Shock waves and explosions

280 Pages·2004·1.203 MB·English

by  P.L. Sachdev

CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 132
SHOCK WAVES
AND
EXPLOSIONS
© 2004 by Chapman & Hall/CRC
CHAPMAN & HALL/CRC
Monographs and Surveys in Pure and Applied Mathematics
Main Editors
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© 2004 by Chapman & Hall/CRC
CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 132
SHOCK WAVES
AND
EXPLOSIONS
P. L. SACHDEV
CHAPMAN & HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
© 2004 by Chapman & Hall/CRC
C4223 disclaimer.fm Page 1 Monday, May 3, 2004 2:05 PM
Library of Congress Cataloging-in-Publication Data
Sachdev, P. L.
Shock waves and explosions / P.L. Sachdev.
p. cm. — (Chapman & Hall/CRC monographs and surveys in pure and applied
mathematics ; 132)
Includes bibliographical references and index.
ISBN 1-58488-422-3 (alk. paper)
1. Differential equations, Hyperbolic—Numerical solutions. 2. Shock
waves—Mathematics. I. Title. II. Series.
QA377.S24 2004
518'.64—dc22 2004047803
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TO THE MEMORY OF MY PARENTS
© 2004 by Chapman & Hall/CRC
Contents
Preface ix
Acknowledgements x
1 Introduction 1
2 The Piston Problem 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 The Piston Problem: Its Connection with the Blast Wave . . 17
2.3 Piston Problem in the Phase Plane . . . . . . . . . . . . . . . 23
2.4 Cauchy Problem in Relation to Automodel
Solutions of One-Dimensional Nonsteady
Gas Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Uniform Expansion of a Cylinder or Sphere into Still Air: An
Analytic Solution of the Boundary Value Problem . . . . . . 33
2.6 Plane Gas Dynamics in Transformed
Co-ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 The Blast Wave 49
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Approximate Analytic Solution of the Blast Wave Problem
Involving Shocks of Moderate Strength . . . . . . . . . . . . . 56
3.3 Blast Wave in Lagrangian Co-ordinates . . . . . . . . . . . . 79
3.4 Point Explosion in an Exponential Atmosphere . . . . . . . . 90
3.5 Asymptotic Behaviour of Blast Waves at a High Altitude . . 99
3.6 Strong Explosion into a Power Law Density Medium . . . . . 103
3.7 Strong Explosion into Power Law Nonuniform Medium: Self-
similar Solutions of the Second Kind . . . . . . . . . . . . . . 110
3.8 Point Explosion with Heat Conduction . . . . . . . . . . . . . 118
3.9 The Blast Wave at a Large Distance . . . . . . . . . . . . . . 130
vii
© 2004 by Chapman & Hall/CRC
viii Contents
4 Shock Propagation Theories: Some Initial Studies 137
4.1 Shock Wave Theory of Kirkwood and Bethe . . . . . . . . . . 137
4.2 The Brinkley-Kirkwood Theory . . . . . . . . . . . . . . . . . 144
4.3 Pressure Behind the Shock: A Practical
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5 Some Exact Analytic Solutions of Gasdynamic Equations
Involving Shocks 153
5.1 Exact Solutions of Spherically Symmetric Flows in Eulerian
Co-ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.2 Exact Solutions of Gasdynamic Equations in Lagrangian Co-
ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.3 Exact Solutions of Gasdynamic Equations with Nonlinear
Particle Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 170
6 Converging Shock Waves 177
6.1 Converging Shock Waves: The Implosion Problem . . . . . 177
6.2 Spherical Converging Shock Waves: Shock
Exponent via the Pressure Maximum . . . . . . . . . . . . . . 183
6.3 Converging Shock Waves Caused by Spherical or Cylindrical
Piston Motions . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7 Spherical Blast Waves Produced by Sudden Expansion of a
High Pressure Gas 195
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.2 Expansion of a High Pressure Gas into Air:
A Series Solution . . . . . . . . . . . . . . . . . . . . . . . . 197
7.3 Blast Wave Caused by the Expansion of a High Pressure
Gas Sphere: An Approximate Analytic Solution. . . . . . . 208
8 Numerical Simulation of Blast Waves 225
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.2 A Brief Review of Di(cid:11)erence Schemes for Hyperbolic
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
8.3 Blast Wave Computations via Arti(cid:12)cial Viscosity . . . . 233
8.4 Converging Cylindrical Shock Waves . . . . . . . . . . . . . . 242
8.5 Numerical Simulation of Explosions Using TotalVariation
Diminishing Scheme . . . . . . . . . . . . . . . . . . . . . . . 253
References 265
© 2004 by Chapman & Hall/CRC
Preface
Ihavebeeninterestedinthetheoryofshockwaves foralongtime. Itstarted
some three decades ago when I published some work on explosions. Since
then my interests have diversi(cid:12)ed but nonlinearity has remained the focus
of all my exertions. I returned to the explosion phenomenon a few years
ago when the Defence Research and Development Organisation (DRDO),
Ministry of Defence, suggested that I write the present monograph.
I have now looked at the phenomenon of explosions and shock waves in
the larger context of hyperbolic systems of partial di(cid:11)erential equations and
their solutions. Application to explosion phenomenon is the major moti-
vation. My approach is entirely constructive; both analytic and numerical
methods have been discussed in some detail. The historical evolution of the
subject has dictated the sequence and contents of the material included in
the present monograph. A reader with a basic knowledge of (cid:13)uid mechanics
and partial di(cid:11)erential equations should (cid:12)nd it quite accessible.
I have been much helped in the present venture by many collaborators
and students. I may mention Prof. K.T. Joseph, Prof. Veerappa Gowda,
Dr. B. Sri Padmavati, Manoj Yadav and Ejan-ul-Haque. Dr. Eric Lord
went through the manuscript with much care. I must particularly thank
Ms. Srividya for her perseverence in preparing the camera ready copy as
I wrote the manuscript, introduced many changes and struggled to make
the material meet my expectations. I am also grateful to Prof. Renuka
Ravindran and Prof. G. Rangarajan for their support.
I wish to thank Dr. Sunil Nair and Ms. Jasmin Naim, Chapman &
Hall, CRC Press for their immense co-operation as I accomplished this most
fascinating piece of work.
Finally, I must thank my wife, Rita, who has permitted me to indulge in
my writing pursuits over the last two decades. She has during this period
borne the major part of family responsibilities with great fortitude.
Financialsupportthroughtheextra-muralresearchprogrammeofDRDO
and from the Indian National Science Academy is gratefully acknowledged.
ix
© 2004 by Chapman & Hall/CRC
x
Acknowledgements
Thefollowing illustrationsandTablesarereproduced,withpermission,from
the sources listed:
Figure2.1: Rogers, M.H., 1958, Similarity(cid:13)owsbehindstrongshockwaves,
Quart. J. Mech. Appl. Math., 11, 411.
Figures 2.2 and 2.3: Kochina, N.N. and Melnikova, N.S., 1958, On the
unsteady motion of gas driven outward by a piston, neglecting the
counter pressure, PMM, 22, 622.
Figure 2.4: Grigorian, S.S., 1958a, Cauchys problem and the problem of
a piston for one-dimensional, non-steady motions of a gas (automodel
motion), PMM, 22, 244; Grigorian, S.S., 1958b, Limiting self-similar,
one-dimensional non-steady motions of a gas (Cauchys problem and
the piston problem),PMM, 22, 417.
Figures 2.5-2.7: Sachdev, P.L. and Venkataswamy Reddy, A., 1982, Some
exactsolutionsdescribingunsteadyplanegas(cid:13)owswithshocks,Quart.
Appl. Math., 40, 249.
Tables 2.1-2.3: Rogers, M.H., 1958, Similarity (cid:13)ows behind strong shock
waves, Quart. J. Mech. Appl. Math., 11, 411.
Figure 3.1: Sakurai, A., 1954, On the propagation and structure of the
blast wave II, J. Phys. Soc. Japan, 9, 256.
Figures 3.2-3.5: Bach, G.G. and Lee, J.H.S., 1970, An analytical solution
for blast waves, AIAA J., 8, 271.
Figures 3.6-3.9: Laumbach, D.D. and Probstein, R.F., 1969, A point ex-
plosion in a cold exponential atmosphere, J. Fluid Mech., 35, 53-75.
Figures 3.10 and 3.11: Raizer, Yu.P., 1964, Motion produced in an inho-
mogeneous atmosphere by a plane shock of short duration, Sov. Phys.
Dokl., 8, 1056.
Figure 3.12: Waxman, E. and Shvarts, D., 1993, Second type self-similar
solutions to the strong explosion problem, Phys. Fluids A, 5, 1035.
Figures 3.13-3.15: Reinicke, P. and Meyer-ter-Vehn, J., 1991, The point
explosion with heat conduction, Phys. Fluids A, 3, 1807.
Figure 3.16: Whitham, G.B., 1950, The propagation of spherical blast,
Proc. Roy. Soc. A, 203, 571.
Table 3.1: Taylor, G.I., 1950, The formation of a blast wave by a very
intense explosion, I, Proc. Roy. Soc. A, 201, 159.
© 2004 by Chapman & Hall/CRC
Acknowledgements xi
Tables 3.2-3.4: Sakurai, A., 1953, On the propagation and structure of the
blast wave I, J. Phys. Soc. Japan, 8, 662.
Figures 6.1-6.3: Chisnell, R.F., 1998, An analytic description of converging
shock waves, J. Fluid Mech., 354, 357.
Figures6.4and6.5: VanDyke, M.andGuttmann,A.J.,1982, Theconverg-
ing shock wave from a spherical or cylindrical piston, J. Fluid Mech.,
120, 451-462.
Table 6.1: Fujimoto, Y. and Mishkin, E.A., 1978, Analysis of spherically
imploding shocks, Phys. Fluids, 21, 1933.
Tables 6.2 and 6.3: Van Dyke, M. andGuttmann, A.J., 1982, Theconverg-
ing shock wave from a spherical or cylindrical piston, J. Fluid Mech.,
120, 451-462.
Figures 7.1-7.3: McFadden, J.A., 1952, Initial behaviour of a spherical
blast, J. Appl. Phys., 23, 1269.
Figures 8.1-8.4: Brode, H.L., 1955, Numerical solutions of spherical blast
waves, J. Appl. Phys., 26, 766.
Figures 8.5-8.12: Payne, R.B., 1957, A numerical method for a converging
cylindrical shock, J. Fluid Mech., 2, 185.
Figure8.13: Sod.,G.A., 1977,Anumericalstudyofaconvergingcylindrical
shock, J. Fluid Mech., 83, 785-794.
Figure 8.14: Liu, T.G., Khoo, B.C., and Yeo, K.S., 1999, The numeri-
cal simulations of explosion and implosion in air: Use of a modi(cid:12)ed
Hartens TVD scheme, Int. J. Numer. Meth. Fluids, 31, 661.
© 2004 by Chapman & Hall/CRC