Determination of Test Time in Shock Tube Chizhikov

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ISSN 0020-4412, Instruments and Experimental Techniques, 2008, Vol. 51, No. 5, pp. 748752. Pleiades Publishing, Ltd., 2008.Original Russian Text A.S. Chizhikov, 2008, published in Pribory i Tekhnika Eksperimenta, 2008, No. 5, pp. 118122.

748

INTRODUCTION

The state of the art in development of means for

diagnosing processes in pulsed flows turns a shock tubeinto a simple and convenient tool for laboratory inves-tigations of various gas-dynamic subjects. Apart fromauxiliary equipment, a shock tube consists of two sec-tions separated by a diaphragm. The sections with high-and low-pressure gas are called the driver and drivensections, respectively. When the diaphragm is broken, ashock wave is produced. The surface separating bothgases is called the contact surface. The flow pattern issubstantially simplified by the fact that a shock wavepropagating in an immobile gas can be considered asnearly plane and the flow behind its front as one-dimen-sional. The possibility of determining the parameters ofthe flow from the initial conditions and one measurable

quantitythe shock wave velocityis an indisputableadvantage of a technique for investigations using anexperimental facility of this type, while the limited life-time of regions with constant parameters is its mostserious drawback.

There are two basic versions of the shock tubedesign, differing in whether the end face of the low-pressure section is fully open or fully closed. Both ofthese versions are used to study pulsed flows. However,in most cases, combinations of them, in which featuresof the flow in each of the designs are combined, mustbe dealt with.

The aim of our study is to determine the limits on the

time of gas outflow from a nozzle that is located at theend of the shock tube and has a channel with a constantinternal cross-section shape.

PROBLEM STATEMENT

In the case of perfect inviscid flow, the distance fromthe shock wave to the contact surface increases withdistance to the diaphragm according to a linear law. Inreality, formation of a boundary layer between the

shock wave and the contact surface results in decelera-tion of the shock wave and acceleration of the contactsurface, which leads to a decrease in this distance. As aresult, the test time may be less than one-half the esti-mated time. (These problems were considered in detailin [1, 2].) A true estimate of this quantity can beobtained only from experimental calibration of theshock tube.

To avoid influence of the boundary layer producedbehind a shock wave in investigation of pulsed flows, aspecial insert is installed directly inside the tube at theoutlet from the low-pressure section. This insert is anozzle set on the end of the low-pressure section andused to fragmentarily cut the central part from the bulkof the gas (Fig. 1). Such a design of the experimental

facility has some other advantages. For example, itoffers a chance to investigate processes of gas outflowfrom channels of different geometries merely by chang-ing nozzles instead of each time designing the shocktube with a required cross-section shape. Among thedrawbacks here are, first, more stringent processrequirements for the coincidence of axes of the con-struction and sharpening of the edges and, second, thenecessity to determine the additional component of theobservation time for this process, which is character-ized by the geometric size of the nozzle.

GENERAL EXPERIMENTALTECHNIQUES

On Determination of Test Time in a Shock Tube

A. S. Chizhikov

Joint Institute of High Temperatures, Russian Academy of Sciences, Izhorskaya ul. 13/19, Moscow, 125412 Russia

Received December 25, 2007

Abstract

A simple analytical solution to the problem of shock tube test time limitation is provided in the casein which the gas flow is investigated behind the incident shock wave from a channel of constant geometrylocated in the low-pressure section of the shock tube. Numerical simulation was performed. The good accuracyof the method is demonstrated by comparing its results to experimental data.

PACS numbers: 07.35.+k

DOI: 10.1134/S0020441208050175

1 23

4

W

Fig. 1.

Schematic diagram of the outlet section of the shocktube: (

1

) low-pressure section, (

2

) nozzle, (

3

) obstacle,(

4

) Kistler-603B sensor, and (

W

) point for measuring thepressure.


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ON DETERMINATION OF TEST TIME IN A SHOCK TUBE 749

In experiment, the limitation of the shock tube testtime due to partial reflection of the shock wave from theend wall is clearly identified from pressure oscillo-grams if the distances to the outlet are rather short,

(0.51.5)

d

(

d

is the equivalent diameter). Figure 2shows the pressure variations at a plane obstacle per-pendicular to the axis of the flow, the flow Mach num-ber behind the shock wave being close to unity. In thiscase, the test time is limited by a value of 352

s. Fromthe shape of the curve, it is apparent that this perturba-tion is of shock nature. As the distance from the outletof the channel increases, this step in the values of

parameters of the effusing gas is gradually smoothed,thus introducing a systematic error in the experimentaldata. It is therefore of principal importance that theobservation time for the process under investigation becorrectly limited when analyzing and processing theexperimental results.

The actual flow pattern formed in the shock tube isan intricate continuum of compressed gas and disconti-

nuity surfaces (Fig. 3), which has defied rigorous math-ematical solution. When defined in this manner, theproblem of determining the test time of the experimentcan be divided into three subsections and consideredsimultaneously: (i) a flow in the region between thewalls of the tube and the insert (a narrow channel),(ii) passing of the incident shock wave into the externalspace, and (iii) formation and propagation of the sec-ondary shock wave in the carrying flow.

A COMPUTATIONAL ALGORITHMAND COMPARISON WITH THE EXPERIMENT

Let us assume that the front of the shock waveformed after opening of the diaphragm in the low-pres-sure section reaches the end wall of the tube at zerotime. (Incidentally, we consider that the speeds of frontpropagation at the central and external gas regions sep-arated by the insert are equal.)

According to the design diagram shown in Fig. 4,the test time of the process in the laboratory system ofcoordinates can be described by the equation

dtl2

Wr'------

l2 l3+

2 Ws'+------------------

l3

Ws------ ,+=

16

900 1000

P*

/

P

0

t

,

s

0

12

8

4

1 2

1100 1200 1300 1400 1500

Fig. 2.

Change in the pressure at the obstacle at the centerof the flow (point W

in Fig. 1) at starting-wave Mach num-berM

s

= 2.01

andL

= 0.5

d

for air at 20

C: (

1

) primary and(

2

) secondary shock waves.

20

0

P

/

P

0

2.5

0

Mach number

Fig. 3.

Mechanism of nonstationary pressure shock occurring inside the channel behind the reflected wave (

M

s

= 2.07, air). TheMach number at the axis is shown with a solid line, the static pressure is shown with a dashed line, the pressure field is presentedwith isolines, and the magnitude and direction of the velocity are indicated with arrows.


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CHIZHIKOV

where is the averaged velocity of perturbation due

to the reflected shock wave,

2

is the velocity of flow,

is the velocity of the secondary shock wave formed

directly in the channel of the nozzle under investigationthrough the pressure gradient, and W

s

if the velocity ofthe primary shock wave.

The unknown quantities in the right part of the equa-

tion are and .

The values of shock velocity W

s

are found experi-mentally by measuring the time interval on base l

,

W

s

= l

/

t

.

The wave Mach number is calculated thereafter,

M

s

= W

s

/

a

1

,

where a

1

is the sound speed in the unperturbed gas.

The relationship for velocity of flow behind theshock wave

2

can be found using the one-dimensionaltheory of normal shock wave. If the gas in the shocktube is in a state of rest, we have

The temperature ratio at the shock wave front is

The value can be determined by measuring the

time interval on segment l

1

. No auxiliary equipment isrequired; only one of the base sensors is used (the near-est to the outlet). The first signal from the sensor isinduced by the incident shock wave front, and the sec-

Wr'

Ws'

Wr' Ws'

22a1

k 1+------------ Ms

1

Ms------

.=

T2

T1-----

P2

P1-----

12-----, i.e.,=

T2

T1-----

kMs2 k 1

2----------- k 1

2-----------Ms2

1+

k 1+

2------------

2

Ms2

-------------------------------------------------------------------- .=

Wr'

ond is due to the perturbation from the nozzle flange(the end wall of the tube). The time interval between thetwo signals from the sensor is

whence it follows that

In the first approximation, it is expedient that thevelocity of the reflected wave be determined from theone-dimensional theory in the case when the end of thelow-pressure section is fully closed, whereas experi-mental data should be used to estimate the correctnessof this approximation. A pictorial presentation of the

relationship between the velocities of the incident andreflected shock waves is given in Fig. 5. Comparison ofthe theoretical curve with the experimental data for thereflected wave shows that complex wave processes inthe region between the walls of the shock tube and theexternal surface of the ring insert affect its propagationvelocity only slightly.

The velocity determined thereby is used thereafterto evaluate the Mach number for the reflected wave,

and the sound speed behind the shock front,

At normal reflection of the shock wave from the flatwall, the gas velocity behind it is zero; the gas loses allits kinetic energy as it passes through the reflected wavefront. Therefore, the region behind the reflected shockwave can be presented as an immobile gas at high val-ues of temperature, density, and pressure.

If the Mach number of the reflected wave is known,we can write

l1Ws------ l1

Wr'------ ,+=

Wr'l1

l1/Ws--------------------- .=

Mr' Wr' 2+( )/a2,=

a2 a1T2

T1-----.=

l l1

l2 l3

A B

ISW

Fig. 4. Design diagram for determining the test time of the process due to reflection of the shock wave from the end wall: (,B)base sensors, (ISW) incident shock wave (in this case,Ms = 1.97, air); the gases under investigation behind the shock and reflectedwaves are indicated with solid and dashed lines, respectively.


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ON DETERMINATION OF TEST TIME IN A SHOCK TUBE 751

where the subscripts 2 and 5 correspond to the parame-ters behind the incident and reflected shock waves,respectively.

On the other hand, by analogy to the dependence forthe initial differential pressure at the diaphragm, whichis compared to experiment in [3], we can also write

Therefore, the Mach number of the secondary shockwave generated by the differential pressure is

and the velocity of this wave in the carrying flow is

The Mach number of the secondary shock waveappears to be considerably smaller than that of the inci-

dent shock wave. Thus, e.g., for air at 20C, = 1.326

P5

P2-----

2kMr'2

k 1( )k 1+

------------------------------------- andT5

T2-----

P5

P2-----

k 1+k 1------------P

5P2-----+

1k 1+

k 1------------

P5

P2-----+

----------------------------,==

P5

P2-----

2kMs'2

k 1( )k 1+------------------------------------- 1

k 1

k 1+------------ Ms'

1

Ms'------ a2

a5-----

2k

k 1-----------

.=

Ms' f k P5/P2,( ),=

Ws' Ms'a2.=

Ms'

corresponds to Mach number of the starting waveMs =

2 andMs = 3 = 1.478 corresponds toMs = 3.

Our experiment was carried out on the shock tubewith medium values of the Mach number; the length ofthe driver section was 2 m, and the length of the drivensection was 4 m (l2 = 95 mm and l3 = 60 mm). The cal-culated observation times for the process due to reflec-tion of the shock wave from the nozzle flange for differ-ent gases are compared in Fig. 6 to the experimentalresults for air; formation of the secondary shock wavein the channel was taken into account in the calculation.

The properties of the gases (101 kPa, 293 K) are pre-sented in the table.

The experimental data are in good agreement withthe calculation; the relative error averaged over 47points is 2.52%. The problem is thought to be solved.

DISCUSSION

The proposed method is based on the one-dimen-sional theory of the shock wave with the followingassumptions.

(1) The velocities of the primary wave inside thechannel and in the region between the external surfaceof the insert and the internal surface of the shock tubeare equal.

This problem is similar to the flow in a narrow chan-nel. While putting aside the spatial form of the bodiesat hand, we can present the flow pattern in this regionas a two-dimensional nonstationary problem of half-wedge flow in a limited half-space. The scheme ofsupersonic gas flow around a half-wedge in a shocktube was investigated in [4]. At zero time of interactionbetween the shock wave and the sharpened edge of thenozzle located inside the shock tube, the shock wave isreflected. According to [5], the following types of waveconfiguration may be realized thereby: simple Mach

Ms'

1200

800

400

01.0 1.5 2.0 2.5 3.0

Ms

Ws/Wr, m/s

1

2

Fig. 5. Propagation velocities of (1) incident Ws and (2)reflected Wrshock waves in the shock tube for air at 20(the experimental data are shown with dots).

750

500

250

0

dt, s

2

1

3

1.0 1.5 2.0 2.5 3.0Ms

Fig. 6. Comparison of the theoretical curve with the experi-mental data: (1) air, (2) carbon dioxide, (3) helium, and ()point corresponding to the experimental oscillogram in Fig. 2.

Table

ParameterGas

He Air CO2

Molecular weight, g/mol 4.0 28.9 44

Ratio of specific heats 1.667 1.402 1.297

Sound speed, m/s 1005 344 271


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CHIZHIKOV

reflection, reflection with a kink on the reflected wave,double Mach reflection, and regular reflection. Thisphase is very short. As the shock wave moves away (ifthe flow is supersonic), an attached shock wave and theMach line are formed at the nose of the wedge.

(2) The gas parameters in the region between theinternal surface of the shock tube and the external sur-face of the insert are equal to the gas parameters in the

region behind the reflected shock wave, as in the one-dimensional case.

In the experiment, the ratio of the inner cross-sec-tional area of the channel to the total cross-sectionalarea of the tube is 0.196 (1/5) and the ratio of the areaof the channels wall to the difference in area betweenthe tubes channel and the inner cross section is0.169 (1/6). The inset in Fig. 5 shows the cross sectionof the shock tube plotted with adherence to proportions:external diameter of the channel, 20 mm; thickness ofthe wall, 3 mm; and equivalent (in area) internal radiusof the tube, 22.6 mm. As a result, if the linear dimen-sions provide similar ratios of areas, the velocity of thereflected wave can be determined with a high degree ofreliability using results of calculations based on theone-dimensional theory. In the case of significant dif-ference, the necessity arises to experimentally deter-mine the velocity of the reflected wave and approxi-mate it thereafter.

(3) A hypothesis is offered that the secondary shockwave is formed in the channel.

At the instant of time when the reflected wave frontreaches the edge of the insert, the differential pressurein the channel gives rise to a secondary shock wave; itis this wave that limits the test time in the experiment.Since this phenomenon occurs inside the shock tubeand cannot be directly observed in an experiment, the

priority in verifying the hypothesis goes to numericalcomputation. A stepwise change in the pressure at theobstacle is one more piece of evidence for this hypoth-esis.

Analysis of the test time in the shock tube not onlyprovides reasonable representation of results of ourstudy, but also acts as an additional criterion of whetherthe design of the experiment is correct. As an example,let us consider the situation when gases with different

physical properties (the molecular weight and the ratioof specific heats) are used to obtain a more powerfulshock wave. If the purity of the work gas in the low-pressure section appears to be inadequate for any rea-son, a shorter test time of the experiment will act as anindicator.

CONCLUSIONS

The solution obtained in our study is suitable forcalculating the limitation of the observation time forprocess of gas outflow from the channel of constantgeometry located in the low-pressure section of theshock tube and used to fragmentarily cut the centralpart of the gas flow, which is insusceptible to effects ofthe boundary layer. Our results can be used in investi-gations on shock tubes and in design of a facility withprescribed parameters.

The technique for determining the test time in theshock tube was developed and tested at the Laboratoryfor Nonstationary Gas-Dynamic Processes of the Rus-

sian Academy of Sciences Joint Institute of High Tem-peratures. The numerical simulation of the process wasperformed using the FlowVision software package.

REFERENCES

1. Mirels, H.,Raketnaya Tekh. Kosmonavtika, 1964, vol. 2,no. 1, p. 114;AIAA J., 1964, vol. 2, p. 84.

2. Geidon, A. and Gerl, I., Udarnaya truba v khimicheskoifizike vysokikh temperatur (Shock Tube in ChemicalPhysics of High Temperature), Moscow: Mir, 1966.

3. Resler, E., Lin, Sh.-Ch., and Kantrovits, A., inMekhan-ika, sborniki perevodov i obzorov inostrannoi period-icheskoi literatury (Mechanics, Collections of Transla-

tions and Reviews of Foreign Periodic Literature), 1953,issue 5, no. 21, p. 33; J. Appl. Phys., 1952, vol. 23,no.12, p. 1930.

4. Naboko, I.M., Issledovaniya po fizicheskoi gazodi-namike (Investigations on Physical Gas Dynamics),Moscow: Nauka, 1966, pp. 172179.

5. Bazhenova, T.V. and Gvozdeva, L.G., Nestatsionarnyevzaimodeistviya udarnykh voln v gazakh (NonstationaryInteractions of Shock Waves in Gases), Moscow: Nauka,1977.

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Determination of Test Time in Shock Tube Chizhikov