JUXE 1966 TffPER 'O)[IC TnrnULE)iT TRAXSPIRATIOX COOLIXG 975

strellm-tran,.piration t'OOling problem," J . .\ero.,pat•e nci. 28, 449-156 ( H>61 ).

II Howe, J . T ., '·&.>me finite difference ~olutior1;:, or the laminar compressible boundary layer showing the effect.- or up:>tream tran'-piration rooling;' .XACA :\[emo. 2-2~.')9..\. (1!)59).

14 Rube:;in, l\I. \Y., Pappa.~, C. C., and Okuno, A. F , "The effect or Ouid injection on the t'Ompr~«ible turbulent boundary layer-prelimiMry te-Lc; on transpiration c·ooling of a Oat plate at M = 2.7 with air as the injected gas,•· NAC.\ Ri1 A.55119 (De­cember 1955).

JU.NE 1966 AlA.A JOURNAL YOL. 4, NO. 6

h ,\/,

Jf.

Jf .. od

'P p' Ph P•

P• p.,. PP

r Be,.' B.

B

sf

Observations of Turbulent Reattachment behind an Axisymmetric Dolvnstream-Facing Step in Supersonic Flow

A xATOL Ro HKo* Axo GERALD J. THoMxEt Douglas Aircraft Company Inc., anla ,lf mica, Calif.

' upersouic flow o.-er a downstream-facing s tep on the circumference of a large, ducted, axi,.)Jnroetric bod) was used lo s tudy flo" rea ttachment. Step h eig ht,, h were 0.25, 1.00, and l.68 in., compared to a body radiu., of6 in. Freestreani :.\Iach nlllnbers -.. ere in the range 2 t o 4 .5. The turbuJeutboundary-la~er thickness just ahead of the "lep Yaried fro m 0.14 Lo 0 .19 in. (momentUJn tluc kuesse" of about 0.01 in.).

urface pressure dis tribution,,. tJuough out the region of separation und reattachment we re m easured , and points of reallacluneut were determined. Comparison of the s hape of the p1·cssure dis tributions for variou ,, !'.tep heights sh o"s that the initial (steepes t ) parts of the re­;.iuuch.menl pres-<uYe r~c, up to the point of reattaclunent, te nd Lo boooroe s upe rimposed ,,lien plotted agains t x/ lt. Do'\\us tream of reattac hn1e nt the c un-e;, bra nch out. e.~hibiting a .de1>endc n cc on geometry and probahl) on initial sheru· la) e r profile. Ln t h e regio n of the initial pre~ure rise (near tbe end of the "dead (Ur"' region) d) nUJnic pressu res a rc low; the p re,sure ri,,e there apparently is balanced by turbulent .,hear ;.tress.

No1nenclnture

!<tep height :\forh number of the base How e.'ternal to the free :-hear

layer 1\Iach number or U1e external flow ju.st upstream of lhe

>leparalion point '.\Jach number of the freestream location or flow revCJbaJ poi II t determined from lhe Orifice

dam terhnique local :mrface pres;urc 1oc11.l surface pressure in the presence or an orifice dam :>tatic pre-<-ure in the base (dead air) region ~tat ic pre:-,ure of lhe Bow external to lhe free ~hear layer,

=Pb surface pre··sure jtt t upstream of the ~eparation point tatic pr~--urc of the freestream

lomtion of the peak in ='Urface pr~.;u.re dk-1ributiomi mell.'lnred in the presence or orifit•e dalll.:i

rt•attachment or Bow rever.;al point test -'ection freestrearo writ Reynolili. number radius of the model forebody at the How separntion

point separation point location or How rever.;al point from lhe :surface Aow

technique local veloci ly in the l>oundary layer

u, velocity ju,t out:side lhe boundar) layer and ahead of the step

x = di•tance nlong the model measured down.slream from the base of lhebtep

y heighl men.sured from the swfoce of the model outward into the air-:slream

o, boundary-layer thickness just ahead of the step a •• = boundary-layer dbplacement thickness just ahead of the

step fJ, = boundary-layer momentum thickne,.s j11.:sta head of lhe

::.tep

Introduction

Tlill importance of the reattachment region in problems of flow separation i well-knmvn. Tbe flow confibrura­

tion in which it has been studied mo~l frequently i" the flow OYer a ba-:e, or down:-tream-facing , t<>p. Although I.his ap­peal", al fir...t. to be a very simple configuration for C:\'J)eri­mental and theoretical attack, at prc:-cnt there is con:-iderablc unrcrtainty a11d controYer;ry about it, both experimentally and thooreticalk

Received )lay 17, 1965, revi.,ion received :\larch 17, 1066. Thi· study was 'pon..;ored by the Douglas Ai.rcr:ift Company Independent Re-earch and Development Program, Account

On the experimental .:;ide, the Ion!!: ::.landing effort to define a bll!<e presRJrc ,.s ~Iach number curve, so far, ha not re­imlted in a dcfinifo·e re."ult. Rernold number effects. often interpreted in t~rm-. of boundal)·:layer thiekne;:,., at the -epa· ration point, ha\•e been e.5pecially difficult to clarify. Thi:::i is due to se\·eral factor.:: lack of appreciation, at fir...t, for the need to e;,"i:abli.~h clearly the boundary-layer conditiollli; lark of appreciation of the importance o[ geometrieal configuration for correlating various re:::il!lt. ; difficu lties of avoiding a tur­bulent-transitional condition in the ordinary range of labora­tory facilities; and, sometimes, difficulties with end effects on two-dimensional models.

0301-016. The authors are indebted to many staff members of the Douglas Aerophy,,ics Laboratory, El Segundo, California, who contributed to the experimental program. Partic:ular thank:> _go to W. E. timith for bk exten.:;ive help in preparing the e."tpcri­inent and report, and for many helprul discu sions.

• Consultant; aL-<o Profos.,or of Aeronautics, Graduate Aero­nautical Laboratories, California I nstitute of Technology. ,Associate Fellow )Iember AIAA.

t Engineer-Scienth.t, Aerophysics Laboratory. As.iociate ?IIem­ber AIAA.

On the theoretical :>ide, the well known E:on--t-Chapman re­attachment criterion1.1 recently has been the subject of critical re-examination; Yarious experimental examples'·~ of

.\. no:-:IJKO .\:\D (;. J. TllO~IKE \ I.\.\ JOPll:X.\ L

I

_l ADC !lmlA..Sl£P llECJtfS Of llH'ID 111! JJC. Qll• V.91.! I' flil:1TA.1.L.Mi SlH't'f

. "

--- IU---'------ 111 -----fCO!IOOT •HE•!!OOT

.Fig . 1 .\xi") mmcu·ic il u c t ed m od el.

reattnt•hml·uL pre ... ,.ure 1i-e ~'fl'llter than the ma\imum ~\·en by the 1·ritcrion ha\e re<lirectt'<I atn•ntion to n ha ... ic incon­-i ... ttmcy in that th('(ir) and to attempt::: at mndifyin~ it.1,6

Jn another approarh, the earlit•r, partially rucr(',..,.ful inte~r:ll method.- :-uch a ... that of Cr&1•0 and Lee-' nr<' ll<'ing im­pro\·e<\.,,li

The pre:-ent paper dc:-cribe- ... ome experiment" on turbulent tlo" OYer a down ... t ream-facing :-tcp. The._'C pro\·ide -ome in­,.j~ht::. into the rrattnd1ment problem and con,.titute a ... et of pre,,. ... ure dktribution ... again"i which theoretical re-ult" can he te ... te<l. The experiment-. were moti\·ated lar~ely by the O.Yailability of a Larii:e facility '' ith high unit Reynolds num­her: the Four-}'oot Tri:-onic Wind Tunnel at the Dougla:, .\ erophy'ic" Laboratory. In thi,., it was po-,,ible to u"e the large axi-ymmetrie model :-hO\rn in Fiu;. 1 and still a,·oid the problem of wake interfrr!'nte from wtwc reflection-. off the :-ide \\all>-. Other adYantages were the following: \\ilh the a"ial t--ymmet ry, problem;; of end effects were eliminated; and with the large ... ize, i:.tcp heiirhb could be small rela.ti\·e to body radiu ... , but ::till rea-.onably large rompared to boundary-layir thirkn~.... TI1e latter effect wa.-. enhanced by u ... inir a ducte<l model, which made it po~ible to ha,·e a much ,.honer length from no,.e to t-tPp than wit11 the Ubual cone-eylindcr body. The lenii:th wa.'i :>till ,uffiricnt to in:-ure tran:'ition well ahead of the -.tep. In thi::- way we hoped to eliminate end-wa.11 cffet"t,. a.nd tran,ition effect... while keeping the boundary­layer thicknCt-- at -.eparation "mall tompared to ... tep heiidit. On the other hand, the axial ~·mmctry introdure" a ~eometri-1·al parameter that \"llrie ... with "'lcp hei~ht.

E~perimental onditionsi

The model 'Ftf(. 1) had a diameter of 12 in. at the ... epam-1 ion .-boulder. with a 1.68-in . ... 1ep down to the eylindrical

u

$ , .,

J .., 0 I q, • ;, o

e a l.Z

i ti 0 e a

• a

OJ 14 II OJ ,: u u,

Fig. 2 Boundar) -la) e r \'elocil) profiles a l '-lCp ,.houlder.

t A detailed d<N·rip l ion nf the e.xµf'rimental -;et up mid re<11lts j ... given in Ref. !'l.

att1·rh<Kl). The -tep hei~ht l'Ou!d lw chau~t~I to 1.00 or U.:.15 in. by in-tnlkttion of ... 1ce\·e" onr the afterbodv. ~lca-ur~ meuh wen' made o\·er the range of free:-trcam :'.iach numher,, from 2.0 to -l.;) al the value" li-ted in Tahir I, in whi<·h other )l(•rtin<·nt t·oudition' are li'tetl. The r<.'frre111·l• eondition' .lf . p I ju-t ahead of the ... tep are ... omcwhat 1lifferent from

m-e-tream ,·3.lues. Thi,. i,., due to the flowfic·ld of the fore-­body. lt would have b('('n preierabh.• to lN' a -traight eylinder for thr outer surface of thi,. iort'bodr, and thu- to ha,·c a uniform flowfield ahead of the -tep, with JJ = JI.,. rk.; I.mt thi" would ha\·e required the inner -.urfac·e (the durt) to be co1ffergent, re ... ultinl! in rhokinu; at the lmwr :\farh number-,.

'fra1i,...itinn, ob:'erYed on -hadmn .. •Taph~, tx·curred in all ca-.e. ... near the end of the conical ,.,ertion of the forebody. The boundary-layer t11ickneb."es ju:-t ahead of the :>tep (Tahir J) wrre determined from Yelocity profile..: (Fiit. 2) obtained by rom·entioual pitot-tube :-un ey:-, u--inu: a Nm,.tant-ener~· a,_ -umption (in all cac;e;; the model -urfal·<• temperature wa. ... within 2~ of the free;tream towl temperature). 'TI1e pro­file..~ -.hown in Fig. 2 were obtained a.t x., the ::,treamwbe po,i­tion of the ... houlder; the probe ju,.t cleared the edge in ordt>r to traver'<.' pa-.t it. A. few profile"' takrn 0.2.5 in. ahead of the ,..boulder were identical with tho:>e in Fie;. 2, t'\l'ept for point ... at Ynlu~ of y 6. < 0.l; the flow in the--e lower llwer,. eYirlentk j,., influen<·e<l h~· the e:-.1mn,ion o\·er thr ,ho1~lder. In th.e talrnla.tion" Of o* and 8, thO-t• poinb WE're ijnlOrCd and tht• profile~ were faired snoothly into the oriJ...rin. The corre­"'Poudinu: "hape factors o• 8, a1?;ree do ... eh· with calc·ulat ion,.. ba..;00 on the theory in Ref. 10. ..

:.\Ieas u 1·cn1ent

.\ n example of a .flowfiekl downstream of th<' ~lep j,, "ho" rt in the rhlirren picture of Fig. 3 taken at J/ = 2.09. 'kett-hed on the pirture i"' a broken line inditating the theoretic·al po-ition of the free ~reamline m the <'orre ... pondin~ inyi,.eid

DOl~STRl~U D l'TAl>C[ fl!(lll $llP ,

Fig. 3 T)l>il'ul Oo-.tield at H , = 2.09, "ilia r..orre!'pondiniz pres,..ure tli.,trihution.

JC~E 1966 OR~ERY..\TTOX'4 OF TURBl'LEXT 1n:.\TT\CR\1£XT !Ii i

Table 1 Su01Jnary of Experilne n t a l Paran:ie lcn.e

.l/

2 OUl: (2 OJ

2 4U7 (2 .ti) 2 !)!J6 ( :l 0 )

a .i. .... u (3 5) :3 !H.'i ( .J 0)

4 460 {4.5)

Ji,

:LO!!

2 .56

3 .HO

4 .37

Uf I

p P« lil in.)

0 1 0 . !:!5

1 02

fl G 30

() . '>7 l :w

h ( in.)

0 250 1 020 l 675 0 250 l 675 0 250 I 020 l 6i5 0 .250 I 675 0 250 1 020 l 675 1 675

0 .HbO () 13-t;3 II 01'.\lb 0 G.360 0 . 097!:! () 72-ID () 1774 0 IO. l 0 7600 0 . 1134 () 7720 0 1S92 0 1152 0 1170

o."• /h

0 136.~ 0 0335 () 020-1 0 1791 0.02G7 0. Z-211 0 0542 0 0330 0 2614. 0 0390 0. 304!1 0 0747 0 0455 0 0;)29

fJ, / h

0 03 9 0 .1>095 0 0058 0 .0415 0 .0062 0 .0402 0 0099 0 0060 0 0382 0.0057 0.0359 0.00 0.0054 0.0052

Par1un-RE:.ittachme1t1 Hcginn l'ter.-

Pb P• JI• (i;/h), (x /lt 1,, (x_h__ (x / h)od ~---~-~~

() 373 U.3!12 0 417 0.274 0 331 (J 214. 0 J9 u 252 0 184 0 . 174 () . 194. U. 132 0 . 122 0 . 111

') -? - · ,_ 2.69 2 .65 3 42 3 2!J 4 12 4 . 17 3 9!l 4 7 4 . 3 5.23 5 . 57 5 65 6.3-1

3 36 3 .73 :1 60 3 . 0-2 3 :~.j

2 '.h :3 0'2 3 24 2 !)

:3.06 3 . -16 3 12 2 97 3 19

2 !}()

3 30

3 '-U 3 ;34 3 25 2 .90 3 1:3 2 :3:! 2 65 :3.10 2 20 2 47 2 30 2 00

.).IJO

:; 4.">

:? • :;11 2 .10

2 ltl

• )lote: Boundary-layer parameter• at .II.., = 2.5, :1.5. anJ L5 obtained by in terpolating and t<Xtrapolating data meaoured at JJ .., - 2, 3, and 4

model, compnterl§ for Lhe measured value of ba~ pre·,ure, Pb = 0.424 p.. Taking p. = Pb lo be the pre,,.:<ure in the ex­l<'rnal flow at the edge of the "hear layer, which i" a ,.,-umed to ha.\'e expanded i."entropically from p., giv~ the corre,;pondinJr Jf, = 2.64. If rontinued to the "houlder, lhi:> line coincide,, \\ith the shear layer vi:-ible on the photograph. Al~o ,-kekhcd is a line indicating l he theoretical position of an oblique shock wa\·e originating; al r in the invllici<I model. It may be com­pared, in the photograph, lo the dark reeompre-,..ion region be~nning :>omewhat up.;;tream of r. AL"<> -illown i- the meas­ured pr~-.ure di triburion along the ~urface and, for com­parison, the pre:;:,;ure jump at r in the inviscid mo<:lel. (The other notation on the figure is explained later.)

Figure 4 i;, a plot of the prc,,.:mrc distribution-. mcarured along the ~urfa('e down.;;tream of the ~tep at three :\farh numbers for three different :>tep heights. The broken line~ (Fig. 4) are the ba~ic pre:;.sure di.:>niburions (for zero step height) due to the nose shape (cf. pre\ious remarks) com­puted by the method of characteristics. The measw·ed prc.."Sure dktribution;:; for the various step heighb all tend toward thi>< ba::-ic di,tribution far dowru;tream. For the larger :,tep height,, the pre~:-i.ire in the reattachment region over~oot..s the ba~ic di;,i:ribution, due to an a\i!!:yrnmetric effect in the cqui,·alent free .. treamline flow. 1: The effect dimini>'hes with deneasing hf R. or with incrca.-.ing ~Iach number; i.e., smaller step height or higher ~Iach number tends to make the flow in the \'icinity of the :.tep more nearly two-dimen. ... ional. Thus, in Fig. 4, for the ,,mallest step (h = 0.25 in.), the pre:-."Ure ri"e" monotonically and fairs into the basic pre ·sure distribution without o\'en•hooc. A.l::;o, with increa:-ing :\lach number, the larger );teps show le ' over ... hoot. • :\ leasurements of circumferential pre;:.:,11re di:s­tribution ga\·e \'Blue"- lha.t were uniform within 0.5% in the ba."(' region and ";thin 2% in the region of ri:.inir pre "ure.

In F ig. 4 it i,, difficult to ::-ee the detail· of the pre' 'mre di~ tributiom~ in the ffi-called dead-air region near the step face; the,,,e are shown in Fig. 5, in which the pre:;.-;ures o\·er the ;;tep face (repre:>ented as - I < x/ h < 0) and -everal '>lep height>­downslream are plotted. The \'alues ~electro for the ba.><e pre.-..-.ure~ are inditated by broken lines. It may he ~n that the.-.c are bia,,ro toward the rnluc ... on the :.tep face, and that there i~ a charactcri.-tic :>mall dip in pre;;;sure just beiore the reat taehment prei;,,ure ri. c bei.,rin '. For each value of base pre~'-w·e Pb ( = p,) , the corre"ponding value of J£. (Fig. 3) wa,, computed b~· o..<..:>uming an isentropic expan:-ion from (.11,, p.). The--e \·alues arr listed in Table I. WC' are not

§ We made 11.:-e uf the free >S treamline calculation., in Ref. l l. \Ye are i11debte<l to J. Xeriko,,, of the Douglas Air eraft Com­

pany, for providing 11;; vdth 1 he-;e re,,ult'-, whit·h nre de>rrihed more fully in Ref. 9.

inrluding here plots of bike pre.""'11re again,,t ~Iach uumher; the...-.e can, however, readily be rletennined from the informa­tion in Table l , in which data for boundary-layer correlation ... also may lw found.

Point of Rcallachme n t

Two technique were u:.e<.I to locate the point oi reathwh­ment. One of the.:se wru: a "urface flow technique u,,ing a coating of titanium dio:ddc (TiO~) in oil. Figure 6 ... how-; a11

example of the flow pattern at Jf_ = 3 for the lar~e -tep, Ji = 1.68 in. The line of flow rever'!l.l is quite well defined.•• The position>< of flow reve1-,;a.l points located in thi;; wn,· ar~ indicated by sf in Fig". 3 and 4 and are li,,tc<l in Table i. 1n the second technique, mall ob"tructions (0.05 in. high and 0.15 in. wide), which we calJ orifite d&Illi:, were cementc<l ju ... t up 'tream of each of the orifice< along the surface ~Fig. 7). Each :::uch orifice and dam is a rough approximation to a .-ur-

11

DI

.:

~ 0 4

~ ll

~

iil ~

u !Iii

O.•

12

OI A Cl M

1 HI r • J!~'tHES 1

- l'BAllCI

0 Ill

•• 0 I QI

b JU

II 11 " Ii ll ~[All DISlfJICE fRllll STEP • l>M:lftS

F ill. 4 ::\lea3ured 1)l'e!'sare di;,u ·ibutio ns .

"*Also evidenl i~ a thr~dimensional pattern thal is -imihu· (fl

that observe<l hy Ginoux11 in laminar two-<limensional flow over 11

-tep, and which is indicative of :::econdary mot.ions in the ret1.l­

tachment region. Their efTeN on the reattaC'hing flow is un­known. "' me detailed pre;sure distribution.,, iu that region indi­cate that these ~econdary motion>' have very ~mall effect on tl1E­static pres.sure distribution, a11d it ~eerm; unlike]~ that the over-all pressure ri~ i" affected.

97 .\.. H.O::;HKO AXD G. J . TnmIKE .U .-U JOl'RXAL

05

: ..

0 I

II 0 209 • 216 0 302 • H9

• t

" !90 0 • • ll 0 f

-00-0- - --co-c50-.~ Q lJl

e ----..--..--om - -;a-...2-- - -0 214

-===~=E==!~11&e

0

0

rft --0-0-~--•1•

"'" " - -L..-.t.---di---- t m • I ~ - 111! l'ltHES

OISlA~CE FRiii SlfDULDE~ ALONG PERIME TE~ • I

face pitot (Prc,,ton) tube: in forward How it should ;;:how a de­crell.1'e of prc .... ure, compared to the clear surface, since the orifice is on the "ba e" side of the dam; whereas in rernrse flow it should :;how an increa$C, the orifice now being on the "fll('e" ide. Thu..:, the s.lreamwii,.e pre:<sure di41ibutions ";th and without orifice dam dlould cro· at the point, of How reYersal.

An example of the re ults obtained is ~hown in Fig. 7. I t may be seen tht\t the effect i;; ,·cry l>mall, an indicat ion of the low dynamic pressures in the reYcr;,e flow region up to the re­attachment point. .\. plot (not :'hown) of the ratio of the two prc,..-.ure5 p and µ' (the cro""o,·er point bein~ at p' / p = 1) impro,·ed the resolution considerably. The re,,ult was quite unrunbiguouo:: for the lowest :\fach number, 2.09, but rather lc:ss arcurate for the other two, due to catter. The method was not applied to the smalle<"t l0.25 in .) step. Locations of flow rever al poin ts delemlined in thb way are indicated by od in Figs. 3 and -1 and are listed in Table 1.

.\.n unexpected re,ult irom the orifice dam technique was the appearance of a peak in the pre&;ure distribution, as in fjg. 7. The decrra~ in pre~ure afier the peak, even though the general oto.tic pressure ii> still rio-ing. indicates that the fio" over the dam has become ~trong enough to produce an nppreciable dynnmit• effect and a Hrong dccrea.--e of the base prei',ure on the dam. Thus, the peak i!' a good indication of the end of the (nearly) dead-air region. The locations of the;e peak pre,,sure point. are indicated by pp in Figs. 3 and -1 and are listed in Table I. It may be seen that they always lie . clo~e to the point.' marked r, which are the theoretical loca-1 ion - of the reattachment points in the corresponding inYiscid

fig. 6 Reattach ment flow pattern obtained u s ing oil-Bow vis ualhation technique; U , = 3.02, h = 1.68 in.

Fig. 5 )Jeas ured pres~ure dis tributions aloog Lhe perime ter of the base region. Broken lines iodicale values used for PblP•·

tep face is represeoted b y n egative •·alucs of x/h.

free streamline model (Fig. 3). On the other hand, the point of Bow re\·ersal, whether sf or od, alway,, occur at filllaller \"alues of x/ h. (The one exception, for the smallest step at JI, = 2.09, may be in error ; the t"orre:;ponding flow pattern wa~ not distinct.) Taken together. these re:.'Ult- 1:-u~est that the real point of "reattachment" (better called the point of flow re­versal) lie. ju:;t inside the region of low dynamic pres.-.ure, which itself corresponds fairly well to the ,;o..called dPad-s.ir region of the free streamline model.

Shape and Scale of Pressure Dis tribution

To study the shape of the pressure di>-tributions of Fig. 4, the curves were replotted against the din1Pn:::ionl~ distance x/ h. In Fig. the distributions are arranged in group of constant ::\Jach number, to bring out the effect of geometry, wherw in Fig. 9 they are arranged in groups of constant h, to display the :\Jach number effect for given geometry.

Due to the crowding together of the,..e curves, the location of the point sf, r, and pp from Fig. 4 are not reproduced in Figs. and 9, but their ranges are indicated in Fig. . . \.n interesting trend is that, the location of r (or pp) tend" to re­main nearly fixed (at a value of x/ h :>lightly greater than 3), but U1e location of sf tends to move toward the step with in­creasing :\Iach number. This should be ' ·iewed with "°me re.-;errntion, since re:.-ults from surface flow techniques are notoriously difficult to evaluate; on the other hand, there i indication of the same trend in od from the orifice dam ex­periments .

12

~ 1 0 0

0

00 0 0 0 0

0 ' '• YffHOUT DAMS e (' llf'H DAMS

4 1a

DOfHSTREAll DISTAHCE fRCM Sl( P, • h

Fig. i Effect o f ori~ free d::u11s on pres­

s ure dbtributiou.

JU~E 1966 OB:3ERYATlON:3 OF TURBULENT REATTACH:.\IE~T 979

11

I 0

01

0,

11

G~

~ 10 n

"" ~ g:

;: as

3 0.6 .. 11 O•

I 0 oi

OS

06

c•

01

66

" " a" 0 ,,o

6

60 0ao

0

oo . 000 A oO

0 <D cg,

t9 "h

~H Oii r::P>H ,

~~ Hst 6 0

0 co ,,,a

~oooo , 0

(j

:/ H•

d H•1

w, 109 o a o o

oooo000Go

o a 0 0

0

0

h (INCHES\ • R 0 011 0 0'1 0 1 01 0110 6 1!8 01!0

10 11

DOlhSTREAM OIST~~Cf fROI STEP,, •

1,

Fig. 8 Effect of s tep height on shapes of m eas u r ed pres­s ur e dis u ·ihutions; ranges of pp, r , a nd sf noled.

In Fig. 9, the effect of :'.\Iach number are compared for fixed geometries. The group for the smallest step height (0.25 in.) is particularly interesting since it includes fiye different rnlues of ~fach number and since h/ R. = 0.04 is small enough that the reattaching flow surely can be regarded as two-dimen­sional. A noteworthy feature is the branchin.,. of the curves just downstream of the reattachment point; ~ break in the pressure distribution. there becomes more pronounced with increasing l\Iach nwnber. The same trend is evident for the other two ya}ues of h.

Figures 8 and 9 are remarkable in the tendencv shown for the pressure distributions to be superimposed o; each other in the region of steepe~t pressure rise, approximately 2 < x/ h < 4. One might expect the length scale of the pressme rise to depend on the thickness to which the shear layer has grown before it reattaches, and the scaling with h eems to imply that the initial thickness is small enough to be unim­portant for the len!,rth scale of the reattachment region (not necessarily for the dynamic, ). On the other hand, the initial thickness o, for our smallest step height i:> comparable to h (Table 1); furthermore, some measurements by Hastings14

included cases in which o. was considerably larger than h, and till the pressure rise region tended to uperimpose on those

for smaller o./h; the base pressures, however, were affected considerably. All this suggests that the dead-air or inner portion of the flow is governed largely by deYelopments along the dividing streamline, i.e. , that an inner shear layer, be­ginning at the eparation point and growing linearly, will be consistent with a scaling that depends mainly on step height

It has been remarked earlier that, up to the reattachment point, dynamic pressures seem to play a small role, and the pres.sure ri e in this region must be balanced mainly by turbu-

11

I 0

Cl

0,

~ ~·

~ 11 o•

i 1 0 02

!ll 0.8

Lt 01

12 oc

1 0 02

OS

06

o.c

01

10

h • I CZ f~CttES

ft ff 0110

11,

0 lO'l

• 156 0 !tl1

• HS 6 liO ... UI

11

OCUSTREA!I OISTA~CE fft(l!I STEP 1 a " F ig . 9 E ITecl of Mach number on shapes of m ea sured

pressw·e dis tributions .

lent shear stress. The breaks in the pre;;sure distribution just downstream of reattachment and, in particular, the bump at x/ h = 3 for JI. = 4.37, (Fig. 9), may be related to the lip shock phenomena discussed by W einbaum 16 and observed by Hama.16

. F~all>:', as noted earlier, the uperposition of the pressure distnbut10ns up to reattachment contrast with the variet.y of developments further downstream, suggesting an inde­pendence from the flow downstream of reattachment· this independence ha been observed more explicitly in e~'J)eri­ments by Bogdonoff et al. 17 and Carriere and irieix, 4 as well a!

in the theoretical results of Crocco-Lees-Reews,6·s where it is attributed to the appearance of a critical condition just after reattachment. The peak pressure ob~erved in the orifice dam experiments (described previously) also occurs at this point, suggesting a large and sudden acceleration of the boundary layer there.

R efer en ces

1 Korst, H. H., "A theory for base pressures in transonic and supersonic flow," J. Appl. )lech. 23, 593-600 (1956); al!'o Kor,-t H. H., " Comments on the effect of boundary layer on sonic flow tluough an abnipt cross-sectional area change," J. Aeronaut. Sci. 21, 568 (1954); also Korst, H. H., Page, R.H., and Childs, )1. E., "A theory for base pre;:sures in transonic and supersonic flow" Univ. of Illinois ?.IE-TX-392-2, Office of Scientific Rernarch, T~ 55-89(March1955).

t .ch3:pman, D. R., Kuehn, D. }.1., and Larson, H. K., "In­vestigation of separated flows in supen;onic and subsonic streams with emphasis on the effect of transition," XACA Ames Research Lab. Rept. 1356 (1958); also, XACA Ames Research Lab. Rl\I A55L14 (June 1956).

•Nash, J. F., "An analysis of two-dimensional base flow in­cluding the effect of the approaching boundary layer," Aeronauti­cal Research Council R&)l 3344 (1963).

.\ . H.OSrTKU ~D G. J . THO:\fKI,<; ALU JOURXAL

I Carril.>re, P. Ull<J ~irie1x, :'.\t., .. n~ultat:> recent:. dan;, r etude cJe:- probl~me:: de mrlange et de reroUcment, 11 11th IntemalionaJ Congre;;:< of Applied :'.\leC'hani<'S l\Iunicb (Augu:.t-September Hl64 1: abo Offi<"e S atio11al D"f:tudes et de R echerC"hes Aero­;-.patiales, Tire a P art 165 < 196-1).

6 ~lcDonald, Tl., "Turbulent ;;hear la~·er reattachment with ~pecial empha.~i>:. 011 the base pre:-sure problem," Aeronaut. Qnttrt. 15, 247-2SO (At1g11,,t 1!)64).

6 Crocco, L. and Lee::, L., .. A mixing theory for the interaction between db.-.ipative flo"--. and nearly bentropic ~treams," J . . \ero~pace, ti. 19, 649-616 (19il2).

1 Lees, L. a11d Hecve-., B. L., '·Supersonic ,:eparated and re­:ll tarhing laminar Bow,:," AT..\A J . 2, 1907- 1920 (1964).

• Heeve;;, B . L. :ind Lee-. L., '·Theon· of laminar uear wake of blunt bodie" in hyper>-onic: flow;• AIA .. .\ J . 3 , 2061-207-1 (1965).

I Thomke, G. J ., .. Separation and reattachment or super.<onic mrbnlenl boimdan· layer behind dowrn<lream facing steps and over ca vi tie<, .. 1>0111.d:i.-. Aircraft Co. Rcpt. • :\1 43062 (:'.\fsrcb 1964-J.

10 Hesholko, E. a11d T ucker, :'.\!., "Approximate calculation of 1he c·ompre,.;ible 111 rlmle11t houudary layer with heat tran:Jer and itrhitrary pr~,11rc irradient, ·· XACA Lewi,, He-e.'\rt'li Center TN ·ll.'H ( Hl5i ).

11 Sim.,, J. L., ··Reiulti< of the computation-. of ~,1per--ouic flow field:< uft of circular cvlindrical bociit"' of revolution by the method of churacteristicS," Army Balli-..tie :'.\li,,.ile Agency ·Rep1. DA-R-19 Oisrch 195 ).

12 Chapman, D. R., "An anaJy,k of ba,.e pre:-.--ure at i:upersoni<.­velocitie:- and ''omparison with experiment," XACA Ame;, Aero­nautical Lab. llept. 10.Jl (1951).

11 Giuoux, J ., "Experimental evidenc·e of three-dimen.•ional per­turbations in the reattachment of a two-dimensional laminar boundary layer at .lf = 2.05," Training Center for Experimental Aerod~,1ami(';i, Belgium, TX-l ( 1 95~) .

u Ha.,,ting-, R. C., "Turbulent flow pa .. t two-dimeiL-.ional ba.-.e­in supersonic :<\reams," Royal Aircraft E.•lablishment T~ Aero. 2931 (AD -rnao11 ) (D ecember 196.'3).

1• \\'einba.11111, K, .. T he rapid e.·q>au.-.ion of a ::uper--oni<· ;.hear How;· ..\vc:o-Everett. Re:<earrh Lab., He-earch Hcpt. 204 ( 1965).

1• Homa, F. R., "E. ... timation of the ,..trength or the lip "hock," AIAA J . 4, 166-167 ( 1966).

11 BoitclonolT, , . :\1., Kepler, C. E., 1111d Sai1lorenzo, E., "A -tudy of ,.J111ck wave turbulent bou11<lary layer iuteraetion at JI=:~." Princeton l"11iv. Aero11autic-al EngineerinJ!: Dept.. Rept. 222 (Hl•i3).