No. 68-53

ATTENUATION OF THE SHOCK IN A SHOCK TUBEDUE TO THE EFFECT OF WALL BOUNDARY LAYER

by

G. F. ANDERSONSoutheastern Massachusetts Technological InstituteNorth Dartmouth, Massachusetts

and

V. SREEDHARA MURTHYBrown UniversityProvidence, Rhode Island

AlAA Paper

No. 68-53

AIAA 61h Aerospace SCiencesMeeling

NEW YORK, NEW YORK/JANUARY 22-24, 1968

First publication rights reserved by American Institute of Aeronautics and Astronautics. 1290 Avenue of the Americas. New York. N. Y. 10019.Abstracts may be published without permission if credit is ;ivento author and to AIM. (Price-AIM Member $1.00. Nonmember $1.501

ATTENUATION OF THE SHOCK IN A SHOCK TUSEDUE TO THE EFFECT OF WALL BOUNDARY LAYER*

- tGordon F. Anderson

Southeastern Massachusetts Technological InstituteNorth Dartmouth, Massachusetts

andV. Sreedhara Murthy:f

Brown UniversityProvidence, Rhode Island

Tr

- T6

= =T=-0-----=T~6-

Radius of the tube

Recovery factor

Exponent 'n' for the driver gas boundary layer

Time reckoned from the burst of the diaphragm

Temperature

Temperature of the wall

Stagnation temperature

Recovery temperature

Velocity

Velocity at the fluid interface in the boundarylayer behind the contact surface

Velocity of the discontinuity at the leadingedge of the boundary layer (= Us for the shock,= -a4 for the expansion fan)

Distance from the diaphragm

Distance of the shock from the diaphragm

Pressure

Exponent 'm'- for driver gas boundary layer,see Eq. (20)

Exponent 'm' for the driven gas boundary layer,see Eq. (19).

Exponent for the density profile:see Eq. (7)

Mach number

Mach number of the shock

Rate of accumulation of mass per unit area ofcross section per unit length along the boun­dary layer 1

Exponent for the velocity profile: ~ = (f) nU

6

x

x s

p

q

n

r

M

Msm

P

m

Abstract

c Specific heat at constant pressurep

c Specific heat at constant" volumev

d = J.:! 2r 2 M6

D = [ 6/ ]1.268P6u6 llw

2f

nm= (nm+m-n) (nm+2m-n).11m

2mgnm = (m-l)(nm+m-n)

List of Symbols

a Velocity of sound

Tb y-l 2 (1-(3)b =T =1 + r -2- M66

c =(3 l.:.!:. M 2r 2 6

The compressible turbulent boundary layer in ashock tube has been analyzed and the attenuation ofthe shock has been estimated. Power law profilesare assumed for both the velocity relative to thewall and the density. The power law exponents aredifferent for the driver and driven gases. Thevalues for the exponents are determined by·applyinga match~ng cond~tion for the boundary layer th~ck­ness on e~ther s~de of the contact surFace. SmaTlperturbations are. assumed in -the evaluation of theshock attenuation due to the effect of-the boundarylayer. The effect oft.re driver-gas boundary layerand first order reflection and'transmission effectsat the contact surface are included in calculationsof the shock attenuation. Numerical calculationsare compared to the results of previous theoreticalinvestigations and are compared to the character­istics of some experime~tal data.

h = 0 l56010-[0.678(n+2)/n]. (n+l)(n+2)nm • 0.268

n0.732

.~nm+m-n)(:+2m-nJExponer,it' 'n' for -thedriven gas poundary, 'layer>

v., .' _ ~-_- - _h' /1.268 --K 0 =-~ . nm . u =u u = -a )nm~ nm+m-n\g udo/uo-f. ,( d2 s' d3 4nm ~ ~ nm

x'

Y

"13

y

6

6 c

Dummy variable for the distance from thediaphragm

Distance from the wall

Exponent for the shape of the fluid interfacein the boundary layer between the diapl)ragmand the contact surface, See Eq. (22)-

cRatio of the specific heats (-E.)c

v

Boundary· layer thickness

Thickness of the embedded driven gas boundarylayer behind the free stream contact surface

*This r<;!search was supported by the National Science Foundation Grants GP-2941 and GP-4825tAssociate Dean of the College of Engineering~Research Assistant

1

Fictitious driver gas boundary layer thicknessbetween the diaphragm and the free stream con­tact surface

Refers to the wave propagating withvelocity (u - a)

Prefix

~ ( ) Increment in the quantity

*o Boundary layer displacement thickness = 1, Introduction

Superscripts

+ Refers to the wave .propagating withvelocity (u + a)

The separation distance between the shock andthe contact.surface, in an inviscid ideal shock tube,varies linearly. with the distance of the shock fromthe diaphragm.· In a real shock ·tube, however, thedriven gas boundary layer between the shock and thecontact surface behaves as an aerodynamic sink andabsorbs mass from"the free stream, thus attenuatingthe shock and accelerating the contact surface. Onthe other-harid; the driver' gas' boundary layerbetween the expansion fan and.the contact·surfaceacts. as an aerodynamic. source, emanating mass intothe free stream and hence accelerating both thecontact surface and the shock. The leading wavefrontof.the expansion fan in·the driver· gas movesessentially at a constant velocity since the highpressure driver gas beyond the expansion fan isunaffected by changes in the rest of the flow. Thewall bound~ry layer is therefore one of the princi­pal causes of non-uniformities in the shock tubeand hence necessitates a thorough investigation inorder to determine the separation' distance' as wellas the shock velocity.

sur ace e if the ro erties of'· the shock:Lnf uencirn §,r:Lven gas and the expansion infJlIe~eddriver gas are identical. This is contradictory toreality, since the identicaJ,. gas section;;;OD the... twosides of the contact surface will have travelled thesame distance under identical conditions and hence ~ 2,.v")"tmust have the same boundary layer thickness. In ~ r!'~ ,,,,-\

/,this report the match:Lng cond:Lt:Lon at the contact '1 liv

surface is employed in the determination of the ,7 ••j" 11ooundarylayer charac!~:!::.istics in both the drJven ,F':?;i<V Jand the driver gases~' A similarity solution may not '" ~,be applicable in the present case, but is a good and t­convenient first approximation for obtaining theaverage properties. In. this respect it is encour- 2aging to note that the experimental study by Martinindicates self-similar velocity profiles along thelength of the boundary layer. The flow near theshock and the expansion fan is different from afully developed turbulent boundary layer and aproper choice of the exponents in the velocity pro-files will average out this behavior.

The turbulent boundary layer has been studied byMirelsl , who assumes the velocity profile to followthe 1/7th power law and uses a quasi-steady approxi­mation by fixing the shock wave and considering thewall to move. This solution involves a discontin-uity in th bounda a er thickness

The effect of a turbulent boundary layer on theseparation distance has been treated by Anderson3taking a 1/7th power law for the velocity profile.Mirels4,S and BrownS have studied the attenuationand non-uniformities in the shock'tube flow byusing linearization methods for small perturbations.The case of large perturbations is treated byMirels 6• The real gas effects and numerical calcu­lations for d~fferen\ driver gases are included inthe work of M:Lrels and MUllen? In the present

. report the perturbations are assumed to be small andhence linearized methods are employed to calculatethe effect of both the driven and the driver gas

Friction force at wall

Coefficient of viscosity at the wall

poUo(udt-x)/llw

2Pouo t/Ilw

Thickness of the overlapping driver gas boun­dary layer above the interface behind thecontact surface.

T -Tr b

=T -T'r 0

T

e Boundary layer momentum thickness =

=fO ~ (1 - ~) dyo Pouo Uo

eu= f: u~ u - u: )dy

Subscripts

1 Undisturbed driven gas

2 Shock influenced driven gas outside theboundary layer

3 Expansion influenced driver gas outside theboundary layer

4 Undisturbed driver gas

c Fluid interface in the boundary layer behindthe free stream contact surface

d Discontinuity

o Free stream,i Value of i gives the corresponding regionm Modification due to the. effect of the embedded

driven gas behind the free stream contactsurface

s ShockThe subscripts for the functions f, g and hdenote the values of n arid m in the correspondingexpr.essions

a

2

The constancy of the pressure acroSS the boundarylayer leads to a simple relationship between thedensity and the temperature

boundary layers.

An experimental study of the turbulent boundarylayer in a shock tube, by Martin2 , indicates that a1/5th power law profile for the velocity in theboundary layer is closer to the measured valuesthan the l/7th power law profile. The shock Machnumbers in Martin's experiments, however, are fairlylow and it is quite possible that the boundary layer'flow belongs to the transition between the laminarand the fUlly developedtu~bulentregimes. Hence,higher Mach numbers at higher Reynolds numbers arenecessary to determine the velocity profile of afully developed turbulent boundary layer.

2. Theory

velocity profile is assumed to be of the form

I

=~:/

and the density profile is ~~ed as

- . / 1 - Ip _ ((l.)-~ i~-~

(3)

(4)

ti11

J!

lii

I1I.1

I~

2.1 Solution of the boundary layer equations

The momentum integral equation for the boundarylayer of a compressible fluid in unsteady flow isS

( 5)

(7)

(8)

(6)

(m-l)(nm+m-n)

1 _ Ib + cn/(n+l) - dn/(n+2)

T =To

82

~~m (9)nm

;S = (nm+m-n)(nm+2m-n)

8u n (10)T = (n+l)(n+2) ,~':

0Iu (11)-0- = n+l

Im

IC{b + cCf) n

or

The friction force 'b at the wall, for compressibleturbulent flow, is g~ven in terms of the skin

The assumption of the power law for the densityprofile is supported by the fact that the experi­mental investigations of Martin2 indicate a similarbehavior. The expressions for the boundary layerparameters may immediately be obtained in terms ofthe exponents in the velocity and the density pro­file shapes.

0*-oP--0- =

The temperature variation in the boundary is givenby 3

where b,'c and d are constants depending on therecovery temperature, wall temperature, freestreamtemperature and the Mach number. The expressionsfor b, c an& d are given in the list of symbols.In order to obtain the approximate power law pro­file, as displayed by Eq. (4), for the density, itis necessary to use an appropriate matching condi­tion connecting the Eqs. (4), (5) and (6). Ones~mple match~ng cond~t~on ~s to equate the areasbetween the y-axis and the two curves representinathe d~str~but~on of dens~ty, i.e.,

(2)

stream, leads to

The assumption of small perturbations, im 1 in in-sign~f~cant var~ations in the .

It is encouraging to note at this stage thatMartin's2 experiments indicate that the streamwisepressure gradient is very small

X

UNAFFECTED DRIVER GASBOUNDARY LAYER

EXPANSION FAN DIAPHRAGM CONTACT SURFACE SHOCK--,.. -r-° __·-o"or:-" r-o_0--r0-~DR~I~VCE:R%G~A~S?z~~~~~¥~'8~d~C~ ~~ 8c ® DRIVEN GAS_® 83 82 0 CD

DRIVEN GASEMBEDDED DRIVEN GASFLUID INTERFACE INSIDEBOUNDARY LAYER

~ BOUNDARY LAYER

FIG. I SHOCK TUBE COORDINATE SYSTEM.

The solution foro the momentum integral equation,Eq. (2), requires the shape of both the velocityand the density profiles. A self-similar solution,for the compressible turbulent boundary layer, maybe obtained by assuming power law profiles for thevariation of the veloc~ty and the density. The

<I

Insertion of the Eqs. (8) to (12) along with Eqs.(3) and (4) in the momentum integral equation, Eq.(2) and subsequent non-dimensionalization yields,

friction coefficient Cf , by Walz9 as~ ' ~,

2Tb 0.246 10-(0.6786 u/6u) 6Cf = =~ R 0.268 '6

P6 6 6 u

(12)

hnm (13)

s,;:u~b;-s..,e-;q~u;;;e;;n~t~l:j:;:;,.:;bie~d:-;e:-;t;;e;;rm;::;i;n~e:;d;.,::b?,y~m;;;a~t;:;c~h~i~n~g~t:.;h~e~b~o~u~n~d~-=-__'ary ayer th~cknesses at t e con ac surface, so i~t the values of n on the two sides average to 7, i . /'which is the exponent for the velocity profile in a ; 1/'0fully developed incompressible turbulent boundary ITlayer. In this way, the details of the boundarylayers are forfeited in favor of an overall confor-mity with the physical reality.

Let k and 1 be the values of nand m respectivelyfor the driven gas boundary layer, and s and q bethe corresponding values for the driver.ga~ boun­dary layer. The dimensionless parameter for theboundary layer thickness of the driven gas is

(18)

( 17)

P 3u31 a4t+x IIIw3

hsq=

=

The corresponding expression for the driver gasboundary layer is

The exponent 1 is the value of m when ~~ubsti­tuted for n,in the Eq. (7) for m, with the constantsb, c, d based on the free stream properties of thedriven gas outside the boundary layer. Similarly,q is obtained from s by using the properties ofdriver gas outside the boundary layer.

(14)+

D

where f nm , gnm and hmn are functions of only n an~and are g~ven ~n tHe I~st of symbols. The express­ions for the dimensionless parameters D, ~ and Tmay also be found in'the list of symbols. Inobtaining the solution for Eq. (13),the wall tem­perature is assumed to be, constant. This'"1'iiiP'1Testhat m ~~ constant for a self similar boundary layerwhere n ~s constant. The general solution forEq. (13) may therefore be written as

(21)hsq(a4/u3+1)

gsqa4/u3+fH=

=

hH (Us;U2-1 )

g'k1us/u2-f U

1 1 - 1 (19)i" b2+C2k/(k+l)-d2k/(k+2)

1 1 - 1 ( 20)= b3+c3s/(S+1)-d3S/(S+2)q

or

The matching condition, Eg~ (21), is appropriatefo~ the case of identical freestream propertiesacross the contact sur-&ce. In actual shock tubes,

The determination of k and s reSl.Yires an appropriatematching condition at the contact surface. For thecase of identical properties across the contactsurface, a su~table matching condition is theequal!ty of the boundary layer thicknesses on the,two sides, Le. ,

The velocity profile exponents k and s may bedetermined by using Eq. (21), if one of them isknown. ~ce neither of k and s is known, a c<:>!1­venient procedure for evaluatin them would be todeterm~ne and s so that their avera e value is 7the exponent or a fully developed turbulent pro­fife. The values of the density profile exponentsP; and q, follow immediately from Eqs. (19) and (20),along with the ,assumption of constant wall tempera...ture.

4

(15)hnmD =

The boundary layer thickness is given by

whe::e 1/J is an arbitrary function and Al,Bl arearb~trary constants. Here, the assumption of con­stant n is supported hy the fact that Martin'sexperiments indicate a constant n along the wholelength of the boundary layer. The bO"yndaxy condi­tion to be applied to Eq. (14) is

IA~ -::: Us 11't.- (). 'I6=0 at x=udt (Le. at ~=O).

Hence, the solution becomes

6 = II w D(1/1.268). ( )

l~~::~:l~;~: :'::0=.0 la oro If n~:6m~~ wer e or t e two boundary layers, for the!;' case of identical free stream properties across the~ constant surface, the solution given by Eq. (15), indicates a discontinuity in the boundary layer

:~~~v~ ~~~~~;:ss:~u1~en~~n~:~~t:u~i:~:'th~h;a:i:~~~I~~~i~~,~~ the two sides of the contact surface will havetra-

f S velled the same distance under identical conditions.This ~iscrepancy is a consequence of the assumption

~ ---:\,..of similarity of the Y~:L.ClI1d ~ensi1y_.p1'"omes,r .~~';rfnd may ~e' r:moved by sacrificing the matching of\,~~ the-aeta~ls ~n the boundary layers at the contact

< /,'." s,:rface. The profile exponents, n and m rna be con--~ s~dered to be differen for the driven and the

~tJ~= ariver gas boundary layers. Ihe~r values may

tj"---,/

(22)=°(x)c

~Icontact surface

Let 0c denote the thickness of the embeddeddriven gas behind the contact surface. The shapeof the fluid interface inside the boundar layer

etween the diaphra(Wl and th!i contact surface, isassumed to be

tac sur ace.

difficult to analyze the development of the ,compo­site boundary layer made of both driven and thedriver gases, an approximate estimation is in order.The shape of the embedded dr1v~ behina-the con­tact-surface, 1S assumed to be self-similar. Theshape is then determined by equating the mass ofdriven gas behind the contact surface at any instant,to the total mass efflux through the plane of thefree stream contact surface starting from the rup­ture of the diaphragm until the instant of interest.The velocity and densit an where inside this em­beddedboun ary layer, are the same as those, at anequal distance from the wall. in the dr1ven gasboundary layer at the free-stream contact surface.The foregoing procedure determines the details ofthe portion below the fluid interface in the bound­ary layer between the diaphragm and the contactsurface. The remaining portion consists of thedriver gas. The velocity and density profiles any­where in this driv'er-gas boundary la er are tak;~ 0

be theport10n, a ove the point corresponding to theveloc1ty e ual to that of the driven a at hefluid interface. 0 the respective profiles at thelocation of interest in that driver gas boundarylayer which is obtained by ignoring the effect ofthe embedded driven gas. Hence, the thickness ofthe driver gas bounda la er varies from its un-

ec e value at the dia hra to zero a he con-~~rictly speaking. the velocity profile expon­

ent for anyone region in the actual flow shouldbe equal to that for the corresponding region ofa possible shock tube flow, which produces allthe way from the shock to the expansion fan. thesame free stream properties as those in the pertin­ent region of the actual flow. The principal draw­back of this method is that it is impossible toestablish, behind the shock. the large Mach numbersattainable behind the contact surface. Hence, theafore-stated direct use of Eq. (21) should be satis­factory.

however, the properties ,of the gases on the twosides of the contact surface are'usually different,and hence, the equality of boundary layer thick­nesses at the contact surface does not apply. In

rsucfi a case the prof1le 1nd1c1es may be obtainedas. follows: The e~nt k is obtained by assumingthe space between the contact surface and the ex-

'~ pans10n fan to be filled with the driven gas, possess1ng the same properties as in the region

between the contact surface'and the shock.. For'/ th~!Lficl;,*hous arrangement, the" matching condition

would be the equalit'y of"'the"boundar la er thick-( nesses a e con ac surface. viz. Eq. (21). This

wirr-give the value or k. S1m11arly, the exponents is calculated by con~dering the reg10n betweenshock and the expansion"'fan to contain the drivergas with properties identical to'those of the dri­ver gas in the region between expansion fan andcontact surface of the actual flow of interest.Once k and s are evaluated, the density profileexponents t and q follow from Eqs. (19) and (20).with the different properties of the correspondingfree streams. Thus. in effect, the E~. (21) is,directly used for obtain~ the profile exponentseven~if the properties of the ases are ~fferenton the wo S1 es of the contact surface.

(26)

(25)=I

P dy dx

1 [2.268t2

(kt+1-k)(1-1) 1.268 (1-k)(t-l)=

*The density P. is given by Eq. (24). and 02 may b~obtained by using Eq. (17) and the definit10n of on.Substitution for P. &* and 0c in Eq. (25) and sub­sequent integration yields the expression to theexponent for the shape of the fluid interface.

The mass of the afore-mentioned embedded drivengas at any instant t must be equal to the total massof the retarded driven gas in the boundary layerwhich has been left behind the free stream contactsurface until the instant t. i.e ••

where S is a constant and 02 is obtained directlyfrom Eq. (17). The velocity and density profiles inthis embedded boundary layer are assumed to be repre­sented by the conditions at contact surface, i.e ••

for

To summarize, the velocities in the actual ideal­ized flow are used in Eq. (21) directly, to obtainthe velocity profile exponents k and s. whichaverage to 7. The density profile exponents 1 andq are then calculated from Eqs. (19). and (20). em­ploying the appropriate local free stream proper­ties. The boundary-layer thicknesses are then ob­tained from Eqs. (17) and (18). The assumption ofconstant, but different prQfile exponents for thedriven and the driver gas boundary layers. and the'subsequent application of Eq. (21). may be ..:;gar9--ed as accounting for both the effect of favorablepressure gradient in the expansion fan, and thecomphcated development of~ouna.aI:Y"Iayersnearthe s~~d the expansion fan.

Effect of the Embedded Driven Gas BoundaryLayer on the Growth of the Driver GasBoundary Layer:

Some of the retarded driven gas in the boundarylayer is left behind the free stream contact sur­face as the shock travels along the tube. Thisretarded driven gas forms. along the wall betweenthe diaphragm and the contact surface, a layerwhose thickness varies from zero at the dia­phragm to the driven gas boundary layer thicknessat the contact surface. Hence. the driver gas inthis region does not flow over a stationary wall,but rides over a fluid surface of varying velocity.This affects the growth of the driver gas boundarylayer ahead of the diaphragm. Since it is very

5

With this information, the driver gas boundarylayer may now be constructed above the fluid inter­face. At the outer edge of the embedded drivengas boundary layer the velocity is given by

free stream perturbations may be obtained as

(32)

= u2 [0.c(x)/02 !" ]l/kcontact surface Momentum, (33)

(27) Ap = IsentropicRelation.

(34)

This is also the velocity of the driver gas justabove the boundary layer interface. The velocityand density profiles for the overlapping driver gasahead of the diaphragm are obtained by taking thatportion, which is above the point corresponding tou=uc ' and belongs to the unaffected driver gasboundary layer.

where the prefix A denotes the perturbation in thequantity following it. Inserting Eq. (34) intoEqs. (32) and (33) and rearranging, we obtain

i.e.,

lisu/u3 = (Ym/03) , plPa

The "acoustic relations Ap± = ±poaoAUo± (+ and­

superscripts refer to the waves propagating withvelocities (uo+ao ) and (uo-ao ) respectively) may beused in Eq. (35) to obtain

and = (30)= (36)

2. a. Attenuation of the Shock due to Boundary layer'.

(38)

Integration of the above Eq. (36) yields

2 ,Ap± (x)

ao r ,

x-x )dx,

m (x t- • (37)2(uo±a

o) p uo±ao

~here [AP2+/AP2-]C is the ratio, of the strength ofthe reflected wave to.that of the incident wave forthe driven gas, at the contact surface, [AP2-!AP2~Sis the ratio of the strength of the reflected waveto that of the incident wave at the shock in themoving driven gas, and [AP2+/AP3+]c is the ratio,of "the strength of the wave transmitted into drivengas to that of the incident wave in the driver gas,at the contact surface. The expressions for theabove ratios may be found in Appendix B. In orderto determine the net perturbation, it is necessaryto calculate mp ' the rate of mass addition, per unitarea of cross section per unit length, to the freestream from the boundary layer. Considering twoplane cross sections at x and x+dx, and calculatingthe rate at which fluid is left behind in between

With reference to Fig. 1, it may be observed thatthe net perturbation at any point A on the shock ismade up of contributions along the characteristiclines BA, DB, EB, FE. i.e.,

[ +j+ A + AP2 B _AP2 I = J A( AP2) +" ------ J A( AP2)

at A B AP2 c E

(31)=

Let mp be the time rate of the mass of the fluidadded per unit area of cross section, per unitlength, at any section x, due to the effect of theboundary layer. The equations satisfied by the

T~e driven gas boundary layer ahead of the freestream contact surface acts as an aerodynamic sink,absorbing the driven gas from the free stream andthereby attenuating the shock. On the contrary, ithe driver gas boundary layer behind the diaphragm ibehaves as an aerodynamic source, emanating mass :into "the free stream and hence effectively acceleraf­ting the shock. The effect of the composite boun­dary layer between the diaphragm and the conta7tsurface depends on the growth of the two const~tu­

ent layers. As a consequence of the finite speedof sound, the disturbances propagate along thecharacteristics, as shown in Fig. 1 (along thelines BA, DB, EB, FE and so on). The interactionsof the disturbances with the contact surface andthe shock should also be accounted for.

where 03 is obtained directly from Eq. (lB).This completes the determination of the compositeboundary layer between the diaphragm and the free­stream contact surface.

The last term in the right member of Eg. (30), viz.03(Uc!Ua)S, gives the point corresponding to u=ucin the driver gas boundary layer which is obtainedby ignoring the effect of the embedded driven gas.The thickness, 03m' of the overlapping driver gasboundary layer is therefore given as

for

6

the two sections, we obtain

mP

(39) 0* = (q-s)(kR,H,-k) [llw2]0.268/1.268(sq+q-s)(~-k) kk~,2 ~2u2J -

(47)

• P2u211/1.268

PIal] .

~llw3jO.268/1.268 1/1.268

+ Ksq ,3 P u I-a"t-xl3 3 ..

*The above expressions for 0 are used in Eq. (39) tocalculate m and the result is then used in Eq. (37)to obtain, ~y direct integration, the perturbationsdue to the different regions. The position of theshock (point Ain Fig. 1) is given by its distanceXs from the diaphragm. The contribution from B to Ais

andfor

0. 1: = K. [ llwil 0.268/1.268Iu

t_xll/1.268 , (40)1 nm,l PiUi__ d

The expression for the boundary layer displacementthickness cO> for the region ahead of the contactsurface, and for that behind the diaphragm, may beobtained by using the definition of 0* and D inEq. (15).

The constant Knm i, as g~ven in the list of symbols,is a coefficient'depending on the profile indiciesn andm, the free stream p~operties and thevelocity of the discontinuity at the leading edgeof the boundary layer. The boundary layer betweenthe diaphragm and the contact surface", is built upof both the driven and the driver gases. Theapproximate velocity anddensify profiles for thiscomposite boundary layer has been given in §2.2.For convenience in estimating the effect of theregion between the diaphragm and the contact sur­face, the boundary layer there is assumed to bemade only of the driver gas, and its thickness istaken to be the sum of the local thicknesses of thedriver and driven gases in the composite boundaryJayer. The velocity and density-profile shapes in this

.fictitious boundary layer are assumed to be thesame as those behind the diaphragm. i.e.,

(/0 )l/Sy dc (41) ,(42)

rj.

O.268/1.268lJ w2lJwl

The contribution from D to P is (~ealizing u3=u2

)

The c~ntribution from E to B is

(44)

° ~ x ~ u2t . (43 )for

~sq+q-s °dc

p/P3

= -(y/o )-l/qdcand

The expressions for 0cand 03m maybe found in Eqs.(22) and (31) respectively. With the above assump­tions the boundary layer displacement thicknessbetween the diaphragm and the free stream contactsurface becomes

~':Thus the expressions for 0 to be used in Eq. (39)are

* ~llW3~ Q. 268/1. 268 . 1/1. 268

= K -- I-a t-xlsq,3 P3u3 4

Ksq,3

7

Using the expression for [lIP2/l1P2+Jat the shock,from Appendix B CEq. B.4), in Eq. (g3) we obtain,

(53)

(52)

=

=

Yl + 1 ['::+J .lIMs = (54)P2 a2 l+M 2

YIMs [2 + _._~JPIal M 3

s

coefficients at the free stream contact surface~The contributions along FE, GF, etc. (Fig. 1) arevery small and hence not included in the above re­lations: There is, however, no difficulty in com­puting them. The perturbation in the Mach numberof the shock may now be obtained by using thepressure ratio across the shock, which is

22yl Ms - (yl-l)

Yl+l.

There immediately follows from this

and finally the contribution from P to B is given by

f: 2+

11<R,', 2(q-s)( kR. +R. -k) aa Pa

t.(lIPa ) = (sq+q-s)(R,-k) a l (a3

+u3

) P2

, 13 1 ']I~ I-d(~ )s s

It is more convenient to express the perturbationin the shock Mach number in the form

(55)liMsliM =s

The calculations indicate that the velocity-pro­file exponent for the driven gas boundary layer isgreater than that for the driver gas boundary layer.This is fortunate, since the driver gas boundarylayer, unlike the driven gas boundary layer, growsinitially under the influence of the favorablepressure gradient near the expansion fan, and henceis more apt to be laminar for a greater distancealong the wall. This conformity with the specula­tion upholds the suggested use of the matching con­dition, Eq. (21).

This completes the calculation of the total pertur­bation in the Mach number of the shock.

3. Numerical Calculations

The perturbation liM *, in the shock Mach numberhas been numerically c~lculated for differentdriver gases and the result is displayed in Figs.(3) and (4). Two different values are consideredfor the temperature ratios between the driver andthe driven gases. The wall temperature is taken to be Tl •

4. Conclusions

( 51)

where,

The net perturbation at the point A is obtained bysubstituting Eqs. (48), (49)·, (50) and (51) intoEq. (38), along with the expressions, as given inAppendix B, for the reflection and the transmission

The results of the numerical computations, asdisplayed in Figs. 2 and 3, comply with the experi­mental findings regarding the behavior of an effi- .cient driver gas. The attenuation of the shock isgreater for a more efficient driven gas. Also, inaccordance with the previous investigators, theneglect of the effect of the driver gas boundarylayer results in a contradiction regarding thebehavior of an efficient driver gas. The calcula­tions of attenuation, using only the driven gas

8

FIG.2 PERTURBATION IN THE MACHNUMBER OF THE SHOCK, T4/TI= I.

Curve No. Driver gas - Driven gas

1 Air - Air2 Helium - Air3 Hydrogen - Air4 Argon - Air

Including the effect of driver gasExcluding the effect of driver gas

boundary layer, indicate lesser attenuation of theshock for a more efficient driver gas. The contra­diction still exists when the driver gas boundarylayer is taken into account, and the compositeboundary layer between the diaphragm and the contactsurface is neglected. This leads to the conclusion,that the composite boundary layer ahead of the 'diaphragm, plays a very important role in the esti­mation of attenuation of the shock.

0.12

The coordinates of the point A are

The suffixes on the coordinates x and t indicatethe corresponding points in Fig. 1.

The expressions for the coordinates of the pointsA, B, P, D and E, which are useful in evaluating thepressure perturbations, are given below.

Characteristic Line Geometries (see Fig. 1)

APPENDIX A

A comparison of the results of the present in­vestigation, with those of Ref. 7 indicates that,for consideration of only the driven gas boundarylayer, the attenuation of the shock is roughly thesame in both the investigations. As it turns out,the attenuations, calculated by using only thedriven gas boundary layer, is roughly the same asthose found in experiments. This agreement tendsto suggest that the driver gas boundary layer haslittle effect on the velocity of the shock. Butthen, as indicated earlier, the neglect of the dri­ver gas boundary-layer contradicts the experimentalobservations regarding efficient driver gases.Hence, it may be concluded that it is essential notonly to account for the driver gas boundary layer,but also to include the effects of the compositeboundary layer between the diaphragm and the contactsurface. Proper incorporation of the real .gas eff­ects and the contact surface diffusion will, mostlikely, amend the qualitatively agreeing results ofthe present investigation to a quantitative agree­ment as well.

attenuate the shock. Thus, beyond Ms =3, with T4=Tl=522°R, for air-air shock, the assumption of idealityis totally invalid. Similar effects may be found inother driver gases also, depending on their respec­tive boiling points. With T4=Tl=522°R, condensationis likely to occur beyond Ms =9 for Helium, beyondMs =16 for Hydrogen, and beyond Ms=2 for Argon. Oneother factor which might be important in the esti­mation of the attenuation is the mixing and the con­sequent spreading of the contact surface. The diff­usion at the contact surface· affects the transmiss­ion and reflection characteristics there.· Added tothe above mentioned factors, there is also.the realgas effects at the high temperatures behind theshock at large Mach numbers. Hence, it is necessaryto account for the afore-stated effects in order toobtain attenuations of the shock which agree withthe experimental observations.

II T4 1T,=1I 2 3

,1 -,- --- ,..-'I --- -;:: ----4_- ---,.." - --

~~.;::" -~" ........... -r--

I i\" M:C~UMB:~ 8 10t'-...

\ \ \ '"14 \ I

1\2 ~31\1

T4/ TI=4,1

4 -- ~~- _2- - ---- --~/ ----: - -- 3-~..:"' --

~:::: .....~,., ......

1"---.. ~........, 2 .\4 6 8 10MACH NUMBER MS

\ \1\

\4 \1

-0.12

0.12

• -0.08U)

:::E<1-0 .04z2... 0<l:m~ 0.04...a::~ 0.08

-0.12

• -0.08U)

:::E<1-0 .04zo... 0<l:ma::::::l 0.04...a::LU0. 0.08

FIG.3 PERTURBATION IN THE MACHNUMBER OF THE SHOCK, T4/TI=4.

x = xA s

The point B is given by,

The point E is given by

t E

a2+u2-us ta 2+u2-usxE xa2 u2+us s a2-u2+uS s

For large Mach numbers, however, the computationspredict an acce ratio Th~s maypar y e due to the assYmPtion of ideality in the~es. At high Mach numbers, the expanded drivergas attains very low temperatures and may condense,thereby tending to attenuate the shock. To give anexample, for air-air operation,. the assumption ofideality leads to a temperature of l6loR.for a Machnumber of 3 and T4=Tl=522°R. The constituents ofair, viz. oxygen and nitrogen condense at tempera­tures close to l50oR. Hence, for air-air shockthere is good possibility of condensation beyond ashock Mach number of 3. Any condensation tends to

=u2(a2+u

2-us )

usa2

xs

9

~-4,,~J667

Confidential Review Report4merican Institute of Aeronautics and Astronautics

1290 Avenue of the Americas, New York, New York 10019

Area Code 212 LT 1·4300

Author(s) G. F. AnJ.e::",s"n '3.n:l V. S!.'eeat.~.r8 Murthy

Date March 21 f 1968

_ Log No. _J2_1....-7,-o5 _

DYes 0 No.

Title AttemI2otion of -t:1e K;ClCk in a Shock Tube Dl1e to the Efft'le"t of Wall' Bound~ry !toyer __

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(PLEASE SEE REVERSE SIDE)

I\

\

~~- ."

The point P is given by

=

From relations (B.l) and

=

we get

(B.a)

The point D is given by Across the shock we must have

APPENDIX B

Interactions at the shock and the contact surface:

Shock: (see Fig. 4)

The pressure ratio across the normal shock is givenby

=

Differentiating the above, we obtain the relationbetween the increments in P2 and Ms.

'where plUS ('+') refers to the faster downstreamwave, which travels at the velocity (u+a) and minus('-') refers to the slower wave, which travels atthe velocity (u-a).

Using Eq. (B.a) and the acoustic relations,Ap±= ±paAu± ,we obtain the ratio of the incrementin the total static pressure behind the shock, tothe strength of the incident wave.

[,p~] 2 (B.4)=AP2 s 1 P2a2 1 + M 2

1 s+---Ma2 PIal

s

xo~_---------------- t'P~ J

1p2a i

=Paaa

(B.7)AP2 c 1

P2a 2+--

Paaa

Using-the acoustic relations, Ap±=±paAu±, in theabove equations, we obtain

Contact Surface: (see Fig. 4)

The incidence of· a wave lIP2 at the·· contact surface,.produces a reflected wave AP2' which travels towardsthe shock and a transmitted wave AP3 , which travelsinto the driver gas. Since there is no discontinuityin the pressure and in the velocity across the con­tact surface, we must have

(B.l)

SHOCK

t

From the shock relations we also have

FIG.4 INTERACTIONS AT THE SHOCKAND THE CONTACT SURFACE

Similarly, if a wave of strength Ap; strikes thecontact surface, a reflected wave liPS travellingback into the driver gas, and a transmitted waveAPt travelling into the driven gas, are formed. Theconditions of no discontinuity in pressure and velo­city across the contact surface, becomes

2 (M 2 - 1)s=

(B.IO)

.2·[lIP~] =lIPa c

Insertion of the acoustic relations, gives

(B.2)2 1

= Y +1 (1 + --2) ~M2 M s

s

Hence, the relation between the perturbations inu2 and Ms becomes

tlU2

a l

10

References

1. Mirels, H., "Boundary Layer Behind a Thin Expan­sionWrv;r~~ving Into Stationary Fluid", N.A.C.A.TN, 3J4,;,3,\~y1956.

~;'-'.<'\'~~

2•. Martin, W.\A., "An Experimental Study of the,Turbulent Boundary Layer Behind the InitialShock Wave in a Shock Tube", Journal of theAerospace Sciences, Vol. 25, 644-652, (1958).

3. Anderson, G. F., "Shock Tube Testing Time"tJournal of the Aerospace Sciences, Vol. 26, No.3, March 1959.

4. Mirels, H., "Attenuation in a Shock Tube Due toUnsteady Boundary Layer Action", N.A.C.A. Tech.Report, 1333.

5. Mirels, H., Braun, W. H., "Non-Uniformities inShock Tube Flow due to Unsteady Boundary LayerAction", N.A.C.A. TN, 4021, (1957).

6. Mirels, H., "Shock Tube Test Time Limitationdue to Turbulent Wall Boundary·Layer", AIAAJournal, Vol. 2, No.1, 1964.

7. Mirels, H., Mullen, J. F., "Small PerturbationTheory for Shock-Tube Attenuation and Non­Uniformity", The Physics of Fluids, Volume 7,No.8, 1208-1218, August, 1964.

8. Howarth, L., "Modern Developments in FluidDynamics, High Speed Flow', OXford at theClarendon Press.

9. Walz, A., "Compressible Turbulent BoundaryLayer", International Symposium of NationalScientific Research Institute, (C.N.R.S.),Marseille, August 28 to September 2, 1961.

11