/}f ~u.s/11,-1_0 ~g

Appl. Sci. Res. 18 September 1967

THE INFLUENCE OF BOUNDARY LAYERGRO\VTH ON SHOCK TUBE TEST TIMES

P. J. :mJSGROVE*) and J. P. APPLETON**

Summary

A theoretical and experimental investigation of the limitation. on shock tubet.est times which is caused by the development of lami;lar and: turbuientboundary layers beh.ind the incident shock is presented. Two theoreticalmethods, of predicting the test time have been developed. In the first alinearised solution of the unsteady one-dimensional conservation equationsis obtained which describes the variations in the average flow propertiesexternal to the boundary layer. The boundary layer growth behind the shockis related to the actual extent of the hot flow and not, as in previous un­steady analyses, to its ideal extent. This new unsteady analysis is conse­quently not restricted to regions close to the diaphragm. Shock tube testtimes are determined from calculations of the perturbed shock and interfacetrajectories. In the second method a constant velocity shock is assumed andtest times are determined by approximately satisfying only the/condition ofrna·; -. .. '1tinuity between the shock and the interface. A critical comparisonis madc c',is and previous theories which assume a constant velocityshock. Test ,i;c;·c, predicted by the constant shock speed theory are gener­ally in agreement ,..-ith those predicted by the unsteady theory, althoughthe latter predicts a transient maximum test time in excess of the finalasymptotic value. Shock tube test times have also been measured over awide range of operating conditions and these measurements, supplementedby those reported elsewhere, are compared \\"ith the predictions of the theo­ries; good agreement is generally obtained. Finally, a simple method ofestimating shock tube test times is outlined, based on self similar solutionsof the constant shock speed analysis.

lYomenclature

aA,B,CD

speed of soundconstants defined in section 5.3.shock tube diameter

*) Department of Applied Physical Sciences, Reading Cniversity, EXGLAND.**) G<:neral ,,10tor5 IXfense Research Laboratories, Santa Barbara, California, U.S.A.

- 116 -

. 1/111111111111111111111111111111111111111111111111TRP00225

SHOCK TUBE TEST n:ifES 117

KI

NPP'qQr5tlTuu*UUoU*vwWx

1]

j"

f-tt'>

pp'(J

= tJ/q17l, boundary layer growth constant, see Appendices A arid Bhot flow length . .constant, = } or! for laminar or turbulent bo'undary layers, re-

--spectively .initial shock Mach number 'at the diaphragmshock ~:rach number at station X s

= (Uo - u2)/a2, hot flow Mach number 'relative to the shockfront . .

= p2a2!P3a3, the ratio of acoustic impedances across the interfacepressure= P e - P2" perturbation 'pressureboundary layer growth, coordinate defined in. §'2= (W - 1 + 5) K" . .radial distance from shock tube axis.boundary layer integral defil1ed by equation A6time= 7/700 , dimensionless ratio of test times ."'.= l/looJ Roshko's dimensionless ratio of hot flo\v lengths·' .. 'axial HOlv velocity in laboratory coordinate system, see figure la= U e - U2, perturbation axial flow velocity .shock velbcity ..initial shock velocity at the diaphragm= U - U0, perturbation shock velocityaxial flow velocity in shock-fixed coordinate system, see figure 1bradial flow velocity= Uo/(Uo - U2), den:sity ratio across the shockaxial distance from shock tube diaphragm'axial distance behveen shock wave and diaphragm= TI/Too , dimensionless ratio of-test-times= lI/loo, Roshko's dimensionless ratio of hot flow lengths= (D/2) .'-- r, radial distance from the shock tube \vall

ratio of specific heatsboundary layer thickness; undefinedboundary layer displacement thicknessboundary layer thickness defined by equation A2characteristic direction defined by dx/dt = (U2 - a2)= (J1;I~ + 1)!(M~ - 1)viscositycharacteristic direction defined by dx/dt = (U2 + a2)density= pe ~ p2. perturbation densityPrandt1 numbershock tube test time= 1v15!(M5 - 1)

\

conditions at the points indicated in figure 2

118 P. J. ~1USGROVE AND J. P. APPLETON

Suffices ,.

1 conditioF.5 in the undisturbed flow ahead of the shock2 conditi'Gl1s immediately behind an unattenuated shock3 conditions in. the expanded driver gas4 conditioJlIs in the undisturbed driver gase conditiOlIS between the shock and the interface, averaged across

the inviscid core flowconditions at the interface

I denotes the predictions of ideal shock tube theory00 asymptotic conditions given when Xs -r 00 and t-r 00

s conditions at or immediately behind the shockw conditions at the shock tube walla, b, b1, C.

d. d1, f, fl,g. gl, j. k, k1

§ L Introdudk!I

Investigatio;, \vhich attempt to explain the discrepancy betweenobserved shock tube performance and that predicted by ideal theoryhave been in progress for about a decade. The majority attributethis discrepancy to the growth of a viscous boundary layer in theflow behind the incident shock which tends to weaken the shockand gives rise to gradients in the flow properties behind it. We shallbriefly consider a few of the more important of the theoreticaltreatments in order to emphasise the novel features of the presentwm~ -

The most sigillficant of the earlier theories, due to Trimpi andCohen [IJ, 1Iirels (2J. Dem'yanov [3J, and Spence and Woods [4J.were primarily concerned with the unsteady motion of the shock.They all considered that shock tube flows may be described by whatare essentially one-dimensional unsteady flow equations and, havinglinearised the equations by assuming small perturbations to theflowo variables, obtained solutions using the method of character­istics.

The analysis of Trimpi and Cohen differed from those whichfollowed later in that it used values of the flow properties whichwere averaged oYer the entire tube cross section. From the solutionof their resulting system of linearised characteristic equations Trim­pi and Cohen were able to show that the shock was attenuatedby unsteady wa\Oes which were continuously generated by the ef­fects of \Oiscosity and heat transfer throughout the entire flow field

IIII

SHOCK TUBE TEST TI?llES I 19

behind the shock. Mirels criticized their formulation on the groundsthat these effects were confined to the wall boundary layers andcould not directly influence-the inviscid core flow. He argued thatthe unsteady waves which were generated in the inviscid coreshould be attributed to the radial component of the velocity at theedge of the boundary layer. However, it must be pointed out thatTrimpi and Cohen have subsequently stated that their averagingprocess implied thick boundary layers, in contrast to the thinboundary layer assumption of iVIirels et seq. 110re recently, Spenceand \Voods [5J, assuming axial symmetry, have carried out a de­tailed derivation of the linearised unsteady conservation equationsand confirmed Mirels' formulation for flows where the boundarylayer thickness is much smaller than the tube diameter. This formu­lation is identical to that used previously by Dem'yanov and bySpence and \Voods [4]. The flow variables are averages of the fluidproperties acro~_the tube external to the boundary layer and arethus of more practical interest than those of Trimpi and Cohen.

Both Trimpi and Cohen's and Mirels' theories lead to elaboratecomputational methods based on linear characteristic networks inthe (x, t) plane. They consider both laminar and turbulent boundarylayer development in the hot flow between the shock and the inter­face and in the cold flow between the interface and the rarefactionwave. However, only a limited amount of agreement has been ob­tained between the predictions of these theories and experimentalmeasurements of variations in the shock strength and the flowproperties behind the shock. Dem'yanov obtained a closed formsolution for the case of laminar boundary layer development be­tween the shock and the interface but neglected disturbances origi­nating in the cold flow. He showed that the growth of the hot flowboundary layer caused the shock to decelerate and the interface toaccelerate. Spence and Woods extended Dem'yanov's basic methodof solution to cover the case of a turbulent hot flow boundary layerand also demonstrated how heat addition at the interface (causedby e.g. combustion) modified the analysis. The experimental shockattenuation data which they present again show only qualitativeagreement with the theoretical predictions.

In 1959 Duff [6J reported some surprising measurements in ashock tube at low channel pressures (PI < 5 mm Hg) which re­vealed that the duration of the hot flow, i.e. the "test time" tended

to a finite limit as {he distance from the diaphragm increased.There was, consequently, a large discrepancy between observedtest times and the predictions of ideal shock tube theory. Duffsuccessfully explained his results in terms of the leakage of·hotshocked gas through the boundary layer past the free stream inter­face. The limiting test time condition was thus obtained 'when themass flux through the shock equalled the mass leakage past theinterface. He deduced that the limiting test time varied as thesquare of the shock tube diameter for laminar boundary layergrowth behind the shock. In a /short research note, Anderson [7Jindependently proposed the same mechanism for the reduction ofshock tube test times and by making a number of simplifying as~

sumptions, the important one being that the shock wave moved atconstant velocity, he was able to predict the variation of shocktube test times with distance from the diaphragm for any givenset of initial conditions.

The theories discussed so far all relate the boundary layer·growthin the hot flow to the axial distance measured from the ideal shocktrajectory. They further relate the boundary layer thickness at theinterface to the ideal shock-interface separation distance. It waspointed out by the present authors [8J that this leads to a con­tinuous overestimate of the boundary layer leakage past the inter­face which ultimately exceeds the mass flux through the shock andresults finally in the prediction that the intfrface overtakes theshock. The variation of test time with distance from the diaphragm,determined from an analysis similar to Spence and Woods', is shownin fig. 9 and it is evident that these earlier theories will only beapplicable in regions close to the diaphragm.

Several successively improved constant shock speed theories havebeen developed since Duff and Anderson first proposed the mecha­nism of boundary layer leakage past the interface; for example seeRoshko [9], Hooker [1 OJ, Mirels [11, 12J, and Ackroyd -[13]. Thesetheories, which. seek to predict only the duration of the hot flow,all attempt to satisfy the condition of mass continuity between theshock and the interface and all describe the same basic .leakagemechanism. Roshko and Smith [14J have recently presented someexperimental tcst time measurements, using both air and argon asthe test gases, which are shown to be in quite close agreement with:,Iirels' predictions provided the initial channel pressure is greater

120 P. J. ),1USGROVE AC\"D J. P. APPLETON

SHOCK TUBE TEST THIES 121

than about 0.1 mm Hg. Ackroyd's theoretical predictions also showgood agreement \vith Duff's 'experimental measurements obtainedat low shock Mach numb~!s in argon. However, none of these theo­ries can properly ,predict the unsteady variations of the free streamflow properties between the shock and the interface since they donot take account of the unsteady motion of the shock.. In the present investigation we have developed the constant shockspeed analysls~ogclE~rwith a linearised unsteady analysis '\ThichaVOIds some of the drawbacks of the previous unsteady treatments.

r In both theories we have related the boundary layer growth aTageneral position between the shock and the interface to the actualtemporal extent of the hot flow.. In this manner we obtained fromthe unsteady analysis asymptotic solutions, appropriate to largedistances from the diaphragm,' which predict finite limiting testtimes in accord with experimental measurements and the pre­dictions of the constant shock speed theory. A simple method isgiven whereby-the unsteady shock and interface trajectories maybe constructed u,sing two equations which relate the perturbed shockand interface velocities at the extremities of the two families of line­ar characteristics in the hot flow. The unsteady variation of theflow properties between the shock and the interface may then bereadily calculated from :;imple linear relationships. Substantialagreement between the, t~st time predictions of the unsteady theo­ry and the constant shock speed theory is generally obtained, al­though at large distances from the diaphragm the unsteady theorypredicts test times'which 'are significantly greater than the finalasymptotic values. Closed form solutions for the variation of theflow properties at the asymptote are obtained from which an overallestimate of the total shock attenuation maybe determined for anygiven set of initial conditions. However, it must be emphasized thatin this investigation we have neglected the effects of cold flowboundaEY}<ty~~th, since the leakage of hot gas past the inter­ra:ceprecfU<I'es any accurate description of the boundary layergrowth in this region. Ultimately the effects of cold flow boundarylayer growth will dominate the shock tube flow even though Mirels[2] and Tnmpi and Cohen [1] sho\v:-~r:;g a very Silliple model forthe cold flow boundary layer growtJi, that its influence is initiallymuch less than that of the hot flowboundary layer. The "asymptotic"limit predicted by the modified unsteady theory should therefore be

where p is the density and u and ware, respectively, the axial andradial components of velocity in the laboratory frame of reference,see fig. la. Integration of (1) across the tube gives

§ 2. The unsteady Iinearised analysis

Consider a shock tube of uniform circular cross section. The un­steady continuity equation in a cylindrical coordinate system is

(2)

(I)op 0 I 0-A + - (pu) + - - (rpw) = 0d ox r or

P. J. ~IUSGROVE AC'D J. P. APPLETON

where the suffix e refers to values of the flow variables averagedacross the inviscid core flow, b is the boundary layer thickness whichis assumed to be very much less than the tube diameter D, andy = !-D - r.

\Ve shall assume that the effect of the unsteady boundary layermay be analysed using the functional form of the solution for flatplate boundary layer growth behind a shock wave travelling withconstant speed Uo. In this investigation we have used the flat platetheory of Rott and Hartunian [15J for laminar boundary layergrowth (see Appendix A) and Spence's [16J theory for turbulentboundary layer growth (see Appendix B). Provided that the influ-

vIewed as a temporary phenomenon, as is evidenced by the variation_of shock speed with distance reported by Duff.

In a parallel experiment?-! investigation shock tube test timeswere measured using 1.77 inch and 5.0 inch diameter sho.ck tubesover a wide range of conditions (2 ~ 11ls ~ 8 and 0.2 ~ PI ~ 200mm Hg). These measurements, supplemented by those reportedelsewhere, have been compared with the predictions of the constantshock speed theories and the linearised unsteady theory. Theagreement is generally good and on this basis a simple methodof estimating shock tube test times is suggested.

122

, .

SHOCK TUBE TEST TDIES 123

u~ Uo--u,

Xs

Ranzfaction .I

Interface Shock.

Diaphragm.

(a) laboratory co-ordinat<2 systQm.

Diaphragm.Rar<2faction InterfacQ. Shock.

Uo

XsI.

D

-C-L~=-----r=----+----==--

(b) .Shock fixed co-ccdinatQ system./

Fig. 1. Flow field in shock-fixed and laboratory coordinates.

ence of free stream variations on the boundary layer growth IS

neglected, (2) may then be rewritten in the form

OPe o· 4 dLl [ oq 5 8q ]. - + - (Peue) = --P2U2- (5 - 1) ---;::- + - -~- ,ot ox D dq ox U2 ot(3)

where the boundary layer thickness Ll = Kr,r with m = ! or ! re­spectIvely for laminar or turbulent boundary layers. The~ffix 2identifies conditions immediately behind an llDattenuated shockwave and the functions 5 and K are defined in the Appendices.EOF a truly constant speed shock the boundary layer growth co­ordin~ (Uot - x). The definition of g for an un­steady shock wave IS less obVious and will be discussed..below. TheadditIonal equations which relate the free stream flow variables arethe momentum equation

where Ug is the shock velocity at g. This definition of~Y.b.eJ:arbITrary srncethe-unearlsecrll-e-thod of analysis treatS' the es­sentially unsteady boundary layer growth as though it were quasi

We now define perturbation quantities p*, p*, and u* relative to·the ideal free stream flow uantities such that Pe = P2 + p*, pe =

= P2 + p* and Ue - U2 + u*. Substitution 0 t ese mto equations(3)-(5) yields the following pair of linear hyperbolic partial differ­ential equations for P* and u*:

(9)

(8)

(7)

(6)

(5)

(4)

(8 8) (Pe)- + 'Ue-~- In --. = o.at ox P~.

(OUe OUe .) opepe -~- + Ue-~-.- + -~- = 0d ex ox

P. J. ~WSGROVE AXD J. P. APPLETON

( oP* +U2 e!* )/P2a~U2+:.2.- cu* =fJt ox U2 oX

. - ~ dL1 [(5 - 1) cq +~ ~J,D dq oX U2 at

au* au* 1 ap*-- + U2--+ - --=0.at ex P2 ex

The energy equation along a particle path reduces to

P*jP2 - yp*jP2 = Constant,

where it should be noted that different particle paths require differ­ent constants on the right hand side.

The boundary layer growth coordinate, q, must be defined beforethe solution of equations (6)-(8) can be obtained. Trimpi and Cohen,}Iirels, Dem'yanov, and Spence and Woods all implicitly identifiedq with the ideal shock-interface separation distance. We define q by_relating the boundary layer thicknessat any position x between theShOck a~d -theint~rf;'~e to the actual temporal extent o:t'ihe hoi.flow at x. Thus, for the general point f shown in fig.2 the Doundary­iayer thIckness is related to (tf - tg). The boundary layer growth co­ordinate is then given by

124

and the energy equation follO\dng a particle path

SHOCK TUBE TEST TIMES 125

Time

t

Distance xi

Fig. 2. C0!lstruction of perturbed shock and interface trajectories.

steady. It would have been equally possible to identify qf with theactual spatial extent of the hot flow ahead of j, i.e. (XI.; - Xf); thiswould have given an identical asymptotic solution at very largedistances from the diaphragm and calculations indicate that thetest times at intermediate distances would be little changed. How­ev'er as the point rcan have no direct knowledge of the shock ve­locity at k it seems more appropriate to relate the boundary layergrowth to the temporal, and not the spatial, extent of the hot flow.The definition of q given by (9) may be rewritten as

x.

q = (Uo + U*)(t - J(Uo + U*)-1 dx) =, o.

x

= (1 + U*jUo)(Uot - x + JU*/Uodx), (10)o

where U* is the perturbation to the initial shock. velocity Uo.Although we may neglect the first and higher order variations ofq with x and t in (6), th-e derivative dLl/dq must be evaluated usingthe full expression for q given by (10). This is necessary because thefirst order quantity J~ (U*/U0) dx is of the same order of magnitudeas the difference (Uot -x) of zero order quantities. Equation (6) is

and

therefore re\yntten in the form

(12)

P. J. :\IUSGROVE AXD J. P. APPLETON

- iJPI]/P2a 21t2 = + ~ a2 (W~ 1) fmqm-l dt, (13)'1.

whereQ = (rY - 1 + S) K, and W = Uo/(Uo - U2) is the densityratio across the shock. The difference form of the characteristicequations corresponding to (7) and (11) may be written as

(8P* . op* )j 1 81£*-A- +U~-A- P2a~1£2 +- _.-. =

ut OX U2 ex

= - ~mKqm-l[S Uo + (1 - S)] =

D U2

4 Q---- mqm-l, (11)

D (W - 1)

where the directions of the characteristic families ~ and'f} are given,respectively, by dx/dt = (1£2 + a2) and dxldt = (1£2 - a2), and ,11£~is the change in the free stream velocity between any two pointson a ~-characteristic. Along the two general characteristics ab andbe which are shown in fig. 2 bounded by the shock and interface,·iJu~ = 1£; - it;, ,1u1]~ 1£; - It~, etc., and the upper and lowerlimits of the integrals on the right hand sides of (12) and (13) arewritten as lc and lb, and lb and la, respectively.

To calculate the perturbation velocity and pressll[e at a generalposition on the interface, point b say, from (13) and the knownconditions at the general point a on the shock, our method requiresprior knmrledge of the locus of all points d in order to evaluate theintegral in (13) along the 1j-characteristic between a and b. Sincethis locus is dependent on the shock trajectory ahead of a, aniterative calculation, using both (12) and (13) to connect the pointsa, b, and c is required. Although such a procedure poses no con­ceptual difficulties, the labour involved may be avoided by ap­proximating the d-locus by the tangent to the shock trajectory

126

SHOCK TUBE TEST THIES 127

U~ ( 1 )" 4 Q m-l- 1 - - = - a2qb (tc - tb) +u2N D (W - 1)

U*'+ _C (A + 24>1'vf2) (16)Uo

where conditions in the cold driver gas are denoted by the suffix 3, and the cold flow boundary layer growth has been neglected. At

the shock front',an additional relationship between u~ and p~ maybe derived from, the Rankine-Hugoniot equations. In terms of theperturbation quantities we can write

U~/U2 = AU~/Uo and p~/p2a2u2 = 24>1Vf2U~/UO, (IS)

where U* = U - U 0 is the shock perturbation velocity, M 2 == (Uo - u2)/a2 and for tn ideal gas A = (M~ + 1)/(M~ - 1),4> == M~/(M~ - 1) and M o = Uo/at. Eqllations (12)-(15) may nowbe combined to give the expressions for the perturbation velocitiesof the shock and inferface:'

(14)(dP/du)b = P~/u~ = -paas,

from a, see af in fig. 2. With this approximation (13) gives a simplelinear relationship between 1(~ and p~ in terms of the known con­ditions at a. In a similar manner, (12) provides a linear relationshipbetween 1(~ and p~ in terms of the known conditions at b. In thisinstance, qf' is identified with the distance (tf' - tg ,) Ug, and thed'-locus is approximated by the straight line joining the points cand f.

Another relationship between u~ and p~ is required to completelydetermine the conditions at the point b. Following previous authors,this is provided by the compatibility relationship at the interface,namely,

and

u~ (1 + _1_) - ~ Q a2qbm - 1(tb _ ta) +1(2 N D (W - 1)

U* ,+ U: (A - 24>J/f2), (17)

where N = p?a?!P3aa is the ratio of the acoustic impedances acrossthe interface.

The shock and interface trajectories are constructed from (16)and (17) in the following manner. \Ve assume that the conditionsat the points b' and a are known from previous cYcles of calculation.The first approximation to the location of the point b is given bythe intersection of the tangent to the interface from b' and the r;­characteristic from a. The first approximation to 1(~ is then calcu­lated from (17). A second approximation to the location of b is ob­tained by using the mean velocity to construct the interface tra­jectory over the interval b'b, and hence an improved value of u;may be determined from (17). This iterative cycle is repeated untilsatisfactory- agreement between successive values of u~ has been ob­tained. The conditions. at the point c are calculated in a similarfashion using (16). The first approximation to the location of c isgiven by the intercept of the shock trajectory from a with the ~­

characteristic from b. In this step-by-step manner the completeshock and interface trajectories are constructed starting from pointson the ideal shock and interface trajectories (which may be takenarbitrarily close to the origin). Knowing the complete trajectories,the intermediate perturbation pressures and velocitiesmay then becalculated from (12) and (13). The perturbation density followsfrom (8), where the constant on the right hand side is evaluatedin terms of the conditions immediately behind the shock on theappropriate· particle path; for example, along the particle paththrough the general point c on the shock trajectory

P. J. )lUSGRO,"E A~D J. P. APPLETON128

[(P*. - P~)/(P* - p~)Jdx/dt=U2 . a~. (18)

The perturbation temperature follows from the perturbationpressure and density and the linearised equation of state;

If it were necessary a more accurate construction of the shockand interface trajectories could be obtained \vith a reduced charac­teristic mesh size by starting the construction on the initial ~- or r;­characteristic from several positions along its length. However, theresults reported here indicate that this refinement is generally un­necessary since the mesh that results from matching directly be­tween the shock and the interface rapidly tends to a finite size thatis small compared with the appropriate shock tube lengths.

It must be emphasized that the linearised anal ·sis resented hereis not valid for wea's ocks. If the shock is weak 1(* cannot be .i\eg~ted by· companson \vlth 1(?, and nonlinear terms such as

SHOCK TUBE TEST TIMES 129

".;-

1£* 01£* jox may not be neglected by comparison with terms such as1£2 ou*jox when substituting the perturbation quantities p*, p*, andu* into (4) and (5).

2.1. The asymptotic solution. The step-by-step characteristic calcu­lation outlined above reveals that as the time from diaphragmrupture tends to infinity the difference between the shock and theinterface velocities tends to zero. Ultimately,

(19)

where the suffix = refers to the asymptotic solution, t -»- =, andthe suffix i has been introduced to identify conditions at the inter­face. At this limit, q is simply the distance behind the shock of ageneral point x in the hot flow region. 'vVe are thus able to integratethe right halld sides of (12) and (13) directly and so obtain thefollowing rel~tionships between u;", p;", andq:

and

u;" - 4 .- Q'qm + u:co

U2 D (W - 1) U2(20)

(21)

where the term (u:'/a2)2 has been neglected by comparison withunity and the suffix s identifies flow conditions immediately behindthe shock. We see from (21) that the variation In p;" is proportionalto the product of small quantities and thus we conclude that thepressure between the shock and the interface is sensibly constantand equal to (P2 + P;co). By making this approximation we cancombine (14) with (15) and (19) to give the following expression forthe asymptotic shock perturbation velocity: '

The perturbation quantities P;co and u;oo may now be evaluated interms of the initial flow properties by combining (IS) and (22); thevelocity variation between the shock and interface then follows from(20). The expression for the limiting, asymptotic hot flow length qco

130 P. J. :'f.USGROVE A:\D J. P ..-\PPLETO:\

is obtained by combining (19) and (20) to give

(UO/1£2 - 1) + (UO/U2 - }.) U:/Uo = ~' (W~ 1) {::;. (23)

Equation (22) sho\vs, somewhat surprisingly, that the asymptoticshock perturbation velocity is independent of both the nature andthe rate of growth of the boundary layer. Furthermore, (19) and(22) show that when N = 0, U: = (1£2 - Uo), i.e. the inteIfacevelocity is constant and equal to its ideal value ·Uz. When N = =,u!oo = (Uo - 112) and'the shockwave is unattenuated. For the lastcase (23) reduces to

4l--Qq1n=OD 00 •

(24)

The ratio of the acoustic impedances N may be rewritten in the form

(25)

(26)

Clearly the limit N -+ = is unrealistic, since it would only be ap­proached if a very hot driver gas were used to propagate very weakshocks. Normally N will vary from order unity for weak shocksdown to zero when the shock :Mach number has reached its theo­reticd ::', ·'m. for a given combination of driver and drivengases. For stro:lg shocks, N -+ 0, the expression for the asymptoticshock perturbation velocity may be written U:/Uo = -(l/W),which for a diatomic gas is approximately -15%.

§ 3. The corrstant shock speed analyses~-'-------- ".

In this section, as its title indicates, variations in the shock velocityare neglected. Unsteady density variations may consequently alsobe negleCted and the shock tube continuity equation (1) can beintegrated across the shock tube radius to give

Di2

J rpv dr = Constant,o

where v is the ~Yial velocity in a shock-fixed coordinate system,see fig. 1b. Then by assuming that radial variations of density andvelocity are small outside the boundary layer (i.e. beyond a thick­ness b which is small compared \\ith the tube radius) and that the

S HOCK TUBE TEST TIMES 131

density is sensibly constant in the free stream between the shockand the interface, (26) leads to the following expression for the hotflow lengthl, '

where the interface velocity relative to the shock is Vi = dlldt. Thefirst boundary layer integral represents the leakage rate of hot,shocked gas past the position of the free stream interface; thesecond represents the rate at which mass accumulates in the bounda­ry layer as the hot flow length increases with time. Equation (27)has the alternative' form

(28)

(27)

<5,

4 v· f(P ') ]___1 --1 dyD Vz pz

o

<51

~- = VZ[1 - ~ f(1z~z- - :;)dY}/- 0

/

where pv varies through' the boundary layer from its free streamvalue PZVi to its valu~ at the wall, pwUo. Except for weak shocks(PwUOfp2VZ) ~ 1, whereas 0 < (vifvz) < 1. Therefore most of theboundary layer mass defect occurs in the region close to the shocktube wall and, to a first approximation, one can neglect variationsin the free stream velocity and assume Vi = Vz. Although itis certainly justified in the region close to the diaphragm the ap­proximation will introduce a small error in the asymptotic solutionwhen t -)- 0 and dl{dt = Vi -7 O. With Vi = Vz, (27) and (28) reduce to

<51

~ =[l-~f(~ -1)dyJ.dt D P2VZo

(29)

Solutions of both (27) and (29) will be obtained using the flat plateboundary layer theories described in the Appendices.

3.1. The similarity solutions. Using the flat plate bounElary layertheories described in the Appendices (29) may be rewritten in the-­form

(34)

(35)

(33)

(30)

P. J. ~ruSGROVE ASD J. P. APPLETON

l- 4[115] - ----,­

(TV - 1)

-21V lx- --- ell + In(l -ll)] - ---(TV - 1) (W - 1)

5 TV;f = - [2 tan- l [1/5 +In(l + (1/5) -In(l _ll/5) -

4 (W - 1)

l = 7J700 and x = 71/760.

In terms of these variables thes~lutions of (30) are

and

for laminar and turbulent boundary layers, respectively.Whereas (31) and (32) express. the variation of hot flow length

with distance, experiments generally measure the test time. How­ever, ,:,,;"~ constant shock speed has been assumed, a hot flowlengLh L at a distance x~ from the diaphragm corresponds to a testtime 7 = IJUoat a distance Xs = x; - I. The corresponding asymp­totic test time is 7 00 = looJU0 and the ideal shock tube test time atXs is 7r = xsJ(fV- - 1) U o. SUbstituting'for I and x~ in the definitionsof T and X gives T = 1 and X = (TV ~ 1) xJTV + [JTV, where thenew dimensionless variables land ;f are defined as

ell r 4 ]-cit = V2 U- ~iJ- Qlm .

Roshko defined dimensionless variables

where Xs is the distance from the shock to the diaphragm station,lr is the hot flow length given by ideal shock tube theory and 100

is the limiting hot flow length given when dlJdt= o. In terms ofthese variables-the solutions of (30Y-are

X = -2[Tt + In(l - Tl)], (31)

·X = U2 tan--:l T l /5 + lnO + Tl/5)- In(1 - P/5) ~ 4Tl/5], (32)- -

132

SHOCK TUBE TEST TIMES .133

for laminar and turbulent boundary layer growth, respectively;The assumption Vi = V2 which led to (29) is not necessary and

\'lith little increase in complexity one can obtain general solutionsof (27) which may be rewritten in the form

~ = V2[1- ~_ WKlJn] [1_+ -~ (S - 1) Klm]-\ _(36)

in terms of the variables ;\; and [ it has the solution'

X=- .2Q

[[t+ln(I-[t)+ (5-1) K[J--K(W -1) 2Q

and

(W-:- 1)

5X=­

4

(37)

---Q [2 tan-1 [1/5 + In(! + [1/5) -In(1 ~ [1/5) -K(J!J\ - 1) .

_ 4l1/5 _ ~ (5 - 1) KlJ _ [ (38)- 5 Q (W - 1)

for laminar and turbulent boundary layer growth, respectively./

Ideal

0-1/--..l-.......L--!..-L..LLL.i,JSf7h.L.-------L--------l

W= = Eqtn 3Oand.Eqtn36

w= 3 Eqtn 36

w= 3 Eqtn30.

~V0-01 0-1 x 1·0 10

Fig. 3. Variation of l with Ii for laminar and turbulent boundary layerdevelopment.

134 P. J. ::>ruSGROVE AXD J. P. APPLETOX

The solutions of (30) and (36), given respectively by (34)-(35)___ and (37)-(38), are shown in fig. 3 over the range of shock strengths

corresponding to 3 ~ W ~ = (With W = 3 and y = 1.4 the shock:Mach number },fs = 2.24). As the shock strength increases andW ~ = these solutions reduce to the general sim;larity for~ givenby (31) for a laminar boundary layer and by (32) for a turbulentboundary layer. It is clear from the figure that the variation of xwith l is insensitive to changes in W, especially for strong shockswhen W is large. (N.ote that for an ideal shock tube X . T andx = l).

3.2. The asymptotic maximum test ti-me. The asymptotic hot flowlength loo, which is related to the asymptotic test time by the ex­pression leo = UOToo> may be calculated from (30) and (36) whendl/dt = O. (Note that Zoo = qoo and that when dl/dt = 0 (30) re­duces to (24), thus demonstrating the equivalence of the unsteadylinearised analysis and the constant shock speed analysis at the

\\1.0 f----\\---i'<-\.----'<-\-+---+--+----+--\------1-----1

\.

1\ " "

2.0 4.0Ms

Fig. 4. Variation of asymptotic hot flow length with shock :\Iach number,laminar boundary layer.

SHOCK TUBE TEST TI~IES 135

Real Air 10"'p,,,,100 mmHq

(\

---EQ.36

Mirets B1-----c--+----p"'--;2:':.~'l--==:::::::===_ EQ.30 .

EQ.36MfretsA

1.0 1---\-l-\r4.-----+-----+-----t------;

ft pa:rHg p'ersq '"

2.0 4.0M.

Fig. 5. Variation of asymptotic hot flow length with shock Mach number,.turbulent boundary layer.

/ ..asymptote). It is found that 100 ce p1D2 for laminar boundary layergrowth whereas 100 ce p}D5/4 when the boundary layer is turbulent.The variation:v~ith shock Mach number of loo/P1D2 and loo!p}D5/4are shown in figures 4 and 5, respectively for ideal air (full lines)and argon (dashed lines). The figures also show the variation ofloolP1D2 and loo!p}D5/4 given by NErels and Ackroyd. The differ­ences between these curves and those obtained from (30) and (36)will be discussed in section 3.3. The chain dotted curves were calcu­lated using real equilibrium air properties; they show that real gaseffects are more significant when the boundary layer behind theshock is turbulent than when it is laminar. In figures 4 and 5 100

is given in ft when !he chan~~l pressu~eis~~inmm Hgand the tube diameter is measured in inch.

3.3. Comparison with previous constant shock speed theories. The firstconstant shock speed analysis was presented by Anderson who con­sidered turbulent hot flow boundary layer growth. He only includedthe effect of mass leakage past the interface, d. equation (27), and

136 P. J. ~IUSGROVEA:c-.-DJ. P. APPLETON

related the interface boundary layer thickness to the ideal hot flowlength. His analysis consequently led to the prediction of negativetest times at finite distances from the diaphragm.

Roshko who considered laminar hot flow boundary layer growth,moreiealistically related the interface boundary layer thickness tothe actual hot flow length. He also only included the effect of massleakage and, in the notation of the present investigation, expressedthe variation of hot flow length ,with time as'

This has the same form as equation (30) and Roshko was thereforeable to derive the similarity solution given by (31). However, hisomission of the boundary layer mass accumulation led to an over.,estimate of the asymptotic hot flow length, too, which is used todefine X and T; his analysiS consequently overestimated test times.Hooker, who also considered only laminar boundary layergrowth,included the effects of boundary layer mass accumulation in hisanalysis but erroneously wrote the accumulation term as

ji 0'-'-

4Vi J(PjP2) dyjvzD instead of 4Vi J (Pjp? - 1) dyjvzD,a a

d. equation (27). The integral in the former of these expressions isvery sensitive to the definition of Oi. IIIirels [lIJ pointed out thatHooker was also inconsistent in that he evaluated the mass leakageterm by assuming Vi = V2 but retained Vi = dljdt when calculatingthe mass accumulation.

Mirels [11] also presented a constant shock speed analysis inwhich he assumed laminar boundary layer growth. Equation (29)is implicit in the early part of his analysis where he assumes uniformfree stream conditions and Vi = V2 in calculating the boundary layerintegrals ; however, he used his o\vn numerical flat plate boundaryhyer solution. Equation (29) has . the alternative form dljdt =

= vz[I .:..- 4bifDJ where Ji is the _' boundary layer displacementthickness at the interface. The values of 3i given by Mirels' nu­mericalboundary layer solution are compared in Table I \'lith thosegiven by the approximate solution of Rott and Hartunian forPrandtl numbers (J = 0.72 and (J = 1.00 and for a range of valuesof _TV. It is e\-ident that the use of a differt:;nt laminar boundary

dl - [4 - ]-- = V2 L,-- -- (TV - 1) Kl~ .dt - D

(39)

\"I

SHOCK TUBE TEST TDIES 137

TABLE I

TV 1.1 1.5 2.5 4.0 6.0

8dR. a~d H.) a = 1.0 1.02 0.99 0.97 0.96 0.96

81 (:\L) a = 0.72 1.02 ·0.99 0.97 0;95 0.94

layer representation will not cause the test times predicted byMirels to differ significantly from those of the present investigation.

In an extension of his analysis Mirels allowed for the influenceof VarIatIOns· in the free stream· flow properties. on the boundarylayer development by cateulatmg the local ra~of boundary layergrowth using flat plate boundary layer theory and the local freestream flow conditions. He concluded that only free stream velocityvariations had an important influence On the boundary layer growth,and so developed an analysis similar to that based on (27). However,he implicitly assumed that the rate of mass accun:l.~1~?:1i9n._~..,~o-

-portlOI1a1 to the raIeor:;;i<l~?c.k~1<_ag~3nd-;v~uat_~cl__.the ra!!2..?fTI1esetwo quantities immediately behind the shock. This is clearlytinreaIis1ic··slnce-c-arffie-asympiotZ"~~;;::·di/dt= 0, t~eOfi1i:lSS accumulation is zero, d. equation (27). The extent by which

lUs analysIs underes.-timates t~e limiting hot fl,2:v length is clearlyshown in fIg. 4 by the difference between the Cl:irvegtven by (36)and Mirels' curve, labelled nErels A. It should be noted that tofacilitate comparisons between the theories, Mirels' curve shownin fig. 4 was derived using Rott and Hartunian's boundary layerrepresentation, see j}Iusgrove [17].

Ackroyd, using Bernstein's [I8J theory for the development ofa quasi steady shock tube boundary layer behind a constant strengthshock, computed the variation of test time with distance bysatisfying the overall condition of mass continuity between theshock and the interlace. He presents solutions for laminar, turbu­lent, and transitional boundary layers in which the growth is re­latedto the distance behind the shock and the local free streamconditions. In terms of the variables i and i, his predicted vari­ation of test time with distance from the .diaphragm is similar tothat shown in fig. 3. Ackroyd's calculated values of loo at low shockMach numbers are appreciably less than those calculated by Mirelsand by the present authors, see fig. 4. This difference is probablydue in part to Ackroyd's use of the approximation P/h = P2/h2

138 P. J. ~[uSGROVE AXD J. P. APPLETON

through the boundary layer, rather than the more usual Pfl = pwflw,

For turbulent boundary layer growth Ackroyd's computationsindicated boundary layer closure, before the asymptotic conditionwas reached, over the entire shock Mach number range which heconsidered, i.e. 1.5 ~ M 8 ~ 6.0. Consequently no comparison withAckroyd is possible in fig. 5. In the present investigation boundarylayer. closure has been calculated to occur before reaching theasymptotic limit only for shock :'Iach numbers M 8 <3; this dis­crepancy is probably due to our assumption of thin boundarylayers. It is therefore probable that asymptotic solutions are un­realistic for turbulent boundary layer growth, except perhaps athigh shock :'Iach numbers. Nevertheless leo is a quantity derivablefrom (30) and (36) \vhich may conveniently be used in the presen­tation of their solutions for a wide practical range of conditionswhere 1< leo and 0 ~ D.

:;\Iirels [12J has also considered the influence of turbulent bounda­ry layer grO\vth on test times and independently obtained the basicsimilarity solution given by (32). Using his own turbulent boundarylayer theory he has evaluated the variation of leo/PfD5/4 with shockMach number. His predictions for constant and variable free streamvelocity, corresE-0~ding respectively to th-e" equation (3D) and (36)~nalyses he~e, are shown in fig. 5ty-the· curves labelled Mirels B..and Mirels A, respectively. The difference between :NErels' B curveand the equation (30) curve is entirely due to the difference be­tween }\firels' and Spence's descriptions of the boundary layer.

It might appear from figures 4 and 5 that test times predictedby the various theories would differ significantly from one another.However examination of (31) and (32) shows that

(40)

For small T, (8l/Cleo )xs ~ 1 and even \\Chen T = 0.5, (cl/8leo )xs isonly 0.2. Test times are therefore relatively insensitive to changesin leo except in the vicinity of the asymptote where 1 -7 leo andT -7 1, \\'hich for most shock tube geometries is only approachedat sub-millimetre initial pressures.

SHOCK TUBE TEST TIMES 139

§ 4. Experimental methods

The experiments were designed to investigate the effect on test"times of independent variations in channel pressure, shock Machnumber, and distance from the diaphragm. :M~easurements werem?-de in 1.77 inch and 50 inch diameter shock ~tllbes for channelpressures in the range 0.2:(; PI :(; 200 mm Hg. In all the experimentsthe drIven gas was alr, but three different drivers were use<r, viz.helium, air, and argon, at pressures up to 100 atmosphere Dia­phmgms, which were of pure aluminium or Melinex, were burstnaturally; only the aluminium diaphragms for the 5 inch tube re­quired scribing.

The shock velocity was measured by using thin film platinumresistance- gauges, Hush-mounted in the shock tube wall, as shockdetectors. In the small tube there were four such detectors spacedat 10 inch intervals just upstream of the test time measuringstation. The, shock transit times between adjacent pairs of de­tectors were' ,recorded using three Racal microsecond counters. Inthe 5 inch shock tube the detectors were spaced at 2 ft intervalsalong the entire tube length, and their shaped outputs were super­posed on the square ,vave raster of a Tektronix type 535 oscillo­scope. The shock transit time between any pair of detectors couldbe measured to withirione microsecond.

The hot flow duration was measured b a staanation thin filmgauge mounted c;entrally in the shock tube. FollO\\'ing Meyer [19 ,a-slgnal directly proportional to the stagnation point heat transferrate was obtained by passing the gauge output through an analoguenetwork. In the present investigation fifty capacitance - resistancestages were used and the network gave the correct analogue forabout 2 milliseconds. The response was such that for a step changein the heat flux to the gauge, the output from the network reached99 % of its steady value within 3 microseconds. Figure 6 shows atypical oscilloscope record. The step increase in the heat transferrate is due to the arrival of the shockwave. Reflection of the inci­dent shock causes the heat transfer rate to exceed its ste.a.c4LY.i!Juefor a short time (~ 10 microsecQDds) until the blunt body flow isestablished aro~d th~, gillJK.~:,.. In an ideal shock tube the heattransfer rate would be constant in the region behind the shockuntil the arrival of the interface. In practice the stagnation en­thalpy of the gas arriving at the gauge increases with time after

Fig. 6. Stagnation point heat transfer rate, 2V1s = 3.15, PI = 0,5 mm Hg,horizontal scale SO [LS/cm, Air/Air.

the passage of the incident shock, due to the continuous shockdeceleration caused by boundary layer growth. This explains thealmost steady rise in the heat transfer rate following the initial step.

Generally. the measured steady heat transfer rate in the ex:­panded driver gas will differ from that in the compressed drivengas. One might therefore expect that the arrival of the interfaceat the gauge station would be evidenced by a discontinuous changein the heat flux, which would increase or decrease according to therelative values of the total temperature and density on either sideof the interface. However, experiments have shown that the firstindication of the arrival of the interface is usually given by theinteraction of the gauge bow shock wave with the interface region.Provided that the acoustic impedance of the driven gas is less thanthat of the driver gas, the interaction of the bow shock wave withthe inf,o:'[",::e will reflect a shock wave onto the gauge. This andsubsequent \vave reflections produce large transient increases in theheat flux, as shown in fig. 6. It may be noted that for air/air oper­ation the ratio of the acoustic impedances, N = pzaz/paaa is alw'aysless than unity. For helium/air operation N is greater than unitywhen J{s < 3.5, and the interaction of the bow shock with theinterface then reflects a rarefaction \yave onto the gauge. (Strictlyspeaking. the products of the interaction of finite waves depend onthe ratio of generalized impedances, which will differ slightly from

P. J. }IUSGROVE _-\~D J. P. APPLETO~

,"

,

"

,

,

, ~ \', ,~

>1,\

:~/' ,"~ '4: ;',,'v,"Vjl '\'j

,

"

,',' , ,','"

140

SHOCK TUBE TEST TIMES 141

the ratio of the acoustic impedances, see for example :\Iusg_rove[17].)

Test times have also been measured using a time resolvedschlieren system which located the shock and interfa,ce as regionsof large axial density gradient. Test times determined from heattransfer rate measurements and from density gradient measure­ments were found to agree closely with one another.

The results of the experiments are presented in § 5 and comparedwith the test times predicted by the various theories.

§ 5. Comparison between measured and predicted test times

5.1. Test times in the 1.77" shock tUDe. Figures 7, 8, and 9 show thevariation of test time with independent variations of channelpressure, shock Mach number, and channel length as measured inthe 1.77 inch diameter shock tube..Test times predicted by the

600

Z' V<- SEC)

400

IDEAL 820

T1HE TOT?AHSITIOH

. _ [ti AIR/AIRfl, - 2.7 'V ARGOll/AIRx =6.32'

D =1.77"

---- EQ.36

---- EIl.30

-- -- U.U.R UNSTEADY

---- - - ACKROYD

100.010.01.0

o '-- L- ---JL.L .......JL- _

0.1p. mm-Hq.

. Fig. 7. Test time variation with channel pressure, JIs = 2.7.

142 P. J. :.\WSGROVE AXD J. P. APPLETON

IDEAL

lAMINAR

TURBULENT

x = 6.32' t. AIR/AIR

P, = 50.0 m", Hq

o=1.77" 0 HElIUM lAIR

Eo..30(All THEORIES ARE

APPROX. COINCIDElIT)500t-----\ +-"'~~--l_-----_._---_1

250 t------+-------'~l::_-----=:""!_=_-::::::=-....d

~I

5.04.03.0

O'---'----.4-- ---''-- -L -J

2.0

Fig. 8. Test L:me \"ariation with shock :i',fach number, turbulent boundarylayer.

linearised unsteady theory and the two constant shock speed theo­ries of the present investigation are also shown. The curves derivedfrom the unsteady theory were obtained from a series of calcu­lations in which the local shock Mach numbers .Ms at the appropri­ate measuring stations were kept constant by varying the initialshock Mach numbers J[o. The values of J[o have been indicatedin fig. 9 at various points along the curve. Figures 7-9 also showthe test time variations predicted from Ackroyd's analysis. For therange of channel pressures considered here Ackroyd only presenteddata relevant to laminar boundary layer development.

The measurements indicate that test times are insensitive to thenature of the driyen gas. Agreement between theory and experi-

_.SHOCK TUBE TEST TDIES 143

20

o

16

o

o Hs~ 2.7

P. ~ 0.5 mmHg

o ~ 1.77u

o

12

o

-------

s

o

4

1001-1-1----j_----j_----t--

200I---ocI==- ~t__.L--~-j_----+_-__,__---I----__t

500,.-----'-r,..--__,__---,..------r------r-------,

't ~SEC)

x. (fEU)

Fig. 9. Test time variation with distance, lVI. = 2.7, D = 1.77"'.

/ment in the figures is seen to be generally good, provided that theboundary layer is either wholly laminar or almost wholly turbuleI).t.The curves in figures 7 and 8 showing the time to transition ~based on Hartunian,Russo, and Marrone's [20J experimental shocktube transltlOn data; they reveal that the poor correlatIOn be:"tween theory arid experiment evident in fig. 7 at channel pressuresof about 5 mm Hg is due to the boundary layer being part laminarand part turbulent. The influence of free stream velocity variations(included in Ackroyd's and the equation (36) analyses) is seen tobe small at all but low channel pressures and large distances fromthe diaphragm. Figure 9 also indicates the variation of test timewith channel length as determined from one of the earlier linearised,unsteady theories which related the interface boundary layerthickness to the ideal hot flow length. The region of applicabilityof the theory is seen to be extremely limited.

In fig. 10 the experimental and theoretical variations of test time\vith shock tube channel length are shown for a shock :Mach number11,18 = 5.0 and a channel pressure PI = 0.5 mm Hg which is low

rEQ.36, l REAL AIR

P. J. "n;SGROVE AXD J. P. APPLETON

........"7'<::....----+--+--+--+---+-1-+-+-1, · H,~ 5.0.,.,. P,=0.5rr.mHq

0=1.77"

144

Fig. 10. Test time variation with distance, M. = 5.0, D = 1.77'".

x (FEET)

enough to ensure a wholly laminar boundary layer. At this highershock Mach number the influence of free stream velocity variationson the boundary layer development, and hence on the test time,is much reduced. The test time variation predicted by the line­arised unsteady analysis was determined from a single calculationof the shock and interface trajectories, starting with an initial shock}Iach number Jl0 = 5.96 at the diaphragm. The local values of theshock }Iach number are indicated along the curve. Surprisingly itis seen that the maxim esJ time is obtained at approximately16 ft rom the diaphragm--and is 35 % larger than the fillaf,aSS'mptotic value. Although there is a slight numerical mstability­-as tnc asymptote is approached ((16) and (17) were solved on adigital computer) the mean value of the test time beyond this pointis in close agreement with the asymptotic test time calculated from(23). The shock and interface trajectories are shown (not to scale)in the inset sketch. The divergent-com-ergent behaviour, which

. 1 . \...----.----~_.~fd b h' -------gJves t le tranSIent maXll1lUm test tune, was oun to e c aracter-istic of all the unsteady linearised solutions which were obtained.

O'--- --'--__L......---'_--'----'----'----'---'--'- --'-__L-----'_--'---'--.J-....L-.J.-l

I 10 100

60

90 1---~--+--t---__i;--+--p,~~/7"5~-e---_t

120 I-----+---I-t--j-_+_-+--+-_+_+_

150

SHOCK TUBE TEST TIMES 145

_ Presumably it is due to the finite time taken by the unsteadywaves irLtravelling between the shockand the interface. It is alsoevident from fig. 10 that the asymptotic test time predicted by thelinearised unsteady theory is less than that predicted by the con­stant shock speed theories. This results from the fact that the rateof boundary layer growth is evaluated in terms of the initial con""ditions at 1110 = 5.96 instead of the more appropriate final con­ditions when irIs = 5.00. An exact unsteady analysis should yieldtest times greater than those given by the present linearised theoryand give an asymptotic test time more in accord with that givenby the constant shock speed analyses.

5.2.Test times in the 5.0 inch tube. Figure 11 shows the test timevariation with distance from the diaphragm station at a shock

3224

p'=2.0 mmHq

0=5.0"

168

300 f----/-,.L-J,,------+-----+-----j

?: Cr Sec)

600 f-------+--r----/'--..);------+-----j

900 1------I-------,f'l------\.-"7"''ir--,.,--'J

1200 r-'-----,.-----,----r--;------;>-,

x (Feet)

Fig. 11. Test time variation with distance, AI. = 4.2, D = 5.0".

r-lach number M s ~ 4 in the 5 inch diameter shock tube. The initialchannel pressure of _2.0 mm Hg was chosen so that the hot flowboundary layer could again be assumed laminar. The correspondingshock velocity measurements are presented in fig. 12. It can beseen that the shock accelerates over approximately the first 12 ftof the tube to Mach numbers well in excess of lVIs = 3.86, which is

..

1,.6 r--------r----.-----.------,r-----,----..........~--...,.

the shock :\Iach number predicted by ideal theory for the measureddiaphragm pressure ratio P41 = 10,800. \Vhite [21J has explainedthis shock acceleration process in terms of the finite diaphragmopening time and has demonstrated that a shock Mach number inexcess of the ideal value will be achieved if the shock is formedfrom an initial isentropic compression wave. Using his simple"formation from compression" model, which assumes that thecharacteristics of the initial isentropic compression intersect at asingle point, the maximum shock l\fach numberforP4/Pl = 1.08 X 104

is calculated to be 4.03. Although the measured maximum value ofthe shock Mach number was approximately 4.35, it seems .likelythat a shock formation mechanism similar to that proposed byWhite is responsible for the observed fast shock speeds.

Stronger-than-ideal shock waves exist only transiently and sub­sidiary waves, resulting from the shock formation process, will re­flect between the shock and the interface and continue to alter theshock strength for some time. The assumption that shock attenu­ation is due solely to boundary layer §!O\\~tll(implicit in the line­

-arised unsteady-rneOfY)ls therefore unrealistic close to the dia-phragm. However, a theotetlcal· shock attenuation cUfvels showninfig.12 in ·which the shock Mach number is matched to the

28

P, =2.0 mmHg

D=5.0"

P. J. ~IUSGROVE AXD J. P. APPLETOX

16 20

x (Feet)

Fig. 12. Shock ~rach number variation with distance.

o

146

4.2

3.8

"- linear Unsteady.~-----

SHOCK TUBE TEST TIMES 147

measured shock :Mach number at a distance of 20 ft from the dia­phragm. Beyond this position it is seen that the rate of shockattenuation given by the linearised unsteady theory is \vell withinthe bounds of experimental uncertainty.

5.3. Comparison of reduced experimental data with theory. Experi­mental test time measurements obtained from four different sourcesare presented in figures 13 and 14 in terms of the dimensionlessvariables i = TI/T00 and [ . T/T00" The wide range of conditionsand shock tube geometries covered by these data is indicated inthe insetTables. It was shown in § 3 that, to a close approximation,the quantities i and [ are simply related by the general sirnilaritysolutions

i = -2[[t + In(1 - it)], (41)

and I\

i = i[2 tan-! [1/5 + In(1 + [1/5) - In(1 - [1/5) - 4[1/5], (42)

which are also shown in figures 13 and 14. The values of loo andhence Too which were used to make the experimental data nondimen­sional could have been taken directly from figures 4 and 5; in fact

. /

simple expressions were used which fit the appropriate curves shownin these figures to within 5 % over the Mach number range 2 <: M s <: 8.These relationships have the form

loglo[l00/pi1 -m)/mDl/ln] = AMs + BjiVIs + C. (43)

The values of the constants A, B, and C which are obtained when

TABLE II

Driven Gas: Real Air Ideal Argon·

Boundary Layer: Laminar, Turbulent Laminar, Turbulent

A -0.0830 -0.0349 -0.0739 +0.0227B +0 + 1.242 +0 +1.720C -0.0147 -0.580 -0.C671 -0.746

loo is measured in ft, PI in mm Hg and D in inch are given inTable II. For the case of a laminar boundary layer the constantswere chosen so that the fitted curve gave closest agreement with

X ='(1/'1;=

Fig. 13. Comparison of reduced experimental data with theory, laminarboundarylayer.

Ackroyd's values of loo/P1D2 at low shock Mach numbers and 'withthe equation (36) values at the higher shock 'Mach numbers.

The bulk of the experimental data shown in fig. 13 is seen to bewell correlated by (41). However, for purposes of comparison, twoX, [ solutions derived from our linearised unsteady analysis at shockMach numbers M o - 6.0 and 1110 = 3.0 have also been shown. Thesolutions, which were also found to scale with P1D2 ,clearly indi­cate a maximum test time about 35 % greater than the asymptotic.The majOlity of the experimental data, obtained in shock tubeswhich used naturally burst diaphragms, shO\v no evidence of suchtransient max.ima. However, the relatively long opening times as­sociated with naturally burst diaphragms lead to extensive shockacceleration regions adjacent to the diaphragm, see fig. 12, whereasthe linearised unsteady theory assumes instantaneous diaphragmrupture and predicts a monotonic shock deceleration. Roshko andSmith attempted to reduce diaphragm opening times in their 17"tube by using a cruciform shaped cutter placed just downstream ofthe diaphragm. A similar arrangement was used for the llrfs = 8.5,

1001001

P. J. ~ruSGROVE AC\n J. P. APPLETON

I III ! "" fMo=60. "I , I I 'Lin Unsteady hel,um/a,r ~---..l\.....~7<! ! I i' jMo:3D

lairl air -;,;.'" . . A 7':. -':'....I! I ~....~ .....o o '":> <KJou,

" ~I I AI I I ! ,

~ Ac"

J I ! , I, ,,-~

I iii I Eqtn.41 ~ .0Ideal

A

I I I II 0

I I11I :/'~~0'1.

Ms I~en D" Source

1 I11I

as. rr?hHo«-00>0" 0 25-3'8 air 0-5-20 1-77 present

'1>"~ .. 5D-5-8 " 0·2-5-0 " "• a 0

O'~Q 4·2 . 2·0 5-0 "

D{>" &4-8·5 1-0-SD 2-0 HOIbecr.e0 "

a 04'2-7'1 " 00545 170 Roshko &

" 55-1>2 0-02-1Cj , Smithargon

" 1·6 " 025-5Q1.13 Dutf

1 *" 85 'air 0·25 p77 present-

01

148

10

001001

SHOCK TUBE TEST TIMES 149

50lO0-5x~t!

1:00

Fig. 14. Comparison of reduced experimental data with theory, turbulentboundary layer.

0-1

'- I II /"--IId~l"",

V ./I i'.. 0-8

I

17 1Eqtn_~ ""l/1/ o 1

""6

V~ <VI:>

1,-

~~[;'0

I/II~b>

-4

~;~ ,p

~ Ms Driven ~mHg D' Sourc<Zgas

Ih 0 2-5-54 air 30-200 1-77 prcs<Znt[]

-2~17 " 2-65 • 100 5-0 •[]

p'> 2-8-4-6 • 2-20 17-0 Roshko

~--- 7\ 3 -1-4-6 argon 3-15 • Smith.

.' I0

o

o

o

/ . . .

PI = 0.25 mm Hg experiments in the 1.7711 tube during the presentinvestigation. As fig. 13 shows the data obtained using the cutterdoes lend some support to the prediction of a transient maximumtest time.

The experimental data shown in fig. 14 are less extensive butthey also are fairly well correlated by the constant shock speedsimilarity solution, given for turbulent boundary layer developmentby (42). There are no experimental results available close to theasymptote as such information would require very long shock tubes.The discrepancy between the experimental results and the similaritysolution is probably due partly to the nonideal diaphragm openingprocess, and partly to the boundary layer not being wholly turbu­lent.

5.4. An example oj a test time calculation. To calculate the test timeat a given shock Mach number, channel pressure, shock tube di­ameter, and distance from the diaphragm station, the value of

loglo[loofP~l-m)!mDI!m] must first be determined from (43). As anexample a specific set of conditions at which Roshko and Smithhave measured the test time will be used. These are JIs = 4.2,PI = 0.076 mm Hg, D = 17", and X s = 54.7'. The test gas \vas air,and at this pressure the boundary layer was assumed to belaminar.Thus from (43) and the appropriate constants listed in Table II weobtain IOgIO[100IPID2] - -0.363 and hence 100 = 9.53 ft. If thesound speed in air is taken to be 1117 Hfs, then the limiting testtime 1"00 = 2.03 ms. The ideal test time corresponding to the shockdensity ratio IV = 4.98 (obtained using Bernstein's [22] shock wavetables) is 1"1 = 2.94 ms and thus i - 1"1f1"00 = lAS. Equation (41)then gives l = 0.602 and hence the predicted test time 1" = 1.22ms. This is in close agreement with Roshko and Smith's measuredtest time of 1.25 ms.

§ 6. Conclusions

A solution of the linearised one-dimensional unsteady shock tubeconservation equations has been developed which includes the effectof boundary layer growth in the hot gas behind the incident shock.Test times predicted by this theory are in general agreement withthose obtained from a constant shock speed analysis similar, in manyrespects, to those of Roshko et seq. Agreement between the testtime predictions of both these theories and a wide range of experi­mental test time measurements has confirmed that mass accumu­lation in the hot flow boundary layer and mass leakage past theinterface are primarily responsible for the discrepancy between ob­served test times and those predicted by ideal shock tube theory.

The linearised unsteady theory was found to predict a transientmaximum test time approximately 35 % greater than the finalasymptotic value and, indeed, some of the experimental measure­ments appear to confirm qualitatively this prediction. The theoryalso leads to an expression for the total reduction in the velocity ofstrong shocks which may be expected at large distances from thediaphragm. Rather surprisingly, this predicted velocity reductionis independent of both the nature and the rate of growth of theboundary layer, and is a function only of the initial conditionsacross the unattenuated shock and the ratio, N, of acoustic im­pedances across the interface.

It is apparent from the experimental measurements presented

ISO P. J. ~1USGROVEAXD J. P. APPLETON

SHOCK TUBE TEST TIMES 151

here that the effect of long diaphragm opening times will be domi-pant in determining the initial. shock and interface trajectories;thus the accuracy with which our unsteady theory can predict thevariation of the pressure, density etc., between the shock and theinterface is not likely to be good. HO\vever the generally closeagreement between the test times given by the unsteady theory andthe constant shock speed theory shows that time dependent vari­ations in the flow properties, resulting from variations in the shockvelocity, do not significantly alter the observed duration of the hotflow. A simple method of estimating shock tube test times is there­fore developed, based on the similar solutions presented in § 3.

Appendix A

Rott and Hartunian [15J consider the case of laminar boundarylayer growth behind a constant speed shock using the shock fixedcoordinate--system shown in fig. 1b. They express the velocityprofile through the boundary layer in the form of a normalizedcomplementary er~or function, viz.

(v - ve}/(Uo - Ve) = (21.Jn) erfc(l;ln), (A 1)

where I; = yliJ, Y = Jg (PIPe) dy, and,1 is a boundary layer thicknessdefined as /'

/ . 00

. iJ = J(v - ve)/(Uo - ve) dY. (A 2)/ 0

II --,

The boundary layer thickness, iJ, at a distance Xs - x behind theshock is determined from the momentum integral equation and maybe expressed in the form iJ = K(xs - x)t, where

K - ~[ Itw Jt(pw)[ Uo _ .J2 + IJ-t (A3)-.In Pw(Uo - ve) pe Uo - Ve

which assumes a linear density-viscosity relationship across theboundary layer with the reference values evaluated at the shocktube wall, viz. Pit = pw,Uw-

If it is assumed that the conditions external to the boundarylayer are those obtained immediately behind the shock then K re­duces to

2 [ I/. Jt( T )t[ U - J-tK = - r

W_2 0 _ .J2 + 1 (A 4)

.J7/; P2(UO - V2) T 1 Uo - V2 •

Similarly the integral on the right hand side of (29) may be written

With the velocity profile given by equation (A 1) the.integrals onthe right hand side of (2) simply become

(A7)

(A5)

(A 6)6,

5 = J (P/pz - 1) d(y/Lt).o

dLt [ cq 5 OqJ= -pzuz dq (5-1) -+--

ox Uz 8t '

P. J. ::.ruSGROVE A~D J. P. APPLETO~

6,

f(~ - 1) dy -:- (W - 1+ 5) Lt.pzvz

o

d1 • dl

~f(Pe - p) dy +~f (Peue - pu) dy =et .. oxo 0

where

152

Note that for an ideal gas with Prandtl number unity the expressionfor 5 becomes

_. T 1 ( ) (y - 1 ) ( U0 - Vz )z5-1--- ../2-1 .T z 2 az

(A8)

In practice the flow conditions external to the boundary layerare not constant. For instance, at very large distances from thediaphragm the hot flow length is constant and, in shock fixed co­ordinates, the velocity external to the boundary layer varies fromVz immediately behind the shock to zero at the interface. An over­estimate of the effect of such a velocity change on the boundarylayer thickness, Ll, may be obtained by calculating the boundarylayer growth on the assumption that Ve = a throughout the hotflo\v. The corresponding expression for J{ is obtained by substitutingV2 = a into (..\..4). The ratio of this value of J{ to that obtaineddirectly from (A 4) is shown in Table A I for several values of theshock density ratio. It can be seen that the influence of free streamvelocity variations on the boundary layer thickness is small exceptfor weak shocks. Therefore the assumption that Ve = V2 is justifiedin the calculation of the boundary layer thickness Ll even when avariable free stream velocity is retained in the integrands of the ex-

SHOCK TUBE TEST TI~IES 153

pressions \vhich represent boundary layer leakage and maSs accumu­lation. Thus the first integral on the right hand of (27) becomes

~,

f p (V Vi ) ( Vi )- --- dy= W-- ,1pz Vz V2 V2

o

and, from (A6), the second integral equals 5,1.

(A9)

Appendix B

Spence and \Voods [5J considered the case of turbulent boundarylayer growth behind a constant speed shock using Spence's [16J flatplate turbulent boundary layer theory. They express the velocityprofile through the boundary layer in the power law form:

(v - ve)/(Uo - ve) = 1 - (ji/,10)1/ n, 0 <y ~,10 (B 1). /.

where ,10 = sg (p/Pe) dy is the transformed value of the boundarylayer thickness. ,Note that Ll = SO' (v - ve)/(Uo - ve)dy = Llo/(n + 1). The skin' friction is expressed in the form tw = pftV; wherePh is the density evaluated at an intermediate reference enthalpy

w

1.31

TABLE~I

2

1.16

4

1.09

8

1.04

f;rtt

f

rrI1tt•

hh = 0.5(hw - he).+ 0.22(lzr - he)and

[V1:/(Uo - ve)J2 = c[LloPe(Uo - ve)//lh]-Zln+l.

The suffix r indicates the recovery enthalpy. This skin frictionrelationship reduces to the Blasius skin friction law for incom­pressible flow when n - 7 and c = 0.02325.

The boundary layer thickness Ll at a distance (xs - x) behindthe shock is derived from the momentum integral equation, fromwhich it is found that ,1 = K(xs - x)4/5 where

(P ~ ) [ If J1/5[U 7 J-4/5K = 0.0389 _" oft . . 0 . ~ _ (B2)pz Ph(UO - V2) Uo-vz 9

Received 2 September 1966

(B3)

P. J. ",ruSGROVE AXD J. P. APPLETON

_ T 1 7 (y - 1 ) ( U0 - V2 )25-1---- .T 2 9 2 az

Acknowledgements

This work was based on the Doctoral thesis of one of us, P. J. Mus­grove, and '''"as supported in part by the European Qffice of Aero­Speverch, United States Air Force, under Contract AF 61(05,)-250 \\;lile the authors were at the University of Southampton.The authors are indebted to Dr. T. A. Holbeche of R. A. E. Farn­borough for making available his unpublished experimental testtime measurements.

Estimates of the error in Ll caused by neglecting the variationsin the velocity V e are somewhat greater than those indicated inAppendix A for a laminar boundary layer. However when theboundary layer is turbulent test times rarely approach the asymp­totic (as channel pressures are of necessity high, giving large maxi­mum test times). Variations in the free stream velocity are thereforerelatively slight and errors in test time predictions arising from thissource are likely to be small.

154

which assumes the linear density-viscosity relationship pp, . Ph,uh

and also assumes that the conditiOJls external to the boundary layerare those obtained immediately behind the shock.

The reduction of the boundary layer integrals in equations (2),(29), and (27) yields expressions formally the same as those givenby equations (AS), (A 7), and (A 9). However, fora turbulentbounda­ry layer with a recovery factor of unity (A6), which detines 5, re­duces to

REFEREXCES

[1] TRDIPI, R L. and N. B. COHE~, A theory for predictL."lg the flow of real gases inshock tubes with experimental verification, X.\C:\ Tx ;)375, 1955.

[2] "'[IRELS, H., Attenuation in a shock tube due to unsteady boundary layer action,NACA Rep!. 1333, 1957.

[3J DDI'YA~OV, Y. A., The influence of the boundary layer on the character of the flowof gas in a tube behind a mO\'ing shock waye, R.A.E. Library Translation Xo. 7961959.

SHOCK TUBE TEST TDIES 155

[4] SPEXCE, D. A. and B. A. \VOODS, Boundary layer and combustion effects in shocktube flows, Proc. XI Symp. of the Colston Res. Soc. p. 153, Butterworths'Sci. Pub.1959.

[5] SPEXCE, D. A. and B. A. \\'OODS, A review of theoretical treatments of shock tubeattenuation, R.A.E. TN Aero 2899, 1963.

[6] DUFF, R. E., Phys. Fluids 2 (1959) 207.[7] A~DERSO~, G. F., J. Aerospace Sci. 26 (1959) 184."[8] ApPLETON, J. P. and P. J. ~IUSGROVE, An investigation of the departure from ideal

shock tube performance. Preliminary results, A.A.S.U. 245, 1963...[9] ROSHKO, A., Phys. Fluids 3 (1960) 835.

[10] HOOKER, W. J., Phys. Fluids 4 (1961) 1451.[II] ~IIRELS, H., Phys. Fluids 6 (1963) 1201.[12] ~!IRELS, H., AIAA J. 2 (1964) 84.[13] ACKROYD, J. A. D., A study on the running times in shock tubes, ARC 24942, 1963.[14] ROSHKO, A. and J. A. S~IITH, AIAAJ. 2 (1964) 186.[15] ROTT, N. and R. HARTUNIA~, On the heat transfer to the walls of a shock tube,

Grad. School of Aero. Eng. Cornell Univ. (OSR-TN-55-422), 1955.[16] SPENCE, D. A., J. Fluid Mech. 8 (1960) 368.[17] MUSGROVE, P. J., The influence of the boundary layer on shock tube running times,

University of Southampton Ph. D. thesis, 1964.[18] BERNSTEIN,.l-.:.r Notes on some experimental and theoretical results for the boundary

layer development aft of the shock in a shock tube, ARC CP 625, 1963.[19] MEYER, R. F., .Further comments on analogue networks to obtain heat flux from

surface temperature measurements, NRC Canada Aero Rept. LR-375, 1963.[20] HARTUNIAN, R. A., A. L. Russo and P. V. 1vIARRONE, J. Aerospace Sci. 27 (1960) 587.[21] WHITE, D. R., J. Fluid Mech. 4 (I958) 585.[22] BERNSTEIN, L., Tabulated solutions of the equilibrium gas properties behind the

incident and reflected normal shock wave in a shock tube, 1 Nitrogen, II Oxygen,ARC CP 626, 1963. /

/

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