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Shock Tube

(Brief introduction on its theory and applications)

by Grazia Lamanna

Institut fr Thermodynamik der Luft- und RaumfahrtUniversitt tutt!art

1.0 Introduction

A shock tube is a device for generating, in a relative simple and low-cost manner, gas flows or gasconditions that are difficult to achieve in other test devices. By its nature, the shock tube produces theseconditions for a very short duration. In its simplest configuration, it takes the form of a long smooth wallsteel pipe, of either circular or rectangular cross-section divided into two compartments separated by adiaphragm of thin material, as shown in Fig. 1.1. he shorter section of the tube is at higher pressure and

is termed the driversection. he longer part of the tube is at lower pressure and is called the drivensection. he gas in the driver and driven section need not to be the same, and they can also be at differenttemperatures.

Figure 1.1. chematic representation of a shock tu"e

!hen the diaphragm is removed rapidly, for e"ample by bursting it, a flow of short duration is establishedin the tube and a compression wave travels into the low-pressure # driven$ section. he front face orleading edge of this pressure wave acts as the head of a %piston of gas&, which drives a pressure pulseahead and rapidly develops into a shock wave, as shown in Fig. 1.'.

Figure 1.2. Transition from a pressure pulse to a shock #ave

At the same time, a train of rarefaction #e"pansion$ waves travels into the high-pressure # driver$ section#see Fig. 1.($. he flow regions induced by these two waves are separated by an interface, also namedcontact surface #or discontinuity$, across which the pressure and speed are e)ual, but the density and

*iaphragm

*riven +ection #ow ressure$*river +ection #igh ressure$

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temperature are )uite different. his difference in density and temperature is due to the opposite actionsperformed by the compression and e"pansion waves, respectively. he first one heats and compresses the%shocked& driven gas, the latter cools down and decompress the e"panding driver gas. Incidentally, weobserve that the contact surface is also the interface between the driver and driven gas, hence differentgases may be present on either side of the contact surface as well. rovided the shock tube is of constantcross section, the shock wave will remain unattenuated with distance, and the pressure and particle

velocity will be uniform along the cross section and constant over a certain region behind the shock. It isthis property of the shock tube to provide both a controlled thermodynamic state and a one-dimensionalgas flow, which makes it an invaluable tool in many investigations.

Figure 1.3. $ave system occurrin! in a shock tu"e upon "urstin! of the diaphra!m% a phenomenolo!ical

representation&

he ob/ective of this report is to provide an understanding of the principles of operation of a shock tubeand an account as to some of the difficulties encountered during the design, reali0ation, and operation of

such devices. o that aim, the report starts with a brief survey on the propagation of finite disturbances ina )uiescent gas, which is an essential preliminary to the understanding of the mode of operation and usageof shock tubes. he difference in the mechanism of propagation between finite disturbances and soundwaves #i.e. small disturbances$ is also emphasised. +ubse)uently, this report illustrates the calculation ofthe e"perimental conditions as functions of the initial thermodynamic state in the driver and drivensection, focusing in particular on the theoretical interdependence among all parameters of a shock tube.+pecial attention is also paid to the generation of the tailored-condition, which is of relevance to manypractical applications.

o conclude this section, we will provide a brief overview of some of the most important investigations,which avail themselves of the shock tube techni)ue. hanks to its general simplicity, versatility, andrelative cheapness, it has been used to study a wide range of physical problems. Although an attempt has

been made to give a balanced account, the survey presented here is by far not complete and reflects theparticular interest of the author. For more details, the reader is referred to the copious literature availableon the sub/ect 1, ', (, 23. 4ne of the first applications of the shock tube is in the development ofinstrumentation for measuring some physical properties of gases. For e"ample, already in 152(, 6eynoldsemployed a shock tube to calibrate pie0oelectric pressure gauges. 4ther typical investigations, which havebeen traditionally conducted in a shock tube, are the reflection, refraction, and diffraction of shock waves,and studies on fundamental flow instabilities. Figure 1.2 shows a shock wave propagating through alayered medium. he interface is perturbed by a #baroclinic$ instability, which rapidly develops into aregion of turbulent flow.

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Figure 1.4. 'ample of a flo# insta"ility studied in a shock tu"e& ourtesy of the *raunhofer Institute for

+i!h peed ,ynamics&

Figure 1.7 illustrates a special case of shock wave diffraction, where a very comple" pressure distributioncan be observed as conse)uence of an interior detonation. his colour schlieren-photograph is part of ane"tensive study on typical elements of a building structure.

Figure 1.5. ide vie# of a shock #ave propa!atin! alon! a staircase& ourtesy of the *raunhofer

Institute for +i!h peed ,ynamics&

+hock tube applications include also studies of break-up of li)uid droplets by a shock wave, determinationof ignition delay times and rate of chemical reactions, nucleation processes, and medical applications,such as lithotripsy. +hock tubes have also been used to study the properties of gases and e"tend theire)uation of state at high-pressure and temperature conditions. Furthermore, by a suitable tuning of theinitial conditions, it is possible to achieve temperatures as high as '8888 9 behind the shock front. In thiscase, the distance behind the shock in which steady-state conditions are attained gives valuableinformation about rela"ation effects and ioni0ation properties of gases. ast but not least, the shock tubecan be also employed as a low-cost variant of an intermittent wind tunnel for the analysis of hypervelocitysuperorbital flows, aerothermodynamic processes, and scram/et propulsion systems, /ust to name a few.Figure 1.: shows schematically the operating principle of such a shock tunnel. he gas in the driversection is compressed gradually by the downstream moving piston, until the bursting pressure of thediaphragm is e"ceeded. At this point, a shock wave propagates into the driven section followed by a

particle flow. As the shock wave reaches the end wall, it is reflected again as a shock wave, leaving behinda high-pressure high-temperature 0one, which may act as a reservoir for the no00le flow. he apparatus isarranged so that the flow from the driven tube passes directly into the no00le, where it can be acceleratedto hypersonic speed. he duration of the flow is e)ual to the time between the arrival of the shock and thecontact surface at the no00le entry.

+hockurbulent Flow

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Figure 1.6. chematic illustration of the operatin! principle of a shock tunnel& ourtesy of *rank

+ou#in! The .ustralian /ational University&

2.0 Theory of compressible flows

his section discusses the basic principles of the theory of compressible flows, with particular emphasison the propagation of finite disturbances and shock waves. his is an essential preliminary to theunderstanding of the mode of operation and usage of shock tubes. !e start off from the e)uations ofmotion of a compressible fluid, using the ;ulerian approach. In the hypothesis of one-dimensional,inviscid flow and perfect gas behaviour, they readp?? 1$, then it is a well-known result in acoustic theory that small disturbances #i.e.sound waves$ propagate at a speed of sound cs3 and, in the hypothesis of isentropic flow, representsthe total variation of sound speed due to the propagation of the disturbance 3. his assertion can beeasily verified in the case of a perfect gas and isentropic flow assumption. In fact, using the isentropicrelation in the form p = constand remembering c= #p>$, we can integrate e)uation #'.($ to

( )

=

11

'

'1

o

o

ppcf . #'.2$

ombining the relations p = const and c= #p>$, the sound speed may similarly be written as( ) '1

=

o

op

pcc . #'.7$

+ubstituting relation #'.7$ in ;). #'.2$, we obtain

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( )occf

=1

'

, #'.:$

which confirms our initial interpretation of the )uantity f. It is noteworthy noticing thatf8 whenppo. hat is, in the limit of small disturbances, there is no variation of speed of sound across the wave #i.e.acoustic waves propagate at uniform speed$.

By introducing the definition off in ;)s. #'.1$ and remembering that

t

c

t

f

(

c

(

f

=

=

and ,

we can rewrite the e)uation of motion #'.1$ as

8=+

+

(

fu

(

uc

t

f#'.$

and8=

+

+

(

fc

(

uu

t

u. #'.C$

In deriving ;)s. #'.$ and #'.C$, we assumed that the flow is isentropic and the gas is initially at uniformtemperature and pressure, so that the energy e)uation takes the form p = const. In this case it holds

(

fc

(c

(

p

(

p

=

=

=

'.

By adding and subtracting ;)s. #'.$ and #'.C$, we obtain

( )( )

( )8=

+++

+

(

ufcu

t

uf #'.5$

and( )

( ) ( )

8=

+

(

ufcu

t

uf. #'.18$

hese relations represent the e)uations of propagation of waves in the direction D with speed u 4 candu - c, respectively. +pecifically, ;). #'.5$ indicates that for a particle traveling with a velocity u 4 c, thevalue of the )uantity f 4 uwill be constant. +imilar considerations hold for ;). #'.18$ as well. he)uantitiesf 4 uand f uare known asRiemann invariants. he curves in the,tspace, along which theseinvariants are constants, are known as characteristics. From ;)s. #'.5$ and #'.18$, we can draw another

important conclusion< during the propagation of a finite disturbance, a particle flow with velocity u isgenerated in such a way that a reversible conversion from internal to kinetic energy takes place along acharacteristic curve.

In order to e"amine in more details the propagation of a pressure disturbance and determine an e"pressionfor the particle velocity u, let us consider a uniform stationary gas where at the instant toa finite pressurepulse is applied, as shown in Fig. '.1a. In time #Fig. '.1b$ the pulse will have separated into two parts,specifically that traveling with speed u c for whichf uis constant and that traveling with speed u 4 cfor whichf 4 uis constant. A particle B, which is initially in the undisturbed region, will eventually be

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overtaken by the disturbance traveling with speed u 4 c#see Fig. '.1c$. At this point, the following tworelations must be simultaneously verified for the particle B< f 4 u 5 constand f - u 5 8. his followsfrom the fact that the particle B belonged originally to the undisturbed gas region, wheref 5 8 and u 5 8,and is overtaken by the whole disturbance. herefore, we obtainc1, where U+is the wave speed of the shock, and c1is the speed ofsound in the undisturbed gas in the driven tube, i.e. 7+is the ach number relative to the stationarydriven gas. he shock ach number76is defined as76 = #U6D u'$>c', where U6 is the speed of thereflected shock #relative to the shock tube$, and u'is the speed of the flow induced by the incident shock#relative to the laboratory reference frame$. hus, 76 is the wave speed relative to the oncoming gasheated by the incident shock #with sound speed c'$. he dependence of 7+ and76as function of the

pressure ratiop21is shown in Fig. (.(.

p2>p

1

p'>p1

7

18

1 7 18 78 188 788

c1>c

2= 1

821

7

Figure 3.3. 0ariation of incident and reflected shock 7ach num"er over a small ran!e of p216 for a shock tu"e#ith air as "oth driver and driven !as and T

?3T

for very strong shocks. For air #= 1.2$, the asymptotic overpressure tends to :.

3.3 Interaction between a shock wave and contact surface

his subsection discusses the refraction of a shock wave as it crosses the interface between two gases #i.e.contact discontinuity$. he simplest case to study is that of normal refraction, where the direction ofpropagation of the incident shock is normal to the plane of the boundary #as in the case of a simple shocktube$. It is possible to distinguish two distinct types of normal refraction H that when there is a reflectedshock and that when there is a reflected e"pansion wave. Figures (.: a$ and b$ show the two casesschematically. Figure (.: c$ shows a third option, the so-called tailored interface, where no disturbance isreflected from the contact surface back towards the rear wall of the shock tube. he %tailored& contactsurface configuration offers a number of advantages when applied to the operations of shock tubes,namely it permits an increase in the testing-time and an improvement in the homogeneity of the workinggas parameters #i.e. it decreases possible contamination effects in the test section caused by the drivergas$.

Figure 3.6. $ave dia!rams for three different

shock3contact surface interactions% a) refracted

shockA ") refracted rarefaction #aveA c) tailored

interface& .ir is used as !as "oth in the driverand driven sections&

'( cc driven gas combination, there e"istsonly one primary shock ach number in correspondence of which the %tailored interface& condition canbe attained. In order to established tailored conditionsin correspondence of a different primary shockach number, it is necessary either to heat up the driver>driven gas or the modify the of the driver gasby modifying its mi"ture-composition.

Figure 3.%. 0ariation of the #ave speed for the incident U