Correlation and Statistical Characteristics of Turbulence Fronts in the Wakes of Hypervelocity Bodies

Transcription

1 Missri University f Science and Technlgy Schlars' Mine Sympsia n Trblence in Liqids Chemical and Bichemical Engineering 97 Crrelatin and Statistical Characteristics f Trblence Frnts in the Wakes f Hypervelcity Bdies R. H. Hwell J. L. French Fllw this and additinal wrks at: Part f the Chemical Engineering Cmmns Recmmended Citatin Hwell, R. H. and French, J. L., "Crrelatin and Statistical Characteristics f Trblence Frnts in the Wakes f Hypervelcity Bdies" (97). Sympsia n Trblence in Liqids This rticle - Cnference prceedings is brght t y fr free and pen access by Schlars' Mine. It has been accepted fr inclsin in Sympsia n Trblence in Liqids by an athrized administratr f Schlars' Mine. This wrk is prtected by U. S. Cpyright Law. Unathrized se inclding reprdctin fr redistribtin reqires the permissin f the cpyright hlder. Fr mre infrmatin, please cntact schlarsmine@mst.ed.

2 CRRELTIN ND STTISTICL CHRCTERISTICS F TURBULENCE FRNTS IN THE WKES F HYPERVELCITY BDIES Rnald H. Hwell University f Missri-Rlla Rlla, Missri James L. French Sperry Rand Crpratin Hntsville, labama BSTRCT Data n statistical wake trblence f the far wake f sphere and cne mdels with velcities frm 9,800 t,500 feet/secnd and range pressres frm 50 t 0 mm Hg. are presented and analyzed. The measrement techniqe sed is that f bserving the trblence edge psitin f the wake n schlieren films. The measred parameters are the wake width; edge rghness; at-crrelatin fnctin; micrscale; e-fld; integral, and average eddy lengths; wake dissipatin parameter; Klmgrff and energy cntaining wave nmber parameters; Lagrangian integral time scale parameter; Strhal nmber; crss-crrelatin fnctin; and wake edge velcity. The reslts f these measrements are cmpared with ther investigatrs, as well as ther methds f bserving the character f the trblence f the far wake. The present data shws reasnable agreement with that f ther investigatrs sing the same bservatin techniqe. Hwever, there is a lack f agreement between the reslts frm sme f the methds f bservatin and/r measrement techniqes. The severest criticism f the present methd f bservatin is fnd t be the need fr a meridinal plane view crrectin factr alng with the lack f a direct relatin f the wake edge psitin t the internal trblence strctre. INTRDUCTIN Hypervelcity bdies mving thrgh the atmsphere generate wakes whse characteristic strctres are inflenced by the shape, size, and velcity f the bdies as well as the pressre and cmpsitin f the atmsphere. Knwledge f the discriminating characteristic strctres and their relatinship t the bdy shape, size, and velcity are essential fr psitive, early radar detectin as well as fr scientific analysis f the case and effect in the wake strctre. Recently,*,3,4,5,6,7 aerballistic ranges have been sed t attempt t classify the effects f the basic bdy parameters n the trblence strctre f the wake behind sbscale bdies. variety f bservatin and/r measrement techniqes have been incrprated in these investigatins, and ne f the difficlt prblems is that f assessing the reslts f each techniqe and its relatin t the reslts frm the varis ther techniqes. The sbject f the present investigatin is the presentatin and analysis f the statistical characteristics f schlieren bservatins f the trblence edge f the far wake (,000-0,000 bdy diameters) behind hypervelcity (0,000 -,000 fps) spheres and cnes. simplified mdel f this type f wake flw indicates that in the near wake regin (x/d <,000) the inviscid - viscid interactins play the dminant rle in the frmatin f the wake strctre while in the far wake, trblent diffsin effects are dminant. The cpling f these regins is manifested in the natre f the trblence which is present at the beginning f the far wake. This mdel f the far wake implies that in a regin f the atmsphere thrgh which a hypervelcity bdy has travelled the flid is in trblent The rigin f these vrtices incldes many srces sch as vrtex generatin dwnstream f the bdy, bndary layer shck-wave interactin, and flw instabilities in the near wake. The discriminating characteristics f far wakes which have been investigated sing this type f bservatin inclde: i) standard deviatin f the measred wake variable, ii) micrscale and integral scale crrelatin lengths, iii) wake width grwth, iv) peridicity f the measred wake variable, v) wake Strhal Nmbers, vi) pwer spectra fnctin, vii) statistical natre (skewness, krtsis) f the measred wake variable. strctre. Several wake bservables have been sed t characterize the far wake Fx and Rngaldier6 sed the signal frm ht-film anemmeters t btain space and time crrelatins f the wake mving past the anemmeters. disadvantage f this methd f bservatin is that the prbe interference with the wake is nt always negligible. Seqential shadwgraphs were sed by Schapker and Uberi and Kvasznay.0 This methd has the advantage f nt interfering with the wake flw, hwever, it is restricted t rather dense atmspheres de t a lss in sensitivity as the density is redced. the wake by se f schlieren phtgraphs. French,5 Clay, et al.,8 and Herrmann, et al.9 bserved This methd has an added advantage in that it can be sed in lwer density atmspheres than shadwgraphs becase f its increased sensitivity. With the shadwgraph r schlieren methd either wake edge psitin r film density can be sed as the measred variable. Electrstatic prbes have been sed by Heckman, et al.5 and Slattery, 4 et al. t measre the electrn r in cncentratins in the far wake. These cncentratins can then be related t ther wake characteristics. s with the ht-film anemmeter, electrstatic prbes can interfere with the wake strctre itself. Radar and micrwave investigatins f the far wake have been cndcted by Zivanvic, et al. and Lane which are free f prbe interference bt intrdce ncertainty as t the natre f the wake in relatin t the measred reflected signals. ther bservable characteristics inclde seqential spark mtin in the wake which yields a wake velcity distribtin;5 bservatin f the wake chemilminescence;5 interfermetric bservatins;^ ^ electrn beam traverse f wake; and hlgraphic interfermetric stdies.^ ll f the abve measrement techniqes pssess certain advantages as well as sme inherent disadvantages. The prblem which persists is the relatinship between the characteristics f the varis bservables and the relatin f these characteristics t the initial generatin f trblence. The sbject f this investigatin is the presentatin f additinal wake data and the cmparisn f these reslts with ther methds f measrement..-range Descriptin EXPERIMENTL TECHNIQUE mtin having a distinct distribtin f characteristic eddy sizes. The majrity f the reslts reprted herein were btained frm the vn The distribtin f the characteristic eddy r vrtex diameters depends n the characteristics f the near wake which in trn are cntingent pn bdy size, bdy Karman Facility 000-Feet Hypervelcity Range G lcated at the rnld Engineering Develpment Center at rnld ir Frce Statin, Tennessee. material, bdy gemetry, velcity, angle f attack and ambient gas cnditins. The basic cmpnents f this range are depicted in Fig.. The test vehicle lancher is a

3 tw-stage pwder-hydrgen gn cnnected t a 000-ft lng, 0-ft diameter factred by Bensn-Lehner Crpratin). test range with a cntrlled inch in the plane f the magnified image. pressre (0 phg t atm.). The blast chamber The linearity accracy was within The tpt f the X-Y in the range absrbs the mzzle blast and separates the mdel frm the sabt. psitin f the wake edge was cnverted t pnch card frmat which was finally Cylinders, sjpheres, and cnes with a majr dimensin f 0.5 inch t.5 inches analyzed sing a Cntrl Data Crpratin 604 cmpter. made f plastic, steel, cpper, r alminm have been fired at velcities frm 7,000 t 4,000 feet/secnd. Fig. 3 is a sketch f the gemetry sed in the measrement techniqe fr series f 43 shadw-graph statins allw psi- the lcatin f wake edge psitin. tin and attitde detectin dring the test. drawn near the center f the wake. The remainder f the data was btained frm the 00-ft Hypervelcity Range K at the rnld Engineering Develpment Center. reference line r mean wake axis was The distance t the wake edge, y (pper tline) and yl (lwer tline), was measred perpendiclar t the wake axis This range is a small and repeated at a cnstant increment size 6X. The pictres were read with a scale versin (pilt range) f Range G and has similar perfrmance character- magnificatin f 0X and an increment size f inch. istics. was 9 inches which generated 440 pints n the pper prfile and a like nmber The sample length n the lwer prfile..-mltiframe Schlieren Instrmentatin t the pper prfile, the y^ vales were averaged accrding t a least The primary instrmentatin fr the wake trblence measrements was a sqares methd t find the best straight line thrgh the pints (y = a + bx). high sensitivity, single-pass, schlieren system psitined 88.5 feet frm the test range entrance. Then, at any x, the deviatin e = y - y J The system has a viewfield diameter f 30 inches and was was calclated and frm this e ensemble all f the statistical parameters were determined. perable in either single-frame r mlti-frame mdes. Similar calcla- In the mltiframe mde tins were made fr the lwer tline, and the reslting statistical parameters a mltiflash Strbkin light srce with a variable flashing freqency cvering were averaged with thse frm the pper tline. a range frm 6 t 50,000 frames/secnd was sed in cnjnctin with a high n analysis f the effect f a change f increment size n the wake paraspeed drm camera. With this system as many as 0 phtgraphic frames were meters was given by French^ and shwed that a vale f X = inch cld be btained. Examples f the mltiframe schlieren pictres are shwn in Fig. sed fr all f the wake regin. vale f X fr increment size wld yield fr a 0.5-inch diameter sphere and fr a 0.5-inch base diameter 0 cne. a maximm errr f 0% in X in the very far wake and essentially n change in The length f sample size was als analyzed by French"* and he cnclded, 3.-Measrement Techniqe and Prcedre The methd which was sed fr analyzing the wake edge psitin in schlieren pictres was similar t that established by Schapker.'*' that jst as in the case fr the sample increment size, a change in sample length (t.5 inches, r 5 inches) can affect the micrscale vales, espe- n image f the schlie- ren pictre t be analyzed was magnified and reflected nt a film reader cially in the very far wake bt has little effect n edge rghness r width. screen having tw mveable perpendiclar crss wires (Telereadex Type 9E man- sample length f 30 inches appears t be adeqate t t,000 bdy diameters 86

4 0,66,46,686 3,946 Sht N. 376 Mdel: 0.5~in.~diara Sphere V* - 8,300 ft/sec P 64.5 mm Hg 6,466 5,06 Fig. a. Mltiframe Schlieren Phtgraphs: Hypervelcity Sphere Wake (fr these mdels) and gives vales which appear t be cntins in natre with that reslts frm this type f wake bservatin and measrement techniqe are a minimm f scatter (± 0%). repeatable with a minimm f data scatter. In additin, it can be seen that the wake generated by cnes mving at similar velcities and at the same EXPERIMENTL RESULTS pressre are smaller than thse f spheres at crrespnding x/d psitins The cnditins fr the shts which were sed fr this analysis are given in Table. The angle f attack fr the cne mdels was measred in the shadwgraph films and interplated fr the schlieren lcatin. The data pint symbls sed fr each sht are als given in rder t avid repetitin f the Data frm ther investigatrs have been sed fr cmparisn with the data presented in this analysis and these sht cnditins are given in Table alng generated by cnes de t differences in the angle f attack (p t 8.4 ). Fr larger cne half-angles (sht Ns. 385, 396, 753) the wake width appears t be larger and grwing at a greater rate. It is evident that the data btained tins sch as Schapker and Levensteins and Krmins. It shld be nted that if the wake width were pltted in terms f the drag diameter (m/zc^ verss x/zcp) that the crves fr the cne and sphere with their srces. The dimensinless average wake width in the far wake is given in Fig. 4 fr bth the VKF shts and thse f Schapker. In additin, the wake edge rghness a /d fr bth the VKF shts and thse f Schapker'*' and Levensteins and Krmins ls, there is n apparent effect n the wake width in the VKF ranges exhibit reasnable agreement with thse f ther investiga- cnditins in each f the figres. (shts 385, 4, 396). fr bth spheres and cnes is given in Fig. 5. cllapse t a single crve fr large vales f x/v/c~. Examples f this behavir are given by Slattery, Clay, and Herrmann.^.-Standard Deviatin f Wake Edge The standard deviatin f the wake edge psitin is defined by Eq. : Frm the basic measrements f the wake edge psitin statistical analyses / were perfrmed in rder t find the statistical natre f the sample f measre a e^(x)dx ( ) ments and t relate the wake edge psitin t typical trblence characteristics. DISCUSSIN F RESULTS and represents an indicatin f the rghness f the wake edge. l.-wake Width The reslts given in Fig. 4 fr the wake width f several f the sphere shts (606, 60, 64, 376) with identical size, velcity, and pressre verify The wake edge rghness increases with distance alng the wake and exhibits sme scatter as shwn in Fig. 5. Schapker There is exceptinal agreement between this data and that f and Levensteins and Krmins.

5 m " IB m ] 0,5 ; $ I B ' "" 3,545 L?,885 m W b B *. Jg W# 5,05 MBM 6,865 8,55 Sht. N. 385 Mdel: 0-deg, 0. 5-in.-diam Cne -,500 ft/sec P = 50 mm Hg 0,85,845 Fig. b. Mltiframe Schlieren Phtgraphs: Hypervelcity Cne Wake Fig. See Tables & f r symbl 5 Wake Edge Rghness n ta ti n. There appears t be n discernible effect in ^/d de t pressre r 00velcity differences. 50- Hwever, the edge rghness is always greater fr spheres than fr cnes at the same axial lcatins. a 0- X mining wake edge flctatins. 0- '." V a 5- t this pint it is nt clear whether Re^, M^, r ambient atmspheric prperties are mre imprtant in deter T determine this, hwever, wld reqire a mre systematic prgram f cntrlled sht cnditins. s with wake width, there is n appreciable change in edge rghness fr J i ^ changes in cne half-angles. There is a decrease in the rate f change f the rghness fr sme f the 0 0J 0 V d Fig. 4 0 bdies. This cld be de t a difference in the character f the trblence which is initially generated near the bdy and eventally mves t the wake edge Grwth f Wake Width thrgh trblent diffsin.

6 TBLE VKF SHT CNDITINS Sht N. Mdel d in. P» mmhg V ft/sec ngle f ttack M ReM x 0-5 Material Symbl 376 sphere , C 379 sphere , C-l 38 sphere , C 606 sphere , C 60 sphere , C 64 sphere , C B 098 sphere , Fe V 379 sphere , l cne , "-cne , cne , cne , TBLE. SHT RESULTS FRM THER INVESTIGTRS Mdel d in Pc mm Hg V» ft/sec M Re» xl-5 Reference Symbl sphere sphere cne cne cne cne CT sphere Sme f the scatter and deviatin can be attribted t the methd f btaining the edge psitin frm the schlieren phtgraphs. If the wake axis (see Fig. 3) is lcated incrrectly, changes in wake edge psitin can shw greater deviatins n ne side f the wake than the ther de t the differences in the abslte distances. In additin t this effect, sme differences in wake edge rghness can be attribted t transitin effects in the bndary layer n the bdy as well as >- D Gassian Krtsis * > -sht -sh t -sht > bdy size effects n the transitin lcatin. In an experimental stdy f transitin in the wake f cnes,^ it has been sggested that fr Re^ < x 0^ distance t transitin is dependent pn the abslte size f the mdel. If sch effects exist in transitin measrements, it is reasnable t expect that the - - -t> 00 <> X Gassian L. Skewness > sbseqent trblent strctre may be size dependent. This sggests the desirability f separately stdying size and the sal dimensinless parameters. The mment cefficient f skewness fr the wake edge data defined in Eq. : N _, l (yr y)3 () N 3 is shwn in Fig. 6 fr several f the sphere shts (376, 379, 38). These vales are the average abslte vales fr the pper and lwer wake prfiles. This cefficient assigns a nmerical vale fr the degree f skewness f the distribtin frm the mean vale. cefficient f skewness is zer. Fr a Gassian distribtin, the mment Frm these reslts it is apparent that the data is skewed t the psitive (as expected fr a grwing wake) and that it exhibits a peridic natre alng the wake axis. The mment cefficient f krtsis defined in Eq. 3: Fig. 6 Skewness and Krtsis f Wake Edge Distribtin is the degree f peakedness f a distribtin which has a vale f three fr a Gassian distribtin. The krtsis cefficient is als shwn in Fig. 6 fr the same sphere shts (averages f pper and lwer edges). gain, the average krtsis indicates that the distribtin is nn-gassian and the cefficient appears t be a damped peridic fnctin. 3.-tcrrelatin Fnctin and Crrelatin Lengths sample f the nrmalized atcrrelatin fnctin f the deviatin f the wake edge psitin fr a inch diameter sphere (sht 379) is presented in Fig. 7 as a fnctin f the crrelatin lag increment K. The nrmalized atcrrelatin fnctin is defined as N I J= <y -y) Q (5) (3) RE(5) j) (4)

7 Q ( ) = e(x) e(x+ )dx 3RE( ) H and = crrelatin lag distance = KiX. The reslts presented in Fig. 7 were btained by nmerically integrating Eq. 4 with X = inch. The e-fld length, Le, is the crrelatin lag where RE( ) redces t a vale f /e. This length is a cnvenient well defined pint n the atcrrelatin crve (if it is an rdinary fnctin) and it is generally accepted as a measre f the lngest linear crrelatin distance between the deviatin at tw pints in the flw field sample. The integral crrelatin length is defined as: f^max RE( )d (6) and is representative f the lngest crrelatin distance between tw pints in the flw field sample. This definitin hwever is ambigs when a peridic fnctin f RE( ) exists yielding the pssibility f negative vales fr L^. Fr a peridic crrelatin fnctin, a finite dmain shld be specified fr the integratin limits. The e-fld and integral length are representative f the size f vrtices which cntain the majrity f the kinetic energy in the trblent flid and which in trn degenerate int small scale eddies (measred by X), which eventally dissipate the energy in the wake. Becase f this mechanism f decay the crrelatin lengths shld becme apprximately eqal in the far wake. K Fig. 7 Nrmalized tcrrelatin Fnctin Representative vales f the micrscale length are given in Fig. 8 fr several sphere and cne shts as well as sme f the vales frm ther investigatrs. Fr istrpic and hmgenes trblence the crrelatin fnctin is a symmetrical fnctin and varies between pls and mins ne. When the atcrrelatin fnctin has a vale f near pls ne there is a strng linear relatinship between s(x) and e(x+ ). Therefre the crrelatin lag distance,, represents the distance ver which the deviatin is linearly related. The frther away frm pls ne the smaller is the tendency fr a linear relatin. Hwever, there might be sme ther relatin between the deviatins besides a linear ne. The atcrrelatin fnctin as defined here nly indicates the tendency fr a linear relatinship. Fig. 8 Micrscale Crrelatin Length s expected, the atcrrelatin fnctin decreases tward zer as the crrelatin lag increases. Very little can be said abt the shape f the atcrrelatin fnctin since it depends pn the strctre f the trblence. It may decrease beynd zer t a significantly negative vale r scillate abt zer. There is n apparent similarity r pattern in any f the crrelatin fnctins since at sme lcatins it appraches zer in an scillatry manner (Fig. 7: x/d =,40; 3,00) while at thers it drps well belw zer (Fig. 7: x/d = 6,300; 9,750), and shws n evidence f scillatin. The tre significance f the atcrrelatin fnctin is derived frm the varis crrelatin lengths which can be btained frm the fnctin itself. Near = 0, a parabla can be sed t describe the shape f the crve and the intersectin f this parabla with the - axis is sed t define the micrscale crrelatin length, X. In rder t evalate the micrscale crrelatin length (defined by Eq. 5) a parabla mst be fitted t the atcrrelatin fnctins at = 0. Since this wld be an inexact methd fr evalating X, an apprximate methd was sed. Crrsin and Fistler^ develped an apprximate methd which ses the nmber f zer crssings f the wake edge. Their methd, which assmes a Gassian distribtin fr the deviatins, can intrdce sme inaccracies int these vales fr nn-gassian distribtins f the wake edge deviatins. Intermittency calclatins were made in rder t establish the Gassian natre f the wake edge deviatins.^ The intermittency factr which is defined t be the percentage f the time that the flid at a fixed lcatin is within the trblent wake is given by: The micrscale length is an indicatin f the dimensin f the smallest eddies "n the average" which case a change in the lcal velcity. defined by: X is Pf(y')dy'

8 where P (y') dentes the prbability density f the frnt. The intermittency factrs were evalated"* fr mch f the wake edge data. It was evident frm these reslts that the intermittency factr fllwed a Gassian-type distribtin fr many different shts at varis psitins in the far wake. These reslts spprt the se f the apprximate methd given by Crrsin and Kistler *' fr evalating the micrscale length. The cne appears t exhibit a lwer vale f than the spheres which might be expected fr sharp bdies in cmparisn t blnt bdies. The dimensinless micrscale length,, increases with increasing x/d. In Fig. 9 are representative vales f the e-fld length and the integral length (integrated irrespective f scillatins) fr several spheres (shts 376, 379, 38). There is sme dispersin f the reslts. Hwever, the trend f the reslts is similar t that fr the micrscale length. The magnitdes /d - S H T S H T a - S H T _ t> L Fig. 0 verage Edge Eddy Size f the e-fld length and Lj are slightly higher than X bt bth are nearly eqal in the very far wake. Fig. 0. wake edge. n additinal crrelatin length has been calclated and is shwn in This length, x, is the average axial distance between peaks in the This distance is indicative f the average vrtex size in the wake and its rder f magnitde is the same as that f the bdy, as expected. There appears t be an scillatry behavir f x in the wake which shws an increasing wavelength in the axial directin with a decrease in amplitde. I The x length appears t be a reasnable crrelatin length and has the advantage f nt reqiring the calclatin f RE( ). Varis crrelatin lengths frm varis instrmentatin systems have been cmpared in Fig.. Frm this cmparisn it is apparent that at mst psitins in the wake there is an rder f magnitde difference in the reslts frm varis means f measring crrelatin lengths. This amnt f deviatin shld be reslved by frther investigatin f the far wake as well as mre detailed analyses f the measrement techniqes invlved. 0 - \? I-P m d e l V. (ft/s e c.) gftnrrv Ref. I n s tr m e n t a t i n 0.5" Sph Schl. w a k e e d g e " " 0.38"S ph " d e n s ity " Sph. " " " " Sph E le c t r s t a t i c Q38"Sph " cal H t w ir e " b l n t " " " a n d & y / a a ' s s \ _ 8 'P 8 \ *> \ 0. I Fig. Cmparisn f Crrelatin Length Measrements - 4.-Trblence Qantities Several f the relevant trblence characteristics have been calclated parameter: in rder t relate the wake edge statistics t the internal trblence f the wake. The relatins between frnt statistics and the internal trblence strctre are inherently apprximate, since the wake edge characteristics M Re / ( _ i)- -8 (f5/ - / (7) fr several sphere shts (376, 379, 38) are given in Fig.. It can be bserved that the incmpressible dissipatin rate is apprached in the far wake and that the actal dissipatin is less than the predicted incmpress 0 - See Tables & fr symbl ntatin ible dissipatin t t abt 0,000 bdy diameters. The dissipatin parameter relates the temperatre flctatins at the frnt edge and the velcity s? a & r! f the frnt edge t the edge psitin statistics. d * x/d 0J 0 * Fig. 9 Integral and E-fld Crrelatin Lengths are nly bndary measrements. The details f the derivatins f these relatins are given by Schapker. The dissipatin parameter, which is the entire left-hand side f Eq. 7, was related t wake edge prperties by Schapker.^ Vales fr the dissipatin Fig. Trblence Dissipatin in the Far Wake

9 Fig. 3 cntains the data fr the Klmgrff wave nmber parameter (left hand side f Eq. 8) and its predicted relatin t the wake edge: Mst f these trblence qantities are reasnably well represented by their incmpressible expressins, hwever, better empirical expressins can be btained frm the present reslts. Even s, frther experiments and M ( V d)"5/8 (- )(- ) = 5. x 03 ( % ( ) / rf/8 a p_ i _ d a psitin statistics as given by Schapker3 as well as the energy cntaining (8) calclatins fr cnes and ther size spheres shld be made befre btaining general empirical eqatins. 5.-Strhal Nmber wave-nmber parameter (left hand side f Eq. 9): K d(v Us. d)l/8 (_i)-0. e > p T 0 ^ 3 7 (.) / r 5/4 5/8 s d d The Klmgrff wave nmber indicates the freqency f the small scale energy waves where the dissipatin ccrs while the energy cntaining wave nmber indicates the freqency f the large scale vrtices which cntain the (9) The Strhal nmber, N, fr the wake has been calclated and is shwn st in Fig. 5: freqency X / C /tt n st velcity 7 - r () N is a characteristic f the large scale mtin in the wake. Tw methds st f calclatin were sed in cmpting N. The first sed nly vrtices with * st Q3 n > Sht > -S h t > -S h t 38 slid sym bls Nn> d / p e n s y m b ls N > d _ EN. Ref. 9 * * d ** * 0 8., ' I Fig x/dxi' 3 Wake Strhal Nmber majrity f the kinetic energy. Reasnable agreement exists between the present data and the accepted relatin frm incmpressible trblence thery. These parameters prvide the link between the edge trblence, temperatre flctatins, and internal trblence. These relatins can be made n an empirical basis sing measrements sch as thse presented by Demetriades.3^ a diameter greater than the radis f the sphere and the secnd methd sed 9 nly vrtices with a diameter greater than twice the radis f the sphere. 0 The empirical expressin fr N given by Gldbrg, et al. is: N = 0.9 ( - if ^) () St Keg The Lagrangian integral time scale parameter [left hand side f Eq. 0] and its predicted variatin3 is presented in Fig. 4: is in reasnable agreement with the present data. gain, after abt 0,000 V L T d v/v /4-3/4 h s? 0.6 ( V / () n-/4 d d (0) bdy diameters the Strhal nmber appears t be cnstant as did the crrelatin lengths. This behavir f the Strhal nmber is additinal evidence f the peridic natre f the far wake. The Lagrangian integral time scale is representative f the lngest time 6.-Crss-Crrelatin Calclatins dring which the flid particles remain crrelated. This parameter again The cncept f the crrelatinal fnctin can be expanded t shw the relates the edge trblence t the internal crrelatin fnctin and the relatinship between the deviatin at ne statin (x/d psitin) and that at temperatre intensity (n an empirical basis). sme ther statin. The crss-crrelatin fnctin is defined as:.5.-5 ( )e (x+?)dx (3) - J, r in the nrmalized frm as: RCi j (C) = (4) The crss-crrelatins have been calclated fr each statin in the pper and lwer edges separately fr several shts (39, 379, 376). Frm the shift f the peak psitin f these fnctins the cnvectin velcity at the edge f the wake can be estimated fr each sht. The change in the peak Fig. 4 Lagrangian Time Scale in Far Wake vale f the crss crrelatin fnctin is a measre f the decay f the wake flw field.

10 The reslts frm the wake edge velcity calclatins are given in Fig. 6 alng with measrements frm ther investigatrs. There is mderate agreement between the present data and anther methd f measrement (seqential spark) even thgh there is a great deal f dispersin in the far wake calclatins. viii) The crrelatin lengths frm several different measrement systems have been fnd t be qite similar. Fig. 7 demnstrates the decay f the wake flw field. The rdinate is the maximm crss-crrelatin fnctin and the abscissa is x/d. This cmparisn indicates that as the wake velcity decreases (Fig. 6), there is a lesser amnt f decay. This behavir is indicative f an asympttic type f wake decay, decreasing as x/d increases. 7 -Discssin f Methd f bservatin ne f the severest criticisms f this methd f wake analysis is that the edge f the wake is nt bserved in a single plane bt in a segment f the wake. This type f bservatin injects sme bias int the statistical representatin f the wake edge. This partiality fr the maximm wake edge wld tend t make the "average wake" larger than it actally is. The gemetry f this cncept is shwn in Fig. 8. Schapker'*' shwed that apprximately seventy-five per cent f the tline pints arise frm actal ccrrences within a ±30 segment centered abt the meridinal wake plane. Distribtin f Maximm Vale f the Nrmalized Crss- Crrelatin Fnctin CNCLUSINS The imprtant cnclsins reached as a reslt f this investigatin are: i) n the basis f limited data, the trblent wake width was fnd t increase with an increase in cne angle, and the wake widths fr spheres were generally greater than fr cnes. Film Plane Fig. 8 Wake Crss-Sectin Film-Plane Edge Gemetry CKNWLEDGEMENT This wrk was cndcted at the vn Karman Gas Dynamics Facility at rnld Engineering Develpment Center, ir Frce Systems Cmmand, rnld ir Frce Statin, Tennessee. Fig. 6 Wake Edge Velcity The sccess f any prgram fr the investigatin f the trblent wake behind hypersnic prjectiles in an aerballistic range is dependent n the effrts and activities f a large nmber f perating and spprt persnnel. The athrs express their appreciatin t the members f the erballistics Branch f VKF fr the peratin f the range and the acqiring f the data. The athrs wish t express their appreciatin t Mr. Leith Ptter fr his advice and encragement dring this prject and t the ir Frce Systems Cmmand fr release f the data. ii) The mst reliable crrelatin lengths are the micrscale length, e-fld length, and the average vrtex size, iii) Crss-crrelatin lengths yield a cnvectin velcity fr the wake edge and a wake decay measrement. iv) Empirical relatins between edge characteristics and internal trblence strctre can be frmlated, v) The readings f the wake edge btained frm the schlieren negatives shld be crrected t btain the meridinal plane view. vi) The wake edge psitin distribtin appears t be nn-gassian at sme lcatins. vii) Peridicity exists in the wake edge strctre in sme f the calclated qantities. a b CD d K SYMBLS Crss-sectinal area f bdy Cnstant in mean vale eqatin Cnstant in mean vale eqatin Bdy drag cefficient Diameter sphere r base diameter f cne Crrelatin lag integer, C = KX K, Klmgrff wave nmber d K e Energy cntaining wave nmber Sample length Lg e-fld crrelatin length

11 LI Integral crrelatin length 9 / C d/tt L L r Lt e I b) verage wake width f sample L X Lagrangian integral time scale M» Bdy mach nmber N Ttal nmber f pints sampled N 0 Nmber f zer crssings f wake edge REFERENCES N st Strhal nmber rf(y > Prbability density p. Test sectin pressre q e ( t-crrelatin fnctin Crss-crrelatin fnctin q c i j (5) Re» Bdy Reynlds nmber RE(K) Nrmalized at-crrelatin fnctin R C (5) Nrmalized crss-crrelatin fnctin Re8 V 0/v Static temperatre at trblent frnt Tf T00 Static temperatre in test sectin Schapker, R. L., "Statistics f High-Speed Trblent Wake Bndaries," I J., 4, (966). Levensteins, Z. J.,and Krmins, M. V., "erdynamic Characteristics f Hypersnic Wakes," I J., 5, (967). Heckman, D., Tardlf, L. and Lahaye, C., "Experimental Stdy f Trblent Wakes in the CRDE Free-Flight Ranges," Prceedings f the Sympsim n Trblence f Flids and Plasmas, Plytechnic Institte f Brklyn, 968. Slattery, R. E., Clay, W. G.,and Herrmann, J., "Gas and Electrn Density Flctatins in a Weakly Inized Hypersnic Wake," Prceedings f the Sympsim n Trblence f Flids and Plasmas. Plytechnic Institte f Brklyn, 968. French, J. L., Persnal Cmmnicatin, VKF Gas Dynamics Facility, r, Inc., rnld ir Frce Statin, Tennessee, 968. Fx, J. and Rngaldier, H., "nemmeter Measrements f Velcity and Density in Prjectile Wakes," I J.,, 70 (97). T 0 73 K 7. V» Bdy velcity V Wake velcity at wake edge X xial psitin in the wake 8. X Sample increment size X verage vrtex size at wake edge 9. V Crdinate in radial directin Crdinate in radial directin in lwer prfile 0. y l Y Crdinate in radial directin in pper prfile U Y * Mean vale f crdinate in radial directin in pper prfile Y Mean vale f crdinate in radial directin 6 Wake frnt velcity / bdy velcity e Deviatin f wake edge psitin (Y - Y) 3. e Deviatin f pper wake edge psitin 4, n x/d X Micrscale crrelatin length 5. X Upper prfile micrscale crrelatin length., Witte,. B., Fx, J., Rngaldier, H., "Hlgraphic Interfermetry Measrements f Mean and Lcalized Flctating Wake Density Field fr Cnes Fired at Mach 6 in a Ballistic Range," presented at the I 4th Flid and Plasma Dynamics Cnference, Pal lt, Califrnia, Jne -3, 97, Paper N Clay, W. G., Herrmann, J.,and Slattery, R. E., "Statistical Prperties f the Trblent Wake Behind Hypervelcity Spheres," Phys. Flids. 8, (965). Herrmann, J., Clay, W. G., Slattery, R. E., and Richardsn, R. E., "Sme Statistical Prperties f Trblent Wakes," Preprint f Flid Physics f Hypersnic Wakes, GRD - CP - N. 9, May 967. Uberi, M. S., and Kvasznay, L. S. G., "nalysis f Trblent Density Flctatins by the Shadwgraph Methd," J. ppl. Phys., 6, 9-4 (955). Zivanvic, S., Rbillard, P. E., and Primich, R. I., "Radar Investigatin f the Wakes f Blnt and Slender Hypersnic Velcity Prjectiles in the Ballastic Range," Preprint Flid Physics f Hypersnic Wakes. GRD - CP - N. 9, May 967. Lane, F., "Freqency Effects in the Radar Retrn frm Trblent Weakly Inized Missile Wakes," I J., 5_, 93 (967). Schapker, R. L. and Camac, M., "N Chemilminescent Wake Radiatin," I J (969). Reinecke, W. G., and McKay, W. L., "n Interfermetric Stdy f the Trblent Wake Behind a Hypersnic Sharp Slender Cne," I J, 6^, 04 (968). Dinne, J. G. G.,and Tardif, L., "Mean Density and Temperatre Data In Wakes f Hypersnic Spheres," I J., J5> 707 (970). Pc PQ a e a s Test sectin density 6,.9 x 0-3 gm/cm3 7, Standard deviatin f the wake edge psitin Standard deviatin f the pper wake edge psitin 8, Trblence dissipatin rate 9, Crrelatin lag distance Bailey,. B., Private cmmnicatin, rnld Engineering Develpment Center, rnld ir Frce Statin, Tennessee. Crrsin, S.,and Kistler,. L., "Free-stream Bndaries f Trblent Flws," NC Reprt N. 44, 955. Demetriades,., "Trblent Flctatin Measrements in Cmpressible xisymmetric Wakes," I J (967). Fay, J.., and Gldbrg,., "Unsteady Hypersnic Wake Behind Blnt Bdies," I J.,. 64 (963). V Kinematic viscsity 0, V Kinematic viscsity in test sectin Gldbrg,., Washbrn, W. K., and Flrsheim, B. H., "Strhal Nmbers fr the Hypersnic Wakes f Spheres and Cnes," MP-59 vc Everett Research Labratry Reprt, pril 965.