An interferometric investigation of the diffraction of a planar shock wave over a semicircular cylinder

AN INTERFEROMETRIC INVESTIGATION OF

THE DIFFRACTION OF A PLANAR SHOCK WAVE

OVER A SEMICIRCULAR CYLINDER

by

Jedrzej Kaca

September 1988

_{UTIAS Technical Note 269}

..

,

AN INTERFEROMETRIC INVESTIGATION OF

THE DIFFRACTION OF A PLANAR SHOCK WAVE

OVER A SEMICIRCULAR CYLINDER

Submitted June 1988

September 1988

by

Jedrzej Kaca

©Jedrzej Kaca 1988

UTIAS Technical Note 269

CN ISSN 0082-5263

•

Acknowledgements

The author is grateful to have had the opportunity to study and do research at the Institute for krospace Studies, University of Toronto. My gratitude goes to my teacher and supervi sor, Prof. 1. I. Gl ass, for hi s invaluable help and guidance received during this study. lamalso indebted to R. L. Deschambault, J. Urbanowi cz and all techni cal staff of the UTIAS, for provi di ng ass i stance with the experimental work. Fruit ful di scuss i ons with T. C. J. Hu, D. L. Zhang and Prof. H. Glaz are acknowledged w;th thanks.

This study had not been possible without financial assistance recei ved from the Natura 1 Sci ence and Engi neeri ng Research Council of Canada under grant No. OGP0001647, the U. S. Ai r Force under grant No. AF-AFOSR 87-0124 and from the U.S. Defence Nucl ear Agency under DNA contract No. DNA 001-85-C-0368, which is also acknowledged with thanks •

Summary

When a moving planar shock wave collides with an obstacle~ interesting flow patterns are observed. Such phenomena belong to the nonstationary category of shock wave interactions. Recent advances in numeri cal technology have made it possible to simulate such flows numerically.

The mai n goal of thi s report was to i nvest i gate the accuracy of such numerical simulations. The flow patterns created by the collision of a planar shock wave with a half-circular cylinder were studied experimentally. The UTIAS 10x18cm Shock Tube with a 23cm diameter field of view Mach-Zehnder interferometer was used to obtain isopycnics of the flow field from infinite-fringe interferograms.

1\1 most two hundred experiments were conducted. The shock Mach number was varied from 1.3 to about 7 with initial pressures ranging from 15 torr to 760 torr (1 atm). In all experiments the test gas was ai rand the driver gas helium.

Onlj four types of shock wave reflections were used~ namely regular reflection (RR)~ single-Mach reflection (SMR), complex t~ach-reflection (CMR) and double-Mach reflection (OMR). 1\ thorough analysis and evaluation of the interferograms were made and compared with some previous experimental results and existing numerical simulations.

'"

TABLE OF CONTENTS

AcknoYlledgements

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.

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· .

.

.

Summary

. . .

Notation •

. . .

. . .

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.

.

·

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1.0 INTROOUCTION

. . . .

· . . .

2.0 REVIEW OF TYPES OF SHOCK WAVE REFLECTION5

· . . . .

. . .

.

3.0 UTIAS 10x18CM SHOCK TUBE AND MACH-ZEHNDER INTERFEROMETER AS MAIN

EXPERIMENTAL FACILITIES

.

. .

. .

4.0 MODEL DESIGN AND MOUNTING ••

5.0 PRECISION OF MEASUREMENTS

. . .

.

.

· . . . · . . .

.

.

. . .

.

.

6.0 GENERAL DESCRIPTION OF THE COLLISION OF A MOVING PLANAR SHOCK WAVE

i i

i i i

• vi

1 2 3 4 5

WITH A HALF-CIRCULAR CYLINDER

• • • • • • • • • • • ••

6

7.0 EVALUATION OF THE INTERFEROGRAMS •••

.

. . .

.

.

7

7.1 Eva

1

uat i on of the Interferograms IJsi ng Three-Shock Theory

7

7.2 Evaluation of the Interferograms Through the Number of Fringes

9

8.0 EXPERIMENTAL RESULT5 • • • • • • • •

. . .

.

.

8.1

8.2

8.3

Types of Shock-Wave Reflections Observed • • • • • • • • • •

Transition from RR to SMR and CMR Shortly Af ter Collision

(on the Surface of the Cylinder) • • • • • • • • • • • • • •

Transition from SMR to CMR in the Upper Shock-Wave System

(USWS) • • • • • • • • • • • • • • • •

• • • • • • • •

8.4 Lower Shock-Wave System • • • •

• •

• • • • •

8.5 Detachment Distance

• • • • • • • • • • • • • • • •

8.6

~ifurcation

Effects

• • • • • • • • • • • • • • • •

8.7 Triple Point Paths. • •

• • • • • • •

• • • • •

8.8 Precursor Phenomenon • • ••

• • • • • • • • • • • • • •

8.9 Three-Shock Theory • • • • • • • • • • • • • • • • • • • • •

8.10 Thickness of Shock Waves

• • • • • • • • • • • • • • • • • •

8.11 Density Distribution along the Detachment

D;stanc~

• • • • •

8.12 Vortices • • • • • • • • • • • • • • • • • • • • •

9.0 COMPARISON OF PRESENT EXPERIMENTAL RESULTS WITH PREVIOUS WORK

(BLEAKNEY) • • • • • • • • • • • • • • • • • • • • • • • •••

10.0 COMPARISON OF EXPERIMENTAL RESULTS WITH SOME EXISTING NUMERICAL

11 11 11

13

13

14

15

16

16

17

18

18

18

19

SIMULATIONS (GLAZ, LOMAX) • • • • • • • • • • • • • • • • • • ••

20

11.0 CONCLUSIONS

12.0 REFERENCES

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.

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.

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.

. . .

Continued

iv

21

22

TABLE 1

FIGURES

TABLE OF CONTENTS - CONTINUEO

APPENDIX A. Calculation of pressures from density distribution

APPENDIX B. List of Evaluated Interferograms

APPENDIX C. Basic Features of Successful Experiment

Notation

a - speed of sound, detachment distance CMR - complex Mach reflection

DMR - double Mach reflection

I - incident shock-wave

K

- Gladstone-Oale constant L - width of the test section LSWS - lower shock-wave system M - Mach stem

MI _ Mach stem in the lower shock-wave system Ms - incident shock Mach number

p - pressure

R - reflected shock

RI _ reflected shock in the lower shock-wave system RR - regular reflection

S - slipstream

SI slipstream in the lower shock-wave system SMR - single-Mach reflection

t - nondimensional time af ter collision, also a subscript indicating tangent componenet

T - triple point T - temperature

TDMR - terminal double-Mach reflection USWS - upper shock-wave system

v - velocity

V_s - incident shock velocity

(0) (1) ( 2) ( 3) - subscript subscri pt subscript subscript indicating initial indicating values indicating val ues indicating values

values in state (1) in state ( 2) in state ( 3)

a - angle describing position of shock-wave system on the cylinder

~ - inclination angle of À-shock ~o - oncoming flow angle

ö - angle between

I

and tangent to

R

near

T

y - specific heat ratio, also angle between incident shock and tangent to Mach stem at point

T

Ow - wedge angle

w - angle between tangent to Rand to slipstream

S

at

T

•

1.0 INTRODUCTION

Since Mach (Ref. 1) discovered shock waves in the 1870's, the interest in high speed aerodynamics and gasdynamics did not cease to increase. But until World War lI, research in this area had a mostly "academic" character, with little or no practical applications. But World War II with its great need for faster aircraft, the invention of jet engines, and finally the explosion of the first nuclear bomb made the study of shock waves of great practical importance.

Since then shock-wave research s~it into two main directions: one, studying stationary shock waves (i .e., which do not change their position wi th regard to the frame of reference, for exampl e the shock -wave system associated with an airplane flying at constant speed), and the other dealing with moving (nonstationary) shock waves with respect to the regardect object, for example, a shock wave createdby an ex?losion moving along the surface of the earth. At the beginning the first direction of research prevailed, stimulated by development of still faster aircraft and, ultimately, space flights. Mthough some important research on nonstationary shock waves were al ready made in the 1930' s and 1950' s (see Refs. 2 and 3), some major 'l'«>rks came 1 ater.

In the past 30 years, the research capabilities of shock tubes improved and the interest in nonstationary gasdynamics increased. In addi ti on, the practi cal appl i cati ons of nonstati onary shock waves became of greater importance for both ci vil and mil itary needs. One of the many examples is the design of a nuclear reactor where, in case of an explosion which generates very powerful shock waves, the safetyaspects of the reactor, which has to withstand enormous pressures, are of prime importance.

The studi es of shock -wave i nteracti ons en ab 1 ed the constructi on and improvement of shock tubes, which in turn accelerated the research on nonstationary shock waves. Furthermore, the collision of planar shock waves with different types of obstacles, which give complex flow fields behind the shock, were si mul ated numeri cally (see Refs. 4-7). Thus a new need for experimental studies of such flows was created: a need to verify experimentally the numerical simulations.

The main goals of this study were as follows:

a) to observe and investigate flow patterns af ter collision of a planar shock wave with a half-circular cylinder,

b) to evaluate the density and pressure distributions of the flow field using infinite-fringe interferometric techniques for the case of a collision of a planar shock wave with a half-circular cylinder,

c) to compare the experimental results with some existing numerical simul ati ons.

We should keep in mind that if such numerical simulations were proven to be precise and accurate enough, then a great part of the experimental work could be eliminated and replaced by numerical simulations, saving time

and money. However, it is firmly believed, that the ~timate proof for any new analysis will always be a real physical experiment.

2.0 REVIEW OF TYPES OF SHOCK WAVE REFLECTIONS

There are basically five known types of shock wave reflections (see Figs. 1-5):

a) regular reflection (RR) b) single-Mach reflection (SMR) c) complex-Mach reflection (CMR) d) double-Mac~ reflection (DMR)

e) terminal double-Mach reflection (TDMR)

l\1though regular reflection (RR) and single-Mach reflection (SMR) were already identified by Mach in 1878, the study of shock wave reflections did not begin in earnest until the 1940's.

von Neumann (Refs. 1, 2) proposed a criterion for the termination of RR in nonstationary flows, known as the "detachment" criterion, which states that, Mach reflection (SMR) occurs whenever the streamline deflection angle 02' through the deflected shock wave exceeds in magnitude the maximum deflection angle 0l through the incident shock (see Fig. 6). Later, Smith at Pri nceton Uni versity, under the supervi si on of Prof. W. Bl eakney, conducted an experimenta 1 i nvest i gat i on of nonstati onary shock -wave reflections, providing foundations for later studies. But soon some questions arose as to the validity of von Neumann's criterion. Bleakneyand hi s coll egues (Ref. 3) tri ed to resol ve thi s di sagreement found by Smi th (Ref. 32), but did not succeed. Soon af ter, White (Ref. 31), was able to identify double-Mach reflection (DMR) and also suggested a criterion for the termination of SMR. Thus, at this point in time, the basic types of shock wave reflections were already identified.

Regular reflection (RR) consists of two shocks, i.eo, the incident shock and reflected shock (see Fig. 1), both originating in one common point on the wall, cal led the intersection point P. When the coordinate system is attached to the intersection point

P,

simple two-shock theory can be used to determine the flow properties around point P. The directions of the flow in RR a re s hown in F i g. 6.

When the wave angle <1>0 is sufficiently great, the intersection point P will separate itself from the wall and single-Mach reflection will occur. SMR is characterised by three major waves, i.e., incident shock (1),

refl ected shock (R) and, the so-call ed, Mach stem (M). All th ree shocks intersect at one common point, called the triple point T. Between the Mach stem (M) and the reflected shock (R) a slipstream (5) is located, across which the pressure is continuous and all other flow parameters di sconti nuous. With the frame of reference attached to the poi nt T, the flow becomes stationary and three-shock theory may be used to calculate flow properti es in the angul ar regi ons around T. A general vi ew and a detailed scheme of SMR are shown in Figs. 2 and 7.

With the flow behind the reflected shock, relative to the triple poi nt T, bei ng supersoni c, the refl ected shock wave (R) near it becomes a

2

'a!

straight line. Such a shock configuration is referred to as a complex-Mach reflection (CMR) (see Fig. 3). The main difference between SMR and CMR is the reflected shock (R), which in case of SMR is curved all along, and in the case of CMR has a straight part near the tri pl e point T that ends at a point cal led the kink K. A detailed diagram of CMR is shown in Fig. 8. The kink K becomes a second triple point TI when a band of compression waves converges near K to form a new shock MI. The flow behi nd the refl ected shock (R), relative to the second triple point TI, is supersonic, thus two systems of three shock interactions exi st (around Tand TI). Such a shock-wave system, is then label led a double-Mach reflection (DMR) (see Fig. 4), and until now represents the most compl icated case of shock reflection. The flow properties around the second triple point TI (see detailed diagram on Fi g. 9) [sta t~s (4) and (5

p,

can be obta i ned by pass i ng through ei t her the second refl ected shock (R ), or through the refl ected shock (R) and a second Mach stem (MI). Terminal double-Mach reflection is characterised by a straight sector of the refl ected shock attached to the model surface (see Fig. 5 and Ref. 26).

As excell ent 1 iterature al ready exi sts on shock wave refl ecti ons, more on this subject can be found in Refs. 8-12.

3.0 UTIAS 10x18CM SHOCK TUBE AND MACH-ZEHNDER INTERFEROMETER AS MAIN EXPERIMENTAL FACILITIES.

All experiments were conducted using the UTIAS lOx18cm Shock Tube with Mach-Zehnder Interferometer set up in the infinite-fringe mode.

The UTIAS 10x18cm Shock Tube consists mainly of a long metal tube divided into two parts: a driver anrl a channel (see Fig. 10). Both parts are separated by a mylar (cold run) or steel (combustion run) diaphragm and, prior to an experiment, each of them usuallycontains a different gas (helium in the driver and air as a test gas in the channel) at different pressures - higher in driver and lower in channel.

Near the end of the channel, at a well-determi ned di stance from the diaphragm, the test section is situated, where a model is mounted. The test section is equipped with two parallel, high-quality windows to enable optical observation of the flow arourid the model.

When the diaphragm is. suddenly removed (broken) , a shock wave propagates along the channel. To measure the shock speed, a set of pressure transducers (stati ons) is mounted along the way of the shock wave. Each transducer is connected through an amplifier to an electronic time counter. When the shock arrives at the first station, an electronic signal from it starts all remaining counters. The shock propagating along the channel stops the counters, one af ter another. Knowi ng the di stance between the counters and reading the time the shock needed to travel from one station to the next one, one can easily calculate the shock speed between them. As the initial temperature in the channel is also well known (and as a consequence the speed of sound), the shock Mach number in each experiment is easily obtainable. The test section windows are in the field of view of the Mach-Zehnder interferometer, equi pped wi th a camera and a monochromati c 1 i ght source, so the pi ctures of the flow can be taken. However, to obta in a clear picture, a light source Il1.Jst be activated at the right time, when

the shock wave is on the model. To achieve it, there is a special de1aying e1ectronic circuit wnich activates the light source at a certain time (de1ay time) af ter arriva1 of the shock wave at the first station. This de1ay time is set up manually prior to each experiment. The basic instrumentation for recording the shock-wave speed and activating the light source, is shown in Fig. 11. More details about the construction and operation of the UTIAS 10x 18cm Shock Tube can be found in Refs. 12, 13 & 24.

The Mach-Zehnder Interferometer consists of a ruby, laser light source, a sp1 itter, two focusing mirrors, a camera, a compensation chamber and adjusting mechanisms (see Fig. 12). A monochromatic light beam, coming out of the laser, is sp1 it by a sp1 itter into two independent beams - one going through the test section, and the other passing through the compensation chamber, where the test gas is undisturbed and has the same pressure as the channe1 i niti ally. When the flow occurs, the density fi el d around the model i s not uni form and the beam trave11 i ng through the test secti on wi 11 be di storted, wi th di storti on proporti ona1 to the 10ca1 density. Then, the beam from the test section is combined with the undisturbed light from the compensation chamber to create fringes, which are then focused on the photographic film in the camera. As aresult one gets a picture of the flow (interferogram) with light and dark fringes representing the density di stribution around the model.

There are two basic modes of operation of the Mach-Zehnder interferometer, i.e, a finite-fringe mode (Fig. 15) and an infinite-fringe mode (Figs. 13 & 14). In the first case, the fringes prior to the experiment are parallel straight 1ines (see Fig. 15) and af ter firing the shock tube, the shock wave is visib1e as a shift of fringes (see Fig. 13). To get a density distribution, a thorough count of fringe shift must be done for each point of the flow field (Refs. 3,13 & 22). In the infinite-fringe mode, fringes prior to an experiment are set up such that a single fringe covers the field of view (Fig. 14). In such a case, the shock-wave system is visib1e direct1y and the density distribution is determined from the fringes. To get density va1ues in the flow field, it is enough to eva1uate densi ty on one fri nge and proceedi ng from one fri nge to another knowi ng the densi ty gap between nei ghbouri ng fri nges (i n the present w:>rk thi s gap was 3xl0-2 _{kg/m3 ). All experiments in the present work were done in the} infinite-fringe mode. In all cases the light source wave1ength was 6943A and durat i on of the pul se (exposure time) was about 15 ns. f.'ore details about operation of the Mach-Zehnder interferometer may be found in Refs. 13 and 14.

4.0 MODEL DESIGN ANO MOUNTING.

Three kinds of mode1s were used to conduct all experiments. The 1argest model was 5 cm in diamater and the smallest one on1y 1.5 cm in diameter. Both were mounted to the ceil i ng of the shock tube through an intermediate bolt. In the case of the smaller model, the mounting bolt had a sl i ght1 Y different head, so the model cou1 d he connected to the bol t wi th two small er, i nstead of one 1 arge, screws. The reason for thi sis the si ze of the model, in which there is not enough space to put a large screw. The

fina1 mounting of the bigger model is shown in Fig. 16.

Extreme care must be taken, when mounting the smallest model. lts axi s must he exactly perpendicul ar to the surface of the test section

windows. Otherwise, in the field of view of the camera (and by consequence on all pictures) we will get some strange shape, instead of a half-circul ar cylinder. Although each of the models has a special threaded hole on its side so that it can be connected to the metal plate easing mounting in the test section, in the-case of the smallest model it is not enough and its correct position must be set manually. The most common shape defonnation, caused by careless mounting is shown on Fig. 17.

Äs the field of view in the test section is limited by the size of the windows and there is only one mounting hole in the wall of shock tube, some interesti ng fl ow patterns far behi nd the model coul d not be observed. To overcome this difficulty a third model was designed. It consists basically of two parts: one being a three-quarter circular cylinder and the other being simply a thick planar, metal plate. The cylinder is connected to the pl ate wi th the use of three screws, and the pl ate itsel f has four different mounti ng hol es, so that it can be mounted in the test secti on in four different positions. In this way, the possibilities to observe the flow patterns at a later time after collision are greatly enlarged. The general view and mounting of the third model are shown on Fig. 18.

5.0 PRECISION OF MEASUREMENTS

The measurments during the experiments involved pressure, temperature, time and indirectly the incident shock velocity, Mach number and density.

The pressure in the channel (initial conditions) was measured with the use of two techniques; for pressures below 2.67 kPa (20 torr) an oil manometer was used, and for pressures above that val ue Wall ace & Ti erman type FA160 dial gauges ~re employed. In the first case the absolute error did not exceed 0.1 torr, and the dial gauges until 200 torr had an absolute error of about 1 torr. For the pressures above 200 torr (up to 800 torr) the error increased to 2 torr.

The ini ti al temperature was obtai ned wi th use of a mercury bul b thermometer cal ibrated in O.l°e interval s. It was assumed that the test gas introduced into the channel achieved thermal equilibrium before firing the shock tube. Usually the time between filling the channel with air and fi ri ng the shock tube was about 5-10 minutes.

The incident shock velocity was obtained by measuring the time intervals between 3-4 stations in the channel, with the use of electronic, digital, Hewlett-Packard counters with an accuracy of ±5IJ.s. With bh~speed of sound cal cul ated from the initial temperature a

=

(y*R*T) • , the incident Mach number Ms was also easily obtainable. For Ms = 1.1 the absolute error turned out to be 0.01 and for Ms

=

10.0 about 0.2z.

Än attempt was made to improve that accuracy by putti ng more accurate (0.1 IJ.s) electronic counters into the system. However, the rest of the system was incompatible with the new counters, resulting in incoherent indications, according to which, the shock wave was accelerating along the channel •

The density jwnp flp from one fri nge to another is gi ven by foll owi ng relation (see the end of chapter 7.1):

A - ;>. op - K*L

so that the error E(t.p) can be calculated from the total derivative i .e. ,

1 1 ;>.

E (t.p ) = x (t.À. x + - x flL)

K L L2

and Kis Gl adstone-Oal e constant assumed here to be 2.26xlO-4 (m3_{/kg) (see} Ref. 32)

The width of the test section L=10.l6cm was fabricated with an accuracy of 1Q-4m and the Ruby laser gives a radiation of wavelength ;>.=6943 A with precision to about 1 A. Taking into account all these values, one can estimate the E(t.p) as:

E(flp) = 4.5xlO-6 _{(kg/m3 )}

which is negligible (less than 0.02%) compared to the value of flp=3 x lO-2 (kg/m3 ).

6.0 GENERAL DESCRIPTION OF THE COLLISION OF A MOVING PLANAR SHOCK WAVE

wtTR

~ RÄLF-CIRCOL~R

CYLINDER

A pl anar shock wave is movi ng along the channel in the shock tube (on the pictures always to the right). In the test section the shock wave encounters an obstacle, in this case a half-circular cylinder. The situation just before coll ision is shown on Fig. 19a) , where the incident shock wave has uniform flow fields, before and af ter it.

Then, the shock wave hi ts the cyl i nder and refl ects, gi vi ng ri se to a regular reflection (RR) flow pattern. The shock system now consists of incident shock (1) and reflected shock (R), a classic case of RR (see Fig. 19b) • As time passes, the shock-wave system cl imbs the model, and at a certain point tor time af ter collision) changes into a single-Mach reflection (SMR) or cooplex Mach-reflection (CMR) (see Fig. 19c). Now the shock system consists of three shocks, as in any case of SMR or CMR. The di fference between SMR (or CMR) created by col1 i si on wi th 'a wedge and SMR (or CMR) created by collision with half-circular cylinder, is 'that in the latter case, the Mach stem (M) is a curved line (except for the early stage of collision), and not a straight line as in the case of a wedge.

Such a shock-wave system grows upward and rightward (the shock is moving to the right), in such a way, that the triple point moves along a st rai ght 1 i ne attached to the model and the Mach stem becomes more and more curved.

Later, as the shock system deve10ps, it overtakes the model with the Mach stem (M) reaching the fioor of the shock tube, which it hits and then ref1ects. First the collision is of the RR type and later, depending on incident Mach number, of SMR or CMR types. This situation is shown in Fig. 19d) (short1y before hitting the ground af ter the model) and Fig. 1ge) and Fig. 19f) af ter collision with the shock-tube bottom, at an ear1y and late deve10pment of the 10wer system.

Now there are two shock-wave systems; one caused by the model (called the Upper Shock Wave System - USWS), and the second one caused by the ref1 ecti on of the Mach stem (M) from the bottom of the the shock tube (called the Lower Shock Wave System - LSWS). The latter one deve10ps into a Mach reflection (MR), with the difference that now the Mach stem (M) in the 10wer system (LSWS) is again a straight 1ine, as in the case for the colli sion of the shock wave with a wedge. The Mach stem in the 10wer system (LSWS) has al so one impo rtant feature, i.e., i t is perpend i cu1 ar to the bottom of the shock tube. The ref1ected shock in the 10wer system (LSWS) is "connected" to the model by the remainder of the origina1 RI (see Fig. 20), being a straight 1ine at the beginning and then curving upward as time af ter collision advances (compare Figs. 20 and 21).

Furthermore, the triple point of the 10wer system (TI) moves a10ng a path, which is 1ess inc1ined to the bottom than the path of the triple point in the upper system (T) (see Fig. 22). As a consequenc:e, these two paths wi11 never cross.

The Mach stem from 10wer system (MI), which is a straight 1ine perpendicu1 ar to the bottom of the shock tube, moves upward until it hits the cei1ing of the shock tube.

When this happens, we again have a straight shock moving rightward, with a uniform, undisturbed, flow field ahead, but very complex flow af ter

it (caused by model, ref1ections from the bottom and the cei1ing, and mutua1 interacti ons between them). Such a shock-wave system 'ttQu1 d move rightward indefinite1y (if the 1ength of the shock tube was un1imited and there was no dissipation) with more and more interactions af ter the main shock.

In addition there are some other interesting phenomena, 1ike 51 i pstream from the upper system (S) penetrati ng the ref1 ected shock wave from the lower system (RI), the existence of a precursor at 10w Mach numbers and many others, which wi11 be described in greater detail in chapter 8.

7.0 EVALUATION OF THE INTERFEROGRAMS

7.1 Eva1uation of the Interferograms Using Three-Shock Theory

Accordi ng to three-shock theory, the flow in the reference frame attached to the tri p1 e poi nt (T), can be assumed stati onary. Thus through simp1e, geometrica1 considerations of the flow velocity vector, in the area around the triple point (T), the flow properties in this particu1ar area can be found. Once havi ng determi ned the fi ow parameters around the tri p1 e point (T), the flow properties (and densities in particu1ar) of the rest of the flow field can be found by following the app1icab1e fringes. To find the flow properties in the nearest neighbourhood of the triple point, we determine the oncoming flow vector

v

(see Fig. 23).

As Vo is known (from the initial Mach number and sound speed in state (0)) and <1>0 is the inclination angle of the triple point path the magnitude of vector ij can be cal cul ated from:

or

Furthermore, knowing the vector

V,

we find the flow velocity vector v1 in state (1) (see Fig. 23) decomposing vector ij in two components: normaT to the incident shock wave

v

_{n and tangential}

v

to it. The tangential component does not change across the shock, thus lor vector v1we can write:

The normal component vn changes and its new value af ter the shock can be easily found from known gasdynamics (see Ref. 19). Knowing the components vn1 and Vtl we can determine the value and the position (angle

a)

of the veloclty vector in state (1)

vl.

All the other parameters in state (1) like ternperature T, density pand pressure p can he calculated from the same gasdynarnics rel ations for a pl ane shock wave.

Next, assuming that very close to the triple point, the reflected shock (R) is a straight line and its angle with the incident shock (1) is known (i .e., measured from the interferogram) we can find the velocity vector in state (2) v2 as follows. We decompqse vector vl into two components: one normal to the reflected shock

v

_{n1 and tangentlal}

v

t

1' as above (see Fi g. 21). To fi nd the

v

_{n2 component and other flow parameters} (T2, P2' P2' a2)' with the difference tnat now the Mach number entering into gasdynamics relations is not M1 = _{v1/a1 but must be taken as M1} = v~1/a1.

Another way of doing this calculation is to make use of the fact that vector 92 must be parallel to the slipstream (S) and angle ~ between the slipstream and reflected shock can be measured directly from the interferogram. In such case only one component is needed to determine vector

v2.

_{Once the components vn2 and vt}2 are known one can easily obtain vector

v2.

Next finding the ~n _{along v2 one determines a2 and T2• Also P2 and} p 2 can be cal cul ated from P1 and P1 based on normal component to the reflected shock (R).

State (3) is cal cul ated in the same way as for states (1) and (2) wi th the di fference that now the angl e y between the Mach stem (M) and

incident shock line must be measured from the interferograms.

From the above description, the use of three-shock theory requires many, though simple, calculations. In addition, although there are no

,

physical obstacles to measure from interferograms angles a, y, 4>0 and ó,

their exact values are difficult to obtain, as the reflected shock (R) and especially the Mach stem (M) increase their curvature dramatically near the triple point.

It was found that, by campari ng pressures in states (1) and (2) (which . should be the same), the results are specially sensitive to the rneasurement of angle y (between the Mach stem (M) and incident shock line (I), see Fi g. 21). To get P2 = P3 the val ue of y shoul d be increased by 30% and sometimes even more. In add~ition the area around the triple point is of ten darkened and bl urred on the interferograms, maki ng these measurements st i1l more di ffi cult.

The problem is that the shock wave on the pictures does not consist of one thin line, but it is almost always a very thick line with singular fringes going out or cooing on to it. As a result, in the nearest neighbourhood of T the areas associated with states (1), (2) and (3) are sometimes difficult to determine, making the whole process tedious and misleading, thus requiring at this point special care and attention.

Having evaluated the flow properties around the triple point, one can find the gas densities (or, as it is of ten done, the density ratios pi/PO with regard to the initial value PO) at other points of the flow field, or more precisely on each fringe, as the fringes represent the lines of constant densi ty - i sopycni cs, the densi ty 11 jumpll ~P from one fri nge to

another is given by the following relation:

where A is the wavelength of the light source, which in this case A=6943A, and L bei ng the depth (i.e. the di stance between two wi ndows) of the test section. In the case of the UTIAS 10><18 cm Shock Tube L = 10.16 cm. The symbol

K

represents the Gladstone-Dale constant, which depends on the wavelength of the light source and the type of gas. For air and a wavel ength of 6440A, K = 2.258*10-4 [m3 /kg]. As K varies very 1 ittl e wi th A, it was assumed that in all our experiments K=2.26*10-4[m3/kg] (see Ref. 20). Thus the density IIjumpll froo one fringe to another is:

~P

= _ _

..:...69;:-.4.;..,;3_x~10.;."..-_1....;°m~ _ _ _ = 30287x 10-6 [kg/m3 ] 2.26x 10-4_[m3_/kg]xO.1016m

or, approximately:

~P = 3x10-2 _{[kg/m3 ]}

7.2 Evaluation of the Interferograms Through the Number of Fringes

As noted above, the eval uati on of the interferograms by three-shock theory has many difficulties and uncertaintities.

Most of these disadvantages can be overcome by evaluating the interferograms through the number of fringes.

This second method is very simple and is based on the fact, that density or the density ratio on the same fringe does not change. KnoWTng the density "jump" fr om one fringe to another, we can find the densities on the next fri nges, or the rest of the f1 ow fi el d. The use of thi s method is also possible in cases where the incident shock Mach number is not known (due to a failure of the equipment) or where it cannot be measured very accurately. In such cases, the strength of the incident shock (or any other shock wave) can be calculated from the following:

pI

PO

nx_6p PO

where n - number of fringes inside the shock line 6p - density jump

Po - initial density in the channel

This feature of the method is especially valuable in cases

where the measurement of the incident Mach number is difficult for

some reasons or not very accurate. The measurernent of the initial

pressure and temperature usually does not pose any problems. Still another advantage of

areas associated with states (2)

because we need not determine

fri nges • • this method and (3) are them at all appears in not easily but only cases where determined, foll ow the However some facts about this method should be kept in mind,

in order to know when its use is not recommended. From the

description above, it follows directly that the smaller the initial pressure, less fringes are in the shock thus easier to count them and the easier is the application of the methode But at the same time, as the pressure goes down, fewer fringes are on the whole flow field

and less details are visible. On the other hand, the higher the

initial pressure the more detailed are the interferograms, but then the shocks have so many fringes inside them that they start to merge

one into another, thus becoming indistinguishable and making their

count very difficult if not impossible.

As an example of the application of this method, evaluation of the strength of the incident shock wave was made in experiment No. 105. As it can be seen on Fig. 24 the incident shock consists of 5 fringes, 50 as po=4.58x_10-2 _kg/m3 _{and 6p=3x10-}2 _{kg/m3 , thus}

and as a consequence P1x_{(PO)-1 =4.27, which is associated with a Mach number} of 3.52. From measurements in the Shock Tube apparatus the i nc i dent Mach

10

number was 3.54, which is very close to the value of 3.52, especially if we take into account that the precision measurement of the Mach number in the Shock Tube was about ±0.02.

In concl udi ng thi schapter we can say that both methods have thei r advantages and disadvantages and should be considered rather as mutually complementary. In the present work the evaluation of interferograms combined both methods. Especially in cases where in doubt, one method was

used to check the results of the other.

8.0 EXPERIMENTAL RESULTS

8.1 Types of Shock-Wave Reflections Observed

From the fi ve known types of shock refl ecti ons, onl y four of them were observed duri ng the experiments, i.e., regul ar refl ecti on (RR), si ngl e Mach-reflection (SMR), complex Mach-reflection (CMR), and double-Mach reflection (DMR), the majority of them being regul ar and single

Mach-reflections.

Compl ex Mach-refl ections occurred whenever the incident Mach number was not less than about 2.78 and the line on which the incident shock lies passed the model. At earl ier stages of coll i si on, even at the same Mach numbers, the shock structure looked like SMR and only af ter careful exami nati on a straight sector of the refl ected shock coul d be seen, i t proved we were deal ing with CMR, not SMR. But even at Mach numbers much greater than 2.78, say about 5 or 6, except maybe for the lower system (LSWS) at later stages of collision, a CMR structure was poorly developed with a straight sector of the reflected shock being relatively short and sometimes difficult to notice. This fact caused CMR to be easily mistaken for SMR. Thus special care and attention in deciding whether we are dealing with CMR or SMR is strongly recommended.

Although the lower system represents a special case and is described in greater detail in chapter 8.4, its reflected shock is connected to the lower triple point (Pi) with a straight, short sector, as in the case of CMR. The rule is again the same as for the upper system, i.e., the incident shock Mach number must not be less than 2.78. _{As for Ms =2.11 or less the} shock structures are clearly SMR, thus the limiting value of Ms for CMR to occur must be located between 2.78 and 2.11. To determine precisely that val~e more experiments are needed in this range of Mach numbers.

8.2 Transition from RR to SMR and CMR Shortly Af ter Collision (on the Surface of the Cylinder)

As the different types of shock-wave reflections for wedges in air were intensively studied before (see Refs. 8-12) it would be interesting to see if and eventually how the resul ts from experiments for wedges eoul d be appl i ed to other types of model s, such as a hal f-ci rcul ar cyl i nder for exampl e.

The types of shock refleetions for the wedges can be well predicted i n adv ance, and they depend on initi al shock Mach number Ms. and wedge angl e e~. Although some differences appear between the results for perfect (trozen) and imperfect ai r (see Ref. 20), these differences become

meani ngful only for Ms greater than about 3.5 (see Fi g. 25) To detenni ne the type of shock refl ect i on on the

cyl i nder ,. imag i ne a wedge asssoci ated with the cyl i nder.

wedge woul d be formed by cyl i nder rad i us, the floor of the tangent to the cylinder surface plane (in 2-0 it would be a (see Fig. 26)

surface of the This imaginary shock tube and a line) at point A In the case of the regul ar refl ect i on (RR) poi nt A woul d be the intersection point P, and in case of single or complex Mach-reflections point A would be the foot (i.e., the beginning) of the Mach stem (M) (see

Fig. 26). .

The irnaginary wedge angle would be then related to a by;

Measuri ng the angl e a from the i nterferograrns and determi ni ng wi th

type of reflection we are dealing with, we eventua11y are able to associate the type of reflection with an imaginary wedge angle Ow and incident Mach

nurnber Ms. The results of such an approach are given in the table bel ow;

Experiment Angl e Imagi nary Type of Incident t~ach

No. a wedge angl e Ow reflection number Ms

47 19 61 RR 2.71 61 36 54 RR 2.81 125 82 8 SMR 1.47 136 40 50 SMR 1.31 176 69 21 CMR 3.92 179 65 25 CMR 3.37 183 25 65 RR 1.85

We can see from the above resul ts that thi s approach is just ifi ed, as the experimental data for a half-ci rcul ar cyl inder fit very we11 with the

data obtained for wedges. Their comparison is given in Fig. 27.

G. Ben-Oor and K. Takayama (see Ref. 11) stud i ed shock -wave

diffractions in air over the concave and convex surfaces (including a half-circular cylinder) to detennine the transition line from RR to SMR. They used the same concept of an associ ated wedge wi th a curvil i near surface

to obtain a continous change of wedge angle Ow. Since seven experiments

conducted in this sUbject obey perfectly the transition line obtained

experimenta 11 y by Ben-Oor and Takayama (see Fi g. 58), it seems that the use

of analogy between diffractions on wedges and a half-circular cylinder is the right approach. Although only 7 succesful experiments were conducted in thi s manner with incident shock t>\1ch nlJTlbers less than 4, there is no reason to suspect that for higher incident shock Mach numbers this approach

12

.'

wou1 d be mi sl eadi ng. However more experiments are needed at higher t>\lch numbers to substantiate this prob1em experimenta11y.

An interesting idea comes to mind when considering good agreement between the resu1 ts for wedges and a ci rcu1 ar cy1 i nder wi th an associ ated imaginary wedge. This idea is as follows: In the case of any given shape of cylinder, wou1d it be possib1e to predict the type of reflection at each point of the cylinder surface (or time af ter collision) by associating with it an imagi nary wedge, simil ar to the case of a ci rcu1 ar cy1 i nder, and usi n'g the resu1ts obtained for wedges? This cou1d be checked.

Note that when the shock system passes the top of the model, i.e., Cl

is greater than 90°, the who1e ana10gy between wedges and half-circu1ar cyl inders disappears. In such cases, the on1y factor determining the type of reflection seems to be the incident Mach number

Ms.

8.3 Transition from SMR to CMR in the Upper Shock-Wave System (USWS) As the shock system passes the model the t>\lch stem M, movi ng downward and expanding rightward, hits the floor of the Shock Tube creating the second shock system, labelled here the lower system. As a consequence in the transition from one type of shock-wave reflection to another, it is always necessary to specify which shock-wave system is being referred to. As the lower system will be described in greater detail in the next chapter, on1y the upper system will be described here.

Only two types of reflections were associated with the upper system, i.e., SMR and CMR. At the same time, the CMR structure seems to be better developed at later stages of collision (i.e., generally speaking, whenever the incident shock 1 ine passes the top of the model), whereas at earl i er stages CMR is so weakly developed that it could be easily mistaken for SMR.

Si nce many experiments were conducted at Ms

=

2.81, it was possible to not i ce that whenever the ref1 ected shock (for some unknown, yet to be determined, reasons) was weaker (the fringes forming R were divergent towards the tri pl e poi nt) the CMR structure, i n i dent i ca 1 run condit i ons, 100ked more 1ike SMR than CMR.

Besides the above mentioned phenomenon, the only factor determining the type of ref1 ect i on in the upper system seems to be the i nci de nt Mach number Ms.

As for Ms

=

2.11 the ref1ection was always an SMR type, and for Ms

=

2.78 one coul d al ready not i ce the weak structure of CMR, thus the 11miting va1ue of Ms for CMR to occur must be located between these two values. The fact that for Ms

=

2.78 the CMR is very poor1y developed suggests that the transition value of Ms between SMR and CMR would be much c10ser to 2.78 than 2.11.

8.4 Lower Shock-Wave System

When the moving upper shock-wave systen overtakes the model, its Mach stem (M) hits the f100r of the shock tube af ter the cylinder and reflects, creating the second shock system, referred to here as the lower shock-wave sytem (LSWS).

First, the reflection is regu1ar, where the incident shock wave (in this case the Mach stem from the upper system) and ref1ected shock (RI) are c1ear1y visib1e (see Figs. 28

&

29).

As time passes and the two shock systems move a10ng and expand, the 10wer system transforms itself from RR into an SMR, a CMR, or a [)tR, depending on the initia1 shock Mach number Ms.

Since at the same time after collision (with identica1 run conditions), the deve10pment of the shock-wave system depends on the size of the model, a more convenient parameter describing the stage of collision was introduced. This parameter is called the nondimensiona1 time af ter collision "t" and is defined as the ratio of the di stance the incident shock travelled since the beginning of the collision, and the diameter of the model. An i11ustration of this definition is given on Fig. 31.

The results of experiments concerning the transition boundaries for the 10wer system are given in Tab1e 1. Their graphica1 representation with incident Mach number Ms as the abscissa and parameter t as the ordinate is shown in Fi g. 30. 1\1 so tentati ve transiti on 1 i nes between the different types of ref1ections in the 10wer system are sketched on the same Fig. 30. Their exact determination, however, requires many more experiments in these transition regions.

8.5 Oetachment Distance

When the shock wave hits the model, it ref1ects, creating a refl ected wave in front of the model.

Although the flow field can be quite complex, the ref1ected shock in front of the model has the same bow shape in all experiments. Thi s bow shape is not only characteristic for the case of a half-circu1ar cylinder, but also for any blunt-shaped body (see Refs. 16,17

&

19).

Near the wal 1 of the shock tube, the ref1 ected shock interacts with the boundary layer, changing its shape and properties. This is known as the bifurcation effect. It wil1 not be dealt with here, as it is the subject of chapter 8.6.

In this chapter we wi11 assume that the reflected shock is a conti nuous curve, i ntersect i ng the f1 oor of the shock tube at one,

we·ll-d~termi ned poi nt. In cases, where the bifurcati on effects di stort the

ref1ected shock, the extrapolation of the regular shape is extended into the distorted area, unti1 the intersection point can be we1l determined.

With the above assumptions, we can now introduce a parameter a, being a di stance between the i ntersecti on poi nt of the shock with the fl oor of the shock tube and the model. The parameter a i s al so known as the "detachment distance" and is shown in Fig. 32.

In the course of experiments, it was determi ned that for subsonic flow the detachment distance is a function of:

a) incident Mach number Ms' and b) the time af ter collision

..

For the same reasons as explained in section 8.4, instead of using time af ter collision to designate the development of the shock system, use the parameter t as defined in the same chapter.

The detachment di stance a, for a fi xed Mach number, i s a 1 i near function of t. The higher the Mach number Ms' the smaller is the inclination angle with the abscissa t (see Fig. 33).

However, taking into account that not a very large number of experiments was done for each Mach number (usually 3-4), caution in this matter is strongly recommended before more detailed studies are done.

8.6 Bifurcation Effects

The interaction of the boudary layer on the wall with the reflected shock (R) causes shock-wave bifurcation effects. The main observable feature of this phenomenon is the "brush effect" at the foot of

R.

Near the wall of the shock tube, the reflected shock begins to divide itself into a À-shock-wave system, consisting of two shock waves, a weak slipstream and the original bow-shock wave (see Figs. 34-36).

Isopycnics, which are closer to the model, are curved and tend to concentrate in the middle of the detachment distance in the region of stagnation point. However at lower Mlch numbers (say below 2.5) and at low initial density (2.5 torr) the concentration of"the isopycnics in the middle of the detachment distance is al most nonexistent (compare Fig. 34 and Fig. 35). Isopycnics going outward from the model (i.e., to the left on the i nterferograms) are almost strai ght, concentrated on the surface of the cylinder and forming a À-shock. Between them (i.e., between these two kinds of isopycnics) the density distribution appears uniform, with only 2-3 isopycnics, and completly uniform at higher Mach numbers and lower initial densities (see Fig. 37). Such a configuration of the flow field can be characterized by three parameters: angle~, band c. Their definition is shown in Fig. 38.

It was found, that angle ~ is independent of time after collision (parameter t), but depends only on incident Mach number Ms. in a non1inear, inverse proportiona1 way. Change of angle ~ with the Mach number l>1s is illustrated on Fig. 39.

Characteri st ic dimens i ons band c dep end both on the stage of the collision and the incident Mach number. Their dependence is somewhat similar; in both cases, functions b=fi(t) and c=f·(t) (index i designates here different Mach numbers) are straight lines, inciined to the abscissa at a certain angle. This angle, in case of parameter c, is slightly decreasing wi th M. The situat i on is the oppos i te for the parameter b. Here the inclinalion angle to the abscissa is increasing with increasing Mlch number Ms'

The bifurcation effects appear significant on1y at Mlch numbers greater than about 2, and at later stages of collision.

Although in Ref. 21 it is suggested that a criterion for bifurcation effects to occur should be the Mach number af ter the incident shock wave Ml.

According to this criterion, the bifuracation effect appears whenever Mi is greater than 1. Unfortunately, for the case of a half-circular cylinder, this effect at later stages of collision is quite well developed, even at Mi smaller than 1 (see experiment No. 115 on Fig. 35), although the perturbation size decreases with decreasing Mi. This is not surprising, as the shock wave for Mi>l produces upstream pressure disturbances in the boundary layer, lifts it, and produces a À-configuration.

8.7 Triple Point Paths

With the passing of time, both lower and upper systems move along the channel together with their triple points (T and T'). As both triple points are speci al poi nts in the shock-wave systems, important informat i on can be gathered from their paths.

Both tri p1 e poi nts, Tand T', as it turned out, move along straight 1 i nes. For the case of the upper system thi s 1 i ne i stangent to the cylinder and inclined to the floor of the Shock Tube at an angle a: = 33°. What is more astonishing is that this angle remains constant regardless of the incident Mach number. In the range of shock-!>1ach numbers from Me, 1.42 c; 5.96, in whi ch all experiments were done, for each Ms the path tne upper triple point T was virtually the same (see Fig. 40).

The situat i on is, simil ar for the lower system. Here, the path of the Triple Point T' begins where the regular reflection changes to SMR or CMR. As it can be seen from Fig. 41, the change into Mach reflection occurs at a nondimensiona1 time af ter collision t~2.0 regard1ess of the Mach number. Although the tri pl e poi nt path vari es a 1 ittl e with Mach number, this change is inconsistent and should be attributed to the error of measurements rather than to a rea1 phenomenon. The Triple Point path in the lower system is inclined to the floor of shock tube at an angle of about 13°, which is not even half of the value for the upper systen. Hence, it is cl ear that the 10wer system wi 11 never overtake the upper one, as thei r paths will never cross.

Note that in order to again have a ~anar incident shock wave af ter collision, the Mach stem from the lower system (M'), being perpendicular to the shock tube floor, must hit the ceiling of the shock tube.

3.8 Precursor Phenomenon

While analysing the interferograms the dark area, usually in the shape of a triangle, ahead of the Mach stem (M) was noticed (see Fig. 42). Th is phenomenon, call ed herei n the precursor phenomenon, has al ready been observed by other researchers (see Refs. 28, 29), but a1ways at high Mach numbers. It used to be explained by radiative heating of the gas ahead of the shock by the high temperature gas behind the shock wave. As a consequence that kind of phenomenon could occur only at re1atively high Mach numbers. However, in this case the precursor phenomenon ahead of the shock appeared in many experiments where radiative heating was nonexistent and the precursor persisted even at very low Mach numbers.

Although no veri fi ed expl anat i on for thi s phenomenon has yet been found, some of its characteristic features can be pointed out from experiments.

First, on1y the Mach stem (M or MI) was preceded by the precursor and with no other shock wave could this phenomenon be associated. Secondly, on1y when the shock system passed over the top of the cyl inder did the precursor appear and continued to be visible until the Mlch stem hit the wall af ter the model. Only at higher Mach numbers, it cou1 d be seen

preceding the Mach stem of the upper system (M) (see Fig. 43)

Higher initia1 pressures usually made the precursor more c1ear and visib1e, its shape a1ways being in the form of a triangle formed by a corner of the Shock Tube wa11 and the Mlch stem, and c10sed by the precursor strai ght sector. It was verifi ed in the experiments that the deeper the corner, the greater probability that the precursor would occur.

Same researchers (Ref. 21) suggest that the precursor appears only when a sound wave catches up and exceeds the compress i on wave, creat i ng ahead of it a region of slightly compressed gas. This happened at very low Mach numbers 1ike 1.35 or 1.60 on1y. Unfortunate1y, this was not confirmed in the present experiments, as the precursor persisted to exist even at very high Mach numbers. Nevertheless, as the model was mounted to the ceiling of the shock tube and helium was used as a driver gas, there was a hypothetical possibility that after the experiment helium, as a 1ighter than air gas, cou1d be trapped behind the model, creating a 1ayer of gas with different sound speed. This wou1d explain the position of the precursor and the fact that it also exists at low Mlch numbers. Another fact supporting this hypothesis is that whenever the model was changed and the shock tube was we11 venti1ated, no precursor appeared. Apparent1y, the remaining amount of helium from previous experiments was removed. This exp1anation, however, is not satisfactory since the free boundary is straight and not diffuse. For a sch1ieren photograph, it could be an opticaleffect, but for interferometry this exp1anation is not credible.

8.9 Three-Shock Theory

The basic assu~ptions and the use of three-shock theory were ment i oned in chapter 7.1. Under1 i ned here are just a few diffi cul t i es

associated with their use.

The main prob1em with this theory is that near the triple point the shocks were very thick and consisted of many 100se1y packed fringes. This prob1em is especially visib1e for the reflected shock R, as it usually bl urrs immedi ate1y before reachi ng the tri p1 e poi nt. That makes

identification of areas associated with states (1) and (2) very confusing. The second problem is posed by the changing curvature of the reflected shock and Mach stem in the nearest neighbourhood of the triple point T. That causes great difficu1ties in the accurate measurements of the ang1es around T. Quite of ten on1y rough measurements bearing a substantia1 error were possib1e. A1so, the slipstream of ten misses the junction of incident shock and the Mach stem (which is supposed to be a triple point), causing confusion in pointing out the areas of states (2) and (3).

In conc1usion, careful attention must be given to the area around-the tri pl e poi nt T, as a mi stake at thi s stage woul d make the eva 1 uat i on of the entire flow field in error.

8.10 Thickness of Shock Waves

On almost every interferogram, the shock waves (except the incident shock) have different thi cknesses along them. The thi ckness of the M:lch stem near the triple point is usually the same as th at of the incident shock wave, but as it approaches the wall, it is getting thinner and thinner. In some cases, this thinning of the M:lch stem towards the wa11 reaches its very mi nimum close to the rear face of the cy1 i nder, and then the thi ckness

starts to increase again (see Fig. 44)

Al so a characteri st i c change in the thi ckness of the refl ected shock was observed. In front of the model, but above the bi furcat i on zone, the

reflected shock is quite thin, thus being relative1y strong. As one ~ approaches the triple point T, the fringes begin to diverge and near T we

have a 1ayer of 100se1y concentrated isopycnics of a compression wave, rather than a c1ear shock wave (see Fig. 45).

8.11 Density Distribution a10ng the Detachment Distance

Interferograms taken duri n9 the experiments showed that i n front of the model, along the detachment distance, the fringes tend to concentrate in the midd1e of the detachment distance.

In other 'fIJrds, the density near the stag nat i on poi nt is qui te uniform, on1y slight1y decreasing outwards the model, but in the midd1e of the detachment di stance it drops sUdden1y. Af ter thi s dramatic drop, the density field is a1most uniform again.

As an examp1 e of thi s phenomenon, experiment tb 96 was ana lysed and the density distribution along the detachment di stance is shown on Fig. 46. On the absci ssa the di stance from the front of the model is marked, and the ordinate shows the va1ues of the densities, expressed as a percentage of its stagnation value.

It was also noticed that for higher incident shock M:lch numbers this sudden drop in density va1ue is shifted slight1y towards the front face of the model.

8.12 Vortices

Basica1ly, two kinds of situations were determined where vortices can be found. The fi rst i s i n the sl i pst ream of the upper system (USWS), whenever the shock system passes the top of the cyl i nder, but not too far af ter (i.e., parameter t does not exceed 2.0, but usua11y does exceed 1.0).

The direction of the flow on both sides of the slipstream in the upper system is the same, but the ve10cities are different, causing a mutual slip of the moving gas in states (2) and (3). As the velocity in state (3) is greater than in state (2) and the curvature of the slipstream is directed to the rear face of the cylinder (see Fig. 47) the shear persists and in such a situation a vortex

win

be formed. Usually it happens in the 10wer part of the slipstream 1ine near the rear face of the cylinder. These flow patterns are quite simi1ar over a wide range of incident Mach numbers.

"

The second kind of situation where vortices can be observed is in the lower shock system, when it (the lower system) is already well developed. Then, the sl i pstream from the upper system penetrates the refl ected shock from the lower system RI, and goes almost to the floor of the shock tube, changing its curvature outwards, towards the rear face of the model. There, near the floor of the shock tube, a vortex is formed. At the same time, the vortex in the sl i pstream above the refl ected shock RI di sappears (see Fi g.

48).

In some cases, when the Mach reflection structure in the lower system is still very poorly developed, the penetration of the slipstream (across RI) from the upper system is very weak or nonexistent. For such "transitional" cases, the vortex is found almost directlyon the reflected shock from the lower system RI, where the slipstream line crosses RI (see Fig. 49).

Interestingly, no vortices were obsereved in the slipstream associated with the lower system, except for higher incident shock Mach numbers (see Fig. 50).

9.0 COMPARISON OF PRESENT EXPERIMENTAL RESULTS WITH PREVIOUS WORK (BLEAKNEY)

Nonstationary shock wave diffractions in air, on different kinds of models, including a half circular cylinder, were studied in the late 1940l

s by Prof. W. Bleakney (see Ref. 27) and others. It is of interest to compare the present results with those obtained by Bleakney and his students.

The shock tube used by Bl eakney had a rectangul ar cross- sect i on 10x45cm and was about 1l.4m long. It was fabricated from dural e1ements 1.125 inch thick, machined but otherwise not polished on the inside surfaces • The opt i cal system was of the same type as in the present work, i.e. a Mach-Zehnder type interferometer, but with a smaller field of vievi (12.5 cm in diameter compared to 23 cm in the present study). The only significant difference in the present work is that B1eakney used the interferometer set up in finite fringe mode (see chapter 3). As aresult to get density distributions, tedious work was required in counting the shift of the fringes for each point of the flow field.

Furthermore, presumably because of the small size of the windows compared to the size of the model, only flow relatively close to the model coul d be observed. Areas situated further behi nd the model were simpl y out of view of the interferometer. Thus in many cases, only a small portion of the shock structure was visible.

Compari ng the resul ts of Bl eakney wi th present \'IOrk (see Fi gs. 51 and 52) certain features are easily noticable:

(1) Isopycnics close to the slipstream agree almost ideally.

(2) Towards the front stagnation point, the isopycnics start to drift apart increasing towards the stagnation region.