PROPAGATION AND REFLECTION OF SHOCK WAVES
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Series on Advances in Mathematics for Applied Sciences  Vol. 49
PROPAGATION AND REFLECTION OF SHOCK WAVES
F V Shugaev L S Shtemenko Department of Physics M V Lomonosov Moscow State University Russia
V f e World Scientific wh
Singapore • New Jersey • London • Hong Kong L
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress CataloginginPublication Data Shugaev, F. V. (Fedor Vasilevich) Propagation and reflection of shock waves / by F. V. Shugaev & L. S. Shtemenko. p. cm. ~ (Series on advances in mathematics for applied sciences  vol. 49) Includes bibliographical references and index. ISBN 9810230109 (alk. paper) 1. Shockwaves. 2. Wavemotion, Theory of. I. Shtemenko, L. S. II. Title. III. Series. QA927.S49 1997 532'.0593dc21 973964 CIP
British Library CataloguinginPublication Data A catalogue record for this book is available from the British Library.
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PREFACE
We vve discuss discuss the me propagation propagation of 01 threedimensional threedimensional shock shock waves waves and and their their reflection from from curved curved walls. walls. reflection A ray method is set forth in the first part. It is based on the expansion of fluid properties in m power series at an arbitrary point on the shock wave front. Continued fractions are used. Results for shock propagation in nonuniform fluids are presented. The second part deals with shock reflection from a concave body. The i m p o r t a n t shock focusing problem is included. The work is supported by both numerical and experimental results. Many interesting features such as formation 01 of aa jet, and the the appearance appearance of of disturbances tion jet, vortices vortices and disturbances on on the the shock shock front front are discussed. discussed. are The The authors authors would would like like to to expess expess their their gratitude gratitude to to Dr. Dr. S.A.Bystrov S.A.Bystrov and and to Dr. O.A.Serov for the discussion of the results and also for to Dr. O.A.Serov for the discussion of the results and also for the the assistance assistance in in conducting experiments. experiments. conducting
vV
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LIST OF SYMBOLS
a1 bap
 Lagrangian variables  components of the second fundamental surface tensor at the wave front  velocity of sound  frequency  covariant components of the metric tensor in the space ala2a?  covariant components of the metric surface tensor on the wave front  velocity of wave propagation  mean curvature of the wave front  Gaussian curvature of the wave front
c / gij gap G H K \_dalidalk
J
M rii — n%
 Mach number  components of the external unit normal to the wave front in a Cartesian coordinate system p  pressure q  2jM2  7 + I S  entropy; area s  distance along the ray T  temperature t  time ul,v?  surface variables vl  components of a particle velocity in a Cartesian coordi nate system 2 w= ( 7  l ) M + 2 xl  Cartesian coordinates xl (a1, a2, a3, t)  Eulerian variables a  angle of incidence of a shock wave /3  angle of reflection of a shock wave
vn
Vlll
Propagation and reflection of shock waves
Tjj
 Christoffel's symbols
Vi
 covariant components of the external unit normal to the wave front in the space a1 a2a3  density
p T=l/p UJ
 cyclic frequency; vorticity
The notation, a denotes covariant surface differentiation. Lower indices 1,2 refer to the state ahead of and behind the front, respectively. Greek indices are used for components of tensors on the surface. Latin indices are used for components of tensors in the space. Brackets [ ] denote the jump of any quantity across the wave front: [(f)] — 02~01An index which occurs twice in a term is to be summed.
CONTENTS
Preface
v
List o f s y m b o l s
vii
Chapter 1. S t r u c t u r e a n d b a s i c p r o p e r t i e s o f s h o c k w a v e s i n g a s e s
1
1. General remarks
1
2. Interaction of shock waves
7
3. Shock tube as an apparatus for obtaining shock waves in the laboratory Chapter 2. S h o c k w a v e p r o p a g a t i o n t h r o u g h a g a s
20 41
4. Basic notions
41
5. Compatibility conditions
44
6. Ray method for calculation of unsteady shock waves
55
7. Path of a particle behind a shock wave
74
8. Distribution of flow parameters behind an unsteady curvi linear shock wave
76
Chapter 3. I n t e r a c t i o n o f a p l a n e s h o c k w a v e w i t h d i s t u r b a n c e s and stability of shock waves
79
9. Linear interaction of shock waves with disturbances
79
10. Propagation of a plane shock wave through a region of nonuniform density (nonlinear case)
84
11. Nonlinear onedimensional interaction of a weak disturb ance with a shock wave 12. Instability of shock waves Chapter 4. R e f l e c t i o n o f a s h o c k w a v e f r o m a c o n v e x b o d y
90 96 105
13. Reflection of a plane wave from a body of arbitrary shape 105 14. Transition from regular to Mach reflection
117
15. Development of flow over a blunt body behind an incident shock wave 120
IX
x
Propagation and reflection of shock waves
Chapter 5. Reflection of a shock wave from a concave body and shock focusing 16. Reflection of a shock wave from a body with rectangular cavity 17. Oscillations of the shock wave reflected from a body with cavity 18. Shock focusing 19. Resonant excitation of vortices behind the reflected shock wave Chapter 6. Propagation of a shock wave through a turbulent gas flow Chapter 7. Propagation of a shock wave through a gasparticle mixture Chapter 8. Laserdriven shock waves Chapter 9. Shock waves in a lowtemperature plasma Appendix A References Subject index
141 141 151 154 164 177 191 197 213 225 233 243
C H A P T E R 1. STRUCTURE A N D BASIC PROPERTIES OF SHOCK WAVES IN GASES
1. G e n e r a l r e m a r k s Shock waves belong to one of interesting phenomena that occur in nature. They arise at explosions, at electrical discharges, at supersonic flight in the atmosphere, in the space (e.g. explosion of supernova). The evolution of a nonlinear compression wave into a shock wave was investigated by Riemann (Riemann 1876) who gave an analytic solution in the case of onedimensional flow. Earlier the propagation of waves with a velocity higher than that of sound was considered by Stokes, Airy and Earnshaw. Further investigations were made by French engineer, former officer of artillery H.Hugoniot (Hugoniot 1889) and by English scientist Rankine (Rankine 1879). Hugoniot showed that the law of energy conservation is incompatible with the constancy of entropy while considering the flow across a shock wave. Shock waves in air were observed for the first time by famous scientist and philosopher E.Mach (Mach 1878). He also discovered the socalled irregular (Mach) reflection which has no analogies in acoustics and optics. Various cases of shock interactions were made possible to investigate after N.E.Kotchine (Kotchine 1926) had solved the i m p o r t a n t problem of breakup of an arbitrary discontinuity. Modern developments in this field were stimulated by the needs of aircraft and spacecraft. It is worth mentioning the works devoted to strong shocks, the shock structure, the use of shock waves for cumulation of energy, shock waves in the space. New experimental techniques with high temporal resolution and numerical methods allowed to investigate very complicated problems. The structure of shock waves can be studied, using kinetic equations. The Boltzmann equation is used widely. The term shock wave has two meanings. Sometimes it denotes a disturbance propagating in a medium at a supersonic velocity. However, this term usually signifies only a comparatively narrow zone where a transition takes place between an undisturbed medium and disturbed
1
Propagation and reflection of shock waves
2
one. The thickness of a shock wave against Mach number is shown in Fig. 1.
Fig. 1. Inverse shock thickness against Mach number (Fisco et al. 1988). Argon data were obtained by Alsmeyer (Alsmeyer 1976). Let us consider in detail the evolution of a nonlinear wave in onedimen sional case. The equations of continuity and of momentum can be written in Lagrangian variables in the following way dx d2x 1 dp 0, 2 = PolP, da dt po da x(a,t), p p(a,t), p = p(a,t), po = Po(a).
(1)
Here p is the pressure, p is the density, po is the density distribution at an initial instant. Below we assume that po is constant. We differentiate the first Eq. (1) with respect to t. Multiplying it by cp/ po, and then adding to the second Eq. (1), and subtracting from it, we have dr dt
cp dr po da
3s
cp ds Po da
= ,+ / * . = ,  / * (2) J pc J pc Here c is the velocity of sound. The quantities r and s are called the Riemann invariants. Eqs. (2) have the solutions in the form of waves (the socalled simple waves): 1) r = Fi(£)> £ — a ~~ cPt/Po, s = const (the wave propagating from left to right); the quantity cp/po is constant along the line £ = const; 2) s — F2(r]), rj = a f cpt/po, r — const (the wave propagating from right to left); the quantity cp/po is constant along the line rj = const. 0,
^7
dt
0,
Structure
and basic
properties...
3
Let us investigate the first case. The derivative of r with respect to a is equal to dL=2_dp da pc da' Hence
(3) wn 9^^:^U^ • /p> w ^='' p2, PN pN > Pi If V2 = va then there is no breakup, and the disturbance propagates as a shock wave. Let v2 «•> be in the range Vb Vf> 0.
(18)
As one can see, Eq. (18) is always valid if M 2 > M 3 > 1. Thus the statement is proved. b) Confluence of two shock waves that propagate in the same direction. If two shock waves move in the same direction they catch up with one another in some period of time. In fact, the first shock moves at a subsonic velocity relative to the gas behind it and the subsequent shock wave moves at a supersonic velocity relative to the same gas. Thus a confluence of waves takes place at a definite instant. An arbitrary discontinuity arises as a result. The solution to the problem depends on the sign of the quantity S x
2
M?l
,
2
M?2l
P3
[2
^"
_ 1
(7+l) + 7"l Pi Pl M ( 7 ) . If 7 > 5/3 then M ( 7 ) > 1. Let 7 < 5 / 3 . Then M ( 7 ) < 1, M i > M ( 7 ) , M i > M * , A(M) > 0. Thus a rarefaction wave moves from right to left if 7 < 5/3. Let 2 > 7 > 5 / 3 . In this case M ( 7 ) > 1. We consider two cases: M i < M ( 7 ) and M i > M(j). (a) If M i < M ( 7 ) then M * > Mx. Let M * > M > Mx. A shock wave moves from right to left in this case. The abovementioned disturbance is a rarefaction wave if M > M* > Mi.
Propagation
12
and reflection
of shock
waves
(b) If M i > M ( 7 ) then M* > Mi. The following condition is valid: M > M*. A rarefaction wave moves from right to left. If M = \{Mx(i
 1) + y/fr  1)M2 + 2 ( 7  1)},
7>5/3
then no disturbance moves from right to left. If 7 > 2 then M * > M\. The flow pattern is the same as in the case (a). c) Refraction of a shock wave at an interface. Let a shock wave fall on a contact discontinuity that separates two uniform regions of a gas at rest, the velocity of sound being different in two regions. A contact discontinuity appears as a result of interaction of the shock wave. In experiments a thin film that separates two different gases is generally used. The thickness of the film is chosen in such a manner that its influence is negligible. The film is destroyed after the passage of the shock wave. As the shock wave arrives at the interface then an arbitrary discontinuity appears. A shock wave propagates forward in all cases. The reflected wave is either a shock or a rarefaction wave. We assume that the adiabatic exponent has the same value on both sides of the contact surface, for the sake of simplicity. Let c\ > C2 In this case the velocity of gas in region 1 is greater than that one in region 2 behind the shock wave that corresponds to the prescribed value of pressure ratio. Consequently, a shock wave reflects from the contact discontinuity. If ci < c2 then the flow velocity in region 1 is less than that in region 2 (for the prescribed value of pressure ratio). The reflected disturbance is a centered rarefaction wave. In other words, if the shock wave comes from a heavy gas to a light one then a rarefaction wave reflects. These results have a clear physical meaning. The contact surface can be treated as a piston that pulls out of region 1. If c\ < c2 then the velocity of the piston is greater than the velocity of the flow in region 1 before the interaction. A reflected rarefaction wave arises. If c\ > c2 then the velocity of the piston is less than that of the flow in region 1, and a reflected shock wave appears. Now we proceed to the case 71 ^ 7 2  The flow pattern depends on the mutual position of the following curves 2
v = fi(p)
=
i 7l
v = f2{p)
=
Pi
c
c2
7 ( 7 i + l)pT + 7 i  l '
T 72
fcl
\Z(72+1)^+72_1
Structure
and basic
properties...
13
If hip) > hip) then a shock wave reflects; if / i ( p ) < hip) then a rarefac tion wave reflects. At hip) — hip) there is no reflected disturbance. Let us consider the ratio / i ( p ) V _ 7 2 f C l \ 2 (72 + c
hip))
7i V 2 /
(7i +
l)£+72l l ) £ + 7 i  l
The quantity P
F
, x
W
=
(72 + 1)^ + 7 2  1
1—TTT1
7'
z
,
= PM
(7i + l)z + 7i  1 is a monotonously increasing (for 71 > 72) or decreasing (for 71 < 72) function of z. The following cases can take place. W % > 72' I > ^ 7 2 ( 7 2 + 1) T h e i n e < l u a l i t y / i ( P ) > /»(P) all values of p > pi. A shock wave reflects. (b) % < 7 ? . % < ^
j
^ + l j  T h e inequality h{p)
< f2(p)
is valid
for
is valid for
all values of p > p\. The reflected disturbance is a rarefaction wave. (c) % > % . §
< y
^
+ l j  (7i < 72, ex < c 2 ). If w < p*, a shock
wave reflects. Otherwise, a rarefaction wave reflects. The value p* is as follows P* = Pi{71  1 ~ (ci/c 2 ) 2 72(72  l ) / 7 i } / { ( c 2 / c i ) 2 7 2 ( 7 2 + l ) / 7 i  7i ~ 1}. If p2 = p* then there is no reflected disturbance. These results enable us to solve some problems connected with the breakup of an arbitrary discontinuity (onedimensional case). a) T h e operation of a shock tube. The shock tube is used for obtaining and studying shock waves in the laboratory and for studying hightemperature kin etics. The very rapid heating to high temperatures, the wide temperature and pressure ranges, the diffusionfree reaction conditions are the main advantages for investigating rate coefficients at high temperature. A shock tube is a cylindrical tube which is divided by a diaphragm into two parts: a lowpressure section and a highpressure one. The diaphragm bursts at some instant, and the driving gas flows into the lowpressure section. A shock wave appears as a result. We assume the bursting of the diaphragm to be instantaneous and the flow to be onedimensional. So we have a breakup of an arbitrary discontinuity. The gas is at rest at the initial instant. So a shock wave propagates in a lowpressure section and a rarefaction wave moves in the highpressure section. Denote as p\ and p± the values of the pressure in the
Propagation
14
and reflection
of shock
waves
driver section and in the driven one, respectively, as p2 and p3 the corresponding values behind the shock wave and rarefaction wave. We use the conditions on the contact surface that separates the cold gas from that one heated by the shock wave, namely Pi = P 3 ,
^2 = ^3
Or
2 C 1 (M'1)_ 2c4 J^fpA2^, (71 + 1 ) M
741
I
(22)
VP4,
We can write El  ElEL 
2
7 i ^ 2  71 + l p i
P4 Pi P4 7i + l By substituting Eq. (23) into Eq. (22), we have
771 4
+ " 11 Ccl4 (M
(23)
pA'
1 M
2 +l ) = 1  ^\2 7 i ^7i  7 i + ^ /^
r VF4
(Pi^~^r
Or 2T4
P4
7i + l
f
2
74 — 1 Cl /
1\1
1
^41
M
pi ~ 27iM  71 +T I " 71 + 1^1 V ~ M J J
'
^
Eq. (24) enables us t o determine the Mach number of the shock wave as a function of the pressure ratio over the diaphragm. The m a x i m u m value of the Mach number (at p^/pi —>■ 00) is equal t o
M m a x = i21±i£l 1+ f 1 + 4 ^  l c ^c 2 X l / 2 2 74  1 Ci I
I
\7l + 1 4 /
I
I
b) The interaction of the shock waves of the same family. If 1 < 7 < 5 / 3 then a shock wave and a rarefaction wave arise as a result of the interaction. A contact discontinuity occurs, too. Let us introduce the following notations. Subscript 0 refers t o the undisturbed state, subscript 1  t o the state behind the first wave, subscript 2  t o the state behind the second wave, subscript 3 to the state behind the rarefaction wave, subscript 4  to the state behind the shock wave that appears after the interaction. We have P4=P3,
V4=V3, Cl
Vl
l
=* /  £  ^ 7
~&
V^V(7l)6+7 + l'
  Q L . /(7 + l K i + 7  l V^V(7lKi+7 +l'
^  £ 0  P i 
h
Structure and basic properties...
15
inally we get
/(7i)6i+7 + l 6) 666 V(7 + i)6i + 7  i v f T  1)66 + (7 +1)6 (7 +1)6 + 7  1 Yl 16 , A/27 / ( 7 + l ) 6 + 7 ^j(i6 2y \ 1 \ ) ' ( 7  1 ) 6 + 7 + 1 'T^TY ( 7  l ) ^ + 7
6 = Pl/P2,
6=P3/P2
(25)
The quantities £i, £2 are known. Thus we have an equation with one unknown quantity, £3. c) The interaction of two shock waves of opposite families. If two shock waves propagating in opposite directions catch up with one another, two new shocks arise after the interaction. A contact discontinuity separates the flow between them. The values of the pressure are equal on both sides of the contact discontinuity and so do the values of the velocity. Let subscript 0 refer to the undisturbed gas, subscripts 1 and 2  to the flows behind the first and second shock waves before the interaction, subscripts 3 and 4  to the shocks after the interaction. We obtain the following equation '(71)6+7 + (71)6+7 +
\ {(16)^^^(666)
4^j^}
+ q^(i6)
(7 + 1)6 + 7  1  o (71)6 +7 + 1 " ' t a
 £0 ~~ Pi '
p _ Po ^ 2 ~ P2 '
xr0£2 ^3 P3 '
The value P4 can be determined, if we know £3. Namely, p± — p i 6 / ( 6 6 ) 2.2. Oblique interaction of shock waves a) Regular interaction of two plane shock waves of opposite families. Let two plane shocks propagate through a gas. The angle between shock fronts is equal to (j) and the flow properties behind the shocks are uniform (Fig. 5). The undisturbed flow is known and so are the velocities of the shocks. The flow downstream the point of intersection is to be determined. The problem is a generalization of the reflection of a shock wave from a rigid wall. The reference frame is placed at the point of intersection of the shock fronts. The flow behind two shocks may be considered as that behind two wedges. The
Propagation and reflection of shock waves
16
planes of symmetry of the wedges are parallel to the velocity of the undisturbed flow ahead of the shock waves.
Fig. 5. Regular interaction of shock waves of opposite families The values of the pressure must be the same on both sides of the contact discontinuity, and so do the directions of the flow velocity. Consequently, we can write
{ q' i
— 2jMfi
Si + 83 = 82 + $4,
— (7 — 1),
M'i = Mi sinwf (i is not to be summed) ,
8{ = arctan < [ ( *n
Mj = M 2 ,
**
1 1 tan LJ;
/,■ = 2 + (7  l)M2.
(26)
The numerical solution to the problem was given by TerMinassiants (TerMinassiants 1962). It is interesting to note that if the angle between the shocks
Structure
and basic
properties...
17
is close to the limiting angle of the regular interaction, the pressure behind the refracted waves exceeds that one at the headon collision of the waves. b) Regular interaction of shock waves of the same family. The problem of the interaction of shock waves of the same family was analyzed by Roslyakov (Roslyakov 1965). T h e flow patterns are shown in Figs. 6 a and 6. The angle of flow deflection behind the shock wave is defined by Eq. 26. In the case of the rarefaction wave (Fig. 6, a) we have S3
=
a r c t a n ( M 2  l ) " 1 / 2 + h arctan((M 3 2  a r c t a n ( M   l ) " 1 / 2  h arctan((M 2 2 
Pi
pt
,
l)1/2/h),
/>=((7 + l)/(7l))1/2,
M3=^{^(/2l)}, & 
l)1/2/h)
i = l ; 2; 3;
U=Po/P4
Fig. 6. Regular interaction of shock waves of the same family. By using the boundary condition across the contact discontinuity, we have 6 6 ^ 3 = £4, S1+62±63
= 64.
(27)
Propagation
18
and reflection
of shock waves
The upper sign in Eq. (27) corresponds to the reflected rarefaction wave, the lower sign corresponds to the reflected shock wave. By eliminating £3 from Eq. (27), we get an equation which contains £4
81{Z1,M) +
S2(Z2,M1(Z1,M))±63(J^
On having determined the unknown quantity £4, we find £3 by using Eq. (27) and then we solve the whole problem concerning the interaction of shocks. The results of calculations are as follows (Roslyakov 1965). There appears a stronger shock at the point of intersection of shocks, a contact discontinuity and a reflected wave which m a y either be a rarefaction wave or a shock wave. The flow pattern depends on the Mach number M . If M < 1.245, 7 = 1.4 then the reflected wave is a rarefaction one. At M > 1.245, 7 = 1.4 the reflected wave m a y either be a rarefaction wave or a shock wave. At high Mach numbers (M > 3, 7 = 1.4), the reflected wave is a rarefaction wave as a general rule. At M > 1.305, 7 = 1.4 there is a range of quantities £1, £2 where regular interaction does not occur. c) Refraction of a plane shock wave at an interface. Let there be a plane interface between two different gases. A plane incident shock wave meets the interface at an angle of incidence a, measured with respect to the interface. The shock wave begins to pass from the first gas into the second one where it becomes the transmitted shock. We consider a regular refraction (Fig. 7). In this case all of the waves move at the same velocity along the interface, and we can write the fundamental law of refraction (Henderson et al. 1991)
mi
mi
=
sin a?
itfri
=
sin at
sin a2
With a continuous increase in the parameter a2 the law m a y be violated, the regular wave system may break u p . In this case the transmitted shock travels ahead of the incident and reflected waves, and an irregular refraction with precursor waves appears, so that Ut sin at
Ui sin a,
U2 sin a2
=
Let us introduce wave impedance z
_
Pi (ui
—
Po UQ)
COS (3i '
where /?,• is the wave angle measured with respect to the disturbed gas interface. The transmitted and reflected wave inpedances are defined in a similar way.
Structure
and basic
properties...
19
Fig. 7. Refraction of a plane shock wave at an interface, i  incident wave; r  reflected wave; t  transmitted wave; MW  Mach wave.
The pressure reflection and transmission coefficients are as follows (Henderson 1988) R =
V2 Pi
_ Z2{Zt

Zj)
PIPO " Zi{z2zty Ptpo _ Zt(Zi  Z2) Zi{ZtZ2)' Plpo
The refraction law m a y also be written as  Ui 
sin ai _ pt Z\ cos /%
\Ut\
sinat
po Zt cos j3t'
It follows from Eq. (28) cos at — (1 — n~2 sin 2
cti)1'2.
The quantity cos at becomes imaginary if a 4 > ac, sm ac — n —
Ut
(28)
20
Propagation and reflection of shock waves
It is obvious that ac only exists for slowfast refraction, n < 1. If a = ac then at = 7r/2. In this case the gas interface is not deflected. In order to solve a problem of regular refraction, we must take into account the boundary conditions along the interface behind the incident and refracted waves: 50 + &i =8t,
p2
=Pt
Each regular or irregular system of waves occurs for definite ranges of values of the parameters (7*, 7*, //,, /i*, &, ctj), /i being molecular weight. In particular, the range of & versus a, was defined for the CO2/CH4 interface (AbdElFattah and Henderson 1978). 3. Shock tube as an apparatus for obtaining shock waves in the laboratory A shock tube is a conventional facility for studying shock waves. Shock tube was invented by Paul Vieille. In 1899 he wrote the famous paper on discon tinuities produced by sudden expansion of compressed gas (Vieille 1899). The simplest shock tube consists of two sections: a driver (highpressure) section and driven (lowpressure) section filled with test gas. The sections are separ ated by a diaphragm. The driver gas flows into the lowpressure section after the burst of the diaphragm and the plane shock wave propagates through the test gas. A contact discontinuity separates the driven gas and the driver one. There are doublediaphragm shock tubes with two driver sections (Tsukahara et al. 1996), diaphragmless shock tubes with a fast action bellows valve, deton ation driven shock tubes in which hydrogenoxygen mixture is used (HR. Yu et al. 1995). Shock tubes are convenient for study of gas kinetics because the flow is nearly onedimensional with practically instantaneous heating of the test gas. Other investigations that use shock tubes are connected with propagation and reflection of shock waves. A simple onedimensional ideal theory of the shock tube assumes the burst of the diaphragm to be instantaneous. Thus the problem is reduced to the breakup of an arbitrary discontinuity. The shock wave is considered to be plane and its velocity constant. The rarefaction wave is assigned to be a centered one. Dissipation and mixing in the region of contact discontinuity are neglected. The shock Mach number M depends on the pressure ratio across the diaphragm (see Eq. (24)). Experiments show that the aforecited formula gives correct values of ve locity for weak shock waves (M < 1.7). However, the calculated values are
Structure
and basic
properties...
21
higher than the measured ones at M > 1.7 by approximately 10%, and they lie below measured values for stronger waves ( M > 5). These discrepancies are caused by the fact that the d i a p h r a g m has finite opening time. When the dia p h r a g m opens, a complex threedimensional flow is developed. This flow was studied experimentally by many authors (Glass et al. 1953; Henshall 1957). After the burst of the diaphragm, a jet of the driver gas appears and compres sion waves arise in the driven gas. They catch u p with one another and form a shock wave whose strength increases. There is a shockformation distance where the velocity of the shock wave reaches its m a x i m u m value. The formation distance increases with the increase of the pressure ratio across the d i a p h r a g m and depends on its opening time. At first the shock front is curvilinear, then it becomes plane due to multiple reflections from the walls of the shock tube. There are theoretical papers that treat the formation of a shock wave in a shock tube. White (1958) assumed that at first a compression wave propagates through a driven gas. Then the compression wave becomes a shock wave at some distance from the d i a p h r a g m . T h e calculated values of shock velocity coincide with the measured ones at M < 5. Ikui (Ikui 1969) proposed a onedimensional multigraded model of shock formation which is a modification of White's model. In accordance with this model, compression waves coalesce with the shock wave. A breakup of an arbitrary discontinuity takes place each time. The number of interactions is infinitely high. The computations were fulfilled u p to the value of the pressure ratio *p equal to 10 7 . The results obtained by Ikui coincide with White's data and that ones calculated by Eq. (24) at the pressure ratio ^ < 10 3 . Kireev (Kireev 1962) and Duntsova (Duntsova et al. 1969) took into ac count the process of diaphragm opening. They assumed the flow to be quasistationary near the diaphragm. Two shock waves separated by a contact discon tinuity arise in accordance with this model. One of the shock waves propagates through a driven gas while the other through a driver one. The latter degen erates into a sonic wave at a later time. The values of the shock velocity were calculated at various points along the shock tube. The velocity of the gas and its pressure are assumed to be constant between the shock wave and contact discontinuity. The flow is assumed to be onedimensional in all of these papers. Petrie et al. (1995) considered axisymmetric flow in the shock tube. The diaphragm was modelled as an opening iris. The rate of the opening was such that the flow area at the diaphragm station was proportional to time until fully open. The contact discontinuity initially has a convex shape but at a later time it becomes concave. The final velocity of the shock wave is higher than that estimated by
Propagation and reflection of shock waves
22
simple onedimensional theory. Experimental studies devoted to the operation of shock tubes can be divided into three groups. The first group includes investigations that are related to shock velocity along the shock tube. As shown by numerous experiments, there are three areas, namely, (1) the region along which the shock velocity increases; (2) the region within which the shock velocity is constant; (3) the region of deceleration along which the shock velocity decreases. The rate of velocity acceleration and the length of each region depend on the nature of the driver gas, the pressure ratio across the diaphragm, the initial pressure in the driven section and so on. The second group contains investigations devoted to the process of the diaphragm opening and its effect on the motion of the shock wave. The full opening time to is equal to 10 — 1000 /is. Its value depends on the size of the diaphragm, its material and initial pressure of the driver gas. Spence (Spence et al. 1964) obtained the following formula for the full opening time of a square diaphragm
Here p is the density of the diaphragm material, 6 is the thickness of the diaphragm, d is the size of a side of the square, k is a proportionality constant. The values of A? are found to be 0.910.93 (Rothkopf et al. 1974, 1976; Simpson et al. 1967). Kireev (Kireev 1962) obtained the following expression for the opening time, by using the theory of elasticity
^'(^"■))'is)' Here
100/is.
(29)
We measured the velocity of disturbances ahead of and behind the second shock wave with the aid of streakcamera records. Then we calculated the sum (112+^2) behind the second shock by using Eq. (29). We compared the measured values of (V2 + C2) with the calculated ones. The data are given in Table 1. The values of the pressure ratio across the diaphragm are also presented and so are the positions that relate to measured values and calculated ones. The calculated values of («2 + C2) agree well with the measured ones. Fig. 19 shows the flow parameters inside the jet against time ahead of the second shock and behind it. The pressure increases as the diaphragm opens. The timederivative ~£ increases by 68%. Compression waves arise as a result. They appear behind the second shock. One can see them at shadowgraphs. If the pressure ratio across the diaphragm is less than the critical value the second shock does
VI
Propagation and reflection of shock waves
V
Fig. 14. Streakcamera record of the flow, the second shock wa¥e is seen, pi = 8.7 kPa.
not appear. Table 2 gi¥es ¥alues of the velocity of disturbances (V), in successive se quence as the diaphragm bursts, for ¥arious initial conditions. Within the dotted frames are gi¥en the ¥alues of the shock wa¥e velocity U if one of the disturbances is assumed to be the shock wave and also calculated ¥alues of the velocity of acoustic disturbances behind the shock wa¥e (tf2 + £2) As is e¥ident from Table 2, the velocity of the disturbances emerging into the driven gas directly behind the presumed shock wa¥e is in good agreement with the cal culated velocity of acoustic disturbances behind the shock wave. Hence, some of the disturbances are weak shock waves (M = 1.031.4) and the others are acoustic waves propagating behind the shock wave. Analysis of the data shows that after the diaphragm has burst, the strength and frequency with which the shock waves appear increase as the bursting time and initial pressure in the
Structure and basic properties...
31
Fig. 15. The reflection of the second shock waYe from the walls of the shock tube (diaphragm 2). Table 1. 1 mm
46 52 57 61
PA/PI
40 65 90 110
(V2 + Co) calc
(v2 + c2) exp
370 380 400 400
340 ± 30 380 ± 40 400 ± 40 410 ±40
\
driYen section decrease and as velocity of sound in the driver section increases. Note that the wave system adopted was constructed under the assumption that the change in flow parameters due to the weak disturbances is sufficiently small, and may be neglected. Within the limits of measurement error, this condition was satisfied: the velocity of each disturbance did not change at a distance of 0.5D2D. How the disturbances observed in the experiment could have arisen will now be considered. As shown by analyzing shadowgraphs of the flow, a set of weak disturbances exists ahead of the second shock wave. These are formed when the jet of driver gas flows past the sharp edges of the leaves of the bursting diaphragm. As is known, weak disturbances propagating counter to the shock front are amplified after interaction with it. The intensity of the acoustic waves that have passed it may be increased a few times. In the driven gas, they must propagate at the local velocity of sound, as observed experimentally. The appearance of shock waves may be represented as follows. As shown by several authors (see e.g. Shtemenko 1972), directly behind the front of the
t,
Propagation and reflection of shock waves
PS
Fig. 16. The position of the second shock wave against time (dia phragm 1); t0 = 440/is; 1  p4/Pl = 230; 2  110; 3  40; ^ 20. second shock the values of all the flow parameters of the driver gas increase a few times at the instant of diaphragm bursting; i.e., in the jet of driver gas, there exists a region of large gradients of density, velocity and pressure, increasing over time. At any instant, the density or pressure gradient may become infinite, with the appearance of a shock wave. The largegradient region moves in the jet of driver gas. Thus, the problem is analogous to the problem of the formation of a shock wave from compression waves of finite amplitude when they propagate. The time for such a formation process to occur will now be calculated in onedimensional flow. Since the compression wave propagates in a nonuniform driver gas, the estimate is performed using a relation between change in density gradient in the compression wave j^{p2x) and the density gradients ahead of (plx) and behind (p2x) the shock wave. This expression is obtained from the compatibility conditions (Shugaev 1983) l_d_ c\ dt
P2x Pi
7+ 1 2 ■
2f)2
11
Plx , PlxPlx
P2x~—
3 /pix "2 \pi
Pi~ Pix \ i cit cu c
+
A v\x \
Ii
da f gig
Pi ) V l ci ) d dt V Pi Here clt is the time derivative of the velocity of sound; vlx is the velocity gradient ahead of wave front. Integration of this expression gives the following value for p2x/pi
Structure and basic properties...
33
Fig. 17. Flow past a slender cylindrical body at a distance of 0.29d from the diaphragm 1, Pl = 3.3 kPa. The time interval between the frames is 12/is.
^ =^ +— Pl
Pl
}?
• (30)
\
J £i12+11 exp (j ( M i  »  » ^ _ s c u g , , ^ dt\ dt + ao
where
1 '
a0 =
'
.
(P3*/Pl)t=0 ~ (Pl*/Pl)t=0 When all the gradients in front of the compression wave are zero, Eq. (30) coincides with the formula proposed by Thomas (1957), which gives the value of p2x in a compression wave propagating in a uniform gas. The compression wave transforms into a shock wave at the instant when p2x  oo, that is
[^r^'riir^* /^(/(^?!?W*)dt dt
^^P
=.
..° (P2,/Pl)t
i
.
= 0(P1*/Pl)t=0'
.
( (
}
Thee time t will now be evaluated express To this end, the raluated from this expression. dependence ters of the gas on the time and distance is linearly ence of the flow parameters
Propagation and reflection of shock waves
34
Fig. 18. The Mach number of the jet against time; a  diaphragm 1, 1P! = 34kPa; 2 12kPa; 5, ^  6 . 7 k P a ; 1, 2, 3 l/d = Q.29\ 4~ 0.54; b  diaphragm %l,2pi = 20 kPa; 5  6 . 7 kPa; 4  3.7 kPa; Z/d = 0.29. approximated: d = fc(x)t + cio(ar), c,t = k(x), vi = a(s)t 4 vio(a), fft = a(^)> vi = £(t)x + A 0 (t), cix = £(*)• Then, taking the obvious relations p\xjp\ = vit/cj, w v n —V2t/c2 — 2t/cl i ^° account, Eq. (31) takes the form
~
and pix/pi
=
(«2t/cl)*=0(vlt/Ci)t=o"
In the conditions of the given experiment, f depends very weakly on t; assuming £0 =const, the integrand is written in dimensionless coordinates: 2T1
(7+!)**§ J?v^
400
J \to L
■.J
t,,
kt0
10
(V2t)t=0 ~
(Vlt)t=0
2
a F
1 2
e x p (  t,UfSo ^ ) r f ( ^
Structure and basic properties...
35
Fig. 19. Flow parameters of the jet (nitrogen) ahead of (a) and behind (6) the second shock wave, 1  p; 2  p\ 3 — v\ 4 ~ c; 5 dp/dt.
where to = d/cio, cio is the initial velocity of sound in the driven section. The integrand is expanded in Taylor series, retaining only the first term. Integration yields 2(kt0)
t\ (7 + l)c 10
2
(32) ({V2t)t=0 ~
(Vlt)t=0)
In our experiments, kt0 « 70m/s 2 ; a/k « 1; the mean value v2t « 4 • 10 6 m/s 2 ; vu « 10 6 m/s 2 , cio lies in the range 100150 m/s. The value of the time t obtained from Eq. (32) lies in the range 2050 ps. The time interval AT between the departure of one shock wave directly from the front of the second shock and the arrival of the subsequent shock wave in the driven gas from the driver gas was determined, using a linear approximation of the shockwave path in the driver gas. As shown by analyzing shadowgraphs of the flow, on which waves moving both in the driven and in the driver gas are seen, this approximation changes the time interval by no more than 10%. It is obvious that AT is the maximum time during which shock waves may be formed in the region of large density and pressure gradients between the front of the second shock discontinuity and the contact surface, i.e., t < AT. Table 3 shows the relation between the shockwave velocity and the time interval AT. AS is evident from Table 3, AT is always larger than the value of t calculated from Eq. (24). Note that, in calculating t, the value vit ~ the
00
Table 2. Initial conditions
Disturbance velocity V V±20
c4 = 350 m/s po = 1.01 x 105 Pa h = 470/zs
330 330330 360 370 360 370 360 350 370 360 370 3*50 390 390 390 400400 390 420420 410 1 1 I U = 360 U = 390 j ac = 402 wae = 378j K±20
C4 = 735 m/s Po = 3.47 x 104 Pa (N,t) M)+{N,t) HN',t
d± 0+
,j*
Gv>°*{
dt At>o A* dt da4'' dt At>o A* dt da The symbol d/dt denotes the timederivative as apparent to an observer .ving moving with the velocity G relative to the medium. The velocity of displacement U of the wave is defined in an analogous manner U = lim ^ , where Am is a distance along the segment of the normal between the two positions of the wave front: one refers to the instant t and the other to the positions of me wave froui: one refers 10 me instant i and une ouner to the instant t + At in the space x11x22x33, x{{ being the Cartesian coordinates (the instant t + At in the space x x x , x being the Cartesian coordinates (the Eulerian coordinates of a particle). There is a simple relation between G and Eulerian coordinates of a particle). There is a simple relation between G and U. At the instant t the wave front intersects the particle whose coordinates U. At the instant t the wave front intersects the particle whose coordinates x\{alll,a222, a33,t) ,i) the At it intersects with are are x[{a x\{a ,a , a 1,i) and and1l at at 2the2 instant instant tt 3+ + At it intersects the the particle particle with the the a2 + Gv22At,a At, a33 + Gv33At,t + At). The distance Am 1 +GvlAt, coordinates x^a At,a coordinates x\ (a +Gv At. a + Gv At. a + Gv At.t + At\. The distance Am is equal to is equal to
Am =
(dx\
Ur
+
r, ~
9
dx[
\ A UjG niAt
da ) >
m m being being the the components components of of the the external external unit unit normal normal in in the the space space As
x 2 3 xx1xx2xx3..
then rr
dx\ xl
V = Cr + ^
+ vln, U = G+ —Hi Hi = = u G+Vi n, «i„ being the normal component of the particle velocity ahead of the t t>i„ wave front. The vector ^  is not orthogonal to the wave front. In fact, we have
dx\ ax\
dx[ ox\
(33)
Propagation and reflection of shock waves
44
Let ga/3 denote the covariant components of the surface metric tensor on the wave front dxl dx1 Multiplying Eq. (33) by the vector dx1
1
which is tangent to the wave front, we obtain 1
dx\ dx{ _ 1 dt dua /q~ x
dxj dx{ dt dua
via,
Qf is not summed.
This quantity is the velocity component tangent to the wave front ahead of it. Generally it is not equal to zero. If the velocity of propagation is a continuous and differentiable function of time then fluid parameters and their derivatives ahead of and behind the wave front must satisfy the socalled compatibility conditions. Geometrical compatibility conditions follow from the existence of a smooth discontinuity front. Kinematical compatibility conditions are derived from the fact that there is no breakup of discontinuity front. The compatibility conditions may be violated only at different instants of time. If the compatibility conditions are violated then the discontinuity front transforms into two fronts. We proceed to the derivation of the compatibility conditions. 5. Compatibility conditions To begin with, we consider geometrical compatibility conditions. The derivative of a fluid parameter (j> has the following values on both sides of a wave front di _ d<j>i da1
d(j)2 _ dfo
9al
Subtracting the former equation from the latter, we have
da1
di a
du
.da
Let us multiply this relation by the term .» + «W£(i
9ml
dg*" 9[] da™
du« dut du*
in d9mi d[<j>] da" (44) + 9' dua dut du7* ' The derivative with respect to ua for the covariant component of the unit external normal can be determined from the following relation dvi oua
d> t( dua v
{
f.i\x
dv "is
idgu
'
oua
oua
As (Kotchine 1965)
Q^=(9ikTls+glkTis)
— ,
then, with Eq. (40) substituted, dv\
du°"=K*lr"krt
dar
da1 du°
l3a 9il9 bvap at ll
8u«
»»
Let us examine the fourth term on the righthand side of Eq. (44). Using the expression of the covariant derivative nm
_
m
L
we obtain d2an duadu^
da"
Y° a
das
dak
sk " • a.+ " r du° " du* du« da™ _ flo^do* (45) bVaVm + dvP du da*daidak (da''
dai
 \L^9^9"Tbaa
X
Qak
\
+ L^{1H9aP9irbai
9 ^9 dai
aj
dak da*
V{
Kg^)
■)}
duadu" dak

+ vk
da'' da* duT du&
3LW
_g«Pg°Tginh
b gaaP naP , Q^?CT  baxT baX 6 b 6 ?A iX a ==^^2 1 _ €T OU"
Here H # is the mean curvature of the shock front, K is its Gaussian curvature. This is the geometrical compatibility condition of the third order. TVT „.~ \A ±i~~ u : ~ ~ ~ ~ „ + ; ~~ i : u ; i u „ „ i:*.: T?: ± „ r „ n i_x Now„ we comnatibilitv consider the kinematicall compatibility conditions. First of all let the external unit normal to us deduce the expression for the timederivative of the the wave front. As is denned, we have 9< ^ ^W = fti"V = 1. We differentiate this identity with respect to t: ^ ■ !^ii /i V / V+ 2+i 2i/,~ 2i/,(48) (48) / ,  ^—  = ==00. . v ; dt 3 dt dt dt 3 A. Let us write a scalar product of two vectors: the unit normal to the wave rfront and the tangent vector to that one j,
da' dai
n
Trhe h e derivative of this expression pression along a ray is equal to da;; a * " \\
dG it UKjr
=U G 9 a dt V r)it I d^ f)n Jt\d^) \
„(I ^y
a
\
_ rLi or
6
da{
U
m UU> , l,da rii
\\
 r)n ^ + i / 9f}ii. ^ ar " "rnJ i' T
,co\
,52)
,5.
Substituting u n g Eq. JC/q. (51) (5i.) and and Eq. r>q. (52) (52) into into Eq. t/q. (50), (50), we we have have k dv dv*
=
9
df ~dT = °
„„ „ a (( ntt
dG dG 8G
1 kk dvf\ dv" da dv™\ dv? \ da daK
+nm nm +
{d^ ^)w{d^ ^)d^dvm ik dx™ dv? dxf 9 kii^r9' ~dV ~w
„ „ ,,,
M , fc. i/¥W
GVV TV Gvv
»i> (53)
(53)
Propagation and reflection of shock waves
50
It is the expression for the time derivative of the contravariant component of the unit external normal. The timederivative for the covariant component of the normal is equal to dUrn
v k dgmk
. +9mkdv
^r
dt or
k
^r
k
dv[ dvn ap 3G da (54) + GI/.I/T' + ni = 9mk9 dt *""** dua duP ' "' dam In an analogous manner, the timederivative for the component of the normal in the frame xlx2x3 is equal to dr>P___ dt ~
af3 g
( dG_ &4 \du
dv%'
da*
~dT
\
(58)
(59)
Taking into account Eqs. (36) and (53), we obtain the following expression for the third term on the righthand side of Eq. (59) d<j> 9
dt ~
da*'
du] dua
(6G \duP
, „ „ap 9 +9 9
dv? du?
(d[]
i^\ir+L
*
(1)
8V? dx? dv duP
G
\ da>
) d&'
(61)
Substituting ¥ §£■ into Eq. (56), we have
~d2f\ dt
2
= GLP +
'df\ dt
dt
(62)
Propagation and reflection of shock waves
52
The second term on the righthand side of Eq. (62) is equal to d_ 8$ H dt
H(^f)=4(
i(1>
°)+
d2[]
dt2
(63)
In consequence of Eq. (63), Eq. (62) yields:
av
L^G2
2
dt
 2G
dLW dt
■^(f + Gg^OWVli) r
a 0 d[] i zdG _z_
dv\ dv,P '
dv,P d^
si
k
zw
+
dv\ dxi dv duP
d2[<j>] dt2 '
(64)
Eqs. (61), (64) are the kinematical compatibility conditions of the second order. They contain the derivatives for the gas velocity ahead of the wave front just as Eqs. (53)(55) do. Now we consider the kinematical compatibility conditions of the third order. We assume that [. We suppose as in the previous cases that [
2 ds(1+ + + B + d d2 "rfT +2^p) ^pJ 2^V o^r [}\ 2di{idy { „ m^r)> []
^d
(
nd
(
/ ^dN\\
d
\
[m^l  ^dl {^dl \yyjl ■ •' V^'oT)) + y(3)f  3e(y n ( 3 )) 2 + 6e(A"  2H2)(B^)2  96e(l  e4){7H2{K
+ 12e{8# 2 ( A  # 2 )  K2}
 H2)  2K2} + ^ ( 4 2 ) ) 4
+ 96e3HBW(4H2  3K)  p(fl( 2 )) 2 (£( 3 ) + y]'))  1 4 e ( 2 # 2  A')y n ( 3 )  e9af}(BW + yW)(flW + y « ) + 4e