloci L.\- concla tion :funcLion 2 3 (fig. 15) may b e consider ed to gin a sort of lateral scale of .d r and /./5(tll ). in. tmbulel1ce in Lh e jet, alLhough lhe r es ult i not associaLed FIGlI P.E 31.- C'ompnl'iso ll of tnms \"c rsc cOl' l'claLi on fun ct ions. I-in ch heated jet. with an,\' parLicular :region in th e jet. The COI1 \TellLi onal expres Ion (25) .9 give a scale, L y = 0.67 cenLimeter. If a later al temperaLme scale i clrfllleci in simil ar rashion , (26) .7 i\. =o.I3 in. .6 .6 th en for th e h eated jet at x/d= 20 it turns out th at 1\ y= O.77 i\. =o.ll in. cen time tel' from th e Junc tion a given in figlll'e 16. (a) (b) .5 Appropriate compal'i on of the e transverse cales m ay be 50 .05 ./0 ./5 0 0.5 ./0. ./5 .20. had if L y is mul tiplied by th e jet-width ratio: 1.15Ly= 0.77 D. r, in. D. r, in. (a) Latera) microsca)c of U-fiUCLu .ltion". centimeter , the same valu e as l\ v. H owever , th e two correla (b) Lateral mieroscale of ,)- f1l1 cL lI ations. tion functions that yield lh ese n et ar ea arc till quite PWU RE 32.-E stimates of mi crosca )~. I-inch jet. 1'/<1 = 20. SPECTRA AND DIFFU IO IN A RO TD TURB LEI T JET 15 wlH' l' e /:;. i' Lhe fandard deviation of the mean-temperatul'c 1= [ - S}(0)],/t=0.43 centimeLel' (28) distribution, and, accordinO' to the the ry of diffusion by continuol! movement, is therefore proportional to the For comparisoll , 1.15}" = 0.32. tanclard deviation of the probability den ity of 1·(t) a well. MEAN THERMAL WAKES BE HI N D LOCAL EEAT SOUR CES pec ifi cally, fo r a mall ~ in a homogeneou tu rbulent fl ow, The rate of spread of th e thermal wake cl ose behind a local (33) ource of heaL was fi]' t used by Schubauer (reference 13) a a mean of m ea lJI'i ng th(' inten i ty of lateral veloci ty fluctua tions V'/ U. A detailed d ivcu sion of thi. technique ha, been ~ imil al'i ty also implies that O'iven by Taylor (rdcJ'el1('e ] 4) and necd not be repeated h er e. OrnaxU",ax (34) The 1'e ults of uch II computatioll, compared with dil'eeL tJ v = w(TJ) (ay) X -meter mea urements of v'{r, arc a fo ll ows: EquaLion (7) may be \\Titten 8 /:;. = onstanL (7a) r max (i n.) Ct).. k. Ctt.... ," -- Then, with equations (32), (34), and (70.), eq uation (3 1) 0 O. IS O. IS o may be tran formed to I 0.22 0.20 ------2 0. 10 I O. ;;0 (35) The X -mcter mcaslll' C'ments w l' C I')' cL 1 fol' Lhr e(f rct. or, wi th equaLion (33), of both 1/ and Ul' upon th e li O'htly un ymmetrical meter. It is possible to get orne additional info1'ma ion abouL the ~w = v' / (TJf) (36) fluctuation field by c mputation from th e t urbulent-heaL TJ mox TJ tran fer equation. In particular, an estimate of th e distribu "inc v' (r",ax icon tant in this apprQ).:imI1Lion, ti.on tJv across a ection of the th ermal wake berund the line ource may be made a fo llow . The steady tUJ'buJ en t-heat-Lran fer equation fol' low velo city (negligible vi eou dis ipation to hat), negligible mole cular h eat conduction, and con tant den ity is, in Carte ian and the boundary condition, w= O at TJ = O, give finally tensor notation, - f}v Vi t 0 = .=- -~ (3 7) (29) \\-hich i cOllYenienLly \\Ti Len in Lhe form For the region in the immedi ate VI '1l1lLy of the jet axi , assume that condition approximate tho e in a homogeneo u tJ v '0 t fi eld of t urbulence; that i , 1'= - 0 and U= Con tant= (38) n ()maxU max ()max ~ max· Then equation (29) become simply In figlU'e 33 thi funcLion i ploLted again L t/~ for t he trav or e }~ inch down tream from the LraiO'ht-wir heat ource - 0'8 0 (- ) 0 (- ) U",az o{" - a~ tJu - at tJv (30) aero the jet cen cr. The assumption of mall tUJ'bul enceieyel irnplie. tJu « oC max and hcnce . / 6 (31) .12 Thi would be a good approximation in tUJ'bulenee far behind a grid placed in a uniform tream, but i certainly rather crude here. The final assumption, that of imilal'ity in the thermal wake, is well-supported by the exp erimental 1'e ult Then let _I .3 .4 .5 .6 .7 .8 tit (3 2) FIGURE 33.-'rhermal wake behind straight-line heat ouroc. T emperature-velocity corr I,,· lion computed rrom 8IoMA" E=O.5 iuch. ""---~~- --- .. ------~./ ~ j i 16 REPORT 1040- rATIO rAL ADVISORY COMl\IITTEE F OR AERO A TI CS DISCUSSIO decrea e more lowly than the product u n' On'. In g neraI, LOCAL ISOTHOPY the meaSUl"emen t of n B uv ems lik a much more specifi c an 1 The mono toni(; d crea e to zero in hear correlation co direct approach than the mea m ement of the power pec trum effi cient with incI' ea in O' frequency cern to be decisive evi of 'lw. Pre um abIy, a ( omowhat more complicated) FOUl'icl' dence for the exi tence of local i otropy at ufficiently high eri c di cussion like that in appendix B could al 0 b carried Reynold number. It i intere ting to note that the pectral out for the power spec trum of uv. region of negligible heal' (nl> 1000 cps in the pre ent par VELOCITY A D TEMPE llATURE Sl'ECT HA ticular del rmination, for example) contain only about 1.5 The apparen t identity of the velocity power spec tra 2 percent of thc turbulcnt kin tic energy in u • Of eoursc, FI (le I) on t he jet axis and in the region of maximum hear i thi i by no mean an indication of the importanc of the only approximate and has b en cI terminecl only lown to exi tenc c of local isotropy in a tmbulent hear flo\\". A mo re leI = 0. 1. There till exi t the possibility of m a ill'able di pertin ent comparison would be with figme 29 , which hows verge nce in the lowes t wave-number range. The good degree in eff ec t di ipation a a Junction of frequency. From this of agreement indicate that, in difru in g from the region of it appears that about 90 pcrcent of the di ipation of tmbu maximum production (ncar tbe ma:\l.mum- hear region) to lent kinetic energy 0 heat takes placc in e entially isotropic the region of maximum di ssipation (on the jet axis), the turbulence . Thi permit the u e of the Taylor expression turbulent kinetic energy has not done any gros migrating for eli sipation in i otropic turbulence (ref rence 25). Of iI) the ,,'ave-number pa ·c. com e it al 0 implie that the isotropic relation between On the other hand, the appal' n t decid ed difference bet\ en longitudinal and lateral microseales, Ax=' 2 A, will be fairly temperature p wer speetra mea LIl'ed on the axi and in the accurate even in th e rc O'ion of high tlLrbulent hear. Fmther maximwn-heat-transfer region eem to indicate uch a more, it impli e a univcr al dimensionless pectral function migration . However, the con id erable ca tter at the highes t for all tlll'bulen t now , in the high-frequency reO'ion. For mea ured frequencies renders definite conclusion impossible. t urbulence at thi R eynold Humber, it appears that a u11i Somewhat more specifi c conchl ions can be drawn from the wrsal part or the pectrum eXl l only for le l> 7.9, which i compari on between one-dimen ional velocity and tempera tme pec tra. On the jet axis, for example, in spite of di tinct well beyond the point of slope-*. In fact, the" poin " -% diff erence bet"'ee n these two pectra, it turn out that within in this pectrum i ju 1, at lei = 1.0 . the experimental catter (which is c n i lerable) the tlu'ee At thi point a few remark on the appropriate type of limensional power pectra may he much more nearly iden measurement for veri6cation of local isotropy may be in tical. The fact th at they did in fact come out to be identical order . In par ticular, a careful distinction mu t be made oYC'r a wiel e range of way number when computed from th between the heal' pectrum nR u. a pre ented in this r eport empirically fi tte ] Von Karman formula must certainly be and the power spectrum of the randomly flu ctuating quantit regard ed as pure chance. This is true not only becau e of the uv 'whi ch mi O'ht be measUl" ed with a multiplyin g circuit experimental un cer tainty, but also becau e t he e pec tI-a followed hy a freq L1 eney analyzer. were mea urecl in two irnilar bu 1, different fl ow , who Local isotropy pecirie. that restriction Lo a sufficientl y characteristi c length probably eliA'ered b. 15 perce nt. small domain in a tmbulell t hear [JO\\' how up i otropy in Kl EMATlC AND THEHMAL SALES he various tatistical propertie that are studied \\"i thin that domain. I t implies that the frequ ency (or wave-number) From the extrapolated zero-wave-number intercept of the vector must be large in magni tude. 01 ad y then a good one-dimen ional pec tra, the following longitudinal scale in dication of i otropy is zero correlation bet\\" en orthogonal were obtained a t J.Jd= 20 : velocity-iluctuation componen t; thi means that the hi O'hest L x = 4.1 5 centimeter frequency parts of u , Z', and. w are LUlcorrelated with each other. H ence it j clear that jf U nV n decreases to zero with Ax=2.4 centimeters increa ing frequency fa tel' than the product u n' v,,' cl ecrea e to zero, local isotropy exi t. In terms of co effi.cient, this Thi Lx is ] 5 percent greater than the unheated-j et value, merely r equires that nR"v deerea e to zero eventually. to allow for the greater wid th of the heated jet. Thus, Now con ider the flu tuating qu antity uv. In a turbulent L x/Ax= 1.7 . In a homogeneous, isotropic field of velocity and shear now u v ~ O , so that uv co n i ts of it direct-culTent emp rature fluctuations, it t urn out (reference 24) that, if component wi th uperimpo ed random fluctuation . incc the three-dim nsional pow r spe tra of velocity and tempera the cO IlY entional elec tronic tec hnique eliminate the direc t ture are proportional, L x/Ax= ] .50. I t may also be noted ("LUTent, the qnantity to be analyzed would be uv- uv a a that, if the mea ured ratio wer in an iso tropic fi eld, L y = function of tim e. If local i otropy w re present, the lower }~L x and Ay=Ax, 0 that L y/Ay=O. 5. The ideal value would frequencies of 'lL and v would be rec tifi ed in the multiplying be 0.75. Actually, the in tegrals oJ the tran v r COlT lation proce ,and therefore the oscilloO'ram and power spec trum fLm ction R y and S y are ('o n iderably Ie than the cales that of uv-u v would have relatively great emphasis on the high would b expec ted, according to these relation , in a homoO'e frequencies. In other word , if the naively measur d power neou isotropic turbul nce. pectrum of lIV were u ed a an indication of local isotropy, it On the oth er hand, the relative values of longitudinal and would ho'" a tr nd oppo ite to that of U n V n ; that is, it " 'ould lateral kinematic micro cale follow the i otropie r lation, j PECTRA AND DIFFU ION IN A RO ND 'rURBULEN'r JET 17 A,,= .y'h , at lea within the experimental llUcertainty. Thi In the ection entitled "Tran vel' e Correlation Func is on the order of ±25 percent in the ca e of the parabola tion " under "Analysi of Results," it was found that in the "fi tted" at the vertex of R y . heated jet Lv~ A y. On the other hanel, it is well-known that Unfortuna tel.\" the temperature spectrum on the jet axis the lateral rate of lransfer of heat i appreciably area tel' than is not extended uffieiently far to permit compu tation of the lateral rate of tran fer of momentum, a wa fu'st found longi tuclinal micro calc lx there. The pectra in figure 13 b~T Ruden (reference 27) from mean-velocity and mean show only that lx i considerably less than Ax; that is, Lx is temperature measurements. Since diffusion is essentially considerably less than 0.44 centimeter. It may be remarked Lagrangian in nature, while L and A arc Eulerian scales, the in passing that i otropy for a calar quantity means equality above 1'e ult are not necessa1'il,Y in eont1'adi tion. The of longitudinal and lateral corl'elation hlUctions. The lateral appreciably greater di Lance ovor which Sy=r£O (a contrasted micro calc, £= 0.43 centimeter, obtained b~T "fitting" a para with R y) may, however, be related to tho fact that the mean bola at the vertex of 11 eems of reasonable magnitude thermal jet diameter i appreciably greater than tho mean relative to 1.15A = 0.32 centimeter. momentum jet diameLer , THANSVEHSE COHRELATION F N CTIONS PHOBABJLITY DENSITY OF v ( l ) AND w(l) Of cour e, the l'eason L y and Ay as determined by integra The moan-temperatme eli t l'ibuiion close behind the tion of function R y and u arc not related isotropicall y to straight-line heat source on the jet axis is effectively sym Lx and Ax i that over most of the range of M the probe are metrical, and closoly 1'0 emblos a Gau sian curve in hapc in decidedly nonisotl'opic tmbulence. Thu , t.here is no (fig. 20); this hows that the probability den ity of v(t) on reason to expect Ly= %Lx or Ay=Ax, when Lx and Ax are the axis is more or less Gau ian, as in i otropie turbulence. computccl from thc spectra. Th mean-lemperature di tribu tions cIo beh inc! 1,h two An examin ation of the behavior of the c two ymmetl'ically ring heat source are decidedly kew. However, some of measUl'ed ('onciation functions how that there i nonzero this skewness eem 1,0 be clue simpl, to the curvature of the correlation over a considerable part of the jet, but that the line source. The1'e[o1'e, the tcmperat ure di tribution across relatively mall cales result from the rather exten ive region the wake o[ a traight wu'c et tangent to the circle r= 1 of negative eOlTclation. Thi behavior is empha ize 1 by a inch wa mca ured. eglecting the effect o[ mean-velocity comparison of Rli with the COlTe ponding function in om gradient, thi cm've (fig. 24) i proportional to the probability typical isotropic tmbulence down tream of I-inch-me h grid den ity of the radial velocity fluctuation v(t) in the shear (reference 26 ). F igme 34 shows the contrast clearly. l'eglOn. It is een to be slightly kew; the skewne factor It i conceivable that lI ch an extended region of negative cOlTelation is characteri lic of tmbulen t shear flow. Ho\\- ever, until omeone e tabli hes this in a shear flow whose transverse extent is very large compared 'wi th the maximum correlation distance, it may be safer to gues that the" execs" i computed du"ectl,\- from this curve. The thermal wake amounl of negative correlation i impl,\' due to a li ah t irregu mea urement of la:amstad and chubauer behind a line lar waving of the :ict a a whole. In reference 9 it wa, ource in a turbulent boundary la.\-er (reporLed ill reference assumed tha.t, sin ce R y aetually goe to zero at larg values 16 ) how a k wne of 0.3. The diffcrences in ian and of D.r, there is no over-all" whipping" of the jet. However, magnitude of the e lwo skewnes factor sugge t lateral such a conclusion doc not appear to be completely war turbulence-Ieyel gradient a he cau e, The gradient in ranted. v' /U are of oppo ite ign in the e two flows. Calculation from figUl'e 25 how that the probability den ity of the tangential fluctuation wet) is . ymmetricaL It may be noted that on th axi of ueh an axially ym metri flow there is no di tinction between radial and tan gential velocity fluctuation; henc figUl'e 20 al 0 applies to w(t) on the ax-i . TEMPERAT R E FL CTUATIO. S REH! D LOCAL HEAT sounCE The extremely high Lempel'atUl'e-fiuctuation levels (tJ ' (e>LO) encounLered in the wake of lhe line heat olll'ee '\: "- arc ea ily understood from a brief con ideration of lhe "- "- nature of tbe temperatUl'e field. Clo e behind the source, "- o~------'~~~~----======~~~--- there i ju t a ingle naITO\\- laminar thermal wake which i being blown in random de\-iaLion from the ~-direction by the turbulent fluctuation. The gro turbulent thermal 2 3 4 wake i imply Lhe wedge- haped region over which thi AT/Ly and y/L 1I relatively nal'l'O\\T wake wandel'. Hence the total thermal FWGRE 34.-Transverse velocity correJationfuuctions. R.='U;U2/Uiin round j t; g= u,u,/u. in isotropiC turbulence. signal at any fixed point in the gross wake con j t imply I l__ ~ _ _ 1 HI~ PORTI O-lO - XATlO TAL A DVI ~ ORY COl\ IMyrTEE FOR AERO TAUTICS of II , cries of pul e8, whero eaell pulsl' cOl'l'es ponds lo a ll In ()'e ne ral, iL s hou ld be emplHlsized t bat l1waSUl'cm ent occasion upon which the lamina r wakc s\\'cpl OVC1" the by CO l1 H'ntional (sm all- pcrturbation) hot-wire an cmome Lry poinL. Obviously, the frequency of oceUlTcn ce of pul os in a £low of Lhi s hi o- h level of turbulen ce cannot b c co n id will decrca e monotonically with iJ1 Cl' Ca ing tmn VOl' (' ered as accuraLe absolute-value m eaSlIl'em enls. Even OJ) eli lance from the center of Lb c g l'o s wake. lhe jct axis, whcrc th e level i a minimum and condition If this typc of tempera Lure s ignal is represented st;h emaLi a re rcla Lively tcady, th er e is no r ea on to believe LhaL cally by p eriodie square pul. e of h eigh L h, width j , and fib olute yalues are bclter Lh a n witllin, say, ± J 0 percent of fundam nlal wave leng Lh 7 (fig. 35), t il 11 it can be ea ily th c "corrcct" valuc . Howcver , relativ e be hay iol' ar e deduced thlll the flu ctuation lev 1 i. undoubtedly deLe rmincd, and c1imensionlc s measures 01 tbe type of correlation coefficient arc more a curaLe th an ab olute values. ;.,r on of the meas ll rcment reported 11 e1'e bav boo ll co rrceLe cl for fin i tc lcnglh of h L-wires. whero iL i' rccalled LhaL rJ = O- O by d efi. niLio ll . SUMMARY OF RES LTS Two pulse spacings of inLer e L ar c F rom m casurem enLs in a ro ul1 cl turbulenL jet ul room tcm perature of t he sh cflr co rrelation coeffi cien L a a function of (1) 7= 2j; th cn 13 //0= 1.0 fr equcn cy, o( yelocity anll temperature flu ctuations wit h find \\'ithout jet Jl ('atiJlg, ancl of the 111.('a ll thermul wakc ' (2) 7->co; t hell i}' jO SOU RCES or }~ nn o n PO\\'C!' s pcctra i a dirc·t m anifestation of thc fa L t hat velocit y a nd h eat Hre vcctor ancl sca la r qlla ntitic's, rcspe(' Aside from the s p c(' i~l e ills tancc mcntion ed earli er ill t hi s tivl'ly. seeL ioll , the so urccs of expcl'im cn la l (' ITOI' Me much th c Sllm(' 4. T hc ratio of longitudinal to laleral sealc (fo r bot h as ouLlill ed 0 11 page 27 a nd 2 of I'de l' en ce 10. Acld it io llnl \-clocity and temperatul'c rl u 'lua tion ) is eon sidernbly la l'gcl' uncer Laint ios ari e in Lhe pecLrum meas urements, espe than \\-ould fol low from i oL ropy. Longitudinal scalc a re cially in th(' high er-frequency I'fln ge, bceausc of (a ) ntpid meas llred on the jct axis, while lateral scalc involve a cha nges in the (,u libration of t hc so und a na lyzer (bund pea k traverse or mo t or th e rully turbulcnt corc o( the jct. I'espon c fl.ga in sL frequency, fi g. 7) (b) light s laLic and 5. The ratio of longituclinul to lateral kinemat ic mi'l'o friction of fluxn1l'lcr bearings. scale on the jet axi i about equal to the i oL ropie alue. 6. The longitudinal t hcrmal micl'oscale (from one-clinwn s ioJllll pO\\'cr s pectra) is lcs than thc long ituclinal kin ematic mi cToscllle, but the II1.tcrul microscalcs (from co rrelat io ll nH' asu l' em ents) ll avc Lhe opposite l'elaLion; th a t i , th 8 t hcl'mal i o- reater Lban the kin ma lic. j --1 7. The probabili ty dens ity of thc radial fluctuation vel) n - .-- thc jet axi s is c A' eet ively G aussian . The probability clen ity -T- in t he hear I'cg ion is sli ghtly kcw . . The temperature-Aucllla tion fie lclul tl)(' \\'n kc b bind a h lo('al heat so uree cons i 1 or a randomly " 'fl ying narrow laminar t hermal wa kc. H en ec lh e temperatme sign al a l n fixed point is a random-pu lse type of function . lt f1u cLua ------tion inlensity is on t hc order of 100 p ere nt on th e cen ter li n e, and incl' ases toward Lhe edge . o r t I 'I - The J OH NS HOPKINS UN IV ER. I'l'Y, FIGl' RE 35.- Ri ll1ulalion or temp ralurc Signal close hehind local heal source. n.\ Ul'IMOHE , ".\I D. , A?/{/?/st 17, 1949. L J Spg TRA A. D DIFFU. IOX I T A IOUND TURB LE T'r .TE'£ 19 APPENDIX A H EAT LOSS FROM A WIRE AT VARIOUS AMBIENT TEMPERATURE In figlll'l:' 3 of refere11ce 20 H rough clleck \Va mflde on t il e point obyiou Iy doc. not repre ent the o\'or-all un certainty; temperature-vllrilltion fi rst t('rm in K in g's (rdel'{'nce 2 ) it simply show. (,Iw range or valuC's that eould be gOLLen hy equation for the steady stn.tic heat los fr om H cylin cler dmwing d~tfe " (,lIt l"C'tlsonnhlt'-]ooking straight lines through perpendicular to a fluid t r('ilm, n t lo\\' Reynolds numbers. the same et of original calibmtion point . From the figure The conven tional form i, it can be een thaL Kino-' equation predict Lhe tempemturr variation of A quite well. The change in R (the slopr of R~R = A+ B-VU (A I ) the calibration lin e in the plot or i2Rj(J? - Ha) against -V=) " are.o small LhaL Lhe exprrimental caltrr is Its great as the whert' change predictrcl for these trmprralurr clifl"erel1ce .. 1.4 and 1.3 R wire resl tane Na wIre re I Lance at ambient fluid temperature wire re I Lance at 0° C A No A l.2 l* wire length r d* wire diameler ---Predicted by Kinq's equation a* temperature 'ocffi cient or change of resistivity of f Ranqe of values obtainable {rom calibratian wir material /.I k* Lhermal conductivity of fluid flt nmbien t lemperfl- lure 'pecific h al of Huicl at ambienL temperature 1.00 density of fluid at ambi nL I empern I lII"e Cllrren t CI,C2 empirical constant- 98~ In reference 20 Lhe cbe 'k on A a a function of temperature wa made by as uming Lhe econd term in eq uation (AI) to B B ·96- be exact in it temperature variation. Th n each mea W' d r calibration point flt fln y velocity and trmprratlll'lrd to a aIur for 1. Thr prr r nt chrck \\'u carried out more completely; a .94 ~ full calihration cun'e was nUl Jor each ambient temprrat ure. From thi, both A and R \I-('rr clr[rrlllinrcl. Figurt' 36 ~ givr t b(' results compared wi t h K ing '. pr('(li('[('(l Yil riit! ion , .921 I I I I 280 320 360 400 440 480 using physitt11 constants from rdt'I'{'n('(' :29. i£ach po int Temperature, OK cOITPsponcL to fl calibn1tion. The \'('rtie,11 liJl(' through II f.'IGl,.TR£ 36.-Variation or hOl-\\ire constants with air telllperatuft'. hI room temper3ture, APPENDIX B MEA UREME T OF H E AR -CO RRE LATIO PECTR M The two voltage ignaI from an id eal ymmetrical Of cow e, tbere would b(' no 10 ]Q o-eneralily if tPlI or if;n X -meler are \I-ere taken a zero. e1=au+ ,BV} The quantity to be mea ured i (Bl) e2=au-,Bv (B3) Supposp Lhat the yelocity flu ctuations lIre periodic: b'or two imple harm nic funC'lion thr correlation co fficirnt I. imply th e co inr of Lhe plll1. c ano-lf'. Thus, (B2) (B4) ub titution of equntion (B2) into equation (BI), followed i l~ _ _ _ ~_ 20 REPOR'l' l040- NATIONAL ADVI ORY OMMITTEE FOR AERONA TI by trigonometric transformation, gives The um a.nd differen ce of Lhe two wire voltage arc '" e, = :L: [(exa " o · ,, + (36 n cos f ,,) (,:0 2 7rnl j e l +e2= 2au = 2a~a " co 1 (exa" sin ,.,+(36 '11 Sill f ,,) in 27rnt] (B5a) (B IO) e\- ez= 2(3 v= 2(3 ~ b .. co '" , e2= :L:, [(exa ll cos "-(36,, cos f n) co 27rnt Fil tering o- ive (ex a lt in "-(36 ,,. in f ll) sin 27r7lt] (B 5b) vVhen the. e two signals arc put eparaLely throuo-h a ll iU TOW banel-pas filter that passe only th e nth h armonic, th e two oUlput \Tollage may be repre ented a Pa age through the vfl,euum thermoco uple givo net = K [( exa n cos " + (3 b" co f ,.) cos 27rnt- ,, (el +ez)2= 2I-{' exza,/} (ex a sin n+ (3b " in f n) sin 27rnt] (B6a) n (Bll) lI(e ,-ez)z= 2K ' (3 2bn2 "e2=K[(exa ll co "-(36 ,, cos f n) co 27rnt (exa " sin "-(36 ,, in f n) in 27rnt] (B6b) ombination of equation (Bll), (B9), ilnd (B4) give the where K is a n attenua tioll faeLol'. final re lilt: For brevi ty, \\Ti te (BI2) (B6c) (B6d) The computation of total-sh ear correlation coefficient from h eal'- oeffi cien t pectrum sugge t it elf as a useful che k These filtered ignal go n ext into the vacuum-thermo poss ibiliLy: couple unit, which put out th e 111 ean- quare values, R"v=u v/u'v' (BI3) with the Fourier seri es for 11 and v, ] - 2 ( "J .J (2 ) ') C D ( ) . ( ) (B14) [{' "ez = ,, - cos- 7rnt - ~ II n CO 27rnt sm 27rnt + D,/ sin z (27rnt) wh ere K' is an OVf' r-all all nuation factor. ancl th e in s tfl,n LaneoLi s (.: I·OSS product can be tran formed to - 2 ' 2 1 d --' 1 Bul, cos "", JI1 ""''2' an 0 IU ""' O, over a arg number of uv= :L::L:anb1l1(cos n co 27rllt - in n sin 27rnt) X \V a velengths. '1'h us, 71= lm= ' (cos f ill os 21rmt - in f m sin 21rmt) The Lime a verag(' or thi expression is (87) _ 1 '" uV = -2:L:a"6,, (cos n (,:0 f ,,+sin t/> " in f n) 71= ' 0 1' T hen, within th e appl'oxima Lion, (B l5) - 2 - 2_ K ' (A z 0 Z+ B 2 D 2) n I - ne2 - 2 It - n. 11 - 11. (B ) Thus, and when the expre sions for A , R, ( I, and Dare SLibslit ut ed, it turn ou t. that ]n term of tb e Fourier ('oeffi eienl , (BQ) The n ece ity of del. rmining a and (3 is ordil1arily avoided R (BI6) wilh a symmetri cill m etol', if only the correlaLion coeffi cient lIv i required. j -~- 1 SPECTRA fL D DIFF ION I I A ROUl D TURB LE T JETI' 21 I I 13. Schubauer, G. B.: .\. Turbulcnce Indicator Utili zing thc DiO'usion I But nFu=an2/t;,a/ and nFv= b,,2/ifb n2 arC' imply Lbe nor of H eal. XACA R p. 52-1, 1935. 14 . Tayl r, >. 1.: Htat iRt ical Thco ry of T\I1"blllcll cc. ] V- Diffusion in malized one-dimensional energy spec( 1'3 0 r 11 iLncl 'V, respec Lively. Therefore, a TurbulenL Air NtTralll . P1'oc. Roy. Soc. (London), . er. A, vol. 151, no. 73, Srpt. 2, 1935, P)J. 46.5-,17 . 1.5. Hkram tad, II. K. , and Schuballcr, G. B. : Thc Application of R lJv=~("F""Fvl / 2)1/ R 1l. (B17) Thcrmal Difr\l~ion to t hc Study of Turbulcnt Air Flow. Phy. n= l llcv., vol. 53, no. ] I, Junc 1, 1938, p. 927. 16. Dryden, lIugh L.: Turbulence and Diffusion. Ind. and Eng. Cbem., vol. 31, no. 4, April 1939, pp. 416-425. REFERE E 17. Wieghardt, K .: Db r Ausbreitung~vorgange in turbulcnten Reibungs chichten. Z. f. a. ;\1. ;\L, Bd. 2 , Heft 11 / 12, Xov./ I. r\:olmogoroff, .\..: The Local 'trucLure of Turbulence in Incom D C. 194 , pp. 346-355. prCR iblc Viscous Fluid for Very I_arge Rcynolds' ),rumber~. 1. . Kova znay, L.: Calibration and ;\Irasuremcnt in Turbulcnce 'amp. rend., acado ci. RSS, vol. 30, no. ~ , Feb. 10, 1941, Re earch by the Hoi-Wire Met hod. XACA Ti\I 1130, 1947. pp.301- 305. 19. cott, H. H.: The Dcgcncrativc Sound Analyzer ..JOll'·. Acous. 2. I U. S. GOVERNMENT PRINTING Of FleE : 1952 I j l J