and a are eval­of the ohserva­

; lragm :co is giV(!ll

jl'e of the ell'iVml ,j (4), the diametei·

~ ,J in inches, lind \ :~2' i.he boundary N:: nity for pCl'fec;t,

: I reference l. "\ get the desirnd

,,' to r JI} ,1--1

Tth

, . (A8)

" (.VO of r/rth Vs 11;1 number M •. ii.cd the relatioll­

~l. = 4 in ar~on, A the data pomi.s ]fn Fig. 11, this, i con,tant Vainc

j

j 1,bI ) Coordinates

liS. (25) and (:.m,

:'~/rth)J ' , :,)

(AD)

. f I

SHOCK-TUBE-FLOW PHENOMENA

7', given in Eq. (AlO),then 1pq. (22) becomes

_~:9~QBr/p~!i_")xo [1 + rlrtll ] 2PI d (pd PI) ':"" 1

= [ __ j(tJDL __ ] t(S"I) - 'SLI

'{In [I In Fig. 10, Eq. (All) has been plotted for a vallie'

of ]),f. = 4 in argon, and in Fig. 1'1, it hus bren plotted for M. = 1.6 in argon.

T n E P H Y SIC S 0 F FL U IDS VOLUME 4, NUMllER 12 DECEMBER, 1961

Effect of Boundary-Layer Growth in a Shock Tube on Shock Reflection from a Closed End*

GEORGE RUDINGER

Cornell Aeronattlical Laboratory, Inc., Buffalo, New York (Received July 10, 1961)

The pressure behind a shock wave propagating in a constant-area duct increases slightly with time as a result of the growing boundary layer. This rise is considerably magnified by the shock reflection from the closed end of the duct. Analysis of the local interaction of the reflected shock with the boundary layer yields information on the shock configuration, but the combined effects of the waves produced by the growing boundary layer along the entire shock tube must be considered to obtain the state of the gas behind the reflected shock. The theory of the flow field behind the incident shock is modified to allow for effects in addition to boundary-layer growth which cause deviations from an idea~ shock-tube flow. Experimental observations of the pressure rise behind the reflected shock obtamed for shock pressure ratios up to about 4 . 5 are in satisfactory agreement with the computed results, and estimates for stronger shock waves are presented.

I. INTRODU€TION

IN the course of a recent investigation, shock waves were reflected from the closed end of a

shock tube, and it was noted that the pressure i)chind the reflected shock exhibited a small, but definite, rise with time. A quantitative explanation of this rise was needed in connection with a study of shock reflection from an orifice. J A better under­standing - of this, phenomenon should also be of general interest since there are numerous shock-tube applications in which shock reflection from a closed or partially closed end is utilized to produce a quantity of heated and compressed gas. Only weak or moderately strong shock waves, having pressure ratios not exceeding 4.5, have been employed in theprcsent experiments, but stronger shocks are briefly discussed.' . .

The usual practice in determining the state of

* An .. nbriclgcd version of this paper ,was presented at the Washington, D. C. meeting of the American Physical Society, April 24-27, 1961.

I G. Ituclinger, J. Basic Engineering (to be published).

the compressed gas behind the reflected shock is to measure one quantity, such as the velocity of the incident or of the reflected shock wave, or the pressure near the closed end, in addition to the initial conditions in the shock tube. Other variables are then computed with the assumption that uniform conditions prevail behind the incident and the re­flected shock waves. Since the nonsteady boundary layer in a shock tube generates pressure waves along the entire duct wherever the flow velocity is substantially different from zero, the state of the gas in these regions cannot be exactly uniform, and the results of the customary calculations depend somewhat on which particular quantity has been measured. Evidence for the nonuniformity of the flow conditions behind the reflected shock has been noted also by other investigators, for instance, Toennis and Greene,2 Strehlow and Cohen 3.4 , (19~?).P, Toennis and E. F. Greene, J. Chern. Phys. 26, 655

3 R. A., Strehlow and A. Cohen, J. Chem Phys 28 "83 (1958). , . . . ,v

4 R. A. Strehlow and A. Cohen, J. Chern Phys 30 257 (1959). . ., .. ,

. 11111111111111111111111111111111111111111111111111 '-------________ TRP00300

~--------------------)

i " , I I

"

·, II

._., ".,.- -'~"""''''-"~"-="==~~''='''""''~~='''~~-~' ----------~'- -r-.I,~ -\ \',

\ GEORGE RUDINGER ~ 1.1;-

l 1464

Bra.')b:", Ziatarich. and Belles. 6 and ,Ga rdincr ,11\",'1 l{i:"t iako\Y:"ky _; These ob"elTa tions indicate the ctfects of the prCSSUl'e waycs that arc generated by the boundary layer, but a quantitative evaluation of these effects on the state of the gas acljaC'('nt to the elOfC'Cj end has, apparently, not yet bC'cn at:('mpted.

.;\Iark" deiicribes a flow model for the interaction of the reflected shock with the boundary layer. If it is laminar, the conditions may be predicted under whieh tJlC l'eftcetcd Hhock remains either a sub­titnn tinIly plane front or develops a complieatcd bifureated interaetion pattern. If the boundary layer beeomes turlmlent, the change of the flow pattern disappears, and the reflected shock again tends to form a plane front. The predicted changes of the flow pattern are in agreement with Mark's obser­vations and have also been verified by Strehlow and Cohen. 4 Although the pressure waves gen­erat.ed by the boundary layer are not eonsidered in Mark's ll,naiysis, his description of the local interaetioll of the shock with the boundary layer

'is sllceessful because these waves produce only eomparatively slow flow changes and thus do not materially affect the local interaction phenomena. On the basis of his n;lOdel, Mark predicts a pressure drop behind the reflected shock. Such a drop has indeed been observed by Holdel' and Schultz,9 but only in thc immedinte vieinity of the closed end of the duct and not at some distance from it. Sehlieren photographs1o indicate that !Jle interaction J:Oi£ion extends over. less than one duct diamete~'. Thus, once the reflected shock has traveled some diskmce, a flow field is estab1isfiea-1ii\VhTC1it~e conditions are a-ffected mainly .£Y. the bQundary cmiclttion of ~ty and .b~_Jhe waves that arc generated bYti'iO~bOu"ndl1ry layer ahead orlhe reflectCcr shoek and penetrate into the region lkhind It. The. weak waves that are produced by the growth of the interaction region are rapidly weakened to an insignificant level because of their spreading in three dimensions. Since the gas behind the reflected shock is substantially brought to rest, no further waves fire generated in this region, and the flow conditions there can be determined by

b U. B. r:lkillncl', J. Chcm. Phys. 31, 268 (Hl59). ' o T. A. Brabbs, S. A. %latarich, and F. E. Belles, J. Chern.

Phys. 33, 307 (J HGO). 7 W. C. Gardiner and G. 13. Kistiakowsky, J. Chern. Phys.

34,1080 (H)G1). 8 H. Mark, NACA 'I'M 1418 (1958). o D. W. Holder and D. L. Schultz, "On the flow in a

rcilcctcd-Hhock tunnel," AoronauticalResearch Council (Great Britain) A. R. C. 22 152 (Hl60).

10 R North, J. Soc. Motion Picture Televis. Engrs. 69, 711 (lH60).

::::=

means of the usual wave-diagram techniques, II pro­",ided the boundary-layer effects just ahead of the

. reflected shock are known. As a result of boundary-layer growth, the strength.

of tEe incident shQck daci'ensQ§ as It propagates afong the duct, but the pressure at a fixed locatiol~ increases slightly with time. VariOlIs methods of' analysiS-have been proposed to account for these bounda1'Y-layer effects, and the theoretical and experimental aspects of these phenomena have been rcviewed in some detail by Ji~ml'i(Jh and Wheelel'.12 As long as the thickness of the boundary layel' is small compared to the dimensions of the cr08S section of the duct, the approach of IVlil'els and Braun 13 appears to be the most satisfactory' available at the present time. Si_nceEressure gradients behind· the ineident shock appear greatly magnified 15emnc/ thel'efiected shock. a rcfinemjllit or tllCoi'i'gii'im' tlisoE~ is qescribed in Ses.. II: The computing pro­cedures required to obtain the conditions behind the reflected shock are then presented along with certain simplifications which eliminate the need fol' constructing a complete wave diagram. This analysi:-; requires a decision on whether the boundary layer is laminar or turbulent, and a brief discussion of boundary-layer transition is, therefore, given.

A number of shock-tube experiments have becn carried out to test the validity of the analytieal procedures. The experimental techniques are de­scribed in Sec. III, and a comparison of the experi­mental data with analytical results is given in Sec. IV. A discussion of the findings is presentcd in Sec. V.

II. ANALYTICAL PROCEDURES

A. Modification of the Mirels-Braun Theory

A sketch of the wave diagram for the flow in :t

shock tube which has a driven section with a closed end is shown in Fig. 1, and the basic flow regions are numbered from 1 to 5. The flow conditions within regions 2, 3, and 5 are not uniform, but the weak disturbance waves which result from the boundary-layer effects arc not indicated. Tho flow variables depend on the distance x from the dia­phragm and on the time t elapsed since the breaking of the diaphragm. It is convenient to introduce also the times ti and tT which are measured at some value of ;7t'om the passage of the incident and

II G. Rudinger, Wave Diagrams for Nonsteady Flow in Ducts (D. Van Nostrand Company, Inc., Princeton, New Jersey, 1055).'

12 k J. Emrich and :0. B.Wheeler, Jr., Phys. Fluids 1, 14 (1958).

13 H. Mirels and W. H. Braun, NACA TN 4021 (1957).

f

I I l f I

I

I I I,

I !

EXP WA\

[ CIA

Fro. 1. Wr significan

reflected s sent the : for an ide ditions, th of the dia in t,he ot l'CgiOlls 2 ( prcssure a' be denote(. The pl'esstJ is thus gi behind the of interest fOI' conV0l1 points wi!

The flo' reflected s just ahead tions of t:

HIld by Sll

magnitude by means ( by Mirels the growin weak posit along the s pressure w of the fim 8uperpOse( calculatior cases are

UH. Mi1 16 n. Mil

"

L

1I ':<,. PI'O-

, d of the

:<1 rength, ,'l)lagntes : locat.ion ',hods of lor thes() 'cal and :;\ve heen ;:j'heelel,.l"

J layel' iK liw orOSK 'll'cb and ~LVailahle j" behind

':'Ill behind ",.original , Ill!!; pro­} behind )"lIg wi th llpcd fOI'

'analysis I'y layer

i"sion of ,,\11. -: ve been ,~ lalyt;i('al : are de­',~ expcl'i­.' 'ivml in :'I'()sentc(l

1 .leory

10w in n. cIa oInKed I l'egiollK jnditiolls ;j hu t the ',om the 'I 'J'l1O f1<.HV 'ilhe dta­Ll'eaking 'jlIee also :111; some

'j'llt and , Fl()~(! in

IOllt New

; I·'lllid~ 1,

•• j') 057).

1 ,~~ ~. ~

I

BOFXDA H Y -LA YE,R GROWTH IX SHOCK TUBE 1465

, ~-----x·L----~.~ ,

DIAPHRAGM/ ~SOLENOIO o

TRANSOUCER"P'

FIG, 1. Wave diagram for shock-tube flow showing the Rignificant points and regions required for the analysis.

reflected shock, respectively. Let PI, P2, ... repre­sent the absolute pressure in the various regions for an ideal shock tube. Under experimental con­ditions, the pre""ure" PI and P.l before the breaking of the diaphragm are prescribed, but the pressure in the other regions is slightly modified. Only regions 2 and 5 are of interest here, and the actual prcssure at an arbitrary point in these regio;;-;ill hc denoted by P2(X, ti ) anq p,;(x, tr), respectivelY. The pressure immediately behind the incident.shock iH thus glVeil by P2~ and that immegiately hehind t'l'i'C'l".:ill.iectcd shock In:: ... ~, .9LA few points of interest are marked in Fig. 1 by A, B, 0, .. , for convenient reference, and the preSSUl'e at these points will be denoted simply by PA, P lJ, .,. •

The flow conditions at any point behind the reflected'shock can be computed if the conditions just ahead of this shock are known. Let pertUl'ba­lions of the ideal shock-tube flow be defined by

(1)

and 'by similar equations for other variables. The magnitude of the perturbations may be obtained by means of a theory by MirelsL4

,J5 and its extension by Mirc1s and Braun. L3 These authors consider the growing boundary layer as being equivalent to wcak positive or ilCgative mass sources distributed along the shock tube. Every source element produces pressure waves which contribute to the perturbation of the flow, and all contributions may be linearly, wpCl'posed because of their smallness. 'The necessary en,lculations are rather laborious, and only those cases are treated in detail in which air at room

14 H. Mirels, NAOA TR 1333 (1956) . 16 H, Mirels, NAOA TN 3712 (1956).

temperature is being used both as dl'ivel' 'and as driven gus. The air is assumed to be an ideal dia­tomic gas having a viscosity proportional to' the absolute temperature; the boundary layer is assumed to be either wholly laminar 01' wholly turbulent. For these conditions, the results are presented in the form of convenient charts fl'om which the How condition" for a wide range of shock strengths can be obtained for any point in region 2 provided the corresponding value of Ap2(a;, 0) is known.

Mircb and Braun also pl'ovidegraphs to compute Ap2(X, 0) for prescribed shock-tube conditions, but suggest that, alternatively, this quantity may be determined from a comparison of the ideal with the experimentp,lly observed shock strength accord­ing to Eq. (1) for ti = O. Both methods· should yield the same result, but no such agreement is

'found in the present investigation; the strength' of the observed shocks is always slightly less than £lie ,.,.JJ~ cO'fiii3Utea value. Although the discrepancy is small, ' it should be eliminated because errors of the theory appear greatly magnified in the results for the conditions behind the reflected shock.

Improvement of the theory is possible by fi

modification of the concept of an ideal shock-tube How. One may assume that changes of the How variables as a result of boiindary-layer growth are properly described b the boundar -la er the or , but t e actual values of the variables are also affected by unrelated Rhenomena. Tho mOst it);_ portant of these are distortion of the wave pattern resulting from a finite opening time of the shock­tube ~gm and flow losses mtroduceCI by remnant" of the diaphras.m, Flow modifications may arso be produced by mixing of driver and drilren ~s at the contact su!,face and by slight irregularities of the shock-tube wall. All these phenomena are unrelated to the, growth of the boundary layer, and their effects may be combined with the ideal shock-tube How to define a modified ideal flow in such a manner that allowance for the boundary layer according to the Mirels-Braun theory leads to the actually observed shock strength. Thus Eq. (1) must be changed to

P2(X, ti) = P2 + Apeorr + Ap2(X I Ii) , (2)

where A em is a correction term that 1'e resents the combine contributions of all phenomena that eau~ deviations from the ideal flow conditillns e~ ~oundary-Iayer growth, and Ap2(X, t;) repre­sents the boundary-layer effect alone as determined by the Mil'els-Braun theory. Since the correction term is constant for given experimental conditions,

! I, I, \ t t-' I { I' I,

f'

I \ r , ,

? ;

i \ i f (

i r , ,

r f I t

i I I'

t, " I, I

f~ ;i 1: !

t·: " I i I, i: I, !.: "

f ; I

, '''1 i \

1 2 1

" l , ! i ! ! I , ! 1

! I I j 1

l I I , " , 1.

..

146G GEORGE RUDINGER

-r'---'---~

it affects only the level of the pressure, while A])2(:I:, t.) also influences changes of the pressure with position or time. This separation of the per­turbations according to Eq. (2) is much' more important for region i') than for region 2 because of the mentioned amplification of nonulliformities by the reflected shock wave.

Since APeorr represents the combined contri­butions of several )J])rcla,ted processes, its actlml ~le is of little interest. The important_quantity is the sum ])2 + AP"orr whieh determines thCinOCITfieCI weal shoek sY'enB:t!i:Aecordmg to Eq. (2), it ean be determined if the experimental value of the pressurc behind the shock wave P2(X, 0) and the boundary-layer perturbation AP2(X, 0) arc known for some value of x. The calculations are best carried out by iteration.' One starts by making an estimate of the modified ideal shock pressure ratio (P2 + APeo,,)/P! from which the shock Mach number and the flow conditions immediately behind the shoek follow from the Rankine-Hugoniot shock relations. After deciding whether the calculations are to be based on a laminar or a turbulent boundary byer, one may, with the aid of the Mirels-Braun theory, compute AP2(X, 0) for the value of x at which the experimental shock strength ])2 (x, O)/P! is known. The estimate of the modified ideal shock strength must be adjusted until the results are compatible with Eq. (2).

Once the modified ideal shock-tube conditions have been found, the flow variables at any point of region 2 may be eomputed aceording to the procedures described by Mirels and Braun.

B. Calculation of the Conditions behind the Reflected Shock

The disturbance waves that are generated by the boundary layer originate only ill' regions 2 and 3 of Fig. 1. These waves cross the reflected shock into region 5 where the gas is substantially brought to rest, and thus produee the slight variations of the flow 'conditions that are to be determined. Construction of a wave diagram for which the conditions ahead of the reflected shock serve as boundary conditions would yield the desired soluti,9n, but it is possible to avoid this time-consuming procedure and to compute the flow conditions at selected points of region 5 directly on the basis of a few reasonable assumptipns. .

The flow conditions at an arbitrl:J.ry point A in region 5 are completely determined by the properties of the three characteristics that intersect there. These charaot~ristics, shown in Fig .. 1 as

the Jines An, AO, and AD, pl'ojll\.gat,o wHh tho velocities u, ~t + a, and u - a, respectively, Whcl'e u is the flow velocity and a the local speed of sound. For a duct of constant cross section, the pl'opel'ties of the characteristics arc given byll

I SM,iRtJcd, n.r

as/at + u as/ax = 0,

ap ap a [as as] at + (u + a) ax =;ii at + (u + a) a.'"C '

i9 + (u - a) iCl = ..!L [as + (u - a) -~J at ax "fR at ax '

(3)

(4)

(5)

where 8, R, and "f represent the specifie entropy, the gas constant, and the ratio of the specific heats, respectively, and where the Riemann variables P and Q are defined by

P = 2a/("f - 1) + u (6)

and

Q = 2a/("f - 1) - u. (7)

In region 5, variations of the speed of llound al"O sma'Il,. and the gas is substantially at rest so that, .. 1 tl'ie characteristics in the wlltve· diagram may be . appNiXl"mated by straight lines having slopes -dx1iit = 0 anddx/dt-;;.'··±·a:;-;,-coIiSt,Wrlci:C·' aN 'I istlie speed of sound at ~cfOSea- end immediat~ly . aIter shock reflection (see Fig. 1).

A similar simplification may be made fill:... the propagation velocity of .the incident and of the reflected shock wave. The strength of these shocks is modffied by the interaction with the disturbance waves, and the velocity of the gas into which the reflected shock advances is not constant; neverthe­less, it will be assumed that the sbocks also ll,ppca~!.. as straight lines on the wave diagram.

o The entropy of every gas particle increases dis­continuously on crossing the incident and reflected I shock, but, according to Eq. (3), it remains constant • within regions 2 and 5. This equation also . applies i in region 2 because wall shear increases the entropy . 1-

only in the thin boundary layer and affects the main body of the fluid only through isentropic ,t waves.'" Since the strength of the shocks is not, f constant, different particles have slightly different ! entropy levels. It is found, however, that for shock I Mach numbers not exceeding 2, the entropy-grndi. f

,ents III regIOns 2 and 5 Can be neglected without affecting the r~lts (see Sec. V and Fig. 7). Since I :q.o stronger shocks are being considered in . the . present investigation, a considerable further eiolpli­ficatlon becomes possible .. For constant entrol2.Y. within regions 2 and 5, Eq. (3)· is automatically

and

To caler t.he streng!; be determil by mcasuri AR Oil tlinc! ideal shock t,}w Mirels A])2(X, 0). " t.urbations ponent has for a turbu' determine any locatic at the close at the close st.rength of duct, one n t.he strengtl velocity U, flow variabl and that it t.o the statel

The flow detcrmined PA and QA Uiemann Vfl

indicate tha

PA = P

where use h 'Vhen an a flectod shoc: it increases pointed out be neglectel

, simplificatio) of AI> along c.ompute AP

AP = Pc

since the co been deterr, that of fine Since . thes( behind the shock wav! by means I

, I

1

;j <1

J ll:lg:~te with the lpcct.ively, whm'e I· speed of sound, jl, t.he propei'ties !

., . ~

1 I- a) -Q .. ~] I c);'(;'

(3)

(4)

J as] a) D:~ , (5)

pecific entropy, e specific heat.s, nn variables P

j 'j. i

(6)

(7)

j~(~ of sound arc 'J at rest so that, ;\gram may be ,1.having slopes ymst, where a g

>}ld immediately

.~ made for the :It and of the

1)£ these shocks ~ he distm:brmee linto which the '~ant; neverthe­des also appeal' I .

~l •

'} in~reascs dis­jl a:ld reflected 'mams constant; .Ln also applies ~es the entropy illd affects the Lgh isentropie L,hoeks is not. t;htly difTerent 1chat Jor shock }mtropy grad i-

j'cctecl. without

" Fig. 7). Sin eo ~d()l'l~d in tho h;l'th'er simpli­ttnnt entropy ~ au comatically )" ':,

j

--T-- .. ------------------~-....... --........................... ----....... - ...... - ............ -....... --....... ---... -~;.... ... -.......

~

I BOUNDARY-LAYER GROWTH IX SHOCK TUBE 1467

t satisfied, an~Jor region 5, ~qs; (4) and (5) reduce to

I p = const for dx/dt = aE (8)

and

Q = const for dx/dt = -aE' (9)

To calculate the flow conditions in region 5, t.ho strength of the ineident; shock wave must first he determined by some method-most conveniently, by measuring the shock veloeity at some location X.

As ollt;Jincd in the preceding ",cction, a. modified ideal Ilhoek-tnbe flow can then be found for which the Mirels-Braun t.heory yields a perturbat.ion t.P2(X, 0). This theory also indicates that the per­t.urbations are proportional to x", where the ex­ponent has the value 0.5 for a laminar, and 0.8 for a turbulent boundary layer. One can, therefore, determine the strength of the incident shock at

I finy location along the duct; in particular, also

I at the closed end where x = L. The shock velocity at the closed end will be denoted by Ui • From the I strength of the incident shock at the e~d of the

j duct, one may readily find in the usual manner, 11

, the strength of t.he reflect.ed shock, its propagation f velocity UT' t.he value of aE, and that of other

"

flow variables of interest. Note that UT is negative, and that it will be treated as const.ant according

·1 t.(~ the stated assumption;:: . The flow conditions at A (see Fig. 1) can be t ·determined by means of Eqs. (6) and (7) provided t /' A and Q A are known. To obtain the value of these

niemann variables, consider Eqs. (8) and (9) which indicate that

(10)

where use has been made of the condition UD = o· When an advancing characteristic crosses the re­flected shoek wave, the value of P associated with

utilize their charts, it remains only to determine the coordinates of points G and H in the wave diagram in terms of the coordinates XA and tr of point A~

According to the assumptions made, only straight lines appeal' in the wave diagram. Elementary con­siderations, therefore, lead to the relationships for t.he time intervals

til - tK = K[tt - (L - xA)(agl + U;I)]

and

to - tJ = K[t r + (L - xA)(agl - U;I)] ,

where

1 - (Ur/U i ) K = .

1 - (Ur/aE)

(12)

(13)

(14)

The position coordinates are then determined by

XII = XK = L - (til - tK)/(Ui 1 - U;I) (15):

and a corresponding equation for xo(= xJ). Having obtained these coordinates, one may

compute the flow conditions and therefore the values of P and Q at points G and H by means of the charts of Mirels and Braun. Subsequently, P A and QA can be obtained according to the outlined pro­cedure. Equations (6) and (7) then yield aA and UA, and the entropy change 8A - 81 is determined by the strength of the two shock waves. Ot.her flow variables can thus be found without difficulty, and, in particular, the pressure is given by

7JA .(aA )2'Y/('Y-ll . [

c_= - exp PI al

(16)

Results obtained by means of the outlined pro­cedure are given in Sec. IV and compared with experimental observations.

C. Boundary-Layer Transition

it increases discontinuously by an amount t:.P. As In the preceding section, reference is .made to pointed out in'the foregoing, entropy gradients can the charts prepared by Mirels and Braun for the

t be neglected for sufficiently - weak shocks. This computation of the boundary-layer effects behind 'j ~)lifica,tion is eguivalent to neglecting variatillns _ the incident shock wave. These charts are based ! of t:.P along the shock wave. ll One may therefore' on the assumption that the boundary layer is

eOrnpute Ai> as either wholly laminar or whoUy'turbulent. Therefore, a decision about the type of the boundary layer

(11) - must be made before the described computations ~ince the conditions at points E and L have already can be carried out. The considerable literature on been determined. The problem is thus reduced to boundary-layer transition in a sho,ck tube has been that of finding the conditions at points G and H. summarized by Hartunian, Russo, -and Marrone, 16

Since these points are located in the flow field who find that all surveyed data lie reasonably close bohind the incident and ahead of the reflected 16 R. Hartunian, A. Russo, and P. Marrone, "Boundary­shock wave, th~ necessary data can be obtained layer-transition and heat transfer in shock tubes," in 1958

1 Heat Transfer and Fluid Mechanics Institute (Stanford Uni-

ly means of the theory of Mirels and Braun. To versity Pre~s, Stanford, 1958), p. 114. .

1--,-,---~-- .---.--'.'~M".'--"""_~"''"'m'''''''''''''''[;;::'' .~~~~~~~~~~~~~~~~~ '-___ _____~~:.-~,. ~._ •• ~ •• ~ ••• J" .'.\J..;~'>"" ",w*" ": ...... '·>:....:.!.....:.~~.~ •. /··c·

r

•

~

~ ~ ,j

1 ~ ;\ l ~

I

"

¥ ~ 1 ¥ ! 1

! 1

I ! I I I I I I

1 I I I !

! I ,

I I

---- .. _------------------"

1468 GEORGE RUDINGER

1.0

~ E 0.8

w ! 0,6 ... z 9 ... C;; 0.4 :i 0:. ....

0,2

2 3 4 SHOCK MACH NUMBER .II,

FIG. 2. Transition time.in air forpi = 1 atm and Tl = 520 oR, based on reference Hi.

sequently, an estimate about the character of the' bouildary layer may be based on a comp::U'ison of the transition time with the initial value (for tr = 0) of the interval ta - tJ , that is, with the time which elapses between the passage of the incident and the reflected shock wave at the location for whieh the pressure rise is being computed. '

Most of the experiments arc carded out with incident shock waves that' propagate into ail' nt atmospheric pressure and tempemture. A plot of the tml1sitiol1 time for these condit.ions as a functioll of the shock Mach number 111 s is shown in li'ig. 2. It can be seen that the' transition time becomes n small fmction of 1 msec for shock Mach numbers greater than about 1.3, while it increases r~p.i~ly with decreasing shock strength., For other 1111 t.Jn I pressures, the tmnsition time can be readily ob­tained i.rom Fig. 2 since it is inversely prop'Ol'tiol1al lo

t • to Pl'

to a single curve if they arc plotted in terms of 'appropriate variables. With the aid of this curve, it is, therefore, possible to obtain a fair estimat!'l of the transition time tt which is the time during which the bound~£.~er at 'a fixed location in a III. SHOCK-TUBE EXPERIMENTS

duct with smooth walls remains laminar after pa~ Most of the experiments were 'carried out wit.h s~ge of a shock wa'::,e. It should be emphasiz?d, the aid of a simple ~ tube constructed of It

however, that in a duct with rough walls, or WIth cold-dmwn brasstune having an internal diameter imperfect junctions, earlier transition may take d - 3:z:n,u.-'The·length of the driven section was place, while in a smooth tube some additional ti~e 15.5 ft, and the driver section was long eno~lgh may elapse before a turbulent boundary layCl; IS (f2f't) to prevent wave refIections from its end fully established. The transitional stage may last tomtei'fere with the experiments. One to three from a small fraction of the transition time to a layers of photographic film served as diaphrag~l1~ numbcr of times this value.

17

-10

Apparently, no which could'be ruptured by means of a solel101d­systematie study of the dependence of th0se phe- opel'l1ted nee(lIe. The driven chambel' of the sllOek nomena on the flow conditions and .. on the wall tube was made up of three sections having lel1gth~ surface has been undertaken so far. of 12, 2, and 1.5 ft, respectively, with the longcst,

In view of the foregoing, the proper choice of section being placed adjacent' to the diaphragm. the 'charader of the boundary layer ;may not be and the shortest at the closed end. Although the. readily apparent for given experimental conditions. tubes had a smooth finish, no special effOl'ts weJ'e In doubtful cases, it may be necessary to carry out made to have the junctions ncar the closed end tho calculations for both a laminar and a turbulent as smooth as possible. Consequently, this tube may boundary layer. Such computations shoul.d then favor rapid tmnsition to a turbulent boundary ) ay(:J'. be expected, at least, to bracket the expenmental Two condenser-type pressure transducers (HutIH-

data. '. hauser model HH-3) were flush-mounted at 12 In order to d(>cide on th~ character of the bound~ry " and 24 in. from the closed end of the driven sec~ioll;

layer, thc transition time should be compared WIth and th~ output of the one closer to the end wns the intervals fe - tJ and .tll - tIf (FIg. 1).' bl~t photogl'l1phically recorded by means of a cathodc­for small values of t" the mterval til . - tK IS too ray oscilloscope. A single sweep of the beam wnH small for' any. noticea?le. pressure nse' to have triggered by the sigrial from the other tr~ns~uceJ' developed behmd the mCldent sho~k wave. ,Con- with a suitable delay. rhe strength of the mClden/'

17 J. J. Jones, "Resume of experiments co~ducted in the shock wave was determined from the shock-tmvcl high preSSllre shock tube of the Gas DynamICs ;Laboratory time between the transducers which was measurcd at, NASA," il\ i'roceed-ing8 31'd Shock Tube Symp08tum, March 1959 {Air Force Special Weapons Center Report SWR- by means of a microsecond counter, Air at room TM-50-2), p. 1. . . h k b temperature was used both as driver and as driven 18 J. H.. Asbridge, "An 11lte~fer0l!letnc,s~udy of s oc tu e boundary layers," Lehigh Umverslty InstItute of Research, gas, and the latter was, initially, alwl),Ys at atmos-Tech. Rept. No. 14, (1959). h .

, 10 P. Marrone (private communication). p enc pressure.

. __ !....--- -------

Three typic Fig. 3. The pI lire 1 An, 1.8-1, n\ln~bers of 1.: nil in terval of i~ adjusted in / 'fhe pressme negligible for i

pl'onotmced a ~e('.ond, larger. I'()(~ords for tli :;('('.onc\s after rise represen tI!. from ,the conti eon cern to th terminates the

A few expo 'lIllother shock brief perioC\, a higher drivel' I ill the driven Sl

stainless-steel :,oet.ion was m; ;;octjons. Conse leI'S tendency IndeBt botluc\al deseribcd first, woro used and ( operatcd plul1~

ducer (SLM : 2.75 in. from could be dete)')

FIG. 3. Pressure r shock t

~~~~~~~~::~~::::::::::::::::::::.::::::::::==::==::~~~!!~!~~~~~~~~~~~~~~~~~~W~;~4~~t~"~~~·V~A~~1*~ .. ~IM~~w~.~£~~'~i.~Q~; , .••. ~:'.:~:..-::-;::.::_~.;:""~'~ •. ".' .• '. .,-.,.;:.,\;~'~ ~.-~,;." .... _:·,':"'~"j;f~~~ff~ .. (>/ _ JI>:T . ~-: .. ~ .**1* l,:A.A'NIh .. ,·#1<.!t5t N.'ii9..¥4f¥~T\" .. w.~~r"'".~"f.':-~-;""'. .':""'\~.'\'i'3".>t';~~":..~:.~;-t""',~, <,."'.l~."!",.~·. ,., •

----~

~ ~l 1 1

I J 1':1<' (CI' of the 1 -'!;I.

llllpal'l~On ()f

:(~ (for Ir = 0) I t' I' ;. ,line wlId I lillc:idcnt Ilnd .)11 fot· whi('1! i '

--<1 Oil (; wj I/' ! illto ail' at.

\" A plot of 1,:1:-; n fUlwLion 1m in Fig. 2. 1c becomes a ~'('h num))(!I':-; 'a:-;08 rapidly ; othor initial ; readily ob­~l'()POrLional'll

ls ,J i~d out wiLh lr~cted of n ~hl diamet<~I' jiiiection wa~

'I mg enough jom i (,S (ind -jl e to til 1'('0

jdiaphrag~lll' ;j a solenold­Y the :-;hock'

Iii ng lengths • tho IOllgesL 1 diapJlrngm 'i hough Uw

J.l'Tori;t; wero y:losod end !; Lube may :, . jdary InYN, hs (H.u Lis-11 eel' nt· I') "j "'.-,en soc t.i 011;

;~) cnr,J wa:; ~j

, ll. eat\lOde-, J heam wu:.; 'I l'nnsd 1I ce l' .to incidel1 t, 1ock-tmvel '.I llleaHurod I· a~ room I 1113 driven ht atmos-

:1

i i , i

i BOUNDARY-LAYER GROWTH IN SHOCK TUBE 1469

I Throe typical PI'CHHUl'e records are .~ho\Vn in ; Fig. ;~. The prossure ratios of the incident HhockH I :1J'e J .·W, I.S·}, 2.21, corresponding to shock Mach { IIlImhcrH of 1.ln, 1.:31, and 1.4:3. All records covel' ! :rli interval of 10 msec, but the oscilloscope gain i j, Ildju:;tcd in efich case to obtain adequate records . I The pressure rise behind the reflect.ed shoek is i ,;pl(ligdlI6'l'ort11O weakest Huook but becomeH quite

. I pronounced as the shook strength mereases. A I ~eeond, lfil~ressure rise may be note~ on the I n!~ords for t.he two st.ronger shanks sQveral ~nilli-. I .. re()I1~J~ fiHer passage of the refleeted shock. This I ri,<e I'CpreHcnts the reflection of t.he reflected shock ; ,rolll Lhe contact surface (M in Fig. 1); it is of no l I'onr:ern to the pre~ent investigation and merely I terminates the ~seful testing time.. . ~ A few expel'lments were also earned out WIth i alllJfhcr~tube, which became available for a ! ,iriet pel'locGnd ;vhich allowed the nse of both a I ~ligher d~'iver prc~sure and a lower initial prcssure , III the dl'lven sectlOn. It was constructed of a honed, I ,Iaiitless-steel tube (d = 2i in.), and th~ driven I ,eetioJl was made up of two em'dull ali ned 6-ft i ~ecf,jons. onsequently, there might be considerably ll~ndeney toward the development of a tur­t hllirmt boundary layer in this yl,lbe than in the one I dcs(:l'ibed first. Thin, seribed, metal diaphragms i were used and eould be ruptured by a neumatieally I ope·rae p unger. A piezoe ectric pressure trans-1 dllccr-rsLl\1 model J>Z-14) was flush moun~ed i 2.7!i in. from the elo:;ed end. The shoek veloClty I could be detcrmined by measuring the travel time

l'

I I

,~

C'" i

~~ 1 I"

~ \-. i- ~.~-!' '~ I

¥ ~

P, -I o.tm

j: _ 1.49

'i M. -'.19

P, -'a'm

M, • 1.31'

;JP../-- .;) ~ J} - I'atm

PI! _ 2,21 PI

Mr_ 1.43

1----10 mac, -----j

r 1/ II

,/f

""'"

~-4.JJ 'i M, .1.96

P, ·0.13 .. un

~ -1.98

FIG. 4. Pressure records obtained at 2~ in. from closed' end of shock tube; L = 12 ft and d = 2i in.

between two thin-film heat-transfer gauges mounted at 2.75 and 26.75 in. from the closed end. One of these also served as a trigger of the oscilloscope sweep for pressure recording. Air was again used as the driven gas, but nitrogen was used as the driver gas.

Two pressure reeords obtained with this sIwek tube are shown in Fig. 4. The recording time is 5 msec, and the shoek strength is almost the same for both records (shock Maeh number just bolow 2, eOlTesponding to a pressure ratio of about 4.4). However, in one case, the incident /shoek travels into ·air at atmospherie pressure, while in the other, the initial pressure is only about 0.13 atm (100 mm Hg abs) .

The pressure reeords are evaluated by means of a "telereader" which produces an enlarged pro­jeetion on which precise measurements can be made, and the measured veloeity of the ineidcnt shock wave provides a calibration check. The slight non­linearity of the transducer calibration, established

• by a static calibration, is taken into aeeount in the evaluation of the records. With currently available equipment. the absolute v'alue of a pressure can be measured with an accuracy of a few percent, while a eonsiderably higher aecuracy can be achieved in the determination of small pressure changes. To compare an observed pressure rise with computed data, it is therefore preferable to compare the pres­sure ehanges behind the refleetedshoek rather than the pressures themselves. .

IV. COMPARISON OF COMPUTED AND OBSERVED DATA

The computed pressure rise behind the reflected FlO. 3. Prcssure records obtaincd at 12 in. from closed end of a ' shock tube; L ;= 15.5 ft and d = 3.23 in. shock wave is compared iJ;! the following with

'.

! ;j1,;~~~ __ / ___ ~<. __ ~",.~~ .• o>_;. __ ;tml.t"~'~~~~*t"M('··""Ni:'\~f&'<i<l""""""+«""",,y:!'M"':';"';:;;""'''''''''''''''''''''''''''Wi'_''' _"""'·""<",,,""' ..... £ij·it!QtIit<l,~ .. _ .. ·"L· .... 7f!{~ ....... '4i1i;t· ..... ·.rz.E""'}1+ .. ' .... ,;,l·'ii:i"~ ..... I .... """a»""'.::"1,'""""'~',:.; .. ·~mifj:';..';;:-~':::. .. ~·""" .. .JiO_ .. ' ... !1 ........ .,.t .......

;

1470 GEORGE R UDI·NGER

'fABLE 1. Data for experiments of Figs. 3 and 4.' ==~~~========:==========================================

3 4 5 7 8

Diaphragm pressure

ratio· P'/Pl

2

Shock, Mach

number (111 s )cxp

Shock pressure ratio P2/PI

6

Transition time

Time intervalsb

for t, = 0

2.41 4.17

6.64 39,0

38.a

1.19 1.31

1.43 1.96

1.08

Exp. Ideal

1.49 1.84

2.21 4.33 .

4.40

1.53 1. 96

2.40 4.76

4.71

Modified ideale

1,86 lam. 1.92 turbo 2.33 turbo 4,38 lam. 4,68 turb. 4,53 lam.

t/ msee

0.7 0.3

0.2 0.08

0.2

to - tJ tlf - t/\ msee msec

1.65 1.59

1.55 0.31

0.31

0.19

0.22 0.07

0.07

• The driven J(ru! is initially at atmospherie pressuro except for the last experiment for which PI = 100 min Hg absolute. . . b Seo Fig. 1. Intervals arc evaluated for a point loeated 12 in. from the closed enel for the first three and 21 in. for tho last two oxperiments • • Calculated for a bminar or turbulent boundary layer as indicated. •

experimental observations represented by Figs. 3 and 4. Some of the measured and computed data arc collected in Table I. The first column indicates the initial pressure ratio across the shock-tube dia­phragm P4/Pl and the next two columns show the experimentally determined shock Mach numbers and shock pressure ratios. Column 4 lists t.he shock strengths for an ideal shock t.ube as computed from the diaphragm pressure ratio. Column 5 shows t.he modified ideal values computed for a laminar 01' a turbulent boundary layer, as indicat.ed in the table.

'1'he decision whethCl; the calculations are to be carried out for a laminar or a turbulent boundary layer should be based on a comparison between the transition time t, and the time interval to - tJ for t, = O. These times are listed in columns 6 awl 7, re::;pcct.ively. For complet.eness, the int.erval til - t]( is listed in column 8; it is much smaller than the 'other interval, as stated earlier. It will be seen in the following that the boundary layer should be considered as turbulent. in some cases and as laminar in others. Since the modified ideal pressure ratios depend slightly on the character of the boundary layer, two values are listed in column 5 where required. The time intervals in columns 7 and 8 arc only insignificantly affected by the character of the boundary layer so that only one value is needed.

. used for these experiments may favor early bound~ ary-layer t.ransition, indicates that a turbulent. boundary layer should be the propel' choice on which to base the calculations for these two experi· ments. The results are given in Fig. 5, where the pressure change behind the reflected shock, in terms of the initial pressure PI, is plotted as a function of the time elapsed after passage of the reflected shock. The solid lines show the computed pressure rise for the two shock strengths, based on a turbulent. boundary layer, and a ± 1 % deviation from tho absolute pressure value is also indicated. Experi. mental data for the two cases are representeel by t.he different symbols. For the stronger shock wave, the agreement between theory and experiment is excellent, and even for the weaker shock it is qliite sat.isfactory. For the weaker shock, t.he pressure rise is also shown computed on the basis of a laminar boundary layer (broken line). It can be seen that the pressure rise for a turbulent boundary layer

0.8

~O.4 ~ - ~ ~ ~

.:::. ... 0.3

~ '" . ~ 0.2

'" 0: ::> en en ~ 0.1 .. o~~~----______ +-__ ~ ____ _

o 3 TIME t, ,mite

'4

I

i (

I I

For the first experiment, the transition time represents almost one-half of the interval to - tJ.

The. appropriate character of the boundary Jayer is thus quite doubtful, and since the pressure rise behind the reflected shock wave is. extremely small (see top of Fig. 3), t.his record is not evaluated. For t.he seoond and third experiment, the transition time represents only about 19% and 13% of the interval to - tJ , respectively. This fact, combined with the previous statement that the shock tube

FIG. 5. Pressure rise behind reflected shock for second and l' third record of Fig. 3. Comparison between experimental data \ and theory.

is much Iar~ t.he observed ,based on a t1

These res) experiments it. would. be in such a m2 and a lami for the sam Since t.he sl' t.o hold the nrc carried tube descri obtained fo: shown in F last two lin<

The l'eco' a considera t.hough the 1 for the first' and 7 of'l'a laycr shouk shows the Cf

mental resu] agreement. h for a lamina line), it is 8

this curve toward the Apparently, layer transit effect of tral sure become transitIon ti certainty ah ary layer a experimenta calculated pI layer, the f( sible, but tl explanation: ments are tl the closedE have been I

action of the layer wherE may occur.! data could the pressUJ of the shoe sure rise 0

by the bOl in this cal

~ -----------.. ,) r) O.lf)

'It> 0,22 a 0.07

'~ I _______ ~' 07

,j :pcrim~::=~~:"c"

"or early bound. at a tud>l1l(:1)I. 'opel' ehoieo on hese two cxp(\l'i.

'.g. 5, whm'o thn , ;;hoek, in t.OI'rni< : (~ as a fUlletioo 'of the l'()fle<:I(:d npu ted preSStl J'(:

; d on n, turhujpflf •• iation f!'Onl t Iii'

, 'licaLed. Expnri.

reprosent.ed hv her SllO~:k wnv~~. II expCl'lm()IlL IS

~hoCk it i:; qll~t(' :11 ~lC pressure ,I'lhO

'1<1;; of a Jamlllal' ,1n be seon t.hat

,lboundm'Y byel'

1 1

V ,I TURBULENT :+ BOUNDARY

;~ LAVER ~ :.v. 1 j

l '.I:~!..';~R r~

l"k for HO(',oml alld j('x~101'imel\t'1I1 dill"

BOUNDARY-LA YER GROWTH IN SHOCK TUBE '1471

is much larger than for a laminar one, and that the observed data agree mueh better with the eurve based on a turbulent boundary layer.

These results make it desirable to extend the ('xperimcnts to stronger shock waves. In particular, iL would bedesirablo to vary the initial pl'essuros in ~u('h a manner that tho effects of both a turbulent, IJ.nd :1 bfninar boundary b~'er ('ould he :"tudi0d :.): til<:.' ~<1m~' stl<.,tlgth of dl(' lw:,t(kut :;rw('k '\~,n',

Since t.he shock tube originally used is not designed to hold the required pressures, these experiments are carried out with the aid of the second shock tubc described in Sec. III. The pressure records, obtained for a sIwek Mach number close to 2, are shown in Fig. 4. Further details are listed in the last two lines of Table 1.

The rccol'd for the higher initial pressure shows Il considerable rise behind the reflected shock, al­though the pressure seems to remain almost constant for the first ~, msec. From a comparison of columns 6 ~lIld 7 of Table I, it would seem that the boundary Iaycr:;hould be considered as turbulent. Figure 6 ohows the comparison of the calculated and experi­mental rosults. Except for the first millisecond, the agrccment is quite satisfactory. If the pressure rise fot' a laminar boundary layer is also plotted (broken line), it is soen that the experimental data follow

( IhiH curve rather closely at first and then shift 1 toward the curve for a turbulent boundary layer. ~ Apparently, these data show the effect of boundary-

laycr transition, but it would then follow that the effect of transition in this smooth tube on the pres­,lIl'C becomes s{gnificant only after more than eight transition times have elapsed. In view of the un­l'(~ltainty about the transitional stage of the bound­ary laycr and the good agreement between the ('xpcl'imental data for the first 0.7 msec and the !·:tlculated pressure rise based on a laminar boundary layer, the foregoing explanation seems quite plau­,jilIe, but the possibility, at least, of a different (';;planation should be kept in mind. These measure­ments aro taken at about one duct diameter from Ihe closed end so that the observations may still have Leen made within the region of direct inter­adiOll of the reflected shock with a laminar boundary laycr where, as mentioned earlier, a pressure drop , Illay oecur.s

•n The observed dip in the experimental

dab. could thus also represent a superposition of the pressure drop owing to the direct interaction of the shock with the boundary layer and the pres­sme rise owing to the disturbance waves produced by the boundary layer along the entire shock tube; in this case, however, the agreement between the

, 2.0

"" ~ ~ ~rq 1.&

" .::' ~ ~ 1.0 0:

It'~O'PI 0

0

O,~ 1.0 LG 2.0

TIME I" "'lie

FIG. 6. Pressure rise behind reflected shock for up!'er record of Fig. 4. Comparison between experimental data and theory.

I experimental and computed data for small values of tr would be accidental.

For'the low-pressure record of Fig. 4 (last line in Table I), the transition time represents a major portion of the interval ta - tJ , and the prcssure rise is therefore computed for a laminar boundary layer. The result is almost the same as ti},e cor­responding curve in Fig. 6, that is, the pressure risesaoout 2% during the first mllhseqQJii;LOA1tl~-­the record seems ~9)lldicate no rise at all, sU<l,h a small rise could easily be masked..Qx the rather high noise level-a consequence .of the necessary hlgIlOsCi11Oscope gain at low pressures. The pos­sibility again exists that the pressure rise owing to boundary-layer growth is counteracted by the effects of the direct interaction of the shock with the boundary layer near the closed end.

V. DISCUSSION

Once the calculations for the described examples have been carried out, it becomes possible to check the validity of the assumptions made in the, course of the analysis; namely, whether it is permissible to neglect the flow velocity in region 5 with respect to the speed of sound in this region, and whethet, the paths of the characteristics and of the reflected shock in the wave diagram (Fig. 1) may be con­sidered as straight lines~ The calculations for all examples show that tbe flow velocity behind the J:eflected shock never reaches 2% of the local speed of sound which varies by less than 5% between p~1r,an<rE of Fig. 1. As the reflected shock travels upstream into the 'noW. in region 2, it in­creases in strength, but encounters an increasing flow velocity. These two effects partly cancel each other, and the net result is a decrease of I Uri;

I

p: t,

i i'

i I

I

to

1472 GEORGE RUDINGER

~ 0.20 0.'"

~ S? '}'" j O.I~

'" '0.10 V>

ir

'" n:

iil '"

O.o~ 0: .. '" ::

/ ." \ zoo.

~

~ ~

I~O •

100 ~

\i ... '" 0:

°l~ ----~----~--~4----~OO

SHOCK MACH NUMBER M,(L) ,

Fro. 7. Pressure and temperature rise at closed end of duct during first millisecond nfter shock reflection; based on It turbulent boundary layer, JJ = 20 ft, d = 3 in. and PI = 1 atm. The broken line represents the pressure rise if the entropy gradients are neglected.

that is, the shock slows down relative to the duct in spite ~ of its increasin)' strength. Although this be lavior applies to all cases for which the calcu­lations have been carried out, a general proof for it has not been found; the actual variations of flow velocity and shock strength are rather small, and 1}.9 changes of U r in excess of 5% have been noted over a distance of 1 ft:..These observations amply justify the simplifying assumptions made.

The most striking feature of the ressu'e rec rds in Figs. 3 an 4 is the marked pressure rise behind the reflected shock compared to the bare I noticeable risen: ea of it. For example, the calculated pressure ahead of the reflected shock for the upper record of l,'ig. 4 increases at a rate of less than 8% per msec; in contrast, the pressure behind the shock increases by about 8% during the first millisecond

,;titer passage of the reflected shock. (The actually observed rise on this record is somewhat smaller because of the dip in the data that has already been discussed in the preceding section.) Since the re­Hected shock wave "amplifies" the small pressure gradients behind the incident shock, these gradients must be properly evaluated. According to Eq. (2), the difference between columns 4 and 5 of Table I represents t.Pcorr/PIJ while that between columns 8 and 5 indicates the shock attenuation C1pz/p, that is caused by boundary-layer effects during the time in which the shock travels from the shoek-tube diaphragm to the point of measurement. The cor­rection term is by no means negligible compared to the boundary-layer term, and th'e importance of using the modified ideal conditions for calculation of the How behind the reflected shock, therefore,

, becomes evident. Although the described experiments are limited

to shock Mach numbers not excceding 2, it is nlso of interest to obtain an estimate of t.hc PI'CSR\Il)' rise behind the reHected shock for stronger incidC'lll. shock waves. Because of the complexity of thr mathematical formulations, a general relatioll~hill between the pressure rise and the conditiollR Oil

which the calculations are based docs not S('('lll fensible. A set of typicnl conditions' is prCSCntl'ti here, based on a shock tube (L = 20 ft., d == a in.) filled initially with ail' at room temperatUl'c nllll atmospheric pressure. For the sake of simplinit.\1 the pressure is calculated at the closed end. Fig\ll\; i shows the relative pressure rise behind the rellC'eh'd shock during the first millisecond,

[P6(1 msec) ~ Pr.(0)l/P6(0) ,

i f ~ ~

l

as a function of the shock Mach number nt till' I end of the duct, 1l1.~(L). In view of the 1'('~IlIt~ t presented in the preceding sectiOll, it mny I~\ ~ assumed that the boundary layer is turbulent, for t . all shock Mach numbers greater than about \':' 1

1~01' strong shock waves, one can no longcr n~~lIn\(' ! that entropy gradients in regions 2 and 5 enn I~\ i neglected. Within each region, the entropy of II }:

gas particle then remains constant, but neighborill~ i particles have different entropy levels. Conveni('lIt 1 relationships for computing the entropy aI, Ilily :i, point of region 2 may be found in refel'cnee It I and the appropriate wave-diagram prOCe(hll·(I~.1l f

based on Eqs. (3) to (5), can be applied .to eompllll' i the conditions in region 5. 1<'01' compal'isoll, Ih., J pressure rise behind the reflect ' , i b~~n compute without taking entropy gmdi(llIl, Ii into account (broken line in Fig. 7), and it cnn Ii!' sCeIltl1attliis SimplIficatIOn IS ot ermlsslli1C'liii"" shock Mach numbers greater t an about 2. It i, " Interesting to note that the pressure rise dlll'ill~ f the first millisecond reaches a maximum of nllllill 17% at shock Mach numbers neal' 8.5 and de(!l'cn,,(', rapidly with further increasing shock strength.

As an incidental result of these calculatioJl~J Ihr 1 f temperature rise behind the reflected shock Will'"

is obtained and is also presented in Fig. 7. It. ill' creases monotonically and reaches about lOOoH ill :~

one msec for JI!I s = 3. Such results arc of cOllsidiil'lIblr i,:.·,·,

interest for various shock-tube applications in wliidl the hot gas behind the reflected shock is uLilizl'" t It should be emphasized, however, that thc bOIllI'" r ary-Ia'yer theory and the calculations of the sllll" t

't:

of the gas behind t.he reflected shock are ba~cd III/ , ~ the assumption that real-gas effects can be neglcetl·" " ~

Also heat losses by conduction and radiation /11\' ~ not taken into account. The results for the stl'Ollgl'f :1'·

t

'a

l-""l~ 1 ' BOUNDARY-LAYER GRO\VTH, IN SHOCK TUBE 1473

':il,g#2, it is al,,) I ,hock WUVCR must (,h.",[o'" be "onsidcrcd a8 t.at.- . also b. mnde at a distaace or seve"al duet diameters ,Jlf the press\~rc i' Live. In any case, the stated results apply only to from the closed end. Such cxpcl'iments would not !'"ongcl' inciden1, .. the particular conditions on which the calculations only test the presented theory in the mnge of shock : plcxity of the are ha,,>{)d. 2o

•21 :Mach numbers greater than 2, but would also yield

1':11 relationship It shol1ld also, he pointed out that at a dosed end, informat.ion as to the importance of the dircet, shock 'I eonditions on the boundary layer must remain laminar at least intcraction with the boundary laycr Hear t,he closed

~; Jons not sem)) fo)' a ;;hol't. time after shock reflection. Consequently, end. . ils is presented WftVOS associated with the direct intt!raction of the )) ft, d = :3 in.) reflocted shock \vi!.h the houndal'y layer may modify '1Iliper~tlll'e. '~nd the pressure rise, as discusscd in the preceding r of slmpll<JlLy, HNJj,j()h. Therefore, experimental ob:30rvations should jd end. Figure 7

: Ie! the l:efieeted

Ilumber at; the : of the rosults

11, it may be '" t.urbulcnt fol' 'Inn about ] .2.

-'l})ftongcr aRsun,w " . ~

OJ , and 0 can b(~

";.. entropy of a ! 'lltneighborillg 'Js. COllvenient; I tropy at any

-- I refercnce 1:3, I proccdures, II

,;; ied to compute

.,'1 lInparison,the ';]1Ock has alRo )'opy gradient,..; 1 and it can be Jpermissible for : about 2. It iH

1'0 rise during Illum of about.

;) and docrease,..; strength.

,lcubtions, the

I'd shock w, ave 1 Fig. 7. It in­iiJout lOO°n, in lof con:;idembl(~ jttio~s in ~\:hich ,,!lck lS utlhzed. ,tat the bound­ls IYf the state ~ :1,;'e based on 11 bo neglected.

1

1 r~diation arc hI' th,e strol1u:el' .'J c,

j I

; 20 In a recent, study by H. Mirels [NASA TN D-291 "(H)(il)I, it is found that real-gas effects have only a minor inflilence 011 the growth of It laminar boundary layer, but this result does not necessarily apply also to a turbulent boundary layer or the transitional stage.

21 The tornperatme increase behind the reHected shock is (,urrcnt.ly being investigated by R. A. Strehlow for shock Much numbers from 2 to 4 in argon (private communication).

ACKNOWLEDGMENTS

The author wishes to express his thanks t.o L. 1\'1. Somers for carrying out most of the shock-t.ube experiments; to P. Manone and A. Russo for valuable di:;cussions; and to C. S. l3ardo for assisting with the experiments.

This work was carried out as part ·of Project Squid, which is supported by the Office of Naval Research.

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