An experimental investigation of shock-wave diffraction over compression and expansion corners

•

AN EXPERIMENTAL INVESTIGATION OF SHOCK-WAVE COMPRESSION AND EXPANSION CORNERS

by R. R. Weynants

AN EXP.ERIMENTAL INVESTIGATION ÖF SHOCK-WAVE DIFFRACTION OVER COMPRESSION AND EXPANSION CORNERS

by

R. R. Weynants

Manuscript received December

1967

APRIL 1968 UTIAS TECHNIpAL NOTE NO.

126

ACKNOWLEDGEMENTS

I wish to thank Dr. G. N. Fatterson, Director of the Institute,

for the opportunity to conduct this work at UTIAS.

The supervision and interest provided by Dr. I. I. Glass

through-out the course of this work are gratefully acknowledged.

Thanks are also due to

Dr.

R. M.

Bowman of the Air Force Institute

of Technology, Dayton, for a stimulating discussion and Prof. A.-K. Oppenheim

of Berkeley for some helpfUl letterso

The assistance of

Mr.

FoJoK. Osborne and

Mr.

D.T. Greenhouse in

carrying out the experimental work and the careful work of the shop staff are

very much appreciated.

This work was supported by the National Research Council of Canada

and the U.S. Air Force Office of Scientific Research under Grant No.

AF-AFOSR-3

6

5-66.

SUMMARY

The dynamical behaviour of shock waves at high Mach numbers in dissociated oxygen, during a sequence consisting of a diffraction over a com-pression corner, the subsequent motion through a constant area tube and a final diffraction over an expansion corner is described in some detail. This type of configuration is required to generat~ a corner-expansion flow over a wall model in a shock tube by means of a liner.

The diffraction over the compression corner was found to yield an interesting irregular Mach reflection phenomenon, brought about by a sub-stantial modification of the flow pattern through the internal energies connect-ed with the high density ratios obtainconnect-ed.

The initial diffraction results in astrong distortion of the shock front. Some aspects of the subsequent decay of this disturbance towards the stable i.e., plane configuration were investigated. It was found th at a very pronounced decrease in stability occurs at high Mach numbers, illustrated by a slow decrease of the axial extent of the shock shapes. Some evidence was obtained in support of the observations made by Bowman on the existence of two distinct shock-distortion regimes. Unfortunately, the present experimental set-up lacked the flexibility to obtain the necessary spatial resolution to investi-gate the transition between the so-called transverse and boundary-layer regimes of shock distortion.

The investigation of the expansion-corner flows were mainly con-cerned with the effects of the shock distortion on the flow uniformity in cases where the disturbances had not yet fully decayed on arrival at the corner, for example, in some of the recent corner-expansion work done by Drewry. A quali-tative assessment shows that the disturbing influence on that investigation was minor. The repercussion on future corner-expansion work in ionized argon was shown to be even less.

'. I"

..

1. 2. TABLE OF CONTENTS NOTATION INTRODUCTION EXPERIMENTAL CONSIDERATIONS

2.1

Geometry of Problem 2.2 Outline of Investigation

2.3

Instrumentation

2.3.1

Shock Tube Facility

2.3.2

Flow Visualization

?

.J. SHOCK WAVE DIFFRACTION OVER eOMPRESSION CORNERS: MECHANISM FGR

GENERATING THREE-SHOCK CONFIGURATIONS

3.1

Interaction of a Shock Wave with a Constriction

3.E

Actual Flow Conditions in a Constriction

I

3.2.1

Initial Mach Reflection

3.2.2

Mot~on of Triple Point

3.2.3

Flow Field Behind Diffracting Shock Wave

4. SHOCK Cill}VATURE PR9DUCED BY :rRANSVERSE WAVES OR THE BOUNDARY

LAYER

4.1

Effects of Transverse Waves on Curvature and Stability

4.2

4.3

4.4

Shock Waves

Influence of Boundary Layer on Shock Curvature Correlation of Experimental Rfns in Oxygen

4.3.1

Interpretation of Meakurements

4.3.2

Infl~ence of Initial Pressure

4.3.3

Decay of Disturbances

Some Comments on the Results

5

.

USE OF ALINER TO GENERATE CORNER-EXPANSION FLOWS

5.1

Some Requirements

5.2

Nonuniformities in Oxygen

5.3

Nonuniformities in Argon

6

.

C ONCLUS I ONS REFERENCES APPENDIX A: APPENDIX B: APPENDIX C: APPENDIX D:

Irregular Mach Reflection

Interaction of Shocks and Vortices Unsteady Bouodary Layers

Analysis of a Three-Shock Configuration

of 1 2 2 2 3 3 3

4

4

6 6

7

7

8

8

9

11

11

12

13

14

14

14

15

15

17

a ai C(M) Ci CL?CT d D h G I Ko Lcrit M Me Mi MI

MM

~ n p r R R(Rl or R2) Rc Re S NOTATION decay origin (Eq. 4.1)

sound speed in region i

function of Mach number (Eq. 3.1) contact surfaces

coefficient for laminar or turbulent bounday layer

diameter

additional shock in Appendix A

half-distance between parallel plates

dimensionless constant (Eq. C.7)

incident shock

dimensionless function (Eq. C.5)

critical transition length

Mach stem

freestream flow Mach number(Eq.

4.6)

flow Mach number in reg ion 2 incident shock Mach number

Mach number of Mach stem

Mach number of reflected shock

decay constant pressure (torr}

radial coordinate

radius

reflected shock

contraction ratio Reynolds number

8. l t T u,v

u

w

x Greek 8ymbols l 5 5* 5 u E Tl t3 t3 _w

e

K À S Smax 1-1 v p

test stations 81 and 82 time

triple point

velocity in a laboratory coordinate system

perturbation velocities (Eq.

4.5)

freestream velocity in a shock-fixed coordinate system relative study velocity with respect to triple points shock velocity

distance

inclination of reflected shock isentropic index

Mach stem angle

displacement thickness boundary-layer thickness

perturbed shock angle (Eq.

4.7)

self-similar coordinate angular coordinate wall angle

total axial extent due to boundary layer

similarity parameter (Fig. 36j

wavelength

shock-shape disturbance amplitude of disturbance viscosity or Mach angle

kinematic viscosi ty (Eq. C.l)

cp

x

w Indices

1,2,3,4

5

e L T

potential function (Eq.

4.5)

geometrical factor (Eq.

4.9)

triple-point path angle

triple-point path angle, corresponding to infinitesimal transverse wave

regions separated by these shock configuration at T uniform 2D region under attached shock wave

freestream conditions laminar turbulent Special notations ai a ..

lJ _{a j} sound speed r~io

Pi

p ..

₌

1. INTRODUCTION

In previous work at UTIAS, done by Drewry (Ref. 1), on corner expansion flow, the expanding region was obtained by means of an initial

con-striction provided by a 20 ft. liner. Drewry's experiments on dissociated

oxygen flows generated by a shock wave, (MI = lf' Pl = 20 torr) , showed the existence of a complex wave system when the shock front arrived at the test section (Fig. 1).

While it was originally felt that ~he whole phenomenon could

be scaled back to a starting process at the expansion corner, the present

investigation shows that t'he orJ.gln can be ~raced back to the very leading

edge of the liner. As such, the observed system is just one stage along the

path that the three-shock configuration is tracing through successive

reflec-tions from the tUbe wall. The configuration starts through Mach reflection at the leading edge (250 angle) of the 20 ft. liner.

An attempt is made to correlate runs un~er varying conditions

of initial shock Mach number and pressure based upon work by Liepmann and

Bowman (Refs. 2 and

3),

where they distinguish two regimes under which the

shock shape is determined by two completely differe~t mechanisms. In the

so-called boundary-layer regime developed by Hartuniau (Ref.

4)

and extended by

DeBoer (Ref.

5),

the shape and curvature of the shock is solely related to

the boundary layer. In the transverse wave regime, under which Drewry's work

apparently took place, the shock shape is continuously shaped by gradually

de-caying transverse waves. Furthermore, an assessment is made of the influence

of these flow disturbances on Drewry's work and it was found to be a minor effect. A similar investigation in argon, carried out with possible future work in mind, shows the mechanisms of decay to be even more effective.

Con-sequently, as far as upstream influence is concerned, corner-expansion flows

as generated by a liner can be used in the present facility.

Another challenging problem was the actual flow configuration that results af ter interaction of the incident shock with the constri2tion. Here, one has to reconcile the requirements of two-dimensional unsteady and two-dimensional steady flows and blend them into the asymptotic solutions as provided by a one-dimensional unsteady flow analysis.

The interaction of the shock with a compression corner (in a two-dimensional terminology) or a convergence (in a one-dimensional terminology) is the more difficult one. T-his note in fact gives a review of the investiga-tions by various authors to date. A series of shadowgraphs showing the Mach reflection of very strong shock waves in oxygen at a 15 degree corner are given. These results and the shock structure immediately afterwards should contribute to an understanding of this problem.

The interaction at an expansion corner will not be treated in detail, although many references dealing with this particular interaction, are mentioned throughout the report.

2. EXPERIMENTAL CONSIDERATIONS

2.1 Geometry of Problem

A descriptio~ of the instrumentation of the

4

in. x

7

in. UTIAS

shock tube will be given in Sec.

2.3.

For the time being, we shall just

con-sider the geometrical tube configuration as sketched in Fig. 2.

A shock wave generated at the diaphragm station strikes the

leading edge of the liner, undergoes Mach reflec~ion over a

25

0 leading edge,

is transmitted with a slight increase in strength (Ref.

6)

and is then finally

expanded over the 15° convex corner. I

Drewry (Ref. 1), while interested in the dissociated oxygen

expansion flow occasionally came across the configuration shown in Figs. 1 and

3,

for runs of MI

=

12 oxygen at2J torr. These are interferograms, shadowgraphs,

and schlieren photographs showing the flow induced by the shock wave

diffract-ing around the corner. If one for instance compares Fig.

3

with the series of

diffraction patterns reproduced by Skews in Ref.

7,

the three shock

configura-tion constitutes a striking difference. The main shock wave is inclined

for-ward and at the discontinuity a transverse shock appears. As a result, a

con-tact surface is generated (Refs.

8

and

9).

Furthermore, a region of marked

density variation (eddying flow) is visible in the main flow. This nonuniformity

is in particular, visible in the interferogram.

2.2 Outline of Investigatiofl

An

apparent step in investigating this configuration is to take

a photograph of the shock before it reaches the corner. The shadowgraph of

Fig.

4

(at the same conditions used in Ref. 1), clearly shows the three-shock

configuration already to exist in front of the corner. It is seen that part

of the original shock wave is highly distorted and the contact surface behind

it becomes unstable and turns into a vortex street. In the following sections

we will refer to the different waves in the manner outlined in Fig.

5.

(See

also list of notations).

Once ~he nature of the phenomenon was accurately assessed, the

following key-points inevitably had to be treated.

(a) What is the mechanism of generation of the three-shock

configura-tion? This is part of the more general problem of the interaction of a shock

wave with constriction. Section

3

deals with these topics. Some irregular

Mach reflection features, analoguous but slightly different from the observations

in CO₂ by Gvozdeva et al (Ref. 10) are given in Appendix A. The runs that

are dealing with these topics are tabulated in Table I.

(b) Given the disturbance, how will it decay?

An

adequate answer

requires that a correlation be found of runs under widely varying conditions

of Mach number and initial pressure. Section

4

considers these problems in

some detail, based on the runs listed in Table 11.

(c) Finally, the next observations require an assessment of their

influence, if any, on the results of Ref. 1, as welT as a more general discussion

on the use of a liner as a means of generating expansion flow$. The relevant

2.3 Instrumentation

2.3.1 Shock Tube Facility

The UTIAS

4

in. x

7

in. Hypersonic Shock Tube, an overall view of which is seen in Fig.

6,

has been described in detail in Ref. 11, including design, construction, and performance. Of particular interest is the geometry of the liner, intended to provide the expansion configuration of Ref. l(Fig. 2). This

4

in. x 1-1/2 in aluminum liner is approximately 19 ft. long and is

clamped rigidly to the upper tube wall. The leading edge startssome 27 ft.

(64

hydraulic diameters) past the diaphragm station.

Under all circumstances the shock shapes on arrival at the lead-ing edge were found to be plane, with slight convex curvature due to the

boundary layer. Because of the limiting resolution of the optical system the detailed shock shapes could not be obtained although all deviations from a plane wave were found to be less than 0.5 mmo

Further liner characteristics are as follows: The leading edge had a 250 wedge; for the expansion work, the downstream end provided a 150 corner. In most of the work reported here, the straight portion of the liner was extended some 2 ft. past the test section. Shock detectors (Atlantic Research type LD-25) , usual~y one foot apart, were mounted in the liner at the observation station which allowed shock speed measurement with a 1.5% accuracy.

For the Mach reflection studies two much shorter models were used (with 150 _{and 25}0 _{leading edges) mounted in the test section. Here too,} gauge stations were provided (Fig.

7).

2.3.2 Flow Visualization

The interferograms were made with the Mach-Zehnder interfero-meter described in Ref. 12. Flow and no-flow pictures had to be made to correct for fringe curvature.

An

interference filter peaked at 5200

R

was placed ahead of the camera to provide monochromatic light.

By blanking off the compensating chamber, the same instrument was used for shadowgraph studies. A simple defocussing of the camera outside the test section gave the desired result. Relatively high initial pressures were used that allowed good sensitivity.

Use was made of the exploding wire light source (Ref. 13) in conjunction with a Kerr Cell shutter (Electro-Optical Instruments, Inc. Model WNT). The 200 nanosecond opening time is sufficient to "freeze" all wave configurations • Kodak Royal X Pan film sheets were used for recording. The shape measurements were done on a Cambridge Universal travel~ing Microscope and ~he obtained total spatial resolution was of the order of 0.05 mmo

Finally, an overall schematic of the electronics involved is shown in Fig.

8.

3.

SHOCK WAVE DIFFRACTION OVER CoMrnESSION CORNERS: MECHANISM FOR GENERATING

THREE-SHOCK CONFIGURATIONS

In this section we try to describe the overall flow configuration

resulting from the interaction of a shock wave with a constriction. The

literature on this topic can be divided in three major categories.

(a) Diffraction of a shock wave at a compression corner. Here the

phenomenon of shock reflection, especially Mach reflection, has received

particular attention.

(b) Flow field in a shock tube resulting af ter the passage of the

incident shock through the constriction. The treatment has been mostly

one-dimensional and it has been found that substantial knowledge can be gained

from this type of analysis, mainly with respect to the transmitted shock

strength.

Cc) Detailed configuration of the transmitted shock front itself. This research is fairly new and it was stimulated by the work of both Whitham

(Ref. 14) and very recently, Bowman (Ref.

3).

The investigations reported here are hoped to contribute to the

understanding of some of the problems mentioned. The diffraction of shock

waves ranging from MI = 5.0 to 12.0 over the compression corner are found to

yield an interesting irregular Mach reflection phenomenon, the detailed

des-cription of which can be found in Appendix A.

In the following subsections we will review briefly some of the

previous theoretical and experimental work, and use this knowledge to describe

the flow field behind the shock. The shock shape itself will be investigated

in Section

4.

3.1 Interaction of a Shock Wave with a Constriction

Shock-wave diffraction over compression corners has been studied

by several authors with varying interest in the details of the interaction.

Laporte (Ref. 15) treats the convergence as an instantaneous area change and

computes the resulting shock strength far downstream. This method does not

predict the details of the flow that actually occurs.

In Ref.

6,

the calculation of the strengthening of the incident

shock is based on the unsteady one-dimensional theory of Whitham (Ref.

14).

It is shown that the shock emerges from the convergence with a strength given

by this theory. The shock strength subsequently is attenuated to the asymptotic

value calculated by Laporte. The results of both theories for inviscid gas

flow are plotted in Fig.

9.

For an area convergence ratio of

0.79

which

corresponds to our configuration, the strength increases by some

4%.

The plot

in Fig. 10, taken from Ref. 1, however does not show a significant difference

in performance with and without liner. This surprising result cannot be

attributed to an error in measuring the Mach number (approximately 1.5% or a

possible error in measuring diaphragm pressure ratio (5%). The latter would

have a minor influence on the logarithmic plot shown in Fig. 10. However,

the increase of viscous attenuation in the smaller channel could account for

this fact, as can be seen from the expression of the relative change in Mach

•

(3.1)

where, C(M) is a function of M (plotted in Ref. 6), dl and d₂ are the greater

~d smaller diameters respectively and 6M is the decrease in Mach number M

due to viscous attenuation in the respective channe1. Applied to our

con-figuration, a change of 2.5% is found, thus leaving a 1.5% discrepancy, which is the order of the experimental error.

pppenheim et al (Ref. 16) give an extended account of the

possible one-dimensio~a1 wave systems involved. They calculate vector polars

for shock waves and rarefaction waves and flow polars relating isentropic

flow canditions through area changes. As such they obtain a wide application

for the familiar (u,p) plane analysis of one-dimensional unsteady gasdynamics. Some of the possible wave systems that can resu1t from the interaction of a shock with a convergence of area ratio Re are shown in Fig. 11. Also

repre-sented there . is the derived wave diagram and extended (u,p) plane under our

standard working conditions. The flow adjusts initially through a steady supersonic deceleration at the constriction followed by an unsteady expansion wave which is swept downstream. Since the net result of the constriction is an increase in shock Mach number, and hence of partiele velocity, the accelera-tion caused by the expansion fan must be greater than the steady deceleraaccelera-tion.

The (p,u) plot illustrates these facts. Since in supersonic flow the conditions

of isentropic flow cannot be satisfied at the constriction, due to the

appear-ance of transverse waves starting at the leading edge, a slight modification

is required. The overa1i effects will be a decrease of the amount by which the rarefaction fan is swept downstream in the time space domain, and a shift

to higher pressures and lower partiele velocities as outlined in Section 3.2.3.

Investigations into the detailed interaction were mostly made

in connection with Mach reflection studies. There are some good review

papers, such as those by Pack (Ref. 17) and by Skews (Ref. 18). We will

e1aborate on three of them that illustrate how an initially unsteady

two-dimensional flow builds up two-two-dimensional steady configurations. It is also

shown that this fits into the one-dimensional analysis.

In a paper on detached shock waves, Griffith (Ref. 19) deals

with the transition from Mach reflection to the two-dimensiona1 steady solution

consisting of a detached shock wave at constant stand-off distance. A

particular case is shown in Fig. 12, corresponding to an area ratio Re

=

0.926.

It follows from Oppenheim's analysis (and a subsequent comment by Rudinger (Ref. 20)) that a slightly smaller ratio would have resulted in the detached

shock moving upstream. This shock is the reflected wave obtained in the

one-dimensional analysis .

Kahane et al (Ref. 21) analysed the complete flow field in the vicini\y of an area change for subsonic flow conditions behind the incident

shock. Their paper provides a good illustration of how the flow adjusts it-se1f to the one-dimensional requirements by creating a reflected shock (with a train of transverse waves behind it). Figure 13 shows the time history of

the process. With increasing time an essentia~y steady flow is gradually

Finally, we ment.ion the work by Bird (Ref. 22) on the

reinforce-ment of a shock wave passing through aconverging channel. A series of streak schlieren records are produced showing the growth of a reflected shock as the envelope of the (x,t) characteristics. These pictures can be compared with the

theoretical predicts by Friedman (Ref. 23) and Rosciszewski (Ref. 24). Figure

14 corresponds to a Mach 2.55 shock hi tting an area constriction Re

=

0.101.

3.2 Actual Flow --Conditions in a Constriction

The flow field becomes very complicated as the shock travels

down the constricted tube. The shock front itself is first diffracted and then

continuously shaped by transverse waves. The path of the intersection of these

transverse waves with the main shock (shock-shock) will prove to be very

im-portant in connection with the stability problem (Sec. 4). In this subsection,

we go briefly over these aspects of the problem.

3.2.1 Initial Mach Reflection

The shadowgraph of Fig. 15 was taken at a distance of 10.5 inches

past the liner leading edge (see inset for overall configuration). The shock

Mach number is MI

=

11.6 and Pl

=

20 torr, in oxygen. The picture is remarkable in many ways and its main features will be dealt with in more detail in

Appendices A and B. The appearance of a second triple-point S is the

character-istic featbU'e of irregular Mach reflection. The very pronounced forward

dis-placement of the shock front near the wall is due to the interaction of the

shock with its self-induced vorticity as can also be seen on Fig. B2. Another

peculiar.ity is the turbulent nature of the originally rolled-up slipstream.

This is thought to be triggered by the interaction of the slipstream with the

corner. The termination of the shock SD at the contact surface also poses an

interesting problem regarding its strength, as it requires a flow adjustment

in order to satisfy the gasdynamical requirement of zero pressure difference

across the slipstream. Lastly, the appearance of the circularly-shaped shock

in the flow field behind the main shock results from the expansion of the Mach

stem over the convex corner of the leading edge. lts function is similar to the shocks in Figs. 38 and

39,

that is, to provide the matching conditions of

the steady-unsteady flow.

It is important to note the following geometrically important

features:

(a) Existence of a three-shock configuration, which creates very

pronounced distortion of the main shock fron .

(b) Triple point moving along a ray inclined at approximately 31

degrees.

The motion of this triple point and the decay of the shock dis-turbance connected to it will be our main concern.

3.2.2

Motion of Triple Point

The three-shock structure will move along the tube and on its way it undergoes. multiple reflections, one. of which .is SLl.OWtl in F.ig. 16. Here

we see a very early stage of the process. The full trajectory of the triple

p0int can be found from Whitham's theory (Ref. 14). Here, information about

area changes is communicated to the shock front i~ the form ofkinematic waves,

that are carriers of changes in shape ~d Mach number of the shock. These waves can catch up to form a discontinuity (a shock-shock) or can spread out

in-definitely. 'l;he anälogy ,,,i th plane waves in gasdynamics is obvious. For

in-stance, when the shock strikes the compression corner, it is diffracted and

carries, as a result, a shock-shock (triple point) along. In the subsequent

expansion, a gradually spreading expansive wave i's communicat,ed to the shock

front. The subsequent interaction of these waves, as well as the effect of

tbeir reflection off the tube walls will cause their decay and restore the

stable plane shock configuration. The: problem itself is amenable to solution

by the method of characteristics, as indicated in Fig. 17 together with the

laws of reflection of a shock-shock. These were indicated in Ref.

25

and it

is shown that the reflecti0rt is not specular, the reflected angle being

smaller then the incident one.

The generated wave system is shown in Fig.

19

.

As in all

supersonic flows, the attached shock and the two-dimensional exp~Dsion fan

interact finally re&Ultip~ in a weakening of the waves till they are of zero

strength and have become Mach waves. (We investigate this case in more detail

in the Rext paragraph.) Likewise, the strength of the kinematic waves will

decrease,' resulting in a decreased shock front disturbance •. In turn this will

mean a weaker transverse wave and as shown in Appendix D, a smaller shock-shock

angle w. This is consistent with our observations. As noted before, the

triple point starts off under under a 31 degree angle for a shock Mach number MI

=

12.0 and Pl

=

20 torr. At 20 ft from the leading edge, this angle is

found to be 17.70 • This value was obtained from a least square fit based on

seven runs, represented in Fig. 16. Qne could point out that it would have

been advantageous to have had the use of a high-speed camera. This w0uld have

cut down considerably the number of runs.

On

the other hand, however, the

scatter gives some idea of the repeatability and the possible phase shift in connection with the shock front oscillations. Finally,we can note that for an infinitesimaldisturbance,w will have a value of 16.10 as shown in Appendix Do

3.2.3

Flow Field Behind Diffracting Shock Wave

When a shock, MI

=

12, moving into oxygen at p,

=

20 torr, hits

a constricyion, the asymptotic one-dimensional solution is as shown in Fig. lla,

which was expalined in Sec. 3.2010 Since this analysis was based on isentropic

flow through the constriction, the presence of transverse waves starting at

the leading edge will require some modifications. The flow Mach number in

region

3

is decreased from the theoretical value of M

=

2092

to a computed

value of 2.42, due to a decrease in the particle velocity and an increase in

temperature both brought about by the transition through the attached shock

wave system. Since the overall result of the constriction will still cause

a strengthening of the transmitted shock, it can easily be seen that requires

a stronger rarefaction wave than the one found by the theoretical analysis.

The computation referred to was performed by means of the method of character

to 1.15 over a distance of about 3 ft. The shock inclinations approach more

and more the Mach angle, which varies from ~ = 19.5 degrees at M

=

3.0 to

~

=

24.5 degrees at M

=

2.42.

The overall flow picture as found by patching together the

different flow fields discussed above, is sketched in Fig. 20. While the shock

moves along in the tube, it sheds transverse waves in the flow, which adjust themselves gradually to the requirements of the two-dimensional steady flow

that eventually will be established. Superimposed on the transverse wave

train one has the rarefaction wave R, the presence of which was explained

be-fore. The influence of this simple wave is by no means hampered by the

existence of these transverse waves. This is similar to the formation of a shock

wave af ter the bursting of the diaphragm by the catching up of a series of

com-pression pulses. In this case the compression pulse passes through the

trans-verse waves attached to the disturbed shock front. The asymptotic behaviour

sketched in Fig. 20(c) needs some clarification. The strength of the

trans-verse wave at the triple point is determined by the time history of the

kine-matic waves on the shock front. The asymptotic two-dimensional steady flow

on the other hand will ~e characterised by its own transverse wave strength brought aboût by the history of the interaction of the gas dynamical waves connected to the leading edge. This possibly could suggest a loosening of the ties between the shock front and its wave train as time progresses, as

indi-cated by the dotted line.

The early arrival of the contact surface (see Appendix C)

illus-trated in (d) will strengthen the shock wave system attached to the leading edge

behind it. However, due to heat transfer, the flow picture in front of it

could also be ffiodified (Ref. 26 and 27). Lastly, the boundary layer under our

running conditions in oxygen can be as thick as 0.4 of the radius (see Appendix C). It lvill constitute a dissipative region for the transverse waves and will

If spread out" these shocks (Ref. 28). (See subsection 4.4.). The boundary layer

will also have an influence on the curv~ture of the main shock as well (see

section 4).

4. SHOCK CURVATURE PRODUCED BY TRANSVERSE WAVES OR THE BOUNDARY LAYER

Liepmann and Bowman (Refs. 2 and 3) were probably the first to

distinguish two regimes where the shock shape is determined by two completely

different mechanisms, which will be outlined briefly in this section.

4.1 Effects of Transverse Waves on Curvature and Stability of Shock Waves

It is known that a shock wave as the property of rapid

adjust-ment to changing geometrical boundary conditions (e.g. Ref. 29). The term

If shock stabilitylf in this regard refers to the phenomenon of equalization of

shock curvature, which is obtained through weakening of the transverse waves

generated at the area changes. Lapworth (Ref. 30) and more recently Bowman

(Ref. 3), initiated the,experimental investigations. Bowman's experimental setup is shown in Fig. 21. An initially plane shock hits an axisymmetric

roof top model and the resulting shock shapes are recorded. All runs were at constant Mach number, MI

=

4.81, but under varying initial pressures. The key conclusions are as follows:

.(1),

layer "- see all initial

Tbe, shock shapes (af ter subtraction of curvature due to boundary

below) .immediately downstream of the disturbance are identical for

pressures (e.g., see Fig. 22 for Pl

=

100 m torr and

3

tarr in argon) .

(2) The rate of decay is independent of the initial pressure. The

magnitude of the disturbance

(ç)

is expressed by the displacement of the point

at the tube center with respect to the one at r

=

.896

R. A sign is allotted depending on the curvature of the shock at the centre (forward is posi ti ve) .

The points 1, 2 and

3

of Fig. 21 correspond to these on Fig. 22. The ampli-tude decay, ~ig. 22, is equally valid for

3

torr as for 100 m torr, and shows

the decay rate of the envelope to be minus one half. The equation of this envelope is given by

A(x + a)-n (4.1)

where n is the decay rate, a fixes the decay origin and A is found from the

initial amplitude. This decay origin is a purely mathematical concept, and

does not have any physical meaning (mathematically the disturbance should be

infinite there}. The value of n is found to be minus one half, and the shock

front itself is seen to oscillate at a fairly constant wavelength.

(3) However, at some distance along the tube, an abrupt change in

decay is noted, which brings the perturbation down to a residual value (see

Fig. 22) comparable to that produced by the boundary layer (see 4.2). The

point of oneset of this decay is pressure dependent, and Bowman proposes the

following experimental rule of thumb.

0.6

Pl R2 _-

<

L "t _{crl -}

< 2.3

Pl R2 (4.2)

where Lcrit gives the distance of the shock past the disturbance (in mm), Pl

\he initial pressure (in torr) and

R

the tube diameter (in mm). At this

loca-tion the periodic behaviour seems to be lost. Equation (4.2) therefare

stipu-lates the limit between the transverse wave regime and the boundary layer

regime. The presumed basis of this rule stems from the following. The

boundary layer is thoughtof as a fictitious wall closing in on the transverse

wave system, shortening the wavelength of reflection until the waves are

com-pletely "choked off". Bowman is thus led to equation (4.1) by stating that

Lcrit is the distance for which the extent of the boundary layer, characterised

by the displacement thickness

( L

)~

5 -

-- 10 Pl

(4.3)

reaches about 1/4 to 1/2 of the tube radius.

to a laminar boundary layer in air or argon.

Equation

(4.3)

roughly corresponds

4.2 Influence of Boundary Layer on Shock Curvature

- Survey of our present knowledge of unsteady boundary layers

is given in Appendix C, where we also deal with the effects of the boundary layer on the present experiments.

In shock-fixed coordinates, (see Fig.

23

for notation) the

d5*

v = _{u e dx}

(4.4)

that acts like a sink and hence decelerates and curves the shock. The analyses

initiated by Hartunian (Ref. 4) and extended by DeBoer (Ref. 5) consist in

solving a potential flow problem, the general approach of which is now briefly

outlined for the case of a shock moving near a wall.

(1 The flow behind a slightly perturbed normal shock can, to

first order, be considered to be vorticity free. Rence a potential solution

can be looked for. Putt-ing

u(x,y)

=

~ (x,y) and v(x,y)

=

~ (x,y)

x y

u and v being the perturbation velocities, and ~ the potential, the latter will

have to satisfy the equation for small perturbations

(1 - M _e 2)- ~ _xx + ~ _yy =

°

with x

>

_{x sh}

y>o

(4.6)

The boundary conditions for the problem are:

v(x,o) is given by equation

(4.4)

u(o,y) = 0, due to the assumption of a small

perturbation of the shock front.

(3) For v(o,y), the following expression can be derived from an

appropriate order analysis of Fig.

24,

v(o,y)

Since E ₌

_d:Y

dxsh , one finds

dxsh

=

dy

(4.8)

The solution of the potential problem provides an expression for v(O,y) that

can be ~sed to integrate Equation

(4.7)

and hence obtain the shock shape.

De Boer applied a similar scheme to a shock moving between two parallel walls

and in a cirGUlar tube. Ris main conclusions can be summarized as follows:

(a) Total axial extent of the shock ( e) .

eL =

x

_L h

(~)~ M"-~

_Re

c

₁ for laminar boundary layers

(4.'9)

1 2w-l

eT

=

1C.r

h

(~e)2

M-

5 for turbulent boundary layers

eis the extend of the shock at the centre with respect to the foot of the wall,

X is a factor dependent on geometry, and at most

3%

different from unit,

h is the half distance between the parallel walls, or the tube radius,

Re is the Reynolds number based on the sound speed al and h,

w is the exponent in the viscosity temperature dependence (= 0.7). 2w-l

1

The values of

CL~-2 and~TM

5 are plotted in Fig.25. From Fig. 25, it can

2w-l

1

---be seen that

CL~-2

and C_TM 5are for M> 4 almost independent of shock Mach.

As aresult, 8 in equation 49 proves to be only dependent on the non-dimensional

quantity Re. The initial pressure hence is the only important parameter, a

result which is very surprising indeed. 8/h is plotted in Fig. 26. As will be

shown below, the laminar portion of ~he boundary layer will have a dominant

influence. At high Reynolds numbers, transition of the boundary layer will occur early and hence the influence of the turbulent part will start to show up. However, increasing Reynolds number will also imply transition to the transverse wave regime, thus invalidating the theory. At low Reynolds number peculiarities at the foot of the shock also have an increasing influence.

(b) An important result from the computations is that the shock

curvature is mainly determined by the boundary layer in the region_immediately

behind the shock. For example, the contribution from the region x > h is less

thaq 1%. Under our working conditions, the Reynolds number ranges from

4 xll0

5

to 105 for Pl

=

2 to 50 ~orr. Assuming the values for oxygen to be

close to those for air, the expected shock shape distortions due to boundary

layer are varying from 0.6 to 0.1 mmo As already mentioned in 2.3.1

measure-ments of the shock shapes before arrival at the disturbing leading edge showed axial extents of this order of magnitude.

4.3 Correlation of Experimental Runs in Oxygen

In this subsection, we try to explain and correlate the shock shapes as found on a series of shadowgraphs in oxygen, taken at two test sections 81 and 82 (Fig. 2). 81 is the interval from 9 to 19 inches past the

leading edge, and 82 extends over the width of the test section (9 inches)

situated at 20 ft or 55 hydraulic diameters past the disturbance. The shapes

were measured with a Cambridge Universal travelling microscope. It should also

be noted that the shock tube was rectangular (4 in x 7 in) and the disturbances were two-dimensional.

4.3.1 Interpretation of Measurements

It is important to elaborate on the basis of the interpretation. As shown before, the shock shape is oscillating with a variable wavelength

determined by a cycle of two reflections of the shock-shock off the walls

(angle w). For instance, for MI

=

12, the wavelength is approximately 3 ft,

and over this distance the shock shape goes through a sequence of stages

sketched in Fig. 27. This would suggest the following interpretation:

(1) As a quantity representative of the disturbance we choose the

axial displacement ~ between the points where the wave meets the walls.

(2) The sign is positive if the shock intersection at the upper wall

is ahead of that of the lower wall (see Fig. 27). It sho~ld be noted here that

this convention loses all validity in the boundary layer regime. Use in the

of w on the Mach number and its cumulative effect along the tube will result in a phaseshif~of the shock shape (at a given distance) or, to put it in another way, the triple point will trace a different path over the test view.

This is shownin Fig. 28 which is a qualitative sketch (e.g. in that it does

not differentiate the variable w's) but with some experimental backing. This

figure shows the inBtance that in an interval such as UV one can expect re-flections to occur of triple points connected to shock waves of 8.5 ~ MI ~ 9.2

and 13 ~ MI ~ 15. Possibly one also could have rjflections related to MI's of

say 6 or 20. It proves that the oscillati0n curve for MI = 9 has over the distanee

from the leading edge to the test section station 82, one more period than the

one for MI = 13.5. On the other hand, since the reflection points for the

shocks in the Mach number rarige mentioned will fall in the test view (which

does not happen for a MI = 12 shock as seen in Fig. 28), we will be able to

measure directly their positive peak values Smax' For shocks of about MI

=

11

one could measure the negative Smax.

A problem then arises if one also wants to know the peak values

of cases for which no reflection occurs in the test view, the Mach 12 case

being a typical example. There, we fall back on similarity properties,

assum-ing that the growing part of the shape proceeds in a self-similar fashion from the last point of reflection. In~Fig. 27 for instanee we assume the pnrtion of

the shock front below the line

RR'

to grow self-similarly starting from R.

The extents of that part of the shock (e.g. a and b) will be proportional to

the distance R and Smax will correspond to the extent achieved at R', the next

reflection point on the opposi te walL 4.3.2 Influence of Initial Pressure

Confirming Bowman's result, the shock shapes in the transverse

wave regime are independent of the intial pressure. This can be seen from Fig. 29, where a comparison is made of shock fron~s corresponding to MI = 11.6

and p~

=

20 and 2.75 torr, immediately af ter the shocks passed over the

com-presslon corner. At a distanèe of 20 ft downstream, the same property still holds. This c~be seen from Fig. 30 which gives the superposition of two

shapes of MI

=

9.2 shocks into oxygen at 37.5 and 55 torr respectively. Figure

31 is a plot of three shock shapes at Mach number MI

=

12.2 and Pl =20,14 and

6 torr. The horizontal scale has been enlarged ten times. In the case of 6

torr, on can see that the discontinuity has disappeared. The shock-shock is

no longer localized, but its influence is still equally strong as proven by

the equality in axial extent.

4.3.3 Decay of Disturbances

1. Rate of Decay

All

evaluation of the decay rate was not attempted, since this

would involve a series of measurements along the tube, instead of the two

stations \-EEd in the present èxperiments. Equ.ation 4.1 suggests that at least

three points should be known to evaluate the three parameters determining the

decay envelope. 8ince, on the other hand, even the peaks of the periodic

.curve do not ne.cessarily lie on this envelope (see the first peak on Fig. 21,

which is the one of the two we measured), any attempts to evaluate n must

involve a more extensive series of runs. This does however not exclude some

stations Sl and S2. The shock shapes backing up the following arguments are

shown in Fig. 32. None of them have been corrected for boundary layer

curva-ture although a typical bou~dary-layer curved shock is also drawn for

compari-son. The six poin~s of Fig. 33 give the peak values ~max close to stations Sl

and S2' Over the same distance, t~e disturbance is reduced by a factor of 10

for the weaker case (which is close to Bowman's standard conditions of M - 4.81),

and only ~ 2 to 3 in the stronger cases. Based on these experiments alone,

it is impossible to determine whether this marked decrease in stability at the

higher Mach numbers is due to an actual change in decay rate, as s~ggested in

some theoretical investigations (e.g. Ref. 31) or that it is only due to a

shift in decay origin with cons~ant decay rate. The results of Ref. 3 favour the l~ter interpretation. The decreased stability is also clearly illustrated

in ~ig. 34, where a comparison is made of three shock shapes at station S2 at

initial Mach numbers MI

=

5.1, 10.2 and 1?8 all moving into 10 torr of oxygen.

From all this, it can be poncluded the Mach number depeBdence of the shock

perturbation is important.

2. Accelerated Decay

In-Ref. 3, a~ accelerated decay was observed at a distance

directly proportional ~o the ~nitial pressure (see Sec. 4.1). It then becomes

of interest to see if the same behaviour could be ascertai~ed from the high

Mach number runs under investigation. For this purpose, we plot the peak

values closest to the test section S (obtained according to the method

out-lined in Sec. 4.3.1) versus Mach

num~er

.

From this we hope to discern whether

Lcrit is smaller or greater than 20 ft. Whereas the investigations of Ref. 3

were done at constant initial pressure but varying lengths, the method rsed

here kept L cons~an~, but varied the initial pressures. The results are

plotted in Fig. 35. Most of the runs could be located in a narrow band,

apparently corresponding to the non-decayed transverse wave regime. Hence,

it seems to be established that no abnormal decay has yet oc~urred at S2 for

all Pl ~ 5 torr. At no time was the boundary layer regime reached. To do so

would have required initial pressures of the order of 1 torr, and in turn would

give fringe shifts too small for a sensitive investigation of the expansion

flow. It should be kept in mind that this indded was the primary interest of

the geometrical configuration. It seems reasona~le to conclude that a region

with greater than normal decay occurs in a relatively small pressure range

4.5 1

It is worthwhile noting \hat Bowman's r~le of thumb would give (see Eq. 4.2)

2.7 > Pl> 0.8

4.4 Some Commen~s on the Results

Whitham theory, as described in 3.2.2, treats the decay of

the disturbances as a phenomenon localized on the shock front, consisting of

shock-shock ~d expansion wave interaction. As a result transverse waves are

created which provide the means for the flow readjustment tothe changed

con-figuration. One could paraphrase this by saying that the geometry conveys to

the shock all necessary information required to build up ~he new flow field

imposed by the new geometry. To explain an abnormal decay (i.e. beyond the

the shock front. The boundary layer is such a mechanism as notedin Ref.

3.

The argument should ~e along these lines: the boundary layer will in fact

change the geometry of the tube, and this new configuration will require another

two-dimensional flow field, which in turn imposes an adaptation of the triple

point, thereby changing the strength of the transverse waves, their inclinations

and deflecting influence and finally the shock-shock angle. A gradually de-creasing decay rate is ~hus predicted, which finally ends in an abrupt change the moment closure is achieved (see Sec. 4.1). Unfortunately, the observation

of the sole shock front in our iRvestigation will never yield the type of information required to really understand the mechanism which is going on.

This fact reduces many of the possible explanations to pure speculatiorr. Therefore, we will just state some points which could be relevant:

1. Concerning the boundary layer, it is felt that the turJulent or laminar

nature should modify the results especially because of their largely differing

boundary layer thicknesses and condit~ons for possible closure. (See Appendix C.) No closure is for instance expe~ted to occur under our working conditions. On the other hand, some additional experimental data are needed to assess the shock - boundary layer interaction, the effective dense part in the latter

being very small even for thick boundary layers.

2. Another activity of the bourtdary layer is more localized and consists in the readjustment of c~vature. It can indeed be expected that the curving

mechanism (see Sec.

4.2)

would be presènt under all circumstances and would

gradually take over as the tranBverse wave-induced curvature decreases. One

should recall that only a very small part of the boundary layer is respo~si­ ble and this action rightly could be considered as localized to the shock front affected by the boundary layer.

In conclusion, one can say that to settle these points an

extensive amou.1t o~ experimental work has yet to be done, which would involve

the study of the erttire flow nORuniformity behind a shock wave. Such a thorough

check would requife taking pictures of both the shock and the reg ion behind it

at different positions down the tube, facilitated by using high-speed laser

photograph (Ref.

32).

To do this, one also would have to provide a more

flexible way of changing the distance from the disturbanee to the test-section.

The combination of high Mach numbers and low pressure should give ideal test

cas s, since there the difference between the transverse and boundary:layer regimes is very pronounced as can be se en from Fig.

35.

Another way of investigating the transverse waves in particu-lar would ~e the use of streak-schlieren photograph, (successfully applied in

Ref.

33),

at least, if by increasing the density, one could achieve the re-quired sensitivity. It should bé mentioned here that in Ref.

33

one can

probably find the first available data on the behaviour of transverse waves associated with the diaphragm opening in a shock tube.

5.

USE OF ALINER TO GENERATE -CORNER-EXPANSION FLOWS 5.1 Some Requirements

Like a shock wave, a corner-expanSiOl'.l wave provides one of the

essential features of high-temperature gasdynamics. The experimentalist will

mainly be concerned with creating the optimum research conditions for investi-gating such a wave flow and here he will quite of ten ge faced with conflicting

requirements. Some of the requirements

encounter

~

d

ih the expansion problem

are as folwws:

1. What is the best method for generating a corner-expansion flow? If one

has to start with a straight. tube, two possibilities come to mind: a liner

and a wedge model. The choice between them will depend on the actual flow

conditions.

2. The wedge model is more advantageous in that lts boundary layer could be

considered to be steady during its interaction with the corner. The boundary

layer is also relat.ively thin çompared to the wall model, which has an ever

growing bo~dary layer.

3.

Measurable ,.characteristic relaxation distances of non-equilibrium flows

as well as the min~mum fringe shifts imposed by optical resolution usually

reqvire high Mach numbers and relatively high initial pressures (especially

for argon) (Refs.

34

and 35~. The pro~lems that arise are gas dependent.

For oxygen: A marked decrease in stability at the higher Mach numbers was

noted above. This is illustrated again in Fig.

36,

which is a schematic,

freely interpreted extension of Fig.

32,

assuming a decay rate of minus orre

half. It shows that only for Mach number MI

<

5 will the shape afthe disturbance

reach on its arrival at Station S2 the distvrbance level due to boundary layer

curvature. At the higher Mach numhers as shown in Fig.

35

only for initial

pressures smaller than say

4

torr could one hope to benefit from some

accelerated decay phenomenon. Under all other conditions an assessment of

"Ehe created flow nonuniformity is required. Since w , the shock-shock

angle, is dependent both on the initial Mach number and the disttlrbing geometry,

a plot as given in Fig.

36

is only vali-d for o~ specific configuration or

geometrically similar one(when a similarity factor is required}. The Mach

number dependence however, should be qualitatively valid for all configurations.

For argon: we will present below some results that show greater stability in

argon. This is a very fortunate situation, since relatively high initial

pressures, and Mach numbers below MI

=

14,

are recommended in order to avoid

non-equilibrium co~ditions upstream of the corner due to radiation effects

(Ref.

36).

5.2 Nonuniformities i~ Oxygen

The three-shockrconfiguration will shed vorticity into the flow

(see Fig. 3) concen}rated as a vortex sheet at first, but gradually spreading

out. 'l'his vorticity will move with the part-icle velocity and hence will show

up periodically in the flow. For example, in Fig.

3,

C2 is the contact surface

created by the three-shock system during its downwards motion before the last reflection at the wall.

In order to assess q~alitatively the inf}uence of the contact

surface , one can comp'ute i ts densi ty jump at the trip1..e point. Figure Dl gives this density variation for the three-shock configuration of the incident shock

right at the ramp of the leading edge. For a MI

=

11.7 shock moving into 20 torr the density difference is 5.5 x 10-5 gjcm

3

corresponding to a non-dimensional

fringe shift of 2.0 (~or À = 5200~). This value corresponds to the maximum ever achieved~ since the strength of the transverse wave will subsequently

decay. In Fig. 39 the fringe shift is seen to be approximately 0.6 af ter 20 ft. ,

i.e. at the test section S2. Hence, the measured values of density given in

Ref. 1 (corresponding to fringe shifts of 8 to 9) should rtot be influenced by

the nonliniformity introduced by the contact surface. Their presence, however,

will WL~ecessarily increase the effective starting time of the expansion.

Another aspect of the perturbing role is due to the dependence of the wavelength

of oscillation~ on Mach number (Fig. 28). It could happen that the reflection

may occur right at or just over the ~orner, in which case both the contact

sur-faces and the transverse shock wave will perturb the flow.

5.3 NonuJiformities in Argon

The number of runs in argon were few, and the method used

con-sisted in fact in a comparison with the oxygen case. The interferograms of the

exprulsion process show hardly arry nonurriformity under the optimum running

con-ditions put forward in Ref. 35. Their small magnitude puts them at the

non-uniformity level that is expected in any real shock tube (Ref. 26). The

shadowgraphs of the shock shapes tend ~o show that this is due to a higher

stabilityof shocks moving into argon (shift in decay origin).

The shock shapes for MI

=

7.8 at Stations Sl and S~, are shown

in Fig. 37. When the values of ~max are irrtroduced in Fig. 36, they are seen

to lie on the MI =

5

curve for oxygen. A direct juxtaposition of the actual

flows in argon and oxygen is made in Fig. 38 and 39. The shadowgraphs of Fig.

38 correspond to a shock of MI

=

13.2 in argon and a MI

=

12 shock in oxygen.

It is seen that the familiar discontinuous three-shock configuration is absent

in argon under conditions for which it is prominent in oxygen. The largely

decreased flow non-uniformity can be seen from Fig. 39, where the two

inter-ferograms are related to a MI

=

1~.8 shock in argon and a MI

=

12.4 shock in

oxygen.

Since no significant interference shows up at S2 for Mach

numbers up to MI

=

16, the argon expansion work could certainly be done with the existing liner.

6.

CONCLUSIONS

The dynamical behaviour of shock waves at high Mach nuffibers in

dissociated oxygen in a sequence consisting of a diffraction over a compression

corner, the subsequent motion through a constant area tube, and a final

diffrac-tion over an expansion corner is described in some detail. This type of

con-figuration was brought about by the geometrical requirement of creating a

corner-expansion flow by means of a liner.

1. The diffraction over a compression corner was found to yield

modification of the flow pattern through the internal energy properties connect-ed with the high density ratios obtained. A significant dependence on the gas

properties was noted (Appendix Ao). Another peculiarity observed consisted in the interactfun:of the shock induced vorticity (localized in a rol led up vortex0

and the shock front itself. An explanation is put forward in Appendix B.

?

On

its arrival in the constant area tube, the shock fron~ is strongly perturbed and this disturbance will subsequently decay to the stable

plane configuration. However, it was found th at a very pronounced decrease of stability (monitored by the gas properties) occurs at high Mach numbers. This

is illustrated by a slow decrease of the axial extent of the shock shape. If the same behaviour holds for the shock perturbations generated by rupturing a diaphragm in a shock tube the present observations should also be helpful in analysing that problem. Some evidence was obtained in support of the observa-tions by Bowman (Eef. 3) on an accelerated decay of the shock shape disturbances. Unfortunately, compared to that investigation, the present setup lacked the

flexibility to obtain the necessary spatial resolution. However, it follows from the experiments that for our Mach number range, the disturbance level for the pransverse wave regime is at least one order of magnitude greater than for the so-called boundary layer regime, thus suggesting the prospect of a more

dramatic illustration of the transit ion phenomenon. The mechanism of transition itself is not yet fully understood. A knowledge of the entire flow field be-hind the shock seems to be required. A description based on the overall shock

tube literature was presented, but also here more experiments are needed.

3.

No deviation from the classical diffractianpattern over expansion corners was noted. Our main concern were expansion flows, where the above

mentioned shock disturbances had not yet fully decayed on arrival at the expan -sion corner, resulting in a residual disturbing diffraction pattern at the shock front. This was known to have been the case in some of the oxygen expansion runs of Ref. 1. lhe main conclusions with respect to the possible repercussions on the measurements of Ref. 1 are:

(a)

(b)

(c)

An increase of the liner length would have reduced the distur-bances hy only a very small amount. ~or instance one could only decrease the disturbance by a ratio of 1.4 while doubling the liner length (as originally suggested).

A qualitative assessment of the nonuniformity created by the shedding of vorticity into the flow shows the influence to be smalle

It was also found that for future work in argon (using the same

method) the possible disturbances were of the order of the non-uniformity flow level expected in any real shock tube and could therefore be ignored.

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