Some aspects of shock-wave research

January 1986

SOME ASPECTS OF SHOCK-WAVE RESEARCH

by

..

4

M~&~i

1986

TECHN!SCtlE HOGESCHOOL DELft LUCHTVAAHT -~i. i:'JI:'-.! fE\'AARTIEClt lEW.

smUOTHi:EK .

I. I.

Glass

Kluyverweg 1 - DEu:T

THIS IS

·

THE "AIAA DRYDEN LECTURE IN RESEARCH"

AIAA-86-0306

DELIVERED AT THE 24th AIAA AEROSPACE SCIENCES MEETING

RENO, NEVADA, JANUARY 7, 1986

UTIAS Review No. 48

CN ISSN 0082-5247

sm1E ASPECTS OF SHOCK-WAVE RESEARCH

by

I. 1. Glass

THIS IS THE "AIM DRYDEN LECTURE IN RESEARCH"

AIAA-86-0306

DELIVERED AT THE 24th AIAA AEROSPACE SCIENCES MEETING

RENO, NEVADA, JANUARY 7, 1986

Submitted November 1985

January 1986

UTIAS Review No. 48

•

Acknowledgements

It is a pleasure to acknowledge the fine research conducted over the years on pseudostationary oblique shock-wave reflections by my former Master's and Ph.D. candidates: R. R. Weynants, C. K. Law, G. Ben-Dor, S. Ando, R. L. Deschambault, T. C. J. Hu, J. Wheeler, J. Urbanowicz, and Research Associates, J.-H. Lee and M. Shirouzu. The present paper could not have been written without their direct and indirect assistlnce. I thank Prof. J. P. Sislian for a critical reading of my manuscript. I owe a particular debt of thanks to Dr. H. t~. Glaz, Dr. P. Colella and their associates for thei r excellent numerical simulations of our experiments. The encouragement and support received over the years from Dr. M. J. Salkind, Dr. G. W. Ullrich, Dr. J. D. Wilson and Dr. A. L. Kuhl is appreciated with thanks.

The careful typi ng of the manuscri pt by Eil een Moffitt and Wi nifred Dillon, the fine figures drawn by Laura Quintero and the careful reproduction of this paper by Carlos Basdeo are very much appreciated.

The fi nanci al ass i stance received from the U. S. Defence Nucl ear Agency under DNA Contract 001-85-C-0368 from the U.S. Air Force Office under Grant AF-AFOSR 82-0096 and from the Canadian Natural Science and Engineering Research Council, is acknowledged with thanks.

Finally, I wish to thank the AIAA for bestowing the honour of presenting the Dryden Lecture on my University, my Institute and me •

Abstract

A few examples are given of shock-wave phenomena on earth and in space

to provide some useful background material. The major portion of the paper

is devoted to a specific shock-wave research problem, namely,

pseudostationary oblique shock-wave reflections in perfect and imperfect

gases. Consideration is given to what has been achieved to date by using

two- and th ree-shock theory to predict what type of reflection results when a planar shock wave Ms, in a shock tube, collides with a sharp compressive wedge of angle, 9w• Experimental (interferometric and other optical) data

are presented in _{(Ms , 9w)-plots for argon, nitrogen, oxygen, air,}

carbon-dioxide, Freon 12 and sulfurhexafluoride, in order to check the

val idity of the analytically predicted regions . and transition lines of the

four types of reflection (RR, SMR, CMR, DMR). Some disagreements are noted

and di scussed. Our interferometri c i sopycni c data are al so compared with

state-of-the-art computational results from a solution of the inviscid Euler

equations using a CRAY I computer. Good agreement was obtained, yet, it

would be important to obtain new data by solving the Navier-Stokes

equations, as well as the rate equations for imperfect-gas excitations, in order to judge the improvement obtained with real-flow interferograms.

TABLE OF CONTENTS

Acknowledgements •

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Abstract •••

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Nomencl ature ••

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1.0 INTROOUCTION •

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2.0 SPECIFIC SHOCK-WAVE RESEARCH PROBLEM - PSEUDOSTATIONARY OBlIQUE

2 3 5 7

SHOCK-WAVE REFLECTIONS • • •

•• • • • • • • • • • • • • • ••

10

2.1 Analysis •

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₁₁ I

2.2 Comparison with Experimental Results in the (Ms' 9w or

9w)-Plane. . • • • . • . • • . • . • • . . • . • . • 16

2.3 Critique •

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₂₄

2.3.1 Comparison of Experimental and Nurnerical Values of

ö

24

2.3.2 Effects of Mach Stem Curvature • • • • • • • • • •

26

2.3.3 Effect of Slipstream Thickness • • • • • • • • • • • ••

27

2.3.4 Effect of Shock-Induced Boundary Layer on the Wedge

Surface •

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3.0 FLOW-FIELD SOLUTIONS • • • • • • • •

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4.0 APPLICATION TO HEIGHT-OF-BURST SHOCK TRAJECTORIES

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5.0 CONCLUSIONS

6.0 REFERENCES •

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APPENDIX A:

ADDITIONAL (Ms,

9W)

AND (Ms,

~)-PLOTS

.

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.

29

33

33

34

a C e • _01R DMR E h "" H I K M, ti' '" t12 t1R Ms P p r r~1 R R, RI RR S, SI SMR T Nomenclature speed of sound wedge corner

specific internal energy complex-Mach reflection double-Mach reflection

.!.

(u2 _{+ V 2) + e, total specific energy} 2

height of burst (HOS), distance between explosive charge centre and ground surface, flow enthalpy

.!.

(u

2 ₊

v

_{2) + h, pseudostationary total enthalpy}

2

incident shock wave in shock tube kink in CMR refl ected waveR first and second Mach stem in DMR

'"

(u

2 ₊

v

2_{)/a2 , self-sirrdlar Mach number (M)}

Mach reflection (SMR, CMR, DMR, TOMR) incident shock wave (1) Mach number static pressure

reflection point; point on RR ~ CMR and CMR ~ DMR lines meet

radial distance of blast-wave front from charge centre radial distance between zero ground range and first triple point

gas constant

first and second reflected shock waves in DMR regular reflection

first and second slipstreams in DMR single-Mach reflection

TD~lR T, TI t U -û' V-u, v

W

Wo

x,

Xl X, _Y VT a y 6 6* e: Tl ed e_2m e_w el w l; <P

x,

Xl P

terminal double-Mach reflection where Xl

=

0 fi rst and second tri p 1 e poi nts i n D~1R

time

(u, v) velocity field

u - l; pseudostationary velocity component in x-direction v - Tl pseudostationary velocity component in y-direction

velocity components in the X and y directions weight of a TNT charge in kg

standard weight of 1 kg TNT

ground range for (HOB) RR and MR, respectively Cartesian coordinates

height of first triple point in (HOB)

velocity-deflection angle in slipstream shear layer specific heat ratio

angle between incident (I) and reflected (R) shock waves boundary-layer displacement-thickness

Mach-stem curv ature angl e

(y - yo)/(t - to) pseudostationary coordinate

the boundary-layer displacement-effect on the difference in the flow angles of the incident and reflected shock waves flow detachment angle

wedge angle

effective wedge angle (e~

=

e

_w + X)

(x - x 0)/ (t - t 0) pseudostati onary coordi nate wave angl e

first and second triple-point trajectory angles flow density

1.0 INTRODUCTION

Shock-wave phenomena on earthl-3 are common and important occurrences in our 1ives. For examp1e, we hear them from the crad1e to the grave as terrori zi ng1y crashi ng or rumb1 i ng thunder fo" owi ng 1 i ghtni ng di scharges

(F " 1 g. 1 • ) 4-7 It is estimated that 1000 thunderstorms occur on earth at any

moment. In the primordia1 state of our earth such shock waves are credited as being enormous1y more efficient in creating the building and rep1icating b10cks of 1ife such as ATP, RTP and DNA,8,2 than the radiation from the sun (Fig. 2). Vet, paradoxically enough, all 1ife is threatened today by the radiation and the very shock waves that might be un1eashed by a thermonuc1ear war (Fig. 3).9-11

The lethal overpressures and winds induced by shock waves are now we11 understood,12,13 even though our eyes cannot see the shock waves generated by 1ightning, firearms or exp10sions in air, since air is a transparent gas. Special optica1 methods (interferometry, sch1ieren and shadow photography) must be used in order to make shock waves visib1e.14 The more intense the exp10sion or rapidity and size of energy release, the higher is the velocity of the resu1ting shock wave, and its attendant increases in pressure, density and flow velocity, and the greater is the possib1e devastation. The shock wave becomes 1ess destructive with time or distance as it compresses and heats the enve10ped air, thereby dissipating its energy, unti1 it becomes a harm1ess sound wave.12-14

At sea level where the mean-free-path is small (6 .6x10-6 cm), the

thickness of the shock 'front is about tenfo1d this va1ue. Consequently, it is quite incredib1e how rapid1y the shock wave changes from its quiescent ambient conditions in front, to pne of destruction behind it through Jllo1ecular collisions. In outer space, where the mean-free-path is very

1 a rge, the shock thi ckness can be thousands of kilometers and the vari ous changes through it are much more gradual.

As i de from the 1 i ghtni ng -thunder process, shock waves on earth are also generated by intense volcanic eruptions and meteor impact. The recent eruption of Washington's Mt. St. Helens is estimated to have had an energy release equivalent to 50 megatons of TNT or about 3000 type bombs used to destroy Hiroshima. The eruption of the Krakatoa volcano in 1883 in Indonesia is estimated to have had an energy release of 5000 megatons of TNT or 100 times as powerful as Mt. St. Helens. Perhaps the greatest eruption in recorded history occurred in Tambora, Indonesia, in 1815, with an energy rel ease of about tenfold th at of Krakatoa. The airborne ash caused "a year without a summer" in 1816 and 12,000 people were killed.15

If a meteor of the size that produced the Arizona Crater ever impacted in a big city, it could be as devastating as a multimegaton bomb,16 as shown in Fig. 4. An energy release of 5 megatons TNT equivalent is estimated to have been released on impact a few 10,000 years ago. Vet, there is geological evidence of craters formed millions of years ago from an energy release of 50 million megatons of TNT, caused by asteroid impact. Such asteroids orbit the sun between Mars and Jupiter. A few orbit the sun to pass inside the orbit of the earth. However, not all are known. It is estimated that perhaps 800 ± 300 such Apollo asteroids, larger than 1 km, may remain undetected. An impact would release an energy of about 100,000 megatons of TNT and leave a crater 20 km wide. As a matter of fact a news

item [Washington (Reuter) 14.2.81J reports that NASA urged the U.S. government to organize "Project Spacewatch" to keep track of such asteroids and meteors, and if necessary to send spacecraft armed with hydrogen bombs to deflect those that may menace to impact the earth.

--- -- -- -

-Vet, we need not associate shock waves only with destruction. Our modern industrial society could not have been built without the use of explosives that produce shock waves. Explosives are used in building highways, canals, tunnels, harbours, railroads and subways; in mining, agriculture, engineering and in the space programs for precise timing, activating and cutting requirements.1 Recently shock waves produced by a high voltage discharge and 5-10 mg of lead azide (PbN₆) have been applied

successfully to medicine in the breaking up of large kidney stones (Fig. 5) and bladder stones to relieve suffering and possibly death.17,18 More recently lasers have been used as an energy source for the same purpose with some promising results.19 It is estimated that 50,000 successful operations have been performed using the spark-discharge machine described in Ref. 17, which is quite expensive. The cost factor (several million dollars) is important for use in the Thi rd Worl d, where the occurrence of such stones are more common. It partly motivated the work in Refs. 18 and 19 to reduce costs by a factor of 10, at least. Basically, the three systems are similar in that the energy discharge (spark, explosive, laser) takes place at the focus of an ellipsoidal mirror and is refocused 'at the second focus located on the kidney stone, as a powerful implosion wave, which breaks up the obstructing stone. It may take many shots to break up a stone. In the case of bladder stones, the explosive is used directly against the stone.

In space, the best example of shock-wave phenomena is the magnetosphere1 flow about planets in our solar system. Our earth is an excellent example (Fig. 6). It becomes a spherical-flow model in asolar wind at hypersonic spe~ds (M ~ 13). The solar wind is composed of ionized hydrogen, helium, some other heavier nucleons and electrons. This plasma is electrically neutral. It blows with a superson;c velocity of 400 km/sec with a bulk number density of about 10 particles/cm3 _{having an average}

kinetic temperature of about 1Q5K. As shown in the sketch the interaction of the ionized particles with the earth's dipole magnetic field distorts it into a streamlined shape with the magnetopause as a boundary, that is, where the ionized gas can no longer penetrate the earth's magnetic field. The co 11 i s i on 1 es s bow shock in front of the 1 i ne of symmet ry is about 30 ea rth radii from the earth's centre. The stagnation point on the magnetopause is about 20 earth radii from the earth's centre on this 1ine. The stand-off distance of the bow shock from the stagnation point is about 10 earth radii

(about 64,000 km). The bow shock itself is many thousands of kilometers thick. The subsonic plasma behind the bow shock probably has a tenfo1d increase in temperature to about 106K. Near the earth the magnetopause cav ity contai ns the famous Van All en radi at i on be1ts.' The magnetopause cavity resembles a comet 'and the tail may extend several million kilometers downwind.

Similar f10ws can exist around a star as shown in Fig. 7. A recent relevant study of this phenomenon is given in Ref. 20. Of the many shock-wave phenomena in space, one of the most interesting is the existence of shock waves in the spiral arms of galaxies21 such as our own Milky Way. Densities can increase by about an order of magnitude making possible the birth of young blue stars as shown in Figs. 8a and 8b.

2.0 SPECIFIC SHOCK-WAVE RESEARCH PROBLEM - PSEUDOSTATIONARY OBLIQUE SHOCK-WAVE REFLECTIONS

Having out1ined very briefly some of the shock-wave phenomena on earth and in space, it is usefu1 to turn to a particu1ar research prob1em, first investigated by E. Mach in 1878,22 and has since occupied many scientists

and engineers and where mY students and I have made some worthwhile contributions over the past decade. The problem is known as "pseudostationary oblique shock-wave reflections". Although much more is known about this problem now than when Mach started it all, there is still a great dea 1 to 1 ea rn. It i s of interest for two mai n reasons , fi rst, the problem involves a great deal of gasdynamics of perfect and imperfect flows (frozen and equilibrium, inviscid and viscous); second, our experimental (interferometric) shock-tube data have provided computational fluid dynamicists, who are working on explosion dynamics, with the means of checking their numerical simulations for accuracy. The latter data provides engineers with pressure, velocity and other flow properties required to build explosion-resistant structures. Such simulations have proved to be extremely good to date considering that the numeri cal solutions solve only the inviscid Euler equations, whereas the experimental data comes from flows which are viscous and can have real-gas effects as well.

2.1 Analysis

Unlike linear waves encountered in acoustics and light, shock waves are nonlinear. For example, the reflected wave angle is different from the angle of incidence and the physical quantities across shock waves such as pressure, density, temperature and flow velocity are highly nonlinear. The simplest method of studying pseudostationary oblique shock-wave reflections is in a shock tube. Here a planar shock wave Ms collides with a sharp compressive wedge of fixed angle

Sw,

and gives rise to regular reflection (RR) and three types of Mach reflections (MR), namely, single (SMR), complex (CMR) and double ~1ach reflection (DMR), as shown in Figs. 9 and

la.

The various symbols are defined on the figures. It is seen that RR consists of

two shock waves meeting at the reflection point P. However, MR consists of three shock waves meeting at the triple point T. It has been shown 23- 25 that such f10ws are self-simi1ar (look the same with time except for an increased sca1e). Consequent1y, the Mach stem grows with time (or distance) a10ng the triple-point trajectory ang1e X. It mayalso be thought of that this ang1e represents an effective wa11 for the incident shock wave I to ref1ect from and produce a ref1ected shock wave R. Th is prov i des the subsequent use of the effective wedge ang1e 9~ = 9_w + X as a usefu1 parameter.

It is seen that regu1ar ref1ection consists of an incident shock wave I, and a ref1 ected shock wave R, whi ch may be attached at the wedge corner C, or detached from it. The stand-off distance s, divided by the distance L, which the incident shock wave I has travelled from the corner, remains constant. The ref1ected shock wave is curved if the flow behind it is subsonic and it may have a straight portion near the ref1ection point if the flow there is supersonic and uniform (see Figs. 9a and 11a).

Mach ref1ection, on the other hand, consists of an incident I, and ref1ected shock wave R, a Mach stem M, and a slipstream S. The three shocks meet at the triple point T, and the slipstream S separates stat es 2 and 3, even though they have the same pressure and flow inc1ination. SMR has a cu rv ed ref1 ected wave. CMR has a kink in the ref1ected wave, which is caused by an interacting compression wave. In DMR the kink becomes sharp and a second Mach shock MI is formed that fina11y extends to the slipstream S. As a result, a second triple point TI, and a second slipstream SI, are a1so formed. In this case the ref1ected portion R is straight and the flow behind it is supersonic. The ref1ected shock R in SMR and RI in all other Mach ref1ection (MR) cases can a1so be attached or detached (see Figs. 9b to 9d and llb to lld).

The question arises, given the initia1 conditions of the gas Po' T 0' y,

the shock Mach number r1s and the wedge angl e

9w,

can one predi ct, a priori, the result i ng ref1 eet i on? I

! The answer is yes. This can be obtained from the following ana1ysis.

It was noted above that such two-dimensiona1 wedge f10ws are se1f-simi1ar. Consequent1y, instead of three independent variables x, y and t, the refl eet i on phenomenon can be descri bed in terms of two independent variables x/t and y/t. The reflection point P, in regu1ar ref1ection, and the triple point T, in ~1ach ref1ection, are chosen as specific points moving at constant velocity with respect to the wedge corner C. 8y attaching frames of reference to these poi nts, the ref1 eet i ons become pseudo-steady (Fig. 9). Consequent1y, the steady-f10w equations of motion can be app1ied to each shock wave in turn such that:

Cont i nu ity : Pl·U l·

=

p.u.sin(~. - S.) J J 1 J (1) Tangentia1 Momentum: p. tan~.

₌

_{Pj tan( ~j}

-

_Sj) 1 1 (2) Normal Momentum: + 2 . 2~ _{Pj +} 2 . 2( _Sj) Pi Piui Sln i

=

p.U. _{J J} Sln ~.

-1 (3) Energy: h i + { u 12

sin2~i =

h _J .

+iu.2sin2(~.

J 1

-

Sj) (4)

where Pi is the flow dens ity of the i nit i al state i n front of the shock wave and Pj is the flow dens ity in the fi na 1 state beh i nd the shock wave, the same for the flow velocity u, the pressure p, the specific flow entha1py h, the wave angle ~, and the flow deflection ang1e S. An equation of state

p

=

pRT (5)

; s al so used for a perfect gas or p

=

p( p, T) for an imperfect gas. In addition the boundary condition for regular reflection must be used so that the flow is parallel to the wedge surface (Fig. 9)

(6)

For Mach reflection, across the contact surface or slipstream, the pressure must be continuous

(7)

and the flow deflections must also be continuous or

(8)

It has been shown by von Neumann26,27 and verified by many experiment-alists that in pseudo-stationary flow a regular reflection wil 1 undergo transition to a Mach reflection when 9₂

=

9_2m, that is, when the wedge angle

9w

produces the maximum possible flow deflection 9_2m• This angle is also known as the flow-detachment angle. For further details see Refs.

14 and 26-28.

As noted above, Mach reflection (Fig. 9) can in turn be divided into SMR, CMR and m1R; and DMR can be subdivided into four subtypes (Fig. 10), depending on whether or not the second triple point trajecory angle Xl lies above, ;s equal to, or is below the first triple point trajectory angle

x.

Finally, there is the terminal double Mach relection (TDMR) case where

Xl = O. That is, the second triple point lies on the wedge surface. The second Mach shock disappears and the reflected shock wave reflects as a regular reflection. Such reflections occur readily in gases with low values of y, such as Freon-1225 or SF_{6 •}28 They are not possible in perfect monatomic, diatomic or triatomic gases, although they are possible when these gases are imperfect and have a low effective y (see Figs. 10d and 1228) •

It is assumed that Sr·1R becomes a CMR when the flow with respect to the first triple point T becomes sonic, th at is, t1 2T

=

1. This assumption is based on experimental observation and computational simulation.29-35 The same heuristic approach is taken for CMR ++ DMR transition and the condition is applied to the kink or second triple point K or TI or M2K

=

1. A change of curvature occurs in the reflected wave R, which steepens into a sharp second triple point TI. This marks the formation of the second t1ach shock MI, and a straight reflected shock R, before W, and a curved reflected shock RI, af ter M'. The flow behind R is supersonic. It should be noted that MI grows in strength and length until it comes close to the slipstream S. Ideally it should terminate there. Then it raises a problem how the contact front could remain stable with a sharp pressure gradient applied to it. Interferometrically, the Mach shock appears to go around the slipstream as a compression wave.36-39 This is also verified by numerical simulation of the interferograms in question for DMR in different gases. 33 ,34

If the above equations and conditions are applied and solved, it is possible to obtain a plot i'n the (M

s' 9w or

9~)_Plane28

delineating the various regions of RR, SMR, CMR and DMR and their transition lines as shown in Figs. 13 and 14, for example, for perfect and imperfect air in vibrational, dissociational and ionizational equilibrium, over a very large initial shock Mach number range 1 < Ms < 20. It can be seen that for

Ms > 10 the curves do not change for perfect air,40 but undulate for equilibrium air as new degrees of freedom are excited.

By assuming that the Mach stem is straight and perpendicular to the wedge surface, from the triple point T, it is possible to obtain an expression for the triple-point trajectory angle x,41 as well as the second

triple-point trajectory angle xl

•42 Unfortunately, the Mach stem is usually

curved. It is concave at the lower shock Mach number Ms' for significant

stem heights, and convex43 at higher

~1ach

number. This introduces an error

in determining X and Xl, especially when X is small.43

I

2.2 Comparison with Experimental Results in the (M_s' 9

w or 9w)-Plane

The i nterferometri c, schl i eren and shadowgram data from the Institute

for Aerospace Studies were obtained in the 10 cm x 18 cm Hypervelocity Shock

Tube. The major instrument for this purpose was a 23-cm diam. field-of-view

~'ach-Zehnder i nterferometer with a gi ant-pul se Q-switched ruby laser with a frequency doubler at 6943A and 3471.5A, having an exposure time of 15 ns. The latter was short enough to photograph the strongest shock wave without blurring.

Figures 15-19 show comparisons of the analytical domains and transition b oun arl es or d . f RR SMR , , CMR an d DMR Wl . th experlmen s 1n argon, . t · 31 oxygen, 41 ,

nitrogen,30 air,36 carbon dioxide 37 and sulfurhexafluoride.38 In addition,

the detailed results obtained by Ikui et a125 for air, CO₂ and Freon 12 are

also shown and discussed.

An examination of Fig. 15a shows the experimental results for RR and MR

in Ar on an (M_{s , ew)-plane.} In our early experiments the major effort

analysis. As more experience was gathered, it became clear that the various transition lines were the sensitive lines to be tested. Consequently, the various analytical regimes look reasonably well validated. Three OMR runs

(1 at 50° and 2 at 40°) lie in the CMR regime. All the other experiments are in their appropriate regimes. It has been found 43 that along with M2T

=

1 for SMR ~ CMR, another condition that must be satisfied is

Ö > 90°. On all graphs, the dashed line represents Ö = 90° and its lower branch lies above the solid line for SMR ~ _{... CMR where M2T}

=

1. The upper branch of the dashed 1 i ne 1 i es bel ow the sol i d 1 i ne, where r~ 2T ;. 1, yet. It can be seen th at two points at

9w

=

40° and 20° are improved to lie in CMR.

When these results are plotted on an _{(Ms , e~)-plane [Fig.} 15(b)], which is more accurate since arelation for X is not required, paradoxically, a poorer agreement is obtained for Ar. Now four DMR runs, one at ew

=

50° and three at

9w

~ 45°, definitely lie in the CMR regime and two other CMR are borderline runs; one CMR run lies in the S~1R regime at Ms ~ 4, all other runs are in their appropriate regimes. Since no runs were made along the RR ... MR transition line, little can be said about the "von Neumann paradox" (where RR persists into the ~1R regime) or possible boundary-layer effects. No real-gas effects occur in Ar for the pressures and Mach numbers of these experi ment s, as the rel axat i on 1 engths are too large and the gas remains frozen with a y

=

5/3.

Figure 16(a) and (b) show the results of N2 and O2 in the (Ms , 9w) and _{(Ms ,} e~)-planes. In Fig 16(a) the six RR runs at the RR ...

t.m

transition are only marginally below it. All runs except a DMR at Ms

=

6 1 i e inthei r appropri ate regimes. The 1 i ne for ö

=

90° is very hel pful in this case, as all the SMR runs lie in their appropriate regime. The same comments apply to Fig. 16(b). It is seen th at as y decreases the NR and

SMR ++ CMR lines lie closer together preventing a clear-cut distinction of the experiments.

Figures 16(c) and (d) show the results for air on (Ms , Sw) and (M

s '

e~)_Planes.36

It is seen from Fig. 16(c) that at the RR

~

.... MR transition line two RR runs lie in the SMR regime; one RR run lies in the CMR regime and seven RR runs lie in the DMR regime. This definitely shows the persi stence of RR into the MR regime. It will be shown subsequently that this is due to the shock-induced boundary layer on the wedge surface. It i s al so known that the CMR ++ DMR 1 i ne must cu", e and meet the SMR H · CMR line at point P on the RR ~ .... MR line, as the distance between the two triple poi nts TT I goes to zero there. As a consequence, the two m1R runs in CMR

now 1 i e in the appropri ate regime. Real-gas effects shown by the equilibrium dashed lines do not explain the results, since the runs were done at different initial pressures. Eight D~1R runs lie in the CMR regime and three SMR runs lie in the CMR regime. Consequently, it appears that better criteria are required for the RR .... MR, SMR .... CMR and CMR + DMR

transition lines. Similar remarks apply to Fig. 16(d). Overall the ö

=

900

line is not helpful in this case.

Figure 16(e) includes the detailed results of Ikui et al for air. It is seen that excellent agreement is obtained for RR at the RR .... MR transition line. For Ms > 2 the runs lie marginally below it. One RR run lies in the SMR regime. Eight S~'R lie in the CMR regime. All CMR and [}.1R runs lie in their appropriate regimes. The perfect gas (y

=

7/5) transition lines appear quite appropriate for their results. No significant evidence of the persistence of RR into MR regimes can be seen. The dashed line for

ö

=

900 is not helpful in this case but a new SMR .... CMR line closer to the

F i g ure s 17 ( a) t 0 ( f ) de al wit h CO 2 i n t h e ( M s , 9w) 0 r

I

Ow) -pl anes. F i g u re 17 ( a ) shows the experiments for transition

lines based on a perfect gas, y

=

1.29. Although such lines are in error, since the vibrational relaxation lengths are short, and an equilibrium flow is appropriate, nevertheless, excellent agreement is obtained, by and large, for all points except two DMR at ~~s ~ 3.5 in the CMR regime and one at Ms ~ 4.7. One CMR at Ms ~ 4 agrees better with the ~~2T

=

1 criteri on, and the ö

= 90° is not helpful in this case. The regular reflection runs at

9w

= 50° are not a test of the RR -+ t~R transition, as they should have been done much closer to or below that line. Consequently, little can be sai dabout boundary-layer effects in thi s case. Somewhat better agreement is obtained in the (Ms, ~)-plane shownin Fig. 17(b).

The correct equilibrium transition is shown in Fig. 17(c). Here the frozen transition lines (y

=

7/5) are also shown for comparison. The agreement is quite good for the RR -+ MR 1 i ne for the frozen case and poor for the rest of the frozen lines. Eight CMR runs lie in the equilibrium Dt1R regime and two S~·1R runs lie in the CMR region. In this particular case the ö = 90° line puts the SMR point at 10° in its proper place. It can be seen

from Fig. 17(a) th at the agreement there is accidentally better.

Figure 17(d) shows an (Ms, ~)-plot of the experimental runs in equilibrium CO 2• Although this is a more accurate plot, since X does not

have to be known, the results are poorer, since nine CMR find themselves in the Dt1R reg i on and one S~1R occu rs in the CMR reg i me • All the DMR runs a re correctly located as well as all the RR runs. Again, the various transition lines crowd together near

9w

=

20°.

Figure 17(e) shows the detailed results of Ikui et a1 25 in the (M_s, 9_{w)-plane. It is clear that there is no agreement w;th CO 2 treated}

very good agreement with the experiments over the enti re Mach number range. Only two RR lie below the RR ~ MR line in the SMR region. There are three Dt1R runs lying in the RR region above the RR + MR line. There is no

evidence of RR persistence or of viscous boundary-layer effects. Four St1R runs at t-_1s

=

2 lie marginally in the CMR region; eleven CMR lie in the D~1R

regime. Sometimes it is difficult to distinguish a CMR from a DMR and perhaps this might account for some of the discrepancy. In this case the ö = 90° line makes it possible for several SMR runs at

Ow

= 10° to lie in their proper regime.

Figure 17(f) shows the results of Ikui et a125 plotted on the (Ms , ew)-plane with their analytical lines for a perfect gas (y

=

1.31). It is seen th at good agreement is obtained with all experimental runs byand large. The RR + MR line does not fit the results as well as for the equilibrium flow (Fig. 17e) since fourteen RR runs lie in the D~1R region. Only two RR runs lie in the SMR regime at Ms ~ 1.5 below the RR + MR line

and two RR in the CMR region. One borderline DMR run lies in the CMR regime at

9w

= 20°. The ~12T = 1 line puts six CMR in the proper regime, unlike

ö

=

90°; the above results are analogous to those of Ando (Fig. 17a).37 Figure 18(a) shows the experimental results for Freon 12, from Ikui et a1 25 plotted on an (M , e )-plane, for their transition lines based on a

s

w

perfect gas (y = 1.141). Again the perfect-gas lines give reasonable agreement. Fou r RR runs 1 i e in the CMR regi on. Many Dt1R runs 1 i e in the CMR regime. About four SMR lie marginally in the Ct1R region and two Ct1R

runs lie in the DMR regime. The dashed line is an experimental best fit for the RR + TDMR. Unfortunately, Ikui et al did not extend the TDMR runs to

higher Ms and lower

9w

to provide a better picture of the TDt~R regime, which can be predicted analytically (see Fig. 18b). Only in the CMR region is there evidence of the persistence of RR. An equilibrium (Ms _, e)

w-..

plot is shown in Fig. 18(b).* It also includes the transition lines for a frozen flow with y

=

4/3, for which the agreement with experiment is poor. This time, more DMR runs lie in the RR region including the TOHR runs. Note that the condition Xl

=

0 only yields a small mt1R region below the RR +-+- t-1R

1 i ne. Consequently, one can only suspect that the detachment criteri on of 9₂ = 9_{2m due to von Neumann is invalid for pseudostationary flow.} The equilibrium SMR ++ CMR is not improved from the one in Fig. 18a, nor does

the criterion ö

=

90° help. The CMR +-+- OMR line does not improve overall

the 1 ocat i on of the runs. Some CMR runs are poorer located and some OMR runs are better located in their respective regions.

Figures 19(a) and (b) show the results for SF₆38 in the (M

s' 9w) and Considerable effort was made in this case to check the transition 1 i nes themselv es. Fi gure 19( a) shows the results for frozen (y

=

4/3) and equilibrium flows. The frozen-flow lines do not represent the experi ments. Two RR runs and two OMR runs 1 i e in the CMR regi on bel ow the RR -+- t~R transition lines. Six RR runs lie right on the RR -+- MR transition

line. There is no serious evidence of RR persistence up to Ms ~ 6.5 and two mt~R at Ms ~ 8. One CMR run lies in the SMR region and five DMR lie in the CMR regime. The M2T

=

1, rather than ö

=

90°, puts five CMR runs in thei r proper place. However, by and large the agreement is good. Similar remarks can be made for the (Ms , ~) -plot of Fi g. 19(b). Here, however, it is clear th at OMR persists into the RR regime over the ent i re range of Ms· Consequently, the _{(Ms ,} 9~) -plot gives poorer rather than better agreement with the RR -+- MR transition line. It should be noted

that if th ree-shock theory was used here rather than two, then X would have

* I am grateful to Dr. J.-H. Lee for computing the analytical curves for Freon-12.

a value from 1° to 3° over the RR ~ MR line which would be pushed upward for better agreement with experiments than the present line. The smal 1 dashed line running from CMR ~ DMR to point P on the RR ~ MR transition line indicates the fact that in actuality this takes place. It makes for better agreement in Figs. 19(a) and (b) in th at the DMR runs at Ms ~ 2 lie in their correct region. Again, the transition lines are too close, especially at 9w ~ 10°, in Fig. 19(b).

Figure 19(c) shows the SF₆ results in the (Ms, Sw)-plane for a perfect gas (y

=

1.093). The experiments are quite well represented on this plot. Two RR runs lie below the RR +-+ MR transition line in the CMR region

and four below in the DMR regime at Ms ~ 2, and four below at Ms ,... 4, showing a persistence of RR. Several DMR lie in the CMR region at 9w ~ 45°, 43°, 40°, 20°, and especially at 10°. All CMR lie properly in thei r regi on at or bel ow the M2T = 1 1 i ne, except two in the SMR regi on at Ms ~ 3.5. The equilibrium plot of Fig. 19(a) is superior since the RR ~

MR line lies lower for lower y than for the fixed (y

=

1.093) line of Fig. 19(c). The SMR ~ CMR and CMR ~ DMR lines are also better in Fig. 19(a) for the same reason, prov id i ng better agreement with experiment overall for the equilibrium plot.

Figure 19(d) provides the (Ms, 9~)-plot for y

=

1.093. It is not as good for overall agreement as Fig. 19(b). Here three DMR lie in the CMR region at Ms ~ 3.5 and several at Ms ~ 2. These would lie correctly in the DMR region if the CMR ~ DMR line were curved to join the SMR ~ CMR line at the RR ~ MR line at point P, as noted previously. On the other hand, several DMR and the two TDMR lie in the RR region. Overall, the equilibrium curves of Fig. 19(b) provide better agreement.

Figure 20(a} shows the extent of the TDMR* over a range of 'Y as well as for vibrational equil i bri urn Freon-12* and SF6 •

t

It is seen that the regi ons are bounded by the two-shock theory RR ~ ~1R _{transition line and the line}

along which Xl

₌

_{O. The region for a perfect gas with y}

₌

_{1.10 is very} small, and occurs for Ms > 9 and

9w -

38°. For a perfect gas with y

=

1.093, the region is much larger and occurs for Ms > 7.5 and

9w

> 37°. For vibrational equilibrium Freon-12 the region is l!Iuch larger still and starts at Ms - 7. For y

=

1.08, the TDMR region is quite large and starts at Ms - 5. Note th at unlike Ikui et al results shO\'m in Fig. 18(a), the

TD~1R experi ment s are supposed to 1 i e in the DMR regi on, not in the RR regime, as they found. It is not clear why this has happened. Even if the th ree-shock theory RR -+- MR 1 i ne was used and X I was i n error by one or two degrees it would not account for this larger discrepancy as can be seen from Fig. 20(b). One can only speculate th at the RR +-+- MR transition line is

not appropriate for pseudostationary flows.

Figure 20(b) shows the regions of TDMR for vibrational equilibrium SF_{6 ,} y

=

1.06 and y

=

1.04. It is seen that the regions become progressively larger. It should be possible to observe TDMR in vibrational equilibrium SF₆ at Ms - 5. Ikui et a1 25 observed TDMR in Freon-12 at Ms -3.8 and

9w '"

46°, and the present analysis shows this to occur at Ms - 7 and

9w '"

37°. Consequently, · this problem wil 1 require a resolution in the nea r futu re.

In summary, it can be stated that the (Ms , ew)-plots provide the experimenter with good engineering results for all gases. Whether a point lies in one region or another sometimes depends on personal judgment when *,t I am grateful to Mr. Masao Shirouzu and Dr.

J.-H.

Lee for computing these curv es, respect iv ely.

the result is not clear cut. Consequently, it is more important to make meaSlJrements of the angles 0, X and w' which are quantitative experimental results, and compare them with their counterparts from analysis using two and three-shock theories, as applicable, in order to provide better tests for the state of the gas frozen or equilibrium. The induced boundary layer causes 0 to be larger and w' to be smaller. Consequently, one can learn a good deal about the boundary layer itself from such measurements. The angle X can be affected by the fact that the inclination of the slipstream S depends on the two velocities on either si de of the ideal contact surface, one in state (2) moving at a higher velocity than that in state (3). Consequently, the velocities in state (2) and in state (3) diverge due to the layer thickness, giving rise to a deflection, a, which could affect X, as well as the relation 9_{1 -} 9₂ = 9_{3 ,} used in the three-shock theory.

2.3 Critique

It was shown above that many experiments in various gases have now been performed on regular and Mach reflections of oblique shock waves in pseudostationary flow. The experimental agreement with the analytical boundari es for such refl ect i ons usi ng two- and three-shock theori es are reasonab 1 e but not preci se enough over the ent i re range of i nci dent shock wave Mach numbers (Ms) and compression wedge angle (Sw). In order to improve the agreement, the assumptions and criteria employed in the analysis were critically examined using the foregoing experimental data.43

Several criteria were proposed to predict analytically the transition boundaries between the various reflections. However, the first complete transition solution in the (Ms, _{9w) or} (t~s,

Sw)

• -pl ane usi ng

two-and three-shock theories two-and the relevant criteria to obtain the transition

30 31

1 i nes were given by Ben Oor and Glass for N₂ and Ar, Ando and Glass for CO₂,37 Lee and Glass 28 for perfect and equillibrium air up to

~\

= 20, Hu

and Glass38 for SF₆ and for Freon-1225 in the present paper.

2.3.1 Comparison of Experimental and Numerical Values of ö

It was noted above that by measuring the angles, ö, X ;J.nd w', it is

possible to obtain quantitative checks with analysis as well as inferring the state of the gas, whether it was frozen or in equi 1 i bri urn. A mi nimum finite length of 1 mm is required to measure the slope of a shock wave from an interferogram. The measured angle is then the average slope within this length. The vibrational relaxation length

"v

at Po

=

15 torr and T 0 = 300 K becomes 1 mm at Ms '" 8 for O_{2 ,} Ms > 10 for N_{2 ,} Ms '" 5 for

CO₂ and Ms ... 2 for SF₆ (Fig. 21). In two and three shock-wave systems for regular and Mach reflections the relaxation lengths do not vary significantly compared to a 102_-107 _{variation in}

_J.v

_{for 2 < r~s < 10}

(Fig. 21). Consequently, the Mach number at which

"v

=

1 mm behind each shock wave in Mach reflection does not differ very much. Therefore, the choice of 1 llITl as a characteristic flow length is quite reasonable. When

the relaxation length is of the same order as the characteristic length, the simplified two- and three-shock theories can no longer be used. It is therefore reasonabl e to apply a frozen

("v »

1 mn) flow or an equil i bri urn

(~ « 1 1lITl) flow assumption. Only vibrational excitation need be

considered in the present experiments as dissociation, electronic excitation and ionization relaxation lengths are much longer than 1 mm, in the present shock Mach number and pressure ranges of the experiments. Ai r with 21% O₂

neg 1 i 9 i b 1 e f 0 r ~1s < 10. The specific heat ratio for frozen-gas calculations are 5/3 for Ar, 7/5 for N2, 02' air and CO 2 (linear molecule) and 4/3 for Freon-12 and SF_{6 •}

In Ref. 43 there are several plots of the variation of angle 0, with shock Mach for fixed effective wedge angle ~ for CO_{2, N2, and Ar and} in Ref. 38 for SF_{6 •} These may be condensed by plotting the quantity k

=

(ox - of)/(oe - of) where 0x is the measured 0, of is the calculated frozen 0, and

Oe

is the calculated equilibrium o. It is seen from Fig. 22 for CO 2, for example, that for ~1s > 4, k ~ 1, that is, 0x = oe and the agreement is for CO 2 in equilibrium, despite the fact that the . experiments in the (Ms, Sw) -pl ane agree best with a perfect gas (y

=

1.29). These results are more reliable: as no transition criteria are required in the calculation of 0; a quantitative comparison can be made for each pair of analytical and experimental points; a measurement of 0 has no ambiguity unlike the classification of the reflection types near the transition boundaries where, for example, a CMR may look like an SMR or a CMR 1 ; k e a DMR • The parameter

ew

I should be used instead of ~ in the comparisons, since ~ can be obtained directly from Eqs. (1) to (4), while 9_{w is derived by assuming that the Mach stem M is perpendicular} to the, wedge surface at the triple point. This is not always the case experimentally, as shown below.

2.3.2 Effects of Mach Stem Curvature

Fi gu re 23 i s a schemat i c di agram showi ng how the Mach stem curv es at an angle E, at the triple point rather than being ideally perpendicular to the wedge surface at the triple point.

The Mach stem was assumed perpendicular to the wedge surface at the triple point in deriving

9w

from ~3 in the three-shock theory. The deviation e: from this idealization is shown for Ar, N₂, air and CO₂ in Fig. 24. It is seen that e: has positive values (0 - 8°) for r1s > 4 and decreases to negative values (_3°) at Ms = 2. Consequently,

ew

as derived from _~3 has an error equal to e:. Therefore, comparisons of quant it i es in r1ach refl ect i on should be made with ~, as long as e:

cannot be predi cted accurately. This error is the mai n reason for the inaccurate prediction of

x,

using the three-shock theory.

The effect of e: on ~3 is shown for CO₂, in Fig. 25, as the variation of D.~3 = ~3 (numeri cal ) - ~3 (experil'1ental) with shock Mach number r"s. The numeri cal results are based on an equilibrium gas, as verified in Fig. 22. It is seen that M3 has a val ue of about -2°, i ndependent of Ms. The results for other gases are similar albeit somewhat more dispersed.

2.3.3 Effect of Slipstream Thickness

In the solution of Eqs. (1) to (4), only the pressures, Eq. (7), and flow directions, Eq. (8), were assumed to be continuous across the slipstream, an idealized surface. In real ity it is a shear layer of increasing finite thickness, where the velocity in state (2) is greater than th at in state (3) and the temperature in state (3) is greater than that in state (2). Near the triple point the layers are laminar, and beyond they become turbulent. Consequently, as noted above, the shear layer thickness gives rise to a deflection of the velocities in stat es (2) and (3). If it

triple point then its possible effect on X should be taken into account in

the discrepancies bet ween the numeri cal and experimental val ues of _X·

Figure 26 shows a plot of

e'

_{w versus}

_9w

with displacement angles a of

1° and 2° , for CO 2 in equi 1 i bri urn at Ms

=

5. The di fference between the

val ues of

a'

_{w and}

_9w

a10ng the ordi nate and abscissa gives the va1ue

of X. It is seen that a value of Cl: of 2° changes X by less than 1° except

at small wedge angles (a_w < 10°). This is insufficient to exp1ain the di screpancy i n X of about 2° i n a typi ca 1 case at Ms = 6 and

9w

= 20°, since an Cl: ~ 6° wou1d be required to explain this discrepancy and it is not

like1y to occur. However, a at small

9w

is significant and shou1d be

taken into account. In addition, the change in

a

_{2 and}

a

₃ due to a must a1so

be taken into account in using the three-shock theory. A 1ess rigorous

attempt to deal with the slipstream diffusion has recently been made in Ref. 44.

2.3.4 Effect of Shock-Induced Boundary Layer on the Wedge Surface

It has been found by many experimenters that RR persists into the MR regime. This has been called the "von Neumann paradox". Recently, several ana1yses 39 ,43,45,46 have shown that this is due to the boundary layer

negative displacell1ent thickness (Fig. 27). Consequently the displaced wall

wd ;s below the actua1 wall wa. The reflected wave R moves over to R'

so that the ang1 e ê between land Ris 1 arger or the refl ected wave angl e w'

is smaller. The condition el - a₂

=

ed must now be used instead of Eq.

(6). This can be checked and has been verified experimentally. Figure 28

compares experimenta1 _{w' for CO 2 with ca1culations for several va1ues of}

ed. The experimenta1 results 1ie between the lines of ed = 0 and -2°.

ad

=

_1° shifts the RR ++ MR von Neumann detachment transition boundary by

~9w =

-0.7° _{for N2 and ai r, and t:.aw}

=

-0.5 _{for CO 2• Recently Sakurai}47

proposed to use a pseudo stat i onary RR ++ MR cri teri on. Howev er, it s effeetiveness is yet to be tested experimentally.

3.0 FLOW-FIELD SOLUTIONS

The foregoing dealt with simplified algebraic solutions49 for RR and HR, which involved mainly the shock waves and the slipstream. In order to compare experimental and analytical solutions of the entire flow field for such sensitive parameters as the density distribution, which can be measured interferometrically, it is necessary to solve the time dependent33,34 or pseudostationary Euler equations as noted below, for a perfect or equilibrium gas. Otherwise it would be necessary to use rate equations for equilibration as well.

Cont i nu ity: P

t + (pu) x + (pV) y

=

0

f10mentum:

Energy:

where p is the density,

~

= (u, v) is the velocity field, E =

i

(u 2 + v 2)+e ;s the total specific energy, e is the specific internal energy, and p is the pressure. The system is closed by specifying an equation-of-state

p = p( p, e) (10)

Such an equation can be obtained from the therma1 equation of state p

=

pRT and the specific interna1 energy for a perfect gas e

=

CvT,

to yie1d

p

=

(y - 1)pe (11 )

where y > 1 is the ratio of specific heats and is considered to be a constant. For gases such as CO₂ and SF₆ in vibrationa1 equilibrium the equation of state has to be modified. For se1f-simi1ar motion, the prob1em has no intrinsic 1ength sca1e. Consequent1y, a new pseudostationary coordinate system can be used such that (1;, Tl) = _{[(x-xo)/(t-t o)'}

(y-yo)/(t-t o)]' where (x o' Yo) are the coordinates of the wedge corner C, and to is the time when the incident shock wave I reaches the corner. As shown by Jones et al, 23 the system of Eq. (9) can be transformed to a pseudo- stationary system (Eq. 12).34

where

'"

( pul I; + (p>Ï) Tl

= -

2 p

(pU2 + p) I; + (puv) Tl = -3 pU

(puv) I; + (",2 + p) Tl

=

-3tN'"

(pUH) I; + (piH) Tl

=

_p(u2 + ;2) - 2pll

'"

u

=

u-I;, v

=

v - Tl,

Il

=

î

(û'2 + ;2) + h

(12)

(13)

and h

=

e + pip is the specific entha1py. A1so

(u, v)

and

Il

are referred to as the self-similar velocity field and self-similar tota1 entha1py, respective1y. In addition

M

is defined by

(14 )

where a

=

sound speed and

M

is called the self-similar Mach number. The system of Eqs. 12 is the steady Euler equations with the addition of source terms. Note that the ratio s/L is constant (Fig. 9c) for given initial conditions, for self-similar solutions of the nonstationary equations just as s i s constant for steady supersoni c flow. In thi s and other ways a change to pseudostationary coordinates is very useful in the analysis of such flow fields and were used in nUl'!1erical simulations of the interferometric experiments on oblique shock-wave reflections.33

Perhaps the best exampl e of showi ng the great strides that have taken place in the reliability of numeri cal simulation of such problems is to compare the results of Schneyer,48 of a decade ago, with the present state-of-the-art simulation by Glaz et al. 34 Figure 29 shows an interferogram of an SMR in air at Ms

=

2.03,

9w

=

27°, Po

=

250 torr and T_o = 300 K. The lines of constant density (isopycnics) over the flow field are also shown in the table. Figure 30 shows Schneyer's results which were the best avai lable at that time. Although the wave system is well represented, the isopycnics are not. Spu ri ous errors and noi se are prevalent. By contrast, Fig. 31 shows the same case treated by Glaz et al, 34 a decade later, which is a very good simulation indeed.

~1any

such cases were treated in air, Ar and SF

6•34 ,35,50 The results are all surprisingly good considering that the simulations utilized the Euler equations for inviscid flow and the interferograms are for real flows with viscosity and real-gas effects. Undoubtedly, repeat simulations using the Nav i er-Stokes equat i ons and equil i brat i on rate equat i ons for the degrees of excitati on i nvolv ed woul d prov ide some important answers whether the costly

It is worth noting that some very worthwhile concepts have been added to pseudostationary oblique shock-wave reflections recently using the Monte-Carlo technique. Here the reflections are simulated at the molecular level by using a large number of particles (> 10,000) and following their positions in phase space.51,52 Of the two papers, the one by Seiler is of greater i nterest si nce he treated both specul ar and diffuse refl ect i ons of the hard-sphere molecules and was therefore able to obtain the boundary 1 ayer i nduced by the i nei dent and ref1 ected shock waves in RR at the reflection point. His results confirm that RR persists into the MR regimes as a result of the induced boundary 1ayer. He a1so showed that the ref1ected wave angle w' is smaller as aresult. However, the method will have to be refi ned to produce sharper shock waves, we1l-defi ned sl i pstreams and more meaningful isolines compared with experimental interferograms and the numerical simulations obtained by G1az et al.33-3S,SO It is a pity that Seiler did not investigate the expansion-compression wave-system model produced by the boundary-1 ayer di spl acement-thi ckness at lts 1eadi ng edge, in order to substantiate if at the molecular level the model can be simulated, since it cannot be seen in optical enlargements of the flow.

A more complex simulation is that of a prob1em where self-similarity cannot be applied. For example, the collision of a Mach reflection with a 900 _{ramp on the wedge surface provides such a case.}24_{,SO Figure 32(a) shows}

an interferogram of a DMR in CO_{2 ,} Ms

=

S.81,

9w

=

200

, Po

=

10 torr,

T_o

=

297

K

before collidng with a 12.7 mm ramp. Figure 32(b) shows the wave system 60 ~s later, af ter the collision had taken p1ace. It is seen that the wave system involving Mach stem-slipstream interactions, boundary-layer interactions, rarefaction and expansion waves, etc., has become very complex. The type of interaction is dependent on the relative heights of Mach stem and ramp. Figure 33 shows the numeri cal simulation. Surprisingly

good agreement is obtained of the main features of the flow using the Euler equations of motion. Many additional details can be found in Refs. 24 and

50.

4.0 APPLICATION TO HEIGHT-OF-BURST SHOCK TRAJECTORIES

Another important application of the two-dimensional oblique shock-wave reflection problem is to apply the plot in the {Ms, Sw)-plane to spherical shock-wave reflections from a height of burst {Fig. 34).53 It is seen th at the spherical reflections resemble those of the wedge problem. It has not been established how far the spherical shock wave would have to be away from the fireball before such an analogy can be made. However, a reasonable figure might be 10 or 20 charge radii. Fi gure 35 shows a nondimensional plot of the shock wave trajectories in the {Ms, Sw)-plane as a function of HOS, charge weight TNT, and atmospheric pressure. It is seen that for a charge \'Iei ght of 1 kg TNT and a hei ght of 1 m, at 1 atm, a shock wave st rength Ms '" 2.9 is produced and it undergoes RR, DMR, CMR and SMR until it becomes a sound wave at Ms

=

1. On the other hand, if the HOB is 2 m, Ms '" 1.5, only RR and SMR will take place. The trajectory fits are based on the best available experimental data.

{Ms, _{9w)-plot has yet to be tested experimentally.} undoubtedly be forthcoming in the near future.

5.0 CONCLUSIONS

Howev er , the This wil 1

1. Although much has been learned about oblique shock-wave reflections since Mach 22 wrote his pioneering paper in 1878, a great deal is still to be known.

2. The presently accepted transition lines for RR +-+- MR, St1R +.~ Cr1R and

CMR +-~ Dt1R prov ide very reasonable engi neeri ng answers, as to the type of reflection that will occur, given Ms,

9w

and y for frozen and perfect gases and for vibrational equilibrium given additionally To• For gases with low y at room temperature such as CO_{2 '} Freon-12 and SF₆ real-gas effects in vibration are very important and must be taken into account. They are not precise enough, and improved transition-line criteri a wil 1 have to be found to account for those experiments that give rise to the "von Neumann paradox" for the RR ++ MR transition.

For the SMR ++ CMR and CMR H D~'R 1 i nes at lower

Sw,

where a number of experiments fall outside the present lines (see Figs. 17 to 19), new criteria will have to remedy this situation.

3. The experimental CMR +-+- DMR line leading to point P on the RR +~ MR

line will have to be replaced by an analytical line (see Figs. 19a, h and d).

4. It is not clear whether the induced boundary-layer slope or thickness is the important parameter to account for the "von Neumann paradox". Some experiments with induced turbulent boundary layers caused by specified roughness should help in a decision on this point.

5. It is not known why the TDMR experiments lie in the RR regime, rather than in the DMR region as predicted from the condition Xl

=

O. Perhaps the von Neurnann RR -+- MR line is not aplicable to pseudostationary

flow.

6. More needs to be known about RR and t1R over rough surfaces , spongy surfaces , surfaces with a thi n layer of He or other gases, and in a dusty-ai r env ironment •

7. Improved computational simulations using the Navier-Stokes equations and real-gas rate equations are required for a comparison with the

It can be seen that much work still remains to be done in pseudostat i onary ob 1 i que shock -wave ref1 eet i ons i norder to answer some of the above-noted prob1ems. Undoubted1y, such work will be forthcoming in the nea r futu re. In addition to the single-wedge problems, the degree of complexity increases using Jrultiple straight wedges, as well as convex and

54 55

concave wedges. '

6.0 REFERENCES

1. G1ass, I. I., lIShoek Waves and ManII, pub1ished by UTIAS, printed by University of Toronto Press, 1974.

2. Glass, I. I., lIShoek Waves on Earth and in Space", Prog. Aerospace Sci., pp. 289-286, 1977.

3. Glass, I. I., "Terrestrial and Cosmie Shock Waves", American Scientist,

65, 4, pp. 473-481, 1977.

4. Fowler, R. G., "lightning", Applied Atomie Physics, Vol. 5, Academie Press, pp. 31-67, 1982.

5. Ribner, H. S. and Roy, 0., "Acoustics of Thunder: A Quasilinear ~10del

for Tortuous Lightning", J. Acoust. Soc. Am. 72(6), pp. 1911-1925, 1982.

6. Rab1e, R. G., "Earth's G10bal Electrica1 Circuit: An lJnsolved t1ystery", Aerospace America, pp. 124-127, Jan. 1985.

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