Blast Waves

Shock Wave and High Pressure Phenomena Founding Editor R. A. Graham, USA Honorary Editors L. Davison, USA Y. Horie, US...

Author: Charles E. Needham

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Shock Wave and High Pressure Phenomena

Founding Editor R. A. Graham, USA Honorary Editors L. Davison, USA Y. Horie, USA Editorial Board G. Ben-Dor, Israel F. K. Lu, USA N. Thadhani, USA

For further volumes: http://www.springer.com/series/1774

Shock Wave and High Pressure Phenomena L.L. Altgilbers, M.D.J. Brown, I. Grishnaev, B.M. Novac, I.R. Smith, I. Tkach, and Y. Tkach : Magnetocumulative Generators T. Antoun, D.R. Curran, G.I. Kanel, S.V. Razorenov, and A.V. Utkin : Spall Fracture J. Asay and M. Shahinpoor (Eds.) : High-Pressure Shock Compression of Solids S.S. Batsanov : Effects of Explosion on Materials: Modification and Synthesis Under High-Pressure Shock Compression G. Ben-Dor : Shock Wave Reflection Phenomena L.C. Chhabildas, L. Davison, and Y. Horie (Eds.) : High-Pressure Shock Compression of Solids VIII L. Davison : Fundamentals of Shock Wave Propagation in Solids L. Davison, Y. Horie, and T. Sekine (Eds.) : High-Pressure Shock Compression of Solids V L. Davison and M. Shahinpoor (Eds.) : High-Pressure Shock Compression of Solids III R.P. Drake : High-Energy-Density Physics A.N. Dremin : Toward Detonation Theory V.E. Fortov, L.V. Altshuler, R.F. Trunin, and A.I. Funtikov : High-Pressure Shock Compression of Solids VII D. Grady : Fragmentation of Rings and Shells Y. Horie, L. Davison, and N.N. Thadhani (Eds.) : High-Pressure Shock Compression of Solids VI J.N. Johnson and R. Chere´t (Eds.) : Classic Papers in Shock Compression Science V.K. Kedrinskii : Hydrodynamics of Explosion C.E. Needham : Blast Waves V.F. Nesterenko : Dynamics of Heterogeneous Materials S.M. Peiris and G.J. Piermarini (Eds.) : Static Compression of Energetic Materials M. Sucéska : Test Methods of Explosives M.V. Zhernokletov and B.L. Glushak (Eds.) : Material Properties under Intensive Dynamic Loading J.A. Zukas and W.P. Walters (Eds.) : Explosive Effects and Applications

Charles E. Needham

Blast Waves With 247 Figures

Charles E. Needham Principal Physicist Applied Research Associates Inc. 4300 San Mateo Blvd, Ste A-220 Albuquerque, NM 87110 USA [email protected]

ISBN 978-3-642-05287-3 e-ISBN 978-3-642-05288-0 DOI 10.1007/978-3-642-05288-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010921803 # Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg, Germany Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

As an editor of the international scientific journal Shock Waves, I was asked whether I might document some of my experience and knowledge in the field of blast waves. I began an outline for a book on the basis of a short course that I had been teaching for several years. I added to the outline, filling in details and including recent developments, especially in the subjects of height of burst curves and nonideal explosives. At a recent meeting of the International Symposium on the Interaction of Shock Waves, I was asked to write the book I had said I was working on. As a senior advisor to a group working on computational fluid dynamics, I found that I was repeating many useful rules and conservation laws as new people came into the group. The transfer of knowledge was hit and miss as questions arose during the normal work day. Although I had developed a short course on blast waves, it was not practical to teach the full course every time a new member was added to the group. This was sufficient incentive for me to undertake the writing of this book. I cut my work schedule to part time for two years while writing the book. This allowed me to remain heavily involved in ongoing and leading edge work in hydrodynamics while documenting this somewhat historical perspective on blast waves. Albuquerque, March 2010

Charles E. Needham

v

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2

Some Basic Air Blast Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Formation of a Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Methods for Generating a Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3

The Rankine–Hugoniot Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Real Air Effects on Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Variable g Rankine–Hugoniot Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Some Useful Shock Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10 11 12 15

4

Formation of Blast Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Taylor Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Sedov Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Nuclear Detonation Blast Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Description of Blast Wave Formation from a Nuclear Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Description of Energy Deposition and Early Expansion . . . . . . 4.5 The 1 KT Nuclear Blast Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Construction of the Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 20 23 23 23 28 33 36

Ideal High Explosive Detonation Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Chapman–Jouget Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Analytic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38 39

5

vii

viii

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5.2 Solid Explosive Detonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 TNT Detonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 High Explosive Blast Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Ideal Detonation Waves in Gasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Fuel–Air Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Gaseous Fuel–Air Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Liquid Fuel Air Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Solid Fuel Air Explosives (SFAE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40 41 48 51 56 57 59 60 63

6

Cased Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Extremely Light Casings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Light Casings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Moderate to Heavily Cased Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Gurney Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Mott’s Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 The Modified Fano Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 First Principles Calculation of Blast from Cased Charges . . . . . . . . . . . 6.5 Active Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 68 69 71 72 75 77 80 81 82 85

7

Blast Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 One Dimensional Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Numerical Representations of One Dimensional Flows . . . . . . 7.2 Two Dimensional Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Numerical Representations of Two Dimensional Flows . . . . . . 7.3 Three Dimensional Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Numerical Representations of Three Dimensional Flows . . . . 7.4 Low Overpressure Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Acoustic Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Non-Linear Acoustic Wave Propagation . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 89 91 92 93 94 94 96 97 99 99

8

Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Boundary Layer Formation and Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Termination of a Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Calculated and Experimental Boundary Layer Comparisons . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 102 103 104 113

9

Particulate Entrainment and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.1 Particulate Sweep-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.2 Pressure and Insertion Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Contents

ix

9.3 Drag and Multi-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Particulate Effects on Dynamic Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Effects of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 122 123 125

10

Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Raleigh-Taylor Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Kelvin–Helmholtz Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Richtmyer–Meshkov Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 132 135 137

11

Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Use of Smoke Rockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Smoke Puffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Painted Backdrops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Overpressure Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Passive Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Self Recording Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Active Electronic Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Density Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Velocity Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Angle of Flow Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Temperature Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Dynamic Pressure Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10 Stagnation Pressure Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.11 Total Impulse Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 140 142 142 144 145 146 147 148 148 149 150 153 154 154

12

Scaling Blast Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Yield Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Application to Nuclear Detonations . . . . . . . . . . . . . . . . . . . . . . . 12.2 Atmospheric Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Examples of Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 159 161 168

13

Blast Wave Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Regular Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Regular Reflection at Non-perpendicular Incidence . . . . . . 13.2 Mach Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Simple or Single Mach Reflection . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Complex Mach Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Double Mach Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Planar Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Single Wedge Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Rough Wedge Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Reflections from Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 172 173 173 175 176 182 182 192 194 198

x

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14

Height of Burst Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Ideal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Nuclear Detonations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Solid High Explosive Detonations . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Range for Mach Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Height of Burst Over Real Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Surface Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Surface Roughness Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Dust Scouring Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.4 Terrain Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Thermal Interactions (precursors) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Free Field Propagation in One Dimension . . . . . . . . . . . . . . . . 14.4.2 Shock Tube Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 Thermal Interactions Over Real Terrain . . . . . . . . . . . . . . . . . . 14.4.4 Simulation of Thermal Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 203 205 216 218 219 222 222 224 227 230 230 232 241 245

15

Structure Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Pressure Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Impulse Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Non Ideal Blast Wave Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Negative Phase Effects on Structure Loads . . . . . . . . . . . . . . . . . . . . . . . 15.5 Effects of Structures on Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 The Influence of Rigid and Responding Structures . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 248 251 254 256 257 261 269

16

External Detonations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

17

Internal Detonations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 17.1 Blast Propagation in Tunnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

18

Simulation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Blast Waves in Shock Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 High Explosive Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Charge Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Use of Exit Jets to Simulate Nuclear Thermal Precursor Blast Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

293 293 294 296 298 302

Some Notes on Non-ideal Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 19.1 Properties of Non-ideal Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Contents

19.2 Combustion or Afterburning Dependency of Non-ideal Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Charge Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.2 Casing Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.3 Proximity of Reflecting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.4 Effects of Venting from the Structure . . . . . . . . . . . . . . . . . . . . 19.2.5 Oxygen Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.6 Importance of Particle Size Distribution in Thermobarics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Modeling Blast Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Non-linear Shock Addition Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Image Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Modeling the Mach Stem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Loads from External Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.1 A Model for Propagating Blast Waves Around Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Blast Propagation Through an Opening in a Wall . . . . . . . . . . . . . . . . 20.5.1 Angular Dependence of Transmitted Wave . . . . . . . . . . . . . . . 20.5.2 Blast Wave Propagation Through a Second Opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

304 304 304 306 306 308 310 312 313 313 314 318 320 320 325 327 328 330

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Chapter 1

Introduction

1.1

Introduction

The primary purpose of this text is to document many of the lessons that have been learned during the author’s more than 40 years in the field of blast and shock. This writing therefore takes on an historical perspective, in some sense, because it follows the author’s experience. The book deals with blast waves propagating in fluids or materials that can be treated as fluids. The intended audience has a basic knowledge of algebra and a good grasp of the concepts of conservation of mass and energy. The text includes an introduction to blast wave terminology and conservation laws. There is a discussion of units and the importance of consistency. This book is intended to provide a broad overview of blast waves. It starts with the distinction between blast waves and the more general category of shock waves. It examines several ways of generating blast waves and the propagation of blast waves in one, two and three dimensions and through the real atmosphere. One chapter covers the propagation of shocks in layered gasses. The book then covers the interaction of shock waves with simple structures starting with reflections from planar structures, then two-dimensional structures, such as ramps or wedges. This leads to shock reflections from heights of burst and then from three-dimensional and complex structures. The text is based on a short course on air blast that the author has been teaching for more than a decade.

C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_1, # Springer-Verlag Berlin Heidelberg 2010

1

Chapter 2

Some Basic Air Blast Definitions

Blast Wave – A shock wave which decays immediately after the peak is reached. This decay occurs in all variables including: pressure, density and material velocity. The rate of decay is, in general, different for each of the parameters. CGS – A system of units based on the metric units of Centimeters, Grams and Seconds. Dynamic Pressure or Gust – The force per unit area caused by the gross motion of the gas. Usually defined as ½ the density times the square of the velocity of the gas. 1 DP ¼ r jU j2 2 Note that this definition makes dynamic pressure a scalar. Mathematically this may be true, but physically the direction of the dynamic pressure is an important characteristic of a blast wave and gaseous flows in general. I therefore prefer, and will use the definition of dynamic pressure to be: DP ¼ 1/2r*|U|*U, this form retains the vector property while providing the proper magnitude of the quantity. Dynamic pressure is sometimes referred to as differential pressure because of the way it is measured. Units are the same as for pressure. Energy Density – see Internal Energy Density Flow Mach Number – The ratio of the flow velocity to the local sound speed. Because this is a ratio, the number is unitless. Although unitless, this should be expressed as a vector, i.e., the direction should be specified. Internal Energy – The heat or energy which causes the molecules of gas to move. This motion may be linear in each of the three spatial dimensions, and may include rotational or vibrational motion. Common units are: ergs, Joules, calories, BTUs, kilotons of detonated TNT. Internal Energy Density – A consistent definition would be the internal energy per unit volume, and would have the same units as pressure. Unfortunately, this term is in common usage as a measure of the internal energy per unit mass of the gas and will be used as such in this book. C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_2, # Springer-Verlag Berlin Heidelberg 2010

3

4

2 Some Basic Air Blast Definitions

Common units are: ergs per gram, Joules per kilogram, calories per gram, BTUs per pound mass. Hertz – Oscillation frequency, 1 cycle/s: 1 Hz. Mass Density – The mass contained in a unit volume. Common units are: grams per cubic centimeter, kilograms per cubic meter, pounds mass per cubic foot. MKS – A system of units based on the metric units of Meters, Kilograms and Seconds. Sometimes referred to as SI or Standard International. Authors note: Before computers were able to use scientific notation, all numbers were stored as fixed point, i.e., there was no exponential notation and numbers were stored as: (nn.nnn). In a 32 bit machine, an artificial decimal point was placed with 5 digits on one side and 4 on the other (plus a sign bit and a parity bit). The smallest number thus represented was 0.0001 and the largest was 107374. All numbers, including intermediate results, had to fit within these bounds. Anything less than the minimum was 0 (underflow) and anything greater than the maximum was infinite (overflow). In order to make hydrodynamic calculations, a system of units was used with Megagram, Kilometer, and Seconds. Thus velocities were in kilometers/second and densities in megagrams/cubic kilometer. Typical velocities and densities were the order of 1 in this set of units. Over Density – The density above or below ambient atmospheric density. Units are the same as density. Overpressure – The pressure above (or below) ambient atmospheric pressure. Units for overpressure are the same as for pressure. (see below) Overpressure is sometimes called gauge pressure or static pressure. Pressure – The force per unit area exerted by a gas having non-zero energy. The force caused by the molecular or atomic linear motion of the gas. Pressure may also be expressed in terms of energy per unit volume. See specific internal energy. Common units are: dynes per square centimeter, ergs/cubic centimeter, Pascals (Newtons per square meter), Joules per cubic meter, pounds force per square inch, Torr, bars or atmospheres (not the same). Reflected Pressure – The pressure caused by the reflection of a shock wave from a non-responding surface. This pressure is a maximum when the incident shock velocity is perpendicular to the surface, but is not a monotonic function of the incident angle. Units are the same as for pressure. Shock Mach Number – The ratio of the shock velocity to the ambient sound speed. Because this is a ratio, the number is unitless. Although unitless, this should be expressed as a vector, i.e., the direction should be specified. SI – System International, see MKS above. Specific Internal Energy – The internal energy per unit mass. Common units are: ergs per gram, Joules per kilogram, calories per gram. Specific Heat – The amount of energy added to a fixed mass of material in order to raise the temperature by one unit. In CGS the units of specific heat are ergs/(g*K).

2.1 Formation of a Shock Wave

5

Specific Heat at Constant Pressure – Cp – The amount of energy added to a fixed mass of material in order to raise the temperature by one unit while holding the pressure constant. Units are the same as specific heat. Specific Heat at Constant Volume – Cv – The amount of energy added to a fixed mass of material in order to raise the temperature by one unit while holding the volume constant. Units are the same as specific heat. Stagnation Pressure – Sometimes referred to as Pitot Pressure, Total Pressure or Total Head Pressure. The pressure measured by a stagnation gauge or Pitot tube. Equal to the sum of the overpressure and dynamic pressure. Units are the same as for pressure. The Symbol g – By strict definition this is the ratio of specific heats of the gas. That is, the specific heat at constant pressure divided by the specific heat at constant volume. We may find it convenient to stray from this strict definition in some cases. Unitless because it is a ratio. Always greater than 1.0 because the Cp is always greater than the Cv of a gas. When the gas is held at constant pressure, energy goes into expansion of the gas (the PdV work done by the gas) as well as heating the gas. g is therefore a measure of the potential efficiency of converting the energy added to a gas into work done by the gas. Temperature – A measure of the energy density of a gas based on the mean translational velocity of the molecules in the gas. Common units are: degrees Celsius, degrees Fahrenheit, degrees Rankine, degrees absolute, Kelvins, electron volts.

2.1

Formation of a Shock Wave

Small perturbations of a gas produce signals which propagate away from the source at the speed of sound in the gas. Such signals propagate as waves, sound waves, in the gas. Single frequency sound waves can be described as being sinusoidal. The pressure of a sound wave oscillates about the ambient pressure with amplitude that is equally above and below ambient. The first arrival of a sound signal may be characterized as a weak compressive wave which smoothly rises to a peak and continuously decays back to ambient, continues smoothly below ambient to the same absolute amplitude as the positive deviation, then returns smoothly to ambient; thus the description as sinusoidal. Each oscillation of the wave is accompanied by a small compression and expansion of the gas and a small positive and negative motion of the gas. These motions take place adiabatically. That is, there is no net energy gain or loss in the gas, no net motion and the gas returns to its ambient condition and position after passage of the wave. The net result of the passing of a sound wave does not change the gas in any way. The frequency of the oscillations does not affect the propagation velocity until the period of the sound wave approaches the collision time between molecules of the gas. A quick calculation can quantify that frequency for sea level condition nitrogen. With Avogadro’s number of molecules in 28 g of gas and a sea level

6


density of approximately 1.2 e3 g/cc, there are about 3.0 e 19 particles per cc. Each particle has an average volume of about 3.3 e20 cc. An individual nitrogen molecular diameter is approximately 2.0 e8 cm. At a temperature of 300 K, at a molecular mean velocity of 5.0 e 4 cm/s, the time between collisions is about 1.5 e8 s. Thus the statement that the propagation velocity of a sound wave is independent of its frequency, holds for frequencies less than 108 Hz. All sound waves travel at the speed of sound of the gas. Superposition of different frequency waves does not alter the propagation velocity. Any sound wave may be constructed by multiple superimposed sinusoids. Each frequency component of a complex wave can be described as above for a single frequency wave. Such decomposition is called a Fourier series representation. The wave train can be represented as a sum of sine and cosine functions, such that the amplitude (A) can be represented by: AðtÞ ¼

X

di sinðWi tÞ þ bi cosðWi tÞ

i

As the amplitude of a sound wave is increased, that is, as energy is deposited more rapidly, the energy cannot be dissipated from the source by sound waves, as rapidly as it is deposited. The result is compression of the gas surrounding the source to the point that the resultant compressive heating increases the sound speed in the local gas. Energy is then transmitted at the local speed of sound, which may be greater than the sound speed of the ambient gas. If the dissipation of the energy caused by the expansion of the gas within the compressive wave does not reduce the sound speed of the front of the wave to that of the ambient gas, the energy accumulates at the front and a shockwave results.

2.2

Methods for Generating a Shock Wave

There are many methods for generating a shock wave. One of the earliest man made shock waves was produced by the acceleration of the tip of a whip to supersonic velocity. The acceleration of an object to supersonic velocity generates a shock wave. An airplane or a rocket creates a shock wave as it accelerates beyond the speed of sound. The point of origin of the shock wave is the leading edge or tip of the object. For simply shaped objects, a single shock wave is formed. The ambient air is accelerated as it crosses the shock front. Thus, at just above sonic velocity, the air behind the shock has a velocity in the direction of motion of the object and the entire object is traveling sub sonically relative to the air in which it is embedded. In the case of an object at constant or decreasing velocity, the shock wave spreads from the object and decays in strength with increasing distance from the object. In Fig. 2.1, the results of a three dimensional hydrodynamic calculation of a guided bomb at supersonic velocity are shown. The velocity of the device is 1,400 ft per second in a sea level atmosphere. This velocity corresponds to a Mach number

2.2 Methods for Generating a Shock Wave

7

Fig. 2.1 Calculated threedimensional flow around a guided bomb at Mach 1.25

of 1.25. Shocks are formed at the nose, the guidance fins and at any sudden changes in body diameter. In addition to the shocks formed, the turbulent wake is clearly seen and extends for many meters behind the device. Sudden deposition of energy in a restricted volume will cause a shock wave when the expansion of the deposited energy exceeds the ambient sound speed. Simple examples of such depositions include the sudden release of confined gasses at pressures significantly above ambient. Compression of gasses by the motion or acceleration of a piston in a tube will generate a shock. Detonation of high explosives or mixtures of volatile gasses are the first common sources to be considered. The high explosive and detonable gas cases are accompanied by significant dynamic pressure caused by the acceleration of the source gasses. The shock waves generated by expanding gasses can and have been analyzed by representing the driving mechanism as a spherically expanding piston. A nuclear detonation, while introducing some mass to the flow, is usually treated as sudden deposition of energy with negligible added mass. Two sources of energy deposition without the addition of mass come to mind; these are lightening or electrical discharge and laser focusing. Some practical limitations of the functioning of mechanisms caused by the formation of shock waves can be mentioned here. The forward velocity of a helicopter is limited because the forward moving blade tip cannot exceed the speed of sound in air. If it does, a shock wave forms and causes serious vibration of the blades. High speed trains which travel through tunnels create shock waves

8


which may cause damage to structures near the exit of the tunnel. The shocks are generated by the train acting as a somewhat leaky piston moving through the confined area of the tunnel. The resulting shock strength is proportional to the sixth power of the speed of the train. This provides a rather sharp cutoff of the practical speed of trains in tunnels which is significantly below the speed that the current technology would otherwise allow. A major contribution to the failure of supersonic transport (SST) is the fact that flying faster than sound creates a continuous shock wave, dubbed a sonic boom, which causes irritation to animals and people as well as property damage.

Chapter 3

The Rankine–Hugoniot Relations

The Rankine–Hugoniot relations are the expressions for conservation of mass, momentum and energy across a shock front. They apply just as well to blast waves as to shock waves because they express the conditions at the shock front, which, at this point, we will treat as a discontinuity. Figure 3.1, below, illustrates the one dimensional form of the equations for the conservation of mass, momentum and energy across a shock traveling at shock velocity U, through a gas having ambient conditions of P0, the ambient pressure; r0, the ambient density; u0, an ambient material velocity (assumed to be zero in this derivation) and T0, the ambient temperature. The properties behind the shock (at the shock front) are P, the shock pressure; r, the density of the compressed gas at the shock front; u, the material velocity at the shock front and T, the temperature of the compressed gas at the shock front. The conservation laws apply in any number of dimensions. For ease of this derivation we will use a one dimensional plane geometry with unit cross sectional area. To derive the conservation of mass equation, the mass of the gas overtaken by ~ This mass is ~ in a time interval t is r0 Ut. the shock front traveling at velocity U ~ compressed to a density r in a volume (U– ~ u ) t. The time cancels and we have the conservation of mass equation: ~ ~ ~ rðU u Þ ¼ r0 U The statement of conservation of momentum and energy are equally straight forward. While the equation of state used is a constant g ideal gas formulation, the application of the conservation equations is much more general and applies to variable gamma gasses. The combination of the conservation equations across a shock is referred to as the Rankine–Hugoniot (R-H) relations.


9

10

3 The Rankine–Hugoniot Relations

Fig. 3.1 The conservation equations across a shock

3.1

Real Air Effects on Gamma

The value of g is the ratio of the specific heat at constant pressure to the specific heat at constant volume. As modes of vibration are excited, energy is absorbed with little increase in pressure. The energy added to the gas goes into vibrational motion of the atoms within the molecules. Thus less energy goes into increasing the PdV work done by expansion of the gas at constant pressure, but does increase the energy added at constant volume, thus reducing the value of g. As energy is further added to the gas, rotational energy of the molecules is excited and energy goes into the rotational motion of the molecules. Dissociation of the gas molecules occurs as energy continues to increase. As energy is further added, the gasses become ionized and the energy is expended in freeing electrons. Air is a mixture of real gasses. For many applications the assumption that air is an ideal gas with a constant gamma of 1.4 is a very good approximation. It is important to understand the limitations of this assumption. When the incident blast pressure exceeds about 300 psi (20 bars), the gamma begins to deviate from the constant value of 1.4. Figure 3.2 shows a fit to (g 1) for air as a function of energy density at a number of densities. This fit to Hilsenrath’s data [1] was developed by Larry Doan and George Nickel [2]. Ambient atmospheric energy density is approximately 2.0 e þ 9 ergs per gram at a mass density of 1.225 e 3 g/cc. The densities in Fig. 3.2 thus range from ten times ambient sea level to 106 of sea level. From this figure, we see that a value of gamma of 1.4 is a good approximation for near ambient sea level energy density for a wide range of mass densities. As air is heated, the value of gamma falls at different rates for different densities. The variations in gamma with increasing energy (temperature) are caused by the excitation of vibrational and rotational states of nitrogen and oxygen, the major constituents of air. If the air is heated further, molecular dissociation occurs and eventually the first ionizations of oxygen and nitrogen occur, thus further reducing the value of gamma.

3.2 Variable g Rankine–Hugoniot Relations

11

Fig. 3.2 Gamma minus one as a function of internal energy density for several values of density

Units in Fig. 3.2 are CGS for both internal energy density and density. The range of plotted energy is from about half of ambient atmospheric to 50,000 times ambient. The fit is accurate to within a few percent from below ambient to about 2.0 e þ 12 ergs/g. The Doan–Nickel representation fails for energies above about 2.0 e þ 12 ergs/g. Above 2.0 e þ 12 dissociation and ionization change the constituency of the gas such that the value of (g 1) should rise toward an asymptotic value of 0.6666 and remain there at higher energies. This rise at very high energy density is because the gas is now approaching the behavior of a fully dissociated monatomic gas. The rise in (g 1) near 1.0 e þ 11, is caused by the dissociation of oxygen. The second rise, near an energy level of 4.e þ 11 is the dissociation of nitrogen and the rise near 1.0 e þ 12 is caused by the first ionization of oxygen. The separation of the curves indicates that above about 1.0 e þ 10 ergs/g (1,500 K) the value of (g 1) is dramatically affected by the density. Figure 3.3 shows the temperature as a function of internal energy density for a similar range of air densities. The two changes in slope at energy densities of 8.0 e þ 10 and 5.0 e þ 11 ergs/gm are the result of oxygen and nitrogen dissociation. The temperature of air below about 1,000 K is independent of the density.

3.2

Variable g Rankine–Hugoniot Relations

Because the equation of state used in the derivation of the R-H was a general g law gas, the R-H relations may be applied to any material which can be represented as such a gas. The R-H relations are a very powerful tool for the study of blast waves and

12


Fig. 3.3 Air temperature as a function of energy density at several densities

shock waves in general. Given the ambient conditions ahead of the shock and any one of the parameters of the shock, all other shock parameters are defined. By combining the R-H relations and doing a little algebra several useful relations can be found.

3.2.1

Some Useful Shock Relations

The overpressure is defined as the pressure at the shock front minus the ambient pressure, i.e.: DP ¼ P P0

(3.1)

We use the overpressure, DP, as one of the main descriptors of the shock front. Using this definition we can derive several other characteristics in terms of the ambient conditions in the gas. These relations may also be used to determine the ambient conditions through which a shock is moving when more than one parameter of the shock front is known. The density at a shock front may be found from the value of g, the ambient pressure and density and the overpressure at the shock front. 2g þ ðg þ 1Þ DP r P0 ¼ r0 2g þ ðg 1Þ DP P0

(3.2)

3.2 Variable g Rankine–Hugoniot Relations

13

An interesting consequence of this relation is that the density approaches a finite value as the pressure grows large. Thus for very high pressure shocks, the density behind the shock approaches a limit of ðg þ 1Þ=ðg 1Þ times ambient density. For a g of 1.4, the ratio approaches 6, while for a g of 1.3 the ratio is 7.667 and for monatomic gasses the ratio is only 4. For air below about 300 psi or 20 bars or 20,000,000 dynes/cm2 a value of g of 1.4 may be used with about 99% accuracy. The above relation then becomes: 7 þ 6 DP r P0 ¼ r0 7 þ DP P0

(3.3)

Similarly the magnitude of the shock velocity can be expressed as: U ¼ C0

½g þ 1DP 1=2 1þ 2gP0

(3.4)

Where C0 is the ambient sound speed. For a g law gas, the sound speed may be calculated using the relation: C0 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi gP0 =r0 for the ambient gas

(3.5)

C¼

pffiffiffiffiffiffiffiffiffiffiffi gP=r for the shocked gas:

(3.6)

and:

For g = 1.4, (3.4) reduces to: 6DP 1=2 U ¼ C0 1 þ 7P0

(3.7)

The magnitude of the material or fluid velocity at the shock front can similarly be calculated from the ambient sound speed and pressure and the overpressure at the shock front as: u¼

DP gP0

C0 1þ

½gþ1DP 2gP0

1=2

(3.8)

For g = 1.4 this equation simplifies to: u¼

5DP 7P0

C0 1 þ 6DP 7P0

1=2

(3.9)

14


The magnitude of the dynamic pressure is defined as ½ the density times the square of the fluid velocity. 1 q ¼ ru2 2

(3.10)

We can therefore combine (3.3) and (3.8) above for density and fluid (material) velocity and find the magnitude of dynamic pressure using the equation: q¼

ðDPÞ2 2gP0 þ ðg 1ÞDP

(3.11)

For g = 1.4 this equation reduces to: q¼

5 DP2 2 ð7P0 þ DPÞ

(3.12)

When a shock wave strikes a solid surface and the velocity vector is perpendicular to that surface, the reflected overpressure at the shock front can be represented as: DPr ¼ 2DP þ ðg þ 1Þq

(3.13)

Thus, the reflected overpressure is a simple function of the incident overpressure, the dynamic pressure and g. For a constant g of 1.4, we can eliminate q and express the reflected pressure in terms of the shock front overpressure and the ambient pressure. The reflected overpressure becomes: 7 þ 4DP=P0 DPr ¼ 2DP 7 þ DP=P0

(3.14)

For an ideal g law gas, we can express the temperature of the shock front in terms of the ambient temperature and pressure and the shock overpressure. The equation is: T ¼ T0

! 2g þ ðg 1Þ DP DP P0 1þ P0 2g þ ðg þ 1Þ DP P0

(3.15)

This simplifies for g = 1.4 to: 7 þ DP T DP P0 ¼ 1 þ T0 7 þ 6 DP P0 P0

(3.16)

Another property of a shock which can be calculated using these conservation laws is the stagnation pressure. The stagnation pressure is a measure of the total

References

15

energy density in the flow at the shock front. The pressure, overpressure and temperature are static properties of the gas. They are functions only of the random molecular motions within the gas. They are independent of the mean motion of the gas. The stagnation pressure includes the kinetic energy of the stream wise motion of the gas. The stagnation pressure is the sum of the overpressure and the dynamic pressure. Measurement of the stagnation pressure is accomplished by inserting a probe into the flow such that the pressure sensing element is oriented opposite to the direction of the flow. The insertion of the probe causes a reflection of the shock. Any material striking the pressure sensor must therefore pass through the reflected shock front and is partially stagnated before reaching the probe. The measurement of stagnation pressure is therefore a function of the Mach number of the flow. The flow Mach number can be expressed as: M ¼ u=C or M2 ¼

u2 C2

(3.17)

Substituting the equation for the sound speed (3.5) this becomes: M2 ¼

u2 r gP

(3.18)

When the value of M2 is less than 1 the stagnation pressure can be calculated as:

Pstag

g g1 2 ð g 1Þ ¼P 1þM 2

(3.19)

When M2 is greater than 1, the relation becomes:

Pstag

1 2 n ðgþ1Þog 3ðg1 Þ M2 2 ¼ P4n 2 o 5 2gM g1 ðgþ1Þ gþ1

(3.20)

References 1. Hilsenrath, J., Green, M.S., Beckett, C.W.: Thermodynamic Properties of Highly Ionized air, SWC-TR-56-35. National Bureau of Standards, Washington, DC (1957) 2. Doan, L.R., Nickel, G.H.: A Subroutine for the Equation of State of Air. RTD (WLR) TN63-2. Air Force Weapons Laboratory, (1963)

Chapter 4

Formation of Blast Waves Definition of a Blast Wave

Figure 4.1 below is a cartoon representing a typical parameter found in a blast wave at a time after the shock has separated from the source and a negative phase has formed. This may represent the overpressure, the overdensity or the velocity at a given time, as a function of range. The blast wave is characterized by a discontinuous rise at the shock front followed by an immediate decay to a negative phase. The positive phase of a blast wave is usually characterized by the overpressure and is defined as the time between shock arrival and the beginning of the negative phase of the over pressure. The negative phase may asymptotically approach ambient from below or, more commonly, end with a secondary blast wave which in turn may have a negative phase. In general the over pressure, over density and velocity will have different positive durations. In some cases the positive duration of the dynamic pressure is used as the positive phase duration. The end of the positive phase of the dynamic pressure is determined by the sign of the velocity. The density may be below ambient, but if the velocity is positive, the dynamic pressure will be positive. Remember from the definition of dynamic pressure, the vector character is important; this is the first example. As a blast wave decays to very low overpressures, the signal takes on some of the characteristics of a sound wave. The positive duration of the pressure, density and velocity approach the same value. The magnitude of the peak positive pressure and the peak negative pressure approach the same value. The lengths of the positive and negative phases approach the same value and the material velocity approaches zero.

4.1

The Taylor Wave

The Strong Blast Wave, or Point Source generated Blast Wave have been investigated in detail and solutions provided for special cases of constant g gasses with specified initial density distributions. These solutions became especially important during the development of nuclear bombs in the early 1940s. The initial conditions for the version of this problem which is most applicable to a nuclear detonation places C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_4, # Springer-Verlag Berlin Heidelberg 2010

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18

4 Formation of Blast Waves Blast Wave Parameter vs. Range at a Fixed Time

parameter

Peak Value

End of Positive Phase

Range Negative Phase

Arrival

Fig. 4.1 Cartoon of a blast wave

a finite total energy at a point in a uniform density gas having a gamma of 1.4. (air) The analytic solutions have been provided by Sir Geoffrey Ingram Taylor in 1950 [1], by Hans Bethe [2], Klaus Fuchs, John von Neumann and others in 1947 with a comprehensive analysis of the solution by Leonid Ivanovich Sedov in 1959 [3]. The assumption for this solution is a finite energy source generating a shock wave that has a very high pressure compared to the ambient pressure (infinite shock strength) propagating in a constant gamma compressible fluid. The solutions presented by Sedov include three different geometries (linear, cylindrical and spherical) and three different density distributions: a constant density, a density varying as a power (depending on the geometry) of the radius and a vacuum. A clear and complete explanation of the derivations, and the analytic solutions including comparisons with numerical solutions can be found in [4]. I will illustrate only the spherical solution for the constant density initial conditions. Other solutions are derived and tabulated in [4].

4.2

The Sedov Solution

The solutions presented by Sedov provide analytic solutions which may be readily evaluated using modern Personal Computer (PC) software. I include here the solution provided by Sedov in [3]. This solution, in spherical coordinates, can be used as a validation point for the evaluation of computational fluid dynamics (CFD) codes. Table 4.1 contains a tabulation of Sedov’s original solution to the spherical geometry case for the strong blast wave. This is a self similar solution, which means that the solution is valid at all times after the deposition. The table contains Lambda, which is the fraction of the shock radius, and the values for f, g, and h, the fraction of the shock front values for the velocity, density and pressure respectively, evaluated at the several values of Lambda. Figure 4.2 is a plot of the fractional value of the shock front values for the pressure, density and velocity as a function of shock radius fraction. The shock front values are for the case of ambient density equal to 1, gamma ¼ 1.4, and the shock radius is 1 at a time of 1. This results from an initial energy deposition of 0.851072 ergs as the source.

4.2 The Sedov Solution

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Table 4.1 Tabulation of the Sedov solution in spherical symmetry

Lamda (radius) 1.0000 0.9913 0.9773 0.9622 0.9342 0.9080 0.8747 0.8359 0.7950 0.7493 0.6788 0.5794 0.4560 0.3600 0.2960 0.2000 0.1040 0.0000

f (Velocity) 1.0000 0.9814 0.9529 0.9237 0.8744 0.8335 0.7872 0.7397 0.6952 0.6496 0.5844 0.4971 0.3909 0.3086 0.2538 0.1714 0.0892 0.0000

g (Density) 1.0000 0.8379 0.6457 0.4978 0.3241 0.2279 0.1509 0.0967 0.0621 0.0379 0.0174 0.0052 0.0009 0.0002 0.0000 0.0000 0.0000 0.0000

h (Pressure) 1.0000 0.9109 0.7993 0.7078 0.5923 0.5241 0.4674 0.4272 0.4021 0.3856 0.3732 0.3672 0.3656 0.3655 0.3655 0.3655 0.3655 0.3655

Sedov Solution to the Strong Spherical Blast Wave 1.2

1

Velocity Density Pressure

V/V0

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

R/R0

Fig. 4.2 Velocity, density and pressure fraction of the shock front value as a function of shock radius fraction

There are several features to note in this figure. The velocity monotonically decreases from the shock front value to the value of zero at the origin. The pressure has a finite value at the center even though the density goes to zero at the center. This means that the internal energy density (ergs/gm) is not defined at the origin, thus the name “point source.”

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4.3

4 Formation of Blast Waves

Rarefaction Waves

A good description of the rarefaction wave can be found in [5], and includes physical arguments for the impossibility of a rarefaction shock. A rarefaction wave is generated when a gas is expanded, as apposed to a shock wave which is formed when a gas is compressed or otherwise increased in pressure. During shock formation, energy is being transferred from a source to the gas in which the shock propagates. A rarefaction wave is limited to the energy contained in the gas and is the mechanism by which the gas may transfer information about boundaries or discontinuities to the surrounding gas. The leading edge of the rarefaction wave travels at the local speed of sound and the tail of the rarefaction wave is limited to a velocity of Vr ¼ (C0 ½(g + 1)U) where U is the materialpvelocity. ffiffiffiffiffiffiffiffiffiffiffiffi From energy considerations, the velocity U is limited such that jUjb 2 h0 , where h0 is the initial enthalpy of the gas. All hydrodynamic parameters describing the flow (velocity, density, pressure and sound speed) are functions of x/t. Thus in the transition region, between the leading edge and the trailing edge of the rarefaction wave, all hydrodynamic parameters vary smoothly between the leading edge and the trailing edge. For the one dimensional case, the simple shock tube problem (which is an example of the more general Riemann problem) can be used to demonstrate the formation and propagation of the rarefaction wave in its simplest form. This problem is posed as having a tube with a diaphragm dividing two gasses with Pl, rl, Il on the left side of the diaphragm and Pr, rr, Ir on the right, where Pl > Pr . The density, r, the energy, I, and the g of the gasses may differ in any combination so long as the pressure on the left is greater than the pressure on the right and the pressure on the right is greater than zero. When the diaphragm is removed, a shock wave propagates to the right and a rarefaction wave moves to the left. The head of the rarefaction wave travels to the left at the ambient sound speed of the gas on the left, Cl. The tail of the rarefaction wave travels to the right at a velocity of Vr ¼ g þ2 1 Vm Cl , where Vm is the material velocity behind the shock and Cl is the ambient sound speed of the gas on the left. The velocity to the left of the rarefaction wave is zero, the velocity increases linearly with distance to a value of Vm, the material velocity behind the shock. The velocity remains constant at Vm from the tail of the rarefaction wave to the shock front as shown in Fig. 4.6. To the right of the shock front the velocity is again zero. The velocity of the shock front Vs is greater than Vm and is equal to: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ gÞðPl Pr Þ Vs ¼ þ C2r ; 2rr where Cr is the ambient sound speed to the right of the shock front. Vm can be obtained from Vs using the Rankine–Hugoniot relations discussed in the previous chapter.

4.3 Rarefaction Waves

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The pressure in the rarefaction region is equal to Pl at the head of the rarefaction wave and equal to the shock pressure at the tail of the rarefaction wave. The pressure between these two points is given by:

ð g 1Þ V P ¼ Pl 1 2 Cl

2g g1

;

where V is linearly interpolated between the head and tail of the rarefaction wave. Similarly, the density in the rarefaction wave region may be found using the equation: 2 ðg 1Þ V g1 : r ¼ rl 1 2 Cl Thus, there is a complete analytic solution for the case of the simple shock tube problem for all hydrodynamic parameters within the rarefaction region. In fact, an analytic solution exists for the entire domain. In the following example the high pressure gas in the driver has an initial pressure of Pl ¼ 100 bars, a density of rhol ¼ 1.0 e 2 kg/m3, and an energy density of Il ¼ 2.5 e þ 6 MJ/Kg. The driven gas has a pressure of Pr ¼ 0.01 bars, a density of rr ¼ 1.0 e 3 Kg/m3, and an energy density of Ir ¼ 2.5 e þ 3 MJ/Kg. The gasses are assumed to have a constant gamma of 1.4 for these conditions and an initial velocity of zero everywhere. The separating diaphragm is located at a position of 100 m from the origin. Figures 4.3–4.6 show the pressure, density, energy density and velocity as a function of range at a time of 2 ms for the above described initial conditions. The Riemann Solution for Pressure 120

Pressure (bars)

100 80 60 40 20 0 0

50

100

150

Range (M)

Fig. 4.3 Pressure vs. range at 2 ms

200

250

22

4 Formation of Blast Waves Riemann Solution for Density 0.012

Density (Kg/M^3)

0.010

0.008

0.006

0.004

0.002

0.000 0

50

100

150

200

250

Range (m)

Fig. 4.4 Density vs. range at 2 ms

Riemann Solution for Energy

Energy Density (MJ/Kg)

3.0E+06

2.5E+06

2.0E+06

1.5E+06

1.0E+06

5.0E+05

0.0E+00 0

50

100

150

200

250

Range (M)

Fig. 4.5 Energy density vs. range at 2 ms

pressure discontinuously rises at the shock front, remains constant until the range of the tail of the rarefaction wave, then rises smoothly to the initial value of the left side. The density rises discontinuously at the shock front to the Rankine–Hugoniot value for the compressed gas originally to the right of the diaphragm. The discontinuous drop in density marks the contact discontinuity between the gas originally to the left of the diaphragm and the gas originally to the right.

4.4 Nuclear Detonation Blast Standard [7]

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Riemann Solution for Velocity 4.5E+04 4.0E+04

Velocity (m / s)

3.5E+04 3.0E+04 2.5E+04 2.0E+04 1.5E+04 1.0E+04 5.0E+03 0.0E+00 0

50

100

150

200

250

Range (M)

Fig. 4.6 Velocity vs. range at 2 ms

Another example of a strong rarefaction wave is given in Sect. 4.5.2. In that case the rarefaction wave is generated by the sudden expansion of the blast wave when the detonation wave reaches the surface of the TNT charge.

4.4 4.4.1

Nuclear Detonation Blast Standard [7] Description of Blast Wave Formation from a Nuclear Source

Blast wave formation from a nuclear detonation or an intense laser deposition differs from that of a solid, liquid or gaseous explosive in two main ways. First, the mass of the explosive is negligible compared to that of the air in which the shock is propagating and second, the initial energy densities (and temperatures) are generally much higher. There are several sources which can be used to describe the initial deposition and early growth of nuclear fireballs. The formation of a blast wave following a nuclear detonation is described in detail in [6]. I will only cover a brief description of the initial growth and formation of the blast wave to just after breakaway.

4.4.2

Description of Energy Deposition and Early Expansion

A 1 kt detonation in sea level air is used to illustrate the basic phenomena and timing of the formation of a blast wave. Nuclear reactions occurring during the nuclear

24


detonation create a and b particles, g rays and X-rays. Most of this energy is quickly absorbed in the surrounding materials including high explosive detonation products and a steel case and the energy is re-radiated in the form of X-rays. Most of the re-radiated X-rays are absorbed within a few meters of the source in the surrounding sea level air. Thus a nuclear detonation produces air temperatures of 10s of millions of degrees in a region of a few meters radius. This very hot region initially grows by radiation diffusion at a velocity of approximately 1/3 the speed of light. As the temperature of the gasses cools, the radiative spectrum changes and the peak radiating wavelength shifts from X-ray to ultra-violet with an increasing fraction in the visible light wavelengths. The energy in the visible wavelengths has a very long mean free path in ambient air and is radiated to “infinity.” As the fireball continues to cool, hydrodynamic growth begins to compete with the radiation as a mechanism for expanding and cooling the fireball. The fireball grows, compressing the air into a shock wave which separates from the fireball at a pressure of about 70 bars. When the velocity of the shock front begins to outrun the expanding fireball, this time is referred to as shock “breakaway.” This was an event that could be readily observed on high speed photography of low altitude nuclear detonations and therefore became a method of determining the yield of a detonation. By a time of 10 ms, the nuclear and prompt X-ray radiation has been deposited in the air; primarily within a radius of about 4.5 m. A 4.5 m sphere of sea level air has a mass of approximately half a ton into which the energy of 1,000 tons of TNT has been deposited. For this description we assume that the fireball is a uniform sphere of ambient density air at a temperature of just over 300,000 K and a pressure of 40,000 bars. At 10 ms, no significant hydrodynamic motion has occurred and the primary source of energy redistribution is through radiation transport. At such temperatures radiation is a much more efficient method of moving energy than hydrodynamics even though the material velocities exceed 10 km/s. Any compression of the air caused by expansion is quickly overcome by the radiation front traveling at a few percent of the speed of light. This radiative growth phase continues to a time of nearly 200 ms when the fireball is about 10 m in radius and has “cooled” to less than 150,000 K and a pressure of 3,000 bars. At this point, the formation of a hydrodynamic shock begins and continues to be driven by radiative growth. During this phase, the air is compressed by the expansion into a blast wave. Because the mass of air internal to the shock front is equal to the total ambient air mass engulfed by the shock front, any deviation of the density above ambient near the front must be balanced by a region within the shock bubble which is below ambient. This comes from conservation of mass within the shock radius. Radiative driven expansion of the blast wave continues to a time of about 6 ms when a radius of 38 m has been reached. The shock front begins to separate from the radiating fireball and the peak pressure has dropped to about 70 bars. This phenomenon is referred to as shock breakaway. The shock is, for the first time, distinguishable from the fireball. Let us examine the conditions behind the blast wave at this time. Figures 4.7–4.9 show the overpressure, overdensity and velocity at 6 ms. The shock front has reached a radius of 38 m with a peak pressure of about 70 bars. Behind the front, the pressure decays rapidly to 27 bars at a radius of 32 m and


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OVERPRESSURE OVERPRESSURE DYN/ SG CM × 106

80.0 TIME = 6.000E – 03 sec 70.0 60.0 50.0 40.0 30.0 20.0 10.0 –0 2.5

5.0

7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 42.5

RADIUS CM × 102 SAP 1KT STANDARD 50 CM

Fig. 4.7 1 KT Nuclear overpressure vs. range at a time of 6 ms OVERDENSITY

OVERDENSITY GM / CC × 10 –3

12.0

TIME = 6.000E – 03 sec

10.0 8.0 6.0 4.0 2.0 –.0 –2.0 –4.0

2.5

5.0

7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 42.5


Fig. 4.8 1 KT Nuclear overdensity vs. range at a time of 6 ms

remains at a constant 27 bars throughout the interior of the fireball. The overdensity at the shock front has reached a value of more than six times that of ambient air. The mass compressed into the blast wave comes from the interior of the shock radius, resulting in the density falling below ambient at a radius of 35 m, reaching a value of just over 1% of ambient at a radius of 30 m and remaining at that value throughout the interior of the fireball. The material velocity at this time has

26

4 Formation of Blast Waves VELOCITY 24.0 TIME = 6.0005 – 03 sec

VELOCITY CM / SEC × 104

21.0 18.0 15.0 12.0 9.0 6.0 3.0 –.0 2.5

5.0

7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 42.5


Fig. 4.9 1 KT Nuclear explosion, material velocity vs. range at a time of 6 ms

a peak value at the shock front of 2.2 km/s and decays smoothly to a zero velocity at the center. Thus the pressure remains well above ambient at all points behind the shock front; the positive phase of the overdensity ends only 3 m behind the shock with the remainder of the range falling below ambient. The positive duration of the velocity is the radius of the shock, i.e., the velocity remains positive decaying to zero at the center. All of the material within the shock bubble continues to expand. At a time of 50 ms, the shock front has expanded to about 90 m and an overpressure of 6 bars with the material velocity at the shock front of just under 600 m/s. The velocity decay behind the shock remains smooth, continuous and positive; reaching a value of zero at the center. Figure 4.10 shows that the overpressure remains above ambient throughout the interior of the shock bubble, so no positive duration is yet defined. Figure 4.11 shows the density falling below ambient about 23 m behind the shock front. The shock is now well separated from the edge of the fireball which now extends to a radius of 75 m. The velocity of Figure 4.12 remains positive from the shock front through the edge of the fireball. The fireball will continue to expand to a maximum radius of 100 m at a time of 1/3 of a second. A negative phase has formed in all blast parameters by a time of 500 ms. The significance of the formation of a negative phase is that essentially no more energy can reach the shock front from the source region. In order to reach the positive phase, the energy must transit an adverse pressure gradient and velocity field which is moving inward. Even a shock will be trapped in the negative phase because the sound speed is below ambient, the negative velocity and therefore momentum of the gas into which it is traveling must be overcome. The end of the positive phase continues to increase in range at the ambient speed of sound, meaning the following


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Fig. 4.10 1 KT Nuclear overpressure vs. range at a time of 50 ms

Fig. 4.11 1 KT Nuclear density vs. range at a time of 50 ms

shock must travel even further in its attempt to catch the primary shock. Thus once the negative phase has formed in a free field blast wave, the propagating positive blast wave will be indistinguishable from any other blast wave and the propagation will be independent of the source. Figures 4.13 and 4.14 show the pressure and velocity distribution at a time of 0.5 s. The negative phase may contain shocks generated by the source, as in the case of a TNT detonation. The magnitude and timing of these shocks trapped in the negative phase may provide some indication of the origin of the blast wave.

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Fig. 4.12 1 KT Nuclear material velocity vs. range at a time of 50 ms

Fig. 4.13 1 KT Nuclear overpressure vs. range at a time of 0.5 ms

4.5

The 1 KT Nuclear Blast Standard

The nuclear blast standard is a set of equations and algorithms in a computer program which describes the formation and propagation of the blast wave resulting from the detonation of a 1 kt nuclear device in an infinite sea level atmosphere. The model is a fit to the results of first principles numerical calculations using the best available radiation transport physics and computational fluid dynamics methods.

4.5 The 1 KT Nuclear Blast Standard

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Fig. 4.14 1 KT Nuclear material velocity vs. range at a time of 0.5 ms

The computational results are supplemented by nuclear air blast data taken from a wide variety of sources on dozens of above ground nuclear tests. The model is valid from a time of 10 ms to about 1 min. This corresponds to radii from 4.5 m to nearly 20 km. The 1KT standard describes the blast wave parameters for a spherically expanding wave in a constant sea level atmosphere. It describes the hydrodynamic parameters as a function of radius at a given time after detonation. The three basic parameters of Pressure, P, Density, r and Velocity (speed), U, are defined by the fits to these individual quantities. All other hydrodynamic parameters can be derived from these at any point in radius and time. The Energy Density, I, can be derived from the parameters above by using the general variable gamma gas equation of state. P ¼ ðg 1Þ r I or I ¼ P=ðr ðg 1ÞÞ All other parameters such as Dynamic pressure, Q, material flow Mach number, Mm, Temperature, T or any hydrodynamic parameter are likewise derivable. The Dynamic Pressure, Q, is calculated by; Q ¼ 1=2r U U The flow or material Mach number is the local material speed, U, divided by the local sound speed; where the local sound speed is: sffiffiffiffiffiffi gP Cs ¼ ; r where P and r are the local values of pressure and density.

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The basis for the standard is a simple relationship for the peak blast pressure as a function of radius. This equation is valid for distances from about 5 m to many kilometers and is given below. OPp ðRÞ ¼

A B C þ þ h n oi1=2 R3 R2 R R 1=2 R 1n R0 þ 3 exp 13 R0

where R is the radius and OPp is the peak overpressure at the shock front. For CGS units the constants are: R0 = 4.454E4 A = 3.04E18 B = 1.13E14 C = 7.9E9 Some general characteristics of this equation are that the pressure falls off initially as 1/R3 or volumetrically. This corresponds to the early radiative growth period of the expanding blast wave when the pressure is essentially uniform throughout the interior of the shock. The rate of decay then transitions to a 1/R2 form as the shock separates from the fireball and decays as a surface phenomenon. The last term is the asymptotic form and covers the transition from shock to strong sound wave. An interesting note is that the shock never reaches the asymptotic limit. At a distance of 10 km the rate of decay is R1.2 and even at a distance of 1 earth circumference the rate is R1.1. Figures 4.15 and 4.16 show the overpressure obtained from this equation as a function of range. The plot begins at a range of just over 10 m in Fig. 4.15 and extends to a range of just over 5 km in Fig. 4.16. Over this distance the overpressure decays from 3,000 bars to 0.01 bars. Also shown in these figures is the peak dynamic pressure at the shock front. These values were obtained from the Rankine– Hugoniot relations using the variable gamma equations for air. At small distances the dynamic pressure exceeds the overpressure by more than a factor of 8. The dynamic pressure falls more rapidly than the overpressure, primarily because it is a function of the square of the material velocity. The overpressure and dynamic pressure are equal at a pressure of approximately 5 bars at a range of 100 m. The dynamic pressure falls below the overpressure at all distances beyond 100 m. This crossing point of the overpressure and dynamic pressure is a function of the ambient atmospheric conditions only. This is discussed further in Chap. 11 on shock scaling. Below the 5 bar level, the dynamic pressure continues to fall more rapidly than the overpressure. At an overpressure of 0.17 bars, the dynamic pressure is a factor of 17 smaller. This ratio continues to increase as the shock wave decays toward very low pressures. As the shock wave approaches acoustic levels, the material velocity associated with the propagation goes to zero and the dynamic pressure associated with a sound wave is zero. Figure 4.17 below shows the power law exponent of a nuclear blast wave as a function of its peak overpressure. Notice that above 20,000 psi the exponent is


31

Fig. 4.15 Overpressure and dynamic pressure as a function of Radius for a 1 KT nuclear detonation. (high pressures)

approaching 3. Physically this can be interpreted as the energy being uniformly distributed throughout the volume inside the shock front. Thus, because energy is no longer being added to the system, the pressure falls proportional to the volume increase. Radiation transport ensures that the energy is very rapidly redistributed within the expanding shock, thus maintaining the uniform distribution. The exponent remains below three because energy is being engulfed from the ambient atmosphere

32


Fig. 4.16 Overpressure and dynamic pressure as a function of Radius for a 1 KT nuclear detonation. (low pressures)

as the shock expands. In reality, it is possible for the pressure to fall faster than 1/ R3 if the rate of radiated thermal energy loss is greater than the rate of energy being engulfed by the expanding shock front. As the blast wave continues to decay, the rate of decay approaches 1/R2, but this rate is not reached until the relatively low pressure of 1 bar. At this pressure the blast wave has completely separated from the source, a negative phase is well formed for all blast parameters and the decay is independent of the source. At an exponent of two, the pressure is decaying proportional to the surface area of the expanding shock. Decay of the peak overpressure is continuous and approaches acoustic pressures at very large distances. Even at a pressure level of 0.01 bars, the exponent remains


33

Power Law Exponent vs Overpressure 3.0

2.8

2.6

2.4

Exponent

2.2

2.0

1.8

1.6

1.4

1.2

1.0

10–1

100

101

102

103

104

105

Overpressure (psi) Exponent = – Log(p1/p2) / Log( r1/r2) where r2 = 1.001*r1

Fig. 4.17 Power law exponent as a function of peak overpressure

near a value of 1.2. This is consistent with experimental observations from small charge detonations at high altitude and the propagation of the blast wave to the surface. The front remains a shock wave, a non-acoustic, finite amplitude signal propagating to tens of kilometers [6].

4.5.1

Construction of the Fits

4.5.1.1

Overpressure Fit

The next most important parameter is the radius of the shock front as a function of time. For times less than 0.21 s the following equation is used: Rearly ¼ 24210: t 0:371 ð1: þ ð1:23 t þ 0:123Þ ð1:0 expð26:25 t 0:79ÞÞÞ

34


When the time is greater than 0.28 s, the radius is given by: Rlate ¼ ð1:0 0:03291 t ð1:086ÞÞ ð33897: t þ 8490:Þ þ 8:36e3 þ 2:5e3 alogðtÞ þ 800: t ð0:21Þ and when the time is between 0.21 and 0.28 s the two radii are linearly interpolated using the equation:

R ¼ Rlate ðt 0:21Þ þ Rearly ð0:28 tÞ =0:07 The constants in the above equations give the radius in centimeters as a function of time in seconds. Using the equations for radius as a function of time and peak pressure as a function of radius, all shock front parameters, including distance from the burst, can be derived using the real gas Rankine–Hugoniot relations. At early times the pressure at the point of burst remains above ambient for times less than about 130 ms. The pressure decays smoothly and monotonically from the shock front to the center of burst. The value of the pressure at the burst center is a smooth decreasing function of time, reaching zero at 130 ms. The waveform for the overpressure blast wave for times less than 130 ms is very well fit by a hyperbola passing through the shock front and through the pressure at zero radius. After 130 ms, the overpressure at the center falls below ambient, thus forming a well defined positive duration. The pressure decay remains a smooth decreasing function from the shock front value to the minimum found at the burst center. The hyperbola remains the appropriate fit. As time continues to increase, the overpressure at the center reaches a minimum and begins to rise toward zero (ambient pressure). The hyperbola is then multiplied by the asymmetric S shaped curve given by: rn (4.1) GðrÞ ¼ 1 bc ; where the parameters b, c and n are functions of time, pressure and radius.

4.5.1.2

Overdensity Fit

The overdensity waveform differs from that of the overpressure and velocity in that it has a zero crossing, even at very early times (due to conservation of mass). The overdensity has the following time evolution. 1. The Monotonic decreasing phase. The overdensity drops from the peak value at the shock front to a minimum value (negative) and remains nearly constant to the burst center. 2. The Breakaway phase. The shock begins to separate from the hot under dense fireball. The overdensity decreases from the peak, begins to level off, and then rapidly decreases to a


35

minimum value where it remains nearly constant to the center. This nearly constant region becomes well defined and is referred to in the 1kt standard as the “density well.” This region defines the fireball radius at early times. 3. The Late phase. The shock is separated from the fireball. The overdensity decreases from the peak to a minimum value, increases to nearly zero and then decreases rapidly into the density well. In one dimension this density well persists for many seconds. The pressure in the fireball is ambient and the radial velocities are zero, therefore the fireball does not move. Any small pressure gradients are rapidly dissipated at the speed of sound in the hot fireball, so the pressure remains constant and equal to ambient atmospheric pressure. In the real world, the under dense fireball is buoyant and rises rapidly from the burst point. The shock wave remains centered on the burst point. During the monotonic decreasing phase, the overdensity waveform is fit by the function: ODðrÞ ¼ A þ B expðcrÞ;

(4.2)

where A, B and c are functions of time. The breakaway region is represented by a combination of the propagating shock and the fireball or density well. The propagating shock expands beyond the edge of the fireball and the fireball stops growing. The transition from the trailing edge of the blast wave into the fireball must be carefully handled because the fireball can now be treated as a separate entity and may no longer be centered at the burst point. The sound speed within the fireball is about an order of magnitude greater than the sound speed outside the fireball, therefore any changes within the fireball are very rapidly communicated throughout the fireball and the pressure and temperature within the fireball remain nearly uniform. The pressure throughout the fireball region is the ambient atmospheric pressure. The density and temperature gradients at the edge of the fireball are inversely proportional to one another. The magnitude of the density gradient at the edge of the fireball, while large, does not form a discontinuity. The gradient at the edge of the fireball is determined by the temperature gradient that is sustainable in air. There is a physical limit to the temperature gradient in air which is determined by the thermal conductivity and radiative properties of the air. The late time fit has the same form as the late-time overpressure fit. This means that the long-lasting density well is not defined for times greater than 0.2 s. The overdensity waveform can be attached to the density well at late time by interpolating between the “density well” fit and the overdensity waveform fit for times greater than 0.2 s. 4.5.1.3

Velocity Fit

The general description of the evolution of the velocity waveform is similar to that given for the overpressure waveform. Significant timing and shape differences must

36


be taken into account in the fits. There are five points that determine the waveform at a given time. These are: 1. 2. 3. 4. 5.

The peak velocity at the shock front The radius of the shock The radius at which the velocity goes to zero The minimum velocity (negative phase) The radius at which the minimum velocity occurs

The radius of zero velocity becomes defined at a time of about 0.085 ms, much earlier than for the pressure. The early time waveform, prior to 0.085 s, is given by: UðrÞ ¼ Upeak

r Rpeak

a

;

(4.3)

where Upeak is the material velocity at the shock front, Rpeak is the shock radius and a is a function of time. The switch to the late time form, with an established negative phase, takes place at a time of 0.7 s, and follows the same functional form as for the overpressure.

References 1. Taylor, G.I.: The Formation of a Blast Wave by a Very Intense Explosion, Proceedings of the Royal Society, A, vol. CCI (1950) pp.159–174 2. Bethe, H., Fuchs, K., von Neumann, J, et.al.: Blast Wave, Los Alamos Scientific laboratory Report LA-2000, August, (1947) 3. Sedov, L.I.: Similarity and Dimensional Methods in Mechanics. Academic Press, New York (1959) 4. Kamm, J.R.: Evaluation of the Sedov-von Neumann–Taylor Blast Wave Solution, Los Alamos Scientific Laboratory Report LA-UR-006055, December, (2000) 5. Ya Zel’dovich, B., Yu Raizer, P.: Physics of Shock waves and High Temperature Hydrodynamic Phenomena. Academic Press, New York (1966) 6. Glasstone, Samuel and Dolan, Philip, The effects of Nuclear Weapons, A joint publication of the U.S. Department of Defense and the U.S. Department of Energy, 1977. Accession number: ADA087568 7. Needham, C., Crepeau, J.: The DNA Nuclear Blast Standard (1KT), Systems, Science and Software, Inc., DNA 5648T, January, (1981)

Chapter 5

Ideal High Explosive Detonation Waves

5.1

Chapman–Jouget Relations

One common method of generating a blast wave in air is the detonation of an explosive or an explosive mixture. To begin, I will describe the progression of a detonation wave propagating through a spherical charge of TNT, the expansion of the detonation products and the formation of a blast wave in the surrounding gas. (Air in this case). The Chapman–Jouget conditions are a restatement of the Rankine–Hugoniot relations with the addition of energy at the shock front. The difficulty here is that the equation of state for the detonation products is generally much more complex than a simple gamma law gas. The Chapman–Jouget relations state that the propagation velocity of the detonation front, a shock, is equal to the sum of the sound speed and the material speed of the gas immediately behind the detonation front. Referring back to Fig. 5.1, we can write the Chapman–Jouget form of the conservation laws. The conservation of mass equation becomes: rðD uÞ ¼ r0 D

(5.1)

where D is the shock velocity which is the detonation velocity. At the detonation front the detonation pressure is assumed to be large compared to ambient. For sea level pressures this is a very good assumption because the detonation pressure for most high explosives is at least four orders of magnitude greater than ambient. The conservation of momentum equation becomes: P ¼ r0 Du

(5.2)

The conservation of energy equation, assuming that E E0 and P P0 becomes: 1 1 1 ; (5.3) E ¼ Q þ =2P r0 r where Q is the detonation energy per unit mass of the explosive. C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_5, # Springer-Verlag Berlin Heidelberg 2010

37

38

5 Ideal High Explosive Detonation Waves TNT BURN

P × 10–4 P / Po–1 24

Symbols for similarity solution = Pressure = Density = Velocity

–2 V×10–4 D × 10 cm / sec ρ/ ρ0–1 20 24

16

20

16

12

16

12 D

8

12

4

8

V

0

4

P D

–4

0

0

–4

20

8 P 4 V 0 –4 0 0

2

4

6

8

10

12

14

16

18

20

22

24

Radius × 10–1(cm)

Fig. 5.1 Comparison of CFD results with the analytic solution for a TNT detonation wave

5.1.1

Equation of State

The equation of state becomes more complex because of the elastic properties of the explosive. The simplest representation of these two terms for an ideal explosive is the Landau–Stanyukovich–Zeldovich and Kompaneets [1] form of the equation of state (EOS). This equation of state has the form: P ¼ ðg 1Þ r I þ a rb

(5.4)

where the first term represents the gaseous component of pressure and the second term the elastic contribution. g represents the ratio of specific heats for the detonation products, r is the density of the gas, I is the internal energy density and a and b are constants which vary with the elastic properties of the explosive. For a given explosive ambient density and detonation energy, g, a and b are constants. One advantage of this form, besides its simplicity, is that the constants g, a and b can be changed to represent a wide variety of ideal explosives. One property of this EOS is that for the expanded state of the detonation products, the second term goes to zero and the first term is an ideal gas form. There are several advantages of this form of EOS with regard to use in hydrodynamics codes. The function is smooth and has smooth derivatives. The derivative of pressure with respect to density is always positive. This is important because the sound speed is calculated as the square root of ð@P=@rÞs , and the derivative must be positive. This property of positive derivatives is not, in general, true for the popular JWL form. The JWL equation of state has the form

5.1 Chapman–Jouget Relations

39

P ¼ ðg 1Þ r I þ A expðK1 =rÞ þ B expðK2 =rÞ, where g, A, B, K1 and K2 are unknown constants. A and B may be positive or negative. There are many explosives for which either A or B has a negative value (K1 and K2 are always negative). Because of this, I have found several applications of the JWL EOS for which the pressure is non-monotonic with density and the derivative ð@P=@rÞs therefore goes negative. Any requirement for a sound speed is not satisfied under all conditions with this form. Also note that the first, gaseous component, of the equation of state is identical to that of the LSZK form. Because the energy released per gram at the front is a constant, the detonation pressure of any ideal explosive is independent of the charge size, from less than a gram to more than a kiloton. During the detonation, the detonation front has no information about the size of the charge and the detonation wave is self similar in all respects. Self similar means that the density, temperature, pressure and velocity distribution within the charge can be scaled by the detonation front location and are independent of time. Using these facts and the relatively simple form for the LSZK equation of state, it is possible to integrate the equations of motion analytically to define the parameters behind the detonation wave as a function of position relative to the detonation front. The procedure for integration is described in detail in Lutsky (1965) [1].

5.1.2

Analytic Integration

The LSZK form of the equation of state for the detonation products of any solid high explosive is selected for further comment. P ¼ ðg 1Þr I þ arb , where P is the pressure, r is the density, I is the internal energy density and the constants g, a and b must be determined from external data, preferably experimental data. All of the common equations of state for detonation products contain a term with the same form as the first term in the LSZK formulation. For large expansion ratios, this term becomes dominant and treats the products as an ideal gas with a constant ratio of specific heats (g). One method of determining the value of g for the detonation products is to use a mass weighted average value of the gamma for each of the species present in the detonation products. Unfortunately, the value of gamma is highly dependent on the energy density of the products and to a lesser extent on the density of the gasses. None the less, nearly all popular equations of state for detonation products assume a constant gamma gas at volume expansion ratios greater than about 10. Figure 5.1 shows the results of the analytic integration for a TNT detonation at a time of 200 ms, just before the detonation front reaches the outer radius of the charge. In this figure are compared the results of a one dimensional Lagrangian hydrodynamic computational fluid dynamics (CFD) code with the results of an analytic integration with the LSZK equation of state. In this case the charge is

40

5 Ideal High Explosive Detonation Waves

140 cm in radius and has a mass of 18,000 kg or 20 short tons. This was the charge used for the Distant Plain 1-A event, conducted at the Suffield Experimental Station (SES) in Alberta, Canada. Although large, this is a realistic charge size and is intermediate between the more common 250 pound charges and the large 500 ton TNT charges used in other experiments. The plot was made at a time just prior to the completion of detonation. The detonation front is a few cm inside the radius of the charge. The solid curves are the results of the CFD code and the various symbols represent the results of the analytic integration of the motion equations using the LSZK EOS for closure. All of the CFD calculated peaks fall below the corresponding peaks from the analytic solution. This is because the CFD code, as with any shock capturing scheme, smears the nearly instantaneous rise of the detonation front over several computational zones, thus reducing the peaks. The density is plotted as the relative over density ¼ ðr=r0 1Þ, where r0 is ambient atmospheric density = 1.225 e3 g/cc. The pressure is also plotted as the relative over pressure, with ambient pressure = 1.013 e 6 dynes/cm2. The precise numbers are not too important for this demonstration. The detonation parameters are a function of the loading density and will vary accordingly. For this calculation, the loading density for the TNT was 1.59 g/cc. Let me point out some important characteristics of the conditions at this time. The density at the detonation front is only about 36% above the loading density of the cold TNT. This is in spite of the fact that the pressure at the front, the detonation pressure, is just over 200 kbars (about three million PSI). This demonstrates that the detonation products are not very compressible. The peak material velocity is just over 1.8 km/s, even though the detonation velocity is nearly 7 km/s. The great fraction of the detonation velocity comes from the sound speed at the detonation front. This will be important in Sect. 5.2 which discusses formation of blast waves. The velocity decays from the peak, at the front, to zero at a distance of just under half the radius of the charge. The density and pressure are constant inside this radius and nothing is changing because nothing is moving. The density in this central core is only 20% less than the loading density and the pressure is nearly 47 kbars (690,000 PSI).

5.2

Solid Explosive Detonation

The results of the calculations described in the next sections were obtained using a Lagrangian finite difference code called SAP [2]. For this application SAP was used in one dimensional, spherical coordinates. The initial conditions were obtained from the integration of the LSZK equation of state for TNT (see Sect. 5.1.2). The Lagrangian code used the LSZK equation of state for TNT Detonation products and the Doan Nickel equation of state for air (see Sect. 3.1). Because the code uses a pure Lagrangian technique, no mixing of materials is permitted at the detonation product/air interface. The equations solved in SAP are the partial differential

5.2 Solid Explosive Detonation

41

equations for non-viscous, non-conducting, compressible fluid flow in Lagrangian form. These equations are given below. Conservation of Mass

dr dt

þr x0

du ¼0 dx t

Conservation of Momentum du 1 dp þ ¼ 0 ðno gravityÞ dt x0 r dx t Conservation of Energy dI dV þP ¼ 0 ðno energy sources or sinksÞ dt x0 dt t Equation of State (for closure) P ¼ Pðr; IÞ where r = density in g/cc u = velocity in cm/s P = pressure in dynes/cm2 I = internal energy density in ergs/g V = 1/r = specific volume in cc/g x = Eulerian coordinate in cm x0 = Lagrangian coordinate in cm t = time in seconds and where the subscripts denote what is being held constant in each derivative. The finite difference approximations to the above equations, as used in SAP, are obtained in the usual manner. The fluid is divided into a mesh of fluid elements. Pressures, densities, and internal energy densities are defined at zone centers. Velocities and positions are defined at zone boundaries.

5.2.1

TNT Detonation

As the first example, I will use the TNT detonation described in Sect. 5.2. There is an atmosphere of ambient sea level air surrounding the detonating sphere of TNT. In Fig. 5.2, the detonation wave has broken through the surface of the charge, the detonation is complete. Figure 5.2 is taken at a time when the shock has expanded about 10% beyond the initial charge radius. When the detonation wave reaches the

42

5 Ideal High Explosive Detonation Waves TNT BURN V ×10–5 D × 10–2 CM / SEC D/ DO–1

P × 10–4 P / PO–1 12 10

10

14

8

12

6

10

4

8

2

6

0

4

–2

2

–4

0

D 8 6 P 4 2 V

V

0

P D

–2 0

2

4

6

8

10 12 14 16 RADIUS × 10–4 (CM)

18

20

22

24

Fig. 5.2 TNT hydrodynamic parameters at 10% expansion radius

surface of the charge, the air immediately outside the charge is rapidly accelerated. To get an idea of the magnitude of the acceleration, we can use the equation: du 1 dP ¼ ; dt r dr where r is the ambient air density, P is the detonation pressure and r is the radius. If we choose to evaluate the acceleration over the first centimeter of the expansion (0.7% of the radius), the acceleration is 1.6 e 14 cm/s2. An argument can be made that this is about a factor of two too large because the pressure used to calculate the acceleration should be the average of the detonation pressure and the ambient pressure. The reasoning is that the pressure at the detonation front will decrease rapidly toward ambient as the wave expands. In any case the acceleration is about 1.0 e 11 times the acceleration of gravity. When the detonation front reaches the surface of the charge, a rapid expansion occurs. This expansion causes a rarefaction immediately behind the front. This rarefaction wave travels backwards into the expanding detonation products at the local speed of sound. In Sect. 5.2 we showed that the speed of sound at the detonation front was 5.2 km/s. So the initial inward velocity of the rarefaction wave is 5.2 km/s; however, this is relative to the expanding detonation products. The material velocity of the expanding detonation products is 1.8 km/s; therefore, the initial inward motion of the rarefaction wave is 3.4 km/s. Now we will examine what the initial effects of the expansion and rarefaction have on the properties in the detonation products. Referring to Fig. 5.2, taken at a time when the shock radius is 10% greater than the charge radius, we observe that the material velocity has increased from 1.8 km/s in the detonation front to 7.4 km/s


43

and this occurs at the “shock” front. The material velocity is now greater than was the detonation velocity inside the explosive. The ambient sound speed in atmospheric air is .34 km/s. Thus the shock front velocity during this early expansion is about 7.7 km/s or Mach 22. The rarefaction wave has reached a point approximately 10 cm inside the original radius of the charge. The detonation products have expanded about 13 cm beyond the original charge radius. The air that was originally in the 13 cm shell around the charge has been compressed into a shell less than a cm thick and has a density approaching 0.1 g/cc. The pressure at the shock front is less than 0.1% of the detonation pressure and rises to a peak of about half the detonation pressure just inside the rarefaction wave front. The peak density remains nearly as high as it was at the detonation front. We conclude that the drop in pressure is caused by a reduction of the internal energy caused by the acceleration of the surface of the detonation products. Let us examine the energy distribution and how it has changed since the detonation was complete. The energy released by TNT at the detonation front is 4.2 e 10 ergs/g. As the detonation proceeds through the TNT, the compression of the gasses at the detonation front causes further heating. In this example the specific internal energy reaches 6.0 e 10 ergs/gm at the detonation front, while the energy released upon detonation is 4.2 e 10 ergs/g. The kinetic energy density of the moving material at the detonation front is 1.7 e 10 ergs/g. During the early expansion phase, the peak kinetic energy density has increased to 5.5 e 11 ergs/g and the internal energy density at the expansion front has dropped to 3.0 e 9 ergs/g. Figure 5.3 shows the conditions inside the shock front when the shock has expanded to 2.4 times the original charge radius. The rarefaction wave has not yet reached the center of the charge. The velocity in the central 40 cm or so is still TNT BURN –5 D –1 V ×10 cm / sec DO

4 P –1 × 10 Po 6

6

1200

5 P

5

1000

4

4

800

3

3

600

2

2

400

1

1

200

0

0

–1

– 200

D

0

–1

V

0 0

DPV

40

80

120

160

200

240

280

320

360

400

Radius (cm)

Fig. 5.3 TNT hydrodynamic parameters at an expansion factor of 2.4

440

480

44


zero. Because this region has not changed, the density and pressure have the same values that they had at the time the detonation was completed. The expanding surface region has a velocity peak of 6 km/s.; however, this peak occurs some 40 cm behind the shock front. All of the air between the original 140 cm charge radius and the current shock front position has been compressed into a spherical shell about 12 cm thick. The air continues to be compressed and accelerated by the expanding detonation products. This is demonstrated by the increasing velocity immediately behind the shock front. The momentum and kinetic energy of the detonation products is being transferred to the air as the detonation products expand. The peak velocity has dropped from 7.4 km/s in Fig. 5.2 to 6 km/s at this expansion radius (Fig. 5.3). All the material between 2.9 and 3.3 m is being compressed. From this plot it is difficult to see the radius of the detonation products. The time for Fig. 5.4 was chosen just as the rarefaction wave reached the center of the charge. The density and pressure at the charge center have dropped only a few percent. The shock front has expanded to 2.6 times the initial charge radius. The peak material velocity has dropped to 4.8 km/s about 40 cm behind the shock front while the material velocity at the shock front is 4.2 km/s. The material between the shock front and peak velocity is being uniformly compressed. The radius of the detonation products is approximately 350 cm. All of the air originally between the charge surface and 3.7 m is now compressed into a 20 cm thick spherical shell. As the expansion continues, the density and pressure on the interior of the detonation products drops to below ambient atmospheric pressure. Figure 5.5 shows the hydrodynamic parameters at a radial expansion ratio of 4.5 (to 6.25 m). The spherical shell of air is clearly shown between the shock front at 6.25 m and the detonation products at 5.9 m. Because the calculation results shown CYCLE 32030 TIME 1.45833 × 10–4 SEC. TNT BURN P × 10– 4 P / PO –1 6

V ×10–5 D × 10–2 CM / SEC D/ DO–1

6

12

5

10

4

4

8

3

3

6

2

2

4

1

1

2

0

0

–1

–2

D 5 P

0

V

DPV

–1 0

4

8

12

16

20 24 28 32 RADIUS × 10– 1 (CM)

36

40

Fig. 5.4 TNT hydrodynamic parameters at 2.6 radial expansion factor

44

48


TNT BURN

45 CYCLE 48000 TIME 2.99672 × 10–4 SEC.

P × 10–1 P / PO –1 14

V ×10–5 D × 10–1 CM / SEC D/ DO–1

12 10

P

8

6

12

5

10

4

8

3

6

2

4

1

2

0

0

–1

–2

D 6 4 V

DV

2

P

0 0

1

2

3

4

5 6 7 RADIUS × 10–2 (CM)

8

9

10

11

12


here are from a Lagrangian code, no mixing at the air/detonation products interface is allowed. The spike in density is not realistic but does provide a sharp interface marker. Note that at this time and for some significant amount of time previous to this, the pressure gradient and density gradient at the interface have had opposite signs. This condition gives rise to Raleigh–Taylor instabilities that result in mixing at this interface, thus reducing the gradients in the real world. More will be said about this in Chap. 9. The velocity still shows a peak nearly 1 m behind the shock front. All material between the radius of this peak and the shock front is being compressed. The outward momentum of the expanding high density gasses on the interior causes the detonation products to over-expand. Figure 5.6 shows the parameters at an expansion ratio of 11.7. The detonation products continue to expand even though the interior pressure and density are less than ambient. The pressure profile behind the shock front is taking on some interesting characteristics. The shock front overpressure is 25.5 bars. The overpressure drops to a value of 15 bars at the detonation products interface. The slope of the pressure drops from there to about 10 bars just half a meter behind the interface. This point marks the location of an inward facing shock which is moving outward because the velocity of the expanding detonation products is greater than the propagation velocity of the inward facing shock. The density of the detonation products is less than ambient air density except for a thin shell between 13.5 and 14.2 m. The pressure inward from the inward facing shock is also below ambient. Because the velocity at all points interior to the inward

46

5 Ideal High Explosive Detonation Waves CYCLE 99000 TIME 1.37988 × 10–3 SEC.

TNT BURN

V ×10–5 D × 100 CM / SEC D / DO–1 24 12

P × 100 P / PO–1 24 20

20

10

16

16

8

12

12

6

8

8

4

4

4

2

0

0

–4

–2

0

V P D

VPD

–4 0

2

4

6

8

10 12 14 16 RADIUS × 10– 2 (CM)

18

20

22

24


CYCLE 111979 TIME 6.25000 × 10–3 SEC. TNT BURN V ×10–4 D × 101 CM / SEC D / DO–1 6 16

P × 101 P / PO –1 50 40 30 20

V

V

4

12

2

8

0

4

10

D

–2

0

0

P

–4

–4

–6

–8

–8

–12

–10 P D –20 0

4

8

12

16

20 24 28 32 RADIUS × 10– 2 (CM)

36

40

44

48

Fig. 5.7 TNT hydrodynamic parameters at radial expansion of 26

facing shock front are positive outward, the pressure and density of the interior of the fireball continue to drop. When the air shock has reached a distance of 26 charge radii (Fig. 5.7), the inward facing shock is well formed. The center of the fireball has expanded to the point that the pressure and density are less than 1% of the ambient air values and


47

the center of the fireball is cold, only a few degrees absolute. The radius of the detonation products is 22 m. The peak pressure in the outward moving main shock is about 4 bars. The velocity of the interface of the detonation products is very nearly zero and is about to be swept into the tail of the inward moving shock. The interface will continue to move inward until the inward moving shock reflects from the center and passes the interface on its way out. The material velocity at the main shock front is 470 m/s; however the material velocity of the inward moving shock is 800 m/s, nearly twice that of the outward moving shock front, indicating a much stronger shock. The pressure jump at the inward moving front is less than 0.2 bars, indicating that the density and pressure of the interior of the detonation products is indeed small. Figure 5.8 is taken when the main air shock has reached an expansion radius of 34 charge radii. The inward moving shock has reflected from the center of the charge and is now moving outward. The radius of the detonation products has decreased by more than 10% since the inward moving shock passed the interface and continues to move inward. The shock reflected from the center has a peak overpressure of just over 2.1 bars while the main shock has decayed to a peak overpressure of just under 2.4 bars. Because the main shock has separated from the detonation products and a negative phase has formed between the main shock and the reflected shock, the reflected shock will never catch the main shock but will remain trapped in the negative phase. Once a negative phase has formed between the shock and its source, the shock is said to have separated. From that point on the shock has no connection with its source. Reverberating shocks cannot overcome the negative phase and catch the main shock front. It is not possible to distinguish the origin of the shock by

TNT BURN

CYCLE 116203 TIME 1.04167 × 10–2 SEC. V ×10–4 D × 101 CM / SEC. D / DO–1 6 12

P × 101 P / PO –1 24

4

8

2

4

0

0

8

–2

–4

4

–4

–8

–6

–12

–8

–16

20

P

16 12

V

VD

P

0 –4 0

10

20

30

40

50 60 70 80 RADIUS × 10– 2 (CM)

90

100

Fig. 5.8 TNT hydrodynamic parameters at radial expansion of 34

110

120

48


examining any or all of its parameters at a point beyond this range. For a TNT detonation this is a range of about 15 charge radii and an overpressure of about 10 bars. It is for this reason that high explosives can be used to accurately simulate the effects of nuclear blast interactions with structures. The U.S. has conducted high explosive free air detonations of as much as 4,800 tons in a hemispherical geometry to simulate the effects of about an 8 kiloton nuclear detonation on the surface.

5.3

High Explosive Blast Standard

One of the first attempts to provide the peak overpressure as a function of range from TNT detonations was a calculation by Dr. Harold Brode [3] of the blast wave from a spherical charge of TNT. This is the origin of the well known Brode curves. A compilation and fit to experimental blast measurements made by Charlie Kingery and Gerry Bulmash was reported in 1984 [4]. They collected and correlated the data from literally hundreds of other references on experimental data. This is the origin of the widely accepted and used Kingery–Bulmash (K–B) curves. Their fit to the peak overpressure data is an 11th order polynomial as a function of range. The K–B fits for arrival time, impulse, reflected pressure, shock velocity and several other parameters are high order polynomial fits as a function of range. Because these are fits to experimental data, and because there is very little reliable data for blast overpressures above 1,000 PSI, the fit to overpressure approaches 10,000 PSI as an asymptotic limit, even inside the charge radius where the pressure should be three million PSI. The K–B curves provide an accurate representation of the peak blast parameters as a function of range for ranges greater than about three charge radii. More recent applications have required time resolved blast parameters as a function of range. To answer this need, the TNT standard was developed. A fast running model has been developed which produces the hydrodynamic parameters in the blast wave as a function of range at any time after the detonation of a spherical TNT charge. These computer routines are influenced by the 1kt nuclear standard and the model closely follows the description provided in Chap. 4 on the nuclear standard. The TNT standard is based on the calculation of the detonation of a 1kt (two million pound) sphere of TNT in a sea level atmosphere. As with the nuclear standard, the first principle calculations were conducted with a variety of codes using both Eulerian and Lagrangian methods of computation. The fits are not necessarily to any single calculation, but to the results of a “perfectly resolved” ideal calculation. The first fit developed was for the peak overpressure as a function of range. For a condensed high explosive charge, the peak pressure is the detonation pressure and is constant from the charge center to the edge of the charge. Just outside the charge, the peak pressure does not occur at the shock front but in the expanding detonation products. The peak as a function of range is therefore highly influenced by the

5.3 High Explosive Blast Standard

49 TNT Standard Comparisons

1.0e + 06

Peak Overpressure (Psi)

1.0e + 05

Kingery-Bulmash Data TNT Standard Experimental Data

1.0e + 04

1.0e + 03

1.0e + 02

1.0e + 01

1.0e + 00

1.0e – 01 0.1

1 10 Range ft / (lb**1/ 3)

100

Fig. 5.9 Overpressure vs. range for the TNT standard and Kingery–Bulmash compared with experimental data

massive detonation products. In order to fit this behavior, the overpressure as a function of range is divided into several different regions and each region is fit separately. The transition from one region to another must be continuous, but the derivative dP/dr may be discontinuous. The comparison of the peak overpressure vs. range is shown in Fig. 5.9 for the TNT Standard, the Kingery–Bulmash fit to experimental data and a selection of experimental data from many sources. Note that the TNT standard has a discontinuity in the overpressure fit at a scaled range of 0.1536 ft or about 1.14 charge radii. This is the range at which the shock front pressure exceeds the pressure of the expanding detonation products. The pressure in the expanding detonation products falls as the range to the 4.4 power. This is caused by a factor of one over range cubed for the volumetric expansion and an additional factor of 1.4 caused by the conversion of internal energy density (pressure) to kinetic energy of the expanding detonation products. For ranges greater than this, the shock front pressure is the peak pressure. While Kingery and Bulmash site data at higher pressures than are shown in Fig. 5.9, the data above 1,000 PSI in rapidly varying blast waves are very difficult to measure. The variations of the overpressures at a given range in the experimental data should not be considered as errors or as an indication of the size of the error bars on the data. At high overpressures, the measurements are made in the presence of unstable expanding detonation products which can create variations in pressures

50


of more than a factor of two above 1,000 PSI. At the low overpressures, the differences are readily explained by meteorological and terrain variations for the different experiments. The low pressure range on a given experiment may differ by 10–20% on different radials depending on the wind direction and the slope of the land. Many of the experimental points in this plot have been scaled from detonations of several tons of TNT. It is very difficult to find a test range where the terrain is flat and smooth over distances of miles. Scaling is discussed in Chap. 12. The fit to the density as a function of range for the TNT standard differs significantly from the fits in the nuclear case. In the nuclear case, the mass of the device can be neglected and still provide an accurate representation of the density profile. In the case of TNT, the mass of the TNT dominates the density profile. If we assume no mixing at the edge of the expanding fireball, the detonation products expand to a radius of just less than 2 ft for a one pound charge. This means that the average density of the detonation products in the fireball, when the fireball has stopped expanding, is less than half of ambient air density. This also means that the fireball has cooled to an average temperature of about 700 K. When mixing is included, which is the real world situation, the detonation products may extend to nearly twice that radius, but are mixed with cool air in the outer half of the radius. The instabilities and mixing at the detonation product interface are discussed in Chap. 10. In contrast, the equilibrium radius for a 1 KT nuclear fireball is about 50 m or 1.3 ft per equivalent pound. There is little or no instability at the surface of a sea level nuclear detonation and the equilibrium temperature is the order of 5,000 K. Application of the TNT standard to other explosives can be accomplished by using the TNT “equivalency” of the other explosives. Unfortunately there is no single method of establishing the equivalency of one explosive to another. Common methods currently in use include: pressure, impulse and energy equivalencies, each of which vary as a function of range. Pressure equivalency means that the TNT equivalent yield of the explosive is adjusted as a function of radius (or time) so that the shock front pressure of the TNT fit matches the observed peak pressure at a particular range. This equivalency then changes as a function of range. Impulse equivalency has a similar interpretation, with the effective yield being adjusted as a function of radius so that the impulse curves match. Neither of these methods is readily applied because the overpressure and impulse as a function of distance for pressures above a few hundred PSI, is a strong function of the density, detonation energy and detonation velocity of the explosive. The simplest method of determining the equivalency is to compare the total energy released during detonation and use the ratio of that energy to that from a TNT detonation. Figure 5.10 compares the overpressure vs. range for several common explosives that have been scaled using this energy equivalency. Note that all the curves converge for pressures less than about 10 bars. Note also that there is a significant separation at the 10 m range. The overpressure from an ammonium nitrate fuel oil (AN/FO) mixture falls below the pressure for HMX by about a factor of 2. This difference is primarily caused by the fact that the density

5.4 Ideal Detonation Waves in Gasses NUCLEAR/HE COMPARISONS OVERPRESSURE VS. RANGE

108

NUCLEAR HMX PENTOLITE TNT ANFO

107

PRESSURE (PA)

51

106

105

104 101

102 RANGE (M) HE SCALED TO 1 KT NUCLEAR EQUIVALENT

Fig. 5.10 Comparison of the overpressure as a function of range for the energy equivalent of one kiloton of several solid explosives

and the detonation energy of AN/FO are significantly smaller than for HMX. The overpressure range curves for HMX and pentolite meet and diverge at least twice for pressures above 10 bars. All of the solid explosive overpressures fall below that generated by a nuclear detonation for all pressures above 10 bars.

5.4

Ideal Detonation Waves in Gasses

In this section the emphasis is on the generation of blast waves by the detonation of gaseous mixtures. The details of gaseous detonation phenomena, such as the diamond patterns formed in detonating gaseous mixtures, or the question of transition from deflagration (combustion) to detonation (shock induced combustion) will not be addressed. The assumption here, as it was in the discussion of solid explosives, is that detonation occurs. Detonable gasses will burn under a much broader range of conditions. Burning may be limited by the rate at which oxygen is

52


mixed with the detonable gas. One clear example of such burning was the destruction of the Hindenburg where a large volume of hydrogen (seven million cubic feet) was initiated at the exterior surface and a mixing limited burn resulted. The energy release took place over many seconds and did not produce a blast wave. Of the 36 passengers and 61 crew members aboard, 13 passengers and 22 crew died. Many gaseous fuels will detonate when the appropriate mixture ratio with an oxidizer is available. Some of the more common materials which are gasses at room temperature that will support detonation in air are: hydrogen, methane, propane, ethane, acetylene and butane. The mixture ratio at which the gaseous fuels will support combustion is well defined. The fuel to oxidizer ratio takes on a minimum value when the fuel content is the minimum at which combustion will be supported. This limit is reached when there is just sufficient energy released to support the continued heating of the gas mixture to the ignition temperature of the fuel. This is the lean limit. As the ratio of fuel to oxidizer increases it reaches a point at which there is insufficient oxidizer to support the minimum energy release to ignite the neighboring gas. This is the rich limit. When gaseous fuels are mixed with air, the combustion limits come closer together because the inert nitrogen must be heated as well as the reacting gasses. As inert gasses are added to an otherwise combustible mixture, a point is reached beyond which combustion will not be supported at any mixture ratio. The fuel to oxidizer ratio of a mixture that will support a detonation also has rich and lean limits. These are bounded by the combustion limits and are much more restrictive than the combustion limits. The energy released must be sufficient to support the formation of a shock wave of sufficient strength so the compressive heating of the gas mixture raises the temperature above the ignition temperature of the mixture. Thus for detonation the lean limit is greater and the rich limit is smaller than for combustion. As an example of the blast wave generated by a gaseous mixture, the results of a first principles CFD code of a methane oxygen detonation is used. Figure 5.11 compares the results of the hydrodynamic calculation with the analytic solution for a strong detonation wave. For this calculation, the balloon was filled with a near stoichiometric mixture of methane and oxygen. The time of the plot is just prior to the arrival of the detonation at the outer edge of a spherical balloon. The balloon had a radius of 16.2 m and contained approximately 20 tons of the methane/oxygen mixture. The density of the mixture was 1.1 e 3 g/cc or about 90% of ambient air density. The actual balloon in the experiment for which the calculation was made was therefore lighter than air and was tethered over ground zero. The experiment was conducted in Alberta, Canada and corresponded to the yield and height of burst of the detonation described in Sects. 5.1 and 5.2 (20 tons at 85 ft height of burst). The balloon was over ground zero and an early pulse prematurely detonated the balloon. As a result, only self recording data was obtained. All electronic measurements began after the blast wave had passed. The agreement between the calculation and the analytic solution is not expected to be as good as was the comparison with the TNT detonation because the detonation pressure for TNT is 210 kilobars and the detonation pressure for the methane/

5.4 Ideal Detonation Waves in Gasses

METHANE

53

CYCLE 4000 TIME 6 × 12146 × 10–3 SEC.

P × 101 P / PO –1 5

Symbols for Similarity Solution = Pressure = Density = Velocity

V ×10–4 D × 101 CM / SEC. D / DO–1 8 12

4

10

6

3

8

4

2

6

2

4

0

P

2

–2

V

0

–4

–2

–6

1

P D

0 –1

D V

–2 0

2

4

6

8

10 12 14 16 RADIUS × 10– 2 (cm)

18

20

22

24

Fig. 5.11 Comparison of CFD results with the analytic solution for a methane/oxygen detonation wave

oxygen mixture is 38 bars. The assumption for the analytic solution is that the detonation pressure is large compared to the ambient pressure. The TNT detonation pressure clearly satisfies this assumption but the methane oxygen mixture pressure at 38 times ambient is marginal. Figure 5.11 shows that the results of the calculation match the analytic solution very well. This plot is taken at a time just prior to the detonation wave reaching the outer radius of the balloon. The solid lines are the numerical results and the symbols are the analytic solution. Note that the velocity is zero from the origin to about half the detonation front radius. Inside this region the pressure and density are constant except for a small residual from the detonator at the center. Also note that the relative over density inside the balloon is negative because the mixture density is less than ambient atmospheric density. When the detonation wave reaches the ambient air there is no sudden acceleration as there was in the TNT case above. A weak rarefaction wave travels back toward the center of the balloon. Figure 5.12, taken at a time of just over 18 ms., shows the rarefaction wave as it reaches the center. The air shock is well formed at this time with the pressure remaining above ambient from the shock front to the center of burst. A sudden drop in density marks the interface between the detonation products and air. The detonation products have expanded to over 4 times their original volume. All of the air that was initially between the radius of the balloon and the current radius of the shock front has been compressed into a spherical shell 4 m thick with an outer radius of 30 m. By a time of 30 ms, a weak inward moving shock has formed and is converging on the center. Figure 5.13 shows the hydrodynamic parameters as a function of

54


METHANE

CYCLE 11000 TIME 1.86458 × 10–2 SEC. V ×10–4 D × 101 CM / SEC. D/ DO–1 10 20

P × 100 P / PO–1 12 P 10

8

16

8

6

12

6

4

8

2

4

V D

0

0

P

–2

–4

–4

–8

4 V 2 0

D

–2 0

4

8

12

16

20 24 28 32 RADIUS × 10– 2 (cm)

36

40

44

48

Fig. 5.12 Methane/oxygen hydrodynamic parameters at 1.8 expansion factor

METHANE

CYCLE 16213 TIME 3.00000 × 10–2 SEC. V ×10–4 D × 101 CM / SEC. D/ DO–1 4 16

P × 101 P / PO–1 50 40 V

V

30 20

2

12

0

8

–2

4

10

D

–4

0

0

P

–6

–4

–8

–8

– 10

– 12

P –10

D

–20 0

4

8

12

16

20 24 28 32 RADIUS × 10– 2 (CM)

36

40

44

48

Fig. 5.13 Methane/oxygen hydrodynamic parameters at 2.34 radial expansion factor

radius at this time. The sharp drop in density at a range of 31 m marks the interface of the detonation products and air. The pressure and velocity remain continuous across this boundary making it a true contact discontinuity. The inward moving shock can be seen in the mild rise in density and pressure at a radius of 2 m but is most clearly marked by the large inward material velocity at that point. The inward

5.4 Ideal Detonation Waves in Gasses

METHANE

55

CYCLE 25523 TIME 9.00000 × 10–2 SEC. V ×10–4 D × 101 CM / SEC D/ DO–1 3 12

P × 101 P / PO–1 12 10

2

8

1

4

0

0

4 P

–1

–4

2

–2

–8

–3

–12

–4

–16

8 6

D V V

D P

0 –2

0

10

20

30

40

50 60 70 80 RADIUS × 10–2 (cm)

90

100

110

120

Fig. 5.14 Methane/oxygen hydrodynamic parameters at 4.4 radial expansion factor

velocity of this shock is twice the material velocity at the outward moving shock front. The inward moving shock reflects from the center and dissipates rather rapidly in the fireball. By a time of 90 ms (Fig. 5.14) the shock reflected from the center point has passed through the contact discontinuity at 40 m and has divided into a transmitted shock and a reflected shock. The transmitted shock can be seen at a radius of just over 50 m while the reflected shock is near the 35 m radius. The detonation products have expanded and nearly stabilized at their final radius of 40 m. Inside of this radius the density is essentially constant. The peak shock pressure has fallen to only 1.1 bars. The expansion of the detonation products is complete at a radius of 40 m. The initial radius of the balloon was 16 m. If we take the ratio of the cubes of these radii we get 15.6. The average density of the fireball is 7.0 e5 g/cc or a relative over density of 0.94, in good agreement with the calculated density shown in the figure for the interior of the fireball. By this time the blast wave has formed a negative phase outside of the detonation products. The weak transmitted shock is in the positive phase and is slowly catching the shock front. This shock is about 20 m behind the shock front. This weak shock will eventually catch the leading shock but will be so weak that the perturbation will be barely discernable in the pressure vs. range curve. Figure 5.15 is a comparison of the peak overpressure as a function of radius for the TNT detonation of Sect. 5.2 and the methane oxygen detonation described above. Recall that the detonation pressure of TNT is 2.1 e 10 Pa and is two orders of magnitude above the scale on the figure. The detonation pressure (36 bars) of the methane mixture extends to the radius of the balloon (14 m). At this radius, the peak

56


Fig. 5.15 Comparison of peak overpressure from TNT and methane/oxygen detonations (20 tons)

shock overpressure for the methane detonation exceeds that for TNT by over 40%. The methane shock pressure then drops faster than for TNT and falls below the TNT curve before expanding to two balloon radii. The methane curve crosses the TNT curve at a distance of 80 m and remains above the TNT curve to a pressure of 0.1 bars. This figure illustrates the unique behavior of the shock front pressure as a function of radius for various individual explosives.

5.5

Fuel–Air Explosives

Another method of generating blast waves is the use of fuel–air explosives. In these cases the fuel may be gaseous, liquid or solid. In general a fuel–air explosive begins with a container of fuel. The fuel is dispersed into the ambient atmosphere by some mechanism. The dispersed fuel–air mixture is then ignited. If conditions are right,

5.5 Fuel–Air Explosives Table 5.1 Detonation properties for gaseous fuel air mixtures Fuel Chemical Stoichiometric Detonation Detonation formula fuel % energy (ergs/g) pressure (bars) 7.73 5.35E+11 19.4 Acetylene C2H2 Ethylene C2H4 6.53 5.23E+11 18.6 29.5 1.42E+12 15.8 Hydrogen H2 Methane CH4 9.48 5.55E+11 17.4 Propane C3H8 4.02 5.14E+11 18.6

57

Detonation velocity (km/s) 1.86 1.82 1.97 1.8 1.8

that is, the mixture is detonable and the initiator is within the dispersed cloud of detonable fuel, a detonation may occur. The major advantage to explosive fuel air systems is that the device carries only the fuel. In conventional high explosive devices, the fuel and oxidizer must be carried. Thus a fuel air explosive is much more efficient in the sense that it potentially results in more energy being carried to a target for the mass of explosive delivered. Typical detonation pressures for gaseous mixtures are the order of 30–40 bars when the gasses are well mixed near stoichiometric ratios at ambient pressure and temperature. Table 5.1 contains the detonation pressures for a number of detonable gaseous mixtures. The detonation pressure is the maximum pressure that can be achieved by a gaseous mixture. The pressure decays as the distance from the surface of the cloud increases. Because the mixing is not uniform, FAE devices never reach the potential of the theoretical energy available to form blast waves.

5.5.1

Gaseous Fuel–Air Explosives

One example of a gaseous fuel–air explosive is a simple tank of propane. If the tank is broken, ruptures or leaks into the atmosphere, the propane will mix with the ambient air and may form a detonable cloud. The propane molecule is heavier than air, in addition, the propane coming from a pressurized tank will be cold, due to rapid expansion, thereby enhancing the density. Thus a cloud of recently released propane will stay near the ground and, if the tank were large enough, under gravitational pull, may follow the surface contours of the terrain. If the winds are light, the propane may pool in low spots or flow down sloping terrain. All of this motion increases the mixing which may be further enhanced by winds. Only under specific conditions of confinement or congestion is it possible to initiate a detonation of such a cloud from a simple flame. I am aware of only two such accidental explosions in industrial situations in modern history. It is more likely to detonate if the initiator includes a shock source with a spark or flame. To intentionally use propane as a blast generator, careful consideration must be given to the placement and timing of the secondary initiators. Let us consider the question of timing. If the secondary initiator fires too early, the propane will be fuel rich and will not detonate. If the secondary initiator fires much later, the cloud of

58


propane will have dispersed, mixed, heated and the mixture will be too lean to sustain a detonation or even a fire. The placement of the secondary initiator is just as important. If the timing is “right”, the cloud may have drifted to a location such that the detonator is outside the detonable cloud. Light winds may cause the cloud to divide into pockets of detonable concentration. In this case each pocket must be detonated independently. The trick here is to predict where the pockets might form, which is dependent on the prediction of the local wind. Propane or the gaseous cloud formed by the sudden release of Liquid Natural Gas (LNG) will stay near the ground and flow under gravity if the winds are calm. The LNG cloud is dense only because it is cold. As the LNG cloud heats, it will decrease in density, become lighter than air and disperse in the atmosphere. The source of heating the LNG cloud may be the surface over which it is spilled, the structures or foliage engulfed by the cloud or direct solar heating if the spill takes place during the day. The LNG will not detonate in its liquid state and will not detonate after any significant dispersion. Only a small fraction of the LNG will have a detonable concentration at any given time. The initiation source must then be collocated with the detonable part of the cloud. Methane, hydrogen and other gasses which are lighter than air are very difficult to detonate in free air. These gasses simply rise and disperse rapidly. These gasses may collect inside of buildings in rooms or basements, reach a detonable concentration and present a significant hazard. The detonation pressure obtained in the example of Sect. 5.4 was about 36 bars. This was obtained because the detonating gas was near a stoichiometric mixture of methane and oxygen which gives the highest detonation pressure. For a uniform stoichiometric mixture of methane and air, the detonation pressure is 17 bars or less than half the pressure when detonated in oxygen. The same amount of energy is released per gram of methane in both cases but the energy goes into heating the relatively inert nitrogen gas in the air mixture, thus reducing the average energy density. Table 5.1 lists the detonation characteristics of a few common gasses. The values are given for standard sea level atmospheric conditions of P = 1.01325 e 6 dynes/cm2 and a temperature of 300 K. The stoichiometry is based on the sea level air content of oxygen. Note that all of the detonation pressures are less than 20 bars or 300 PSI. This pressure is the highest that can be obtained from any fuel air explosive mixture and this is only obtained under careful confinement and mixing conditions. The rate at which the pressure decays as a function of range decreases as the distance from the initiation point to the surface of the cloud increases. The energy released between the detonation point and the edge of the cloud is a measure of the effective yield of the blast wave moving in a particular direction. Thus the pressure resulting from a detonation with a long run-up (the distance from the detonation point to the edge of the cloud) decays more slowly than from a detonation with a short run-up. More detail on scaling shock parameters is given in Chap. 12.

5.5 Fuel–Air Explosives

59

For fuel air explosives in which the mixing is not uniform and the distance from the initiation point to the edge of the cloud may vary, the peak pressure will be less than the ideal detonation pressure. Remember that the units of pressure are energy per unit volume, thus if the mixing ratio is less than ideal, less energy will be released than is optimal. The peak pressure that can be propagated into the air blast wave will be accordingly smaller. Because the rate of decay of the shock front overpressure outside the detonation region is inversely proportional to the distance between the initiation point and the edge of the cloud, the pressure decay will vary as a function of the azimuthal angle with the irregularities of the cloud geometry.

5.5.2

Liquid Fuel Air Explosives

In the case of liquid fuel air explosives the fuels are initially liquids with low vapor pressures. Some examples include: hexane, heptane, ethylene oxide and propylene oxide. As with gaseous fuel air explosives, the fluids must be mixed with sufficient air and require a secondary initiator. Many studies have been made to find efficient ways of dispersing the liquid in small droplets uniformly into a volume of air with sufficient oxygen that a detonation will be supported. The detonation is then initiated by one or more secondary charges that are dispersed within the fuel cloud and delayed to some “optimal” time. The detonation proceeds through the vaporized fuel releasing energy and vaporizing the remaining fuel droplets. The energy released by the vaporized droplets does not contribute directly to the detonation front pressure, but does support the continuation of the detonation by adding energy immediately behind the front. If the fuel is dispersed in a perfect hemisphere of uniform fuel density at optimum oxygen concentration, a detonation will be supported in all directions so long as the initiation is within the cloud. If the initiation is at the center of the cloud, the blast wave will propagate uniformly in all directions from the initiation point. This means that approximately half of the blast wave energy will be directed upward and away from any ground level targets. Assuming that a detonation is supported throughout the distance, a larger distance between initiator and cloud edge means that more energy is directed along a line from the initiator to the cloud edge. The energy is very nearly proportional to the length of that line. Energy is deposited as the detonation front progresses. Thus the energy deposited is roughly proportional to the distance over which it is deposited. The definition of the optimal shape for fuel dispersal now becomes dependent on the intent of the blast wave generated. For targets on the ground it is more efficient to generate a near cylindrical cloud with a small height and a large radius parallel to the ground. If the initiation point is near the center of such a cloud, most of the energy will generate a blast wave traveling outward and parallel to the ground. The initiator may purposely be placed near one edge of the cloud. In this case more energy will be directed along the line

60


toward the far side of the cloud and a blast wave in that direction will decay more slowly than in other directions. In some sense this provides a method of directing the blast wave energy and resulting in a shock front that is egg shaped. The detonation pressure of either a gaseous or liquid fuel air explosive is reduced from that of a uniformly mixed gaseous detonation described in Sect. 5.4. There are several reasons for this, but the primary reason is the inherent non-uniformity of the mixture. Not all of the cloud will be at the optimal concentration for support of a detonation. As the detonation proceeds through the variable mixed regions of the cloud, the energy release will increase and decrease with the fuel mixture ratio, but will never exceed the optimal detonation pressure. Thus the average detonation pressure will always be less than optimal.

5.5.3

Solid Fuel Air Explosives (SFAE)

SFAEs have the same advantages as gaseous or liquid FAEs in that the majority of the energy released is due to fuel burning in air and the oxidizer does not need to be carried with the fuel. A major difference between SFAE and other FAEs is that a larger proportion of the delivery weight is in the dispersal charge. In this case there is no secondary initiator and the primary dispersal charge provides the energy for the initiation of the solid fuel. A typical SFAE device consists of a central explosive charge surrounded by a solid fuel packed in a relatively heavy case. Figure 5.16 is a diagram of a simple solid fuel air explosive device. It has a steel case (white) filled with explosive (light gray) which is surrounded by solid fuel (dark gray). The detonator is at the right of the diagram, positioned at the hole in the case. The fuel may be a variety of combustible solids ranging from sugar to fine metal powders or flakes. The operation of a SFAE device begins with the detonation of the explosive charge. The blast wave, generated by the explosive, travels through the surrounding fuel compressing and heating it. The shock then reflects from the case and allows further heating of the fuel as the case breaks and fuel dispersal begins. The hot detonation products

Fig. 5.16 A simple SFAE device geometry

5.5 Fuel–Air Explosives

61

Aluminum Particulate Heat Time vs. Diameter for Different Soak Temperatures no Slip

1.0E + 00

2500 K 3000 K 4000 K

1.0E – 01

Time (sec)

1.0E – 02 1.0E – 03 1.0E – 04 1.0E – 05 1.0E – 06 1.0E – 07 1

10 100 Diameter (microns)

1000

Fig. 5.17 Aluminum particle heating time as a function of particle diameter

from the explosive begin to mix with the fuel and continue heating it. Some of the fuel may react with the detonation products prior to any mixing with air. This reaction adds energy and assists with the further heating and dispersal of the fuel. Because the solid fuels are generally particulates, they retain the heat obtained from shock and early chemical reactions. The particulates are generally denser than the surrounding gasses and will slip relative to the gas. As the case breaks, the particulates and some detonation products stream into the air. If the particles are sufficiently hot, they may react with the oxygen in the air, further heating the air and neighboring particulates. The particulates take a finite amount of time to heat. Figure 5.17 shows the heating time for aluminum particles to reach 2,050 K when immersed in a gas of constant temperature. Note that the heating time increases as the square of the diameter of the particle. A 1 mm particle in a 4,000 K bath takes approximately 1 ms to reach 2,050 K. A 10 mm diameter particle takes 100 ms to heat. The particulates also require a finite amount of time to burn and release their chemical energy to the air. Figure 5.18 shows the results of an analytic model, developed under the supervision of the author, for particulate heating based on the assumption of a constant rate of recession of the surface. The rate of recession is a function of the oxidizer concentration and increases as a cubic function of the oxygen concentration. This plot was generated with the assumption that the oxygen concentration was 20% and follows a curve for the burn time proportional to the square of the diameter of the particle. A number of investigators have been examining the burn rate of various sized particles in laboratory experiments. Beckstead, in a paper presented at the JANNAF symposium in November, 2000, [5] summarized the data from a dozen experimenters and plotted the burn times as a function of particle diameter (Fig. 5.19). The best fit to this data gave a relationship of the burn time proportional to the particle

62


10000

Analytic Model for Aluminum Particle Burn Times Assuming 20% Oxygen Concentration

Burning Time (msec)

1000

100 Time = Cons * D2 10

1

0.1 10

100 Diameter (um)

1000

Fig. 5.18 Aluminum particle burn time vs. particle diameter

Fig. 5.19 Experimental aluminum particle burn time vs. particle diameter

diameter to the 1.99 power. Not only is the slope in agreement with the analytic solution, but the mean experimental values agree to within 1%. This is validation of the analytic result stated above.

References

63

The contribution of the particulate burn energy is behind the shock front. If the burn occurs within the positive duration of the blast wave the added energy contributes to the pressure behind the shock front, extending the positive phase duration and increasing the overpressure impulse. The added energy then has the effect of reducing the rate of decay of the peak overpressure with range. If the energy is added after the positive duration, it will not be able to influence the positive blast wave parameters. More of the implications of particulate burn will be discussed in Chap. 18.

References 1. Lutsky, M.: The Flow Behind a Spherical Detonation in TNT using the Landau–Stanyukovich Equation of State for Detonation Products, NOL-TR 64-40, U.S. Naval Ordnance Laboratory, White Oak, MD, February, 1965 (1965) 2. Whitaker, W.A., et al., Theoretical Calculations of the Phenomenology of HE Detonations, AFWL TR 66-141 vol. 1, Air Force Weapons Laboratory, Kirtland Air Force Base, New Mexico, November, 1966 (1966) 3. Brode, H.L.: A Calculation of the Blast Wave from a Spherical Charge of TNT. Research Memorandum, RM 1965 (1957) 4. Kingery, C.N., Bulmash, G.: Airblast parameters from TNT spherical air burst and hemispherical surface burst. Technical Report ARBRL-TR-02555, U.S. Army Ballistic Research Laboratory, April, 1984 (1984) 5. Beckstead, M.W., Newbold, B.R., Waroquet, C.: A summary of aluminum combustion. In: Proceedings of the 37th JANNAF Combustion Meeting, Nov., 2000 (2000)

Chapter 6

Cased Explosives

The previous chapter dealt with bare charges. In this section we will discuss the effects of casing materials in direct contact with the explosive. These casing materials may range from a light paper or cardboard surround to a thick high strength steel case that may have a mass of many times the explosive mass. In the process of studying and understanding the formation and propagation of blast waves, it became clear that very few explosives were detonated in a bare charge configuration. The case or covering material gets in the way of the blast wave. I found that the better the case material was treated in numerical calculations; the better was the agreement with the blast wave data. Even very light casings modify the close-in development of the blast wave. This section is intended to help understand the role of casing materials in the formation and propagation of blast waves. The casing material, in most explosive devices, can be treated as an inert material that contributes no additional energy to the blast wave. The casing, therefore, will absorb some of the energy released by the explosive as it is accelerated. What fraction of the energy absorbed is a function of the case thickness, case material, explosive properties (such as Chapman–Jouget pressure and detonation energy) and the geometry of the device. The next few sub sections describe the effects for three classes of case mass.

6.1

Extremely Light Casings

An extremely light case is defined here as a case that surrounds an explosive charge and has a mass of 3% or less of the charge mass. This ratio is about the equivalent of a soft drink can filled with TNT. Although this ratio appears small, the effects on air blast may be significant. High speed photography of the detonation of carefully machined spherical charges show the close in effects of even a slight amount of mass on the surface of the charge. After the charges were carefully pressed, measured and machined, C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_6, # Springer-Verlag Berlin Heidelberg 2010

65

66

6 Cased Explosives

each charge was marked with a wax crayon to indicate the charge number. The detonator was placed, very carefully, at the center of the charge and the charge was detonated in air. High speed photography followed the early expansion of the detonation products. The wax number on the surface of the charge could be read even after the charge had expanded to over twice its original diameter. That portion of the surface that was covered by wax, expanded at a slower rate than that of the free surface. The developing blast wave was directly affected by the differential between the accelerations of the detonation product surface. Another, nearly as extreme an example, was for a 256 pound cast bare charge which was suspended by a harness made of seat belt material. Figure 6.1 shows the charge being lifted from the shipping container. Note that several layers of seat belt material overlap at the bottom pole of the sphere. When this charge was center detonated about 15 ft above the ground, many non-uniformities (anomalies) were noted in the air blast measurements near ground zero. As a result of these anomalies, the harness was redesigned so that there was no strap mass in the lower quarter of the charge. A circumferential strap was placed just below the equator of the charge and was attached to six straps spaced equally around the charge and joined above the charge. This arrangement provided an unobstructed path for the blast wave to reach the ground to a distance of about twice the height of burst. Figure 6.2 is a sequence of frames from a high speed camera spaced at approximately 12 ms showing the early expansion for the 256 pound charge in the modified harness. Note that the effects of the mass of the straps can be seen in the first frame after detonation in the upper left of Fig. 6.2. The detonation products have expanded to more than twice the charge diameter. The bands of strapping material just above and below the equator have delayed the expansion of the detonation products.

Fig. 6.1 256 pound charge showing lifting harness

6.1 Extremely Light Casings

67

Fig. 6.2 Photo sequence of 256 pound detonation

The vertical strap aligned with the camera is clearly visible. In the next frame in the sequence, middle left, the detonation products have reached four times the original charge diameter and the vertical strap has perturbed the expansion of the detonation products and has had a direct effect on the early formation of the blast wave. The residual effects of the strapping material can be seen throughout the sequence and continue to influence the shock geometry and all of the hydrodynamic parameters of the blast wave. The peak pressure at the shock front is changed, the flow velocity is modified by the additional mass, and the influence of the detonation products is changed in the timing of their arrival in the positive phase of the blast wave. Figure 6.3 continues the photographic sequence to later time. These photos show the reflection of the blast wave from the ground and the interaction of the reflected wave with the detonation products. In this sequence, the shock front is separating from the detonation products. This sequence also clearly shows the instability of the interface between detonation products and air. These phenomena, reflection and instability, will be discussed in later chapters.

68

6 Cased Explosives

Fig. 6.3 Continued Photo sequence of 256 pound detonation

6.2

Light Casings

Light cases are defined here as cases that have a mass between about 3% of the charge mass to about the same as the charge mass. Figure 6.4 shows the results of a first principles CFD calculation of the detonation of a 750 pound cylindrical charge with a light aluminum case weighing about 25 pounds or just over 3% of the charge mass. The cylinder was placed with the axis vertical and the bottom 3 ft above the ground. The detonation was initiated at the top of the cylinder. At this time the shock front has expanded to a range of about 25 ft near the ground. The white dots in a regular array are numerical measuring points or stations used in the calculation to monitor the hydrodynamic parameters as a function of time. Those points are fixed in space and do not affect the flow. The other white dots are massive interactive particles that represent the casing fragments and are accelerated by drag and gravity and fully interact with the fluid flow, sharing momentum and

6.3 Moderate to Heavily Cased Charges

69

Fig. 6.4 Blast wave and fragments from a lightly cased 750 pound detonation

energy. In this plot, high pressures are in blue and the lowest pressures are red with pressure following the standard spectrum. At this time the fragments are well ahead of the shock and had an initial maximum velocity at the time of case breakup, of about 12,000 ft/s. The total kinetic energy of the case material accounted for about 8% of the energy released by the explosive. As the case mass ratio increases from 0.03 toward a ratio of 1, the velocity of the fragments is reduced and the fraction of the detonation energy transferred to kinetic energy of the fragments increases. At just over 3% of the charge mass, the case fragment kinetic energy was about 12%. When the case mass ratio approaches 1, the kinetic energy fraction approaches 0.5 and the fragment velocities decrease to 7 or 8,000 ft/s.

6.3

Moderate to Heavily Cased Charges

Moderate to heavily cased charges have case to charge mass ratios ranging from 1 to 5 or more. At these mass ratios the case becomes a dominant factor in early blast wave formation. The expansion velocity of the case is reduced to levels of 3,000 ft/s and the fragment kinetic energy may exceed half of the detonation energy of the explosive. The average fragment size increases as the case mass ratio increases. For some 2,000 pound class penetrating warheads the larger fragment masses may exceed a kilogram. Figure 6.5 is a simple example of a cylindrical charge with a moderate steel case and heavy end caps. The detonator is at the bottom of the cylinder. The explosive is uniformly initiated at the bottom of the cylinder, generating a plane detonation wave propagating vertically in the explosive. Typical detonation pressures for high explosives are a few million psi (a few hundred kilobars). A steel case has a typical

70

6 Cased Explosives

Fig. 6.5 Simple cased cylindrical charge with detonator and end caps

strength of 50 KSI (3 kbars) and some specially treated steels may approach a strength of 200 KSI (13 kbars). The typical detonation pressure is more than an order of magnitude higher than the strength of the container. We are thus justified in ignoring the material strength when treating the expansion of the case caused by the passage of the detonation wave. Such heavy cases affect not only the total energy available to blast wave formation, but the geometry of the initial energy distribution and the blast wave formation. For example, the end plate on a heavy case may be blown off as a single large fragment. The heavy cylindrical case behaves as a gun barrel and the explosive products are ejected from the end of the case as the detonation proceeds toward the nose. The momentum of the heavy case slows the expansion in the radial direction to about 1 km/s, while the detonation proceeds at a velocity of typically 8 km/s. Thus the angle formed by the initial expanding case is only 7 from the axis of the device. Figure 6.6 gives the pressure contours produced by a very heavily cased device when it was detonated from the tail in a vertical nose down orientation with the nose 1 ft above the ground. Note that the 100 psi contour is far from symmetric and illustrates the effects of the release of energy from the tail and the delay in radial expansion of the case. The extension of the contours near the ground is the result of shock reflection from the ground. As the blast wave expands, the contours become somewhat more symmetric, but even at the 25 psi level, the shape of the contours remain influenced by the initial energy distribution.


71

25

25 psi 50 psi 100 psi

Height Above Ground (ft)

20

15

10

5

0

0

5

10

15 20 Ground Range (ft)

25

30

Fig. 6.6 Pressure distribution following a cylindrical charge detonation

6.3.1

Fragmentation

Let us look at the early case expansion following the passage of a detonation wave for a cylindrical charge in which the detonation products are in direct contact with the surrounding case. Because the detonation pressure is much higher than the material strength, the initial shock travels through the case thickness and begins acceleration of the case material. The high pressure in the detonation products compresses the case material as is starts to expand and keeps the case material in compression during the expansion until the case reaches nearly twice its original diameter. At a radius of about twice the original case radius, the pressure in the detonation products has fallen by more than an order of magnitude. The acceleration of the case has also fallen by more than an order of magnitude and the case begins to form tensile cracks near the outer surface. A simple comparison of the material strength, the detonation pressure and typical case thicknesses can be used to show that the fraction of energy used to overcome the material strength is less than 1% of the kinetic energy of the case material. With the aid of Fig. 6.7, let us examine the consequences of the statement that the case is in compression during its early expansion. First, as the case expands radially, the outer radius of the case expands to some multiple of its initial radius. For this example I will use a factor of two. There is good experimental evidence that

72

6 Cased Explosives Case Radius

Twice Initial Radius

H. E. Detonation Products

Initial Case Thickness T=0

1/2 Thickness just Prior to Breakup T = T1

Fig. 6.7 Cartoon of an expanding heavy cylindrical case

for charges with moderate to heavy cases, even for high strength steel, the case expands to about twice its original radius before tensile cracking is initiated and case breakup occurs. Because the case is in compression, the density of the case material is at or above the ambient density of the case during this early expansion. The outer edge of the case has moved a distance equal to the initial radius of the case. The case has thinned to approximately half its original thickness during the cylindrical expansion. This means that the inner radius of the case material must have moved a distance equal to the initial case radius plus ½ the case thickness which is greater than the distance moved by the outer radius of the case in the same amount of time. This leads to the observation that the inner part of the case is moving faster than the outer part of the case at the time that case breakup begins. Fragments formed from the inner part of the case will, in general, have larger velocities than fragments formed from the outer case material, while larger fragments will have a velocity between the two extremes. Detailed Computational Fluid Dynamic (CFD) code results are presented in Fig. 6.8 for a steel cased device filled with Tritonal, an aluminized TNT explosive. The case mass was approximately equal to the explosive mass. Note that both the highest and lowest speed fragments are small and that the speed of the larger particles narrows toward a mean velocity as the fragment mass increases.

6.3.2

Energy Balance

For ideal explosives, the total energy released is the detonation energy of the explosive. This energy goes into heating the gaseous detonation products. Pressure


73

Fig. 6.8 Fragment speed as a function of fragment mass

is generated locally and this causes pressure gradients which induce motion of the surroundings. The pressure generated by a given amount of energy depends upon the constituents of the gas and their density. To represent this behavior numerically, an equation of state (EOS) is used to describe the partition of the energy between pressure and internal energy in the form of molecular excitation. One simple form of an equation of state for detonation products was given in Chap. 5 as (5.4) and is that of Landau, Stanyukovich, Zeldovich and Kampaneets (LSZK). P ¼ ðg 1Þ r I þ a rb

(6.1)

Clearly there is a problem with this simple representation in that a non-zero pressure may be generated when the internal energy is zero. If a gas has a finite pressure, it can do work on its surroundings. The gas thus transfers some of its energy to its surroundings, however, if the gas has no energy, it cannot do work on its surroundings. The LSZK EOS thus represents a restricted portion of the possible states that detonation products may have. When a normal detonation takes place, the LSZK representation is a good approximation to the behavior of the gaseous detonation products during the detonation and expansion of the products. Immediately behind the detonation front, the energy released is very efficiently converted to pressure. If we artificially represent the pressure from the LSZK EOS as a polytropic gas pressure with a proportionality constant of a, that is, as P ¼ ða 1Þ r I ;

(6.2)

74

6 Cased Explosives effective gamma vs. energy density

effective gamma

1.E+02 density = 1.8 density = 1 density = 1.e–3

1.E+01

1.E+00 1.E+09

1.E+10 energy density (ergs/gm)

1.E+11

Fig. 6.9 Effective gamma as a function of energy density for detonation products

then the conversion of energy to pressure at a constant density is proportional to the value of (a-1). For a typical set of parameters in the LSZK EOS for a near ideal explosive such as TNT, we can show that the pressure generated near the detonation front by a given amount of energy is many times the pressure that would be calculated using an ideal gas where the proportionality constant is the ratio of specific heats. Figure 6.9 shows the effective ratio of specific heats represented by the LSZK EOS. Very similar results would be obtained if other well known forms of EOSs were used. For example, a JWL formulation would give essentially an overlay to these results. This also points out a major shortcoming of the standard forms of equations of state for detonation products. When the detonation products expand by more than a factor of 50 or so, the commonly used EOSs all revert to a constant gamma ideal gas representation of the detonation products. Remember that a factor of 50 expansion means that the detonation products are still at a density of nearly 30 times ambient air density. The equation of state for air takes into account the vibrational and rotational excitation states and the dissociation and ionization of oxygen and nitrogen, simple diatomic molecules. The ratio of specific heats for air thus varies from 1.4 near ambient conditions to a low of 1.1 as the energy density increases. See Fig. 3.1 in Chap. 3 to see the variation in gamma for air. The behavior of the species found in the detonation products of solid high explosives is much more complex than for diatomic molecules. CO2 and H2O are major components of most solid explosive detonation products. There are other more complex molecules such as methane and ethane that should be taken into account by the equation of state. In addition, most explosives are not oxygen balanced and the detonation products contain carbon in the form of soot. These particulates do not contribute to the pressure (gamma ¼ 1.0) but are a component of the detonation products. Thus the effective gamma for detonation products is more complex than that of air and yet most commonly used equations of state use a constant value for gamma for all expanded states.


75

From the above plot, we can see that the energy available to do work on the surroundings is about four times as great near the Chapman–Jouget conditions than it is at the same energy density at an expanded volume. The factor of 4 is found by taking the ratio of the effective gamma minus ones. At an energy density of 1.0 e11, the effective gamma minus one at density 1.8 is 1.4 and at a density of 1.0 e-3 is 0.34 for a ratio of 4.11. As the detonation products expand and cool the fraction of the energy available to do work may increase or decrease, depending on the conditions of the expansion. Thus at early times, the expanding detonation products very efficiently transfer internal energy to the case in the form of fragment kinetic energy. Energy which goes into case and fragment kinetic energy is essentially lost to the available energy to generate air blast. Further, the energy remaining in the gaseous detonation products after expansion is divided between the energy used to raise the temperature of the gas and the fraction which is available to accelerate the surrounding gas, i.e., the production of blast waves. The energy released during the detonation is partitioned between the air blast and raising the temperature of the detonation product gasses for a bare charge. There is also a small (less than 1%) fraction of the energy that is lost in the form of thermal and visible radiation. For a charge which is cased, the energy is partitioned between the case fragment kinetic energy, the detonation products temperature and the blast wave energy.

6.3.3

Gurney Relations

Gurney took advantage of the fact that material strength could be ignored when he developed his equations for predicting the velocity of the expanding case. In his February 1943 report [1], he initially treated two geometric cases, one a sphere and the other a long cylinder. Gurney recognized that fragments exhibited a distribution in their velocities and treated what has been come to be known as the Gurney velocity as the mean velocity of the case fragments. His basic premise is that the fragment mean velocity is a function of the charge to case mass ratio. He uses a straight forward energy argument to come up with the relation: V0 ¼

pffiffiffiffiffiffiffiffiffi 2ER;

(6.3)

where E is referred to as the Gurney energy and is dependent on the properties of the specific explosive being used, and R is a geometric factor. For cylindrical geometry: R¼

C ; M þ C2

(6.4)

where C is the explosive mass per unit length and M is the mass of the case over the same unit length.

76

6 Cased Explosives

In spherical geometry R takes the form: R¼

C ; M þ 3C 5

(6.5)

where C is the explosive mass and M is the case mass. pffiffiffi Because E has units of energy and E has units of velocity, (6.3) can be written as: pffiffiffi V 0 ¼ V1 R ;

(6.6)

where V1 is a velocity characteristic of the explosive. For TNT, Gurney suggests that 8,000 ft/s is a good value for V1. Figure 6.10 is a plot of the velocity from (6.3) and (6.4), for a cylindrical charge filled with TNT as a function of the charge to case mass ratio. The data is from a number of tests using uniform steel cylinders filled with TNT. The fragment velocities were measured using high speed cameras. The measured average velocities from these tests is given in Table 6.1, which is taken from Gurney’s original report. While there may be an argument about how rapidly the velocity goes to zero as the charge mass is decreased, there should be no argument that at zero charge mass, the fragment velocity is zero. At some small charge mass for a very heavy case, the case will not break and there will be no fragments. This is not an interesting case for blast wave propagation and is not further considered. 12000

10000

Velocity (Ft / sec)

8000

6000 Gurney equation data

4000

2000

0 0

1

2

3

4

5

C/M

Fig. 6.10 Fragment velocities as a function of charge to case mass ratio

6

7

6.3 Moderate to Heavily Cased Charges Table 6.1 Measured velocities as a function of charge to case mass ratio

6.3.4

Cylinder C/M 0 0.17 0.2 0.22 0.46 0.8 5.62

77 Data Vel(ft/s) 0 2,600 3,200 3,800 5,100 6,080 9,750

Mott’s Distribution

Another important parameter for cased charges that affects the formation and propagation of blast waves is the way the case breaks after the initial expansion. R.I. Mott [2] worked contemporaneously with Gurney although in Great Britain. His work attempted to define the fragment size distribution from munitions whereas Gurney attempted to define the fragment velocities in terms of explosive and case properties. Mott’s fragment size distribution function is the complement of an exponential distribution function for the square root of fragment weights. Thus pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi Gð Wf Þ ¼ 1 Fð Wf Þ ¼ expð Wf =MA Þ, where Wf is the fragment weight (in pounds) and MA is the fragment weight probability distribution parameter pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( pounds ) which is a function of the explosive type and steel casing geometry. MA is defined as: 1=3 1 þ tc (6.7) d MA ¼ Bt5=6 i c di and is the expected value of the distribution parameter. B is a constant depending on the explosive properties and the casing type, with units of pound1/2/ft7/3. The parameters di and tc are the average case pffiffiffiffiffiffiinside diameter and the case thickness. As the expected value, MA ¼ Eð Wf Þ. The average value of the fragment weight (¼E(Wf)) is twice the square of MA. Thus, EðWf Þ ¼ 2MA 2 . Table 6.2 lists a few of the values for Mott’s constant B and Gurney’s constant V1 used in (6.6). The values for B come from test data using cylindrical mild steel cases with uniform thickness. Use of these constants for other case materials are not supported by experimental data but can provide some guidance for fragment size distribution. Further exploration of the Mott distribution provides some useful equations for evaluating a given case fragmentation. The total number of fragments is the weight of the casing divided by the average fragment weight. Nt ¼ Wc EðWf Þ and the number of fragments with weight greater than or equal to any given weight (Wf) is given by the relation: pffiffiffiffiffiffi! Wf Nf ¼ Nt exp MA

(6.8)

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6 Cased Explosives

Table 6.2 Some Mott and Gurney constants Explosive name Composition

Composition A-3 Composition B Composition C-4 Cyclotol H-6 HMX Nitromethane PBX-9404 Pentolite PETN RDX Tetryl TNT Tritonal

RDX/Al/Wax RDX/TNT/Wax RDX/Binder/Motor Oil RDX/TNT RDX/TN T/Al/Wax HMX (C4H8N8O8) HMX/Binder TNT/PETN PETN RDX TNT/PETN TNT/Al

Density (g/cc)

Specific weight (lb./ft3) 126.0 107.3 99.9

2.02 1.72 1.60

109.8 114.7 70.5 114.7 102.9 109.7 112.6 101.1 101.6 107.3

1.76 1.89 1.13 1.84 1.65 1.76 1.81 1.62 1.63 1.72

Mott constant (lb1/2/ft7/6) B 0.997 1.006 0.895 1.253

Gurney constant (ft/s) V1 9,100 8,800 8,600 9,750 7,900 9,500

1.126 0.964 1.237 1.415

9,600 9,600 8,200 8,000 7,600

This can be rewritten as: pffiffiffiffiffiffi! Wf Wc Nf ¼ exp ; 2 MA 2Ma

(6.9)

the common form of the expression for Mott’s distribution. We can easily divide the fragment size distribution into bins and find the weight or number of fragments in each bin. One example of such a plot is given as Fig. 6.11. Here I have chosen bins starting between 0 pounds and 0.001 pounds and doubled the upper weight limit of each bin. Thus the bin upper limits are 0.001, 0.002, 0.004, 0.008, 0.016, 0.032, 0.064, 0.128, 0.256, 0.512, 1.024, and 2.048 pounds. The weight within each of these bins is then plotted as a function of the average single fragment weight in the bin. This method is used in experiments to assist with evaluation of the fragment size distribution following the detonation of a device. The fragments are laboriously collected, weighed and sorted into bins. The collected fragments are then estimated to be a fraction of the total fragments generated based on geometric factors of the test configuration and the collected weights are extrapolated to the total weight of the case. Typically this method accounts for better than 90% of the total mass, however I have seen data that accounted for less than 85%. Figure 6.12 shows a comparison of the results of an arena test for a heavily cased device compared to results from Mott’s distribution. This shows a typical shortcoming of the formulae proposed by Mott in that the number and weight of large fragments is overestimated at the expense of medium sized fragments. When using Mott’s formulation, I have found that good agreement with experimental data can be found by truncating the high end of the size distribution and reallocating the truncated mass to smaller size bins. This is needed for cases when the thickness of the case is more than about 8% of the diameter.


79

Weight in bin vs. Average fragment size

300

Total bin weight (lb)

250

200

150

100

50

0 0.0001

0.001

0.01

0.1

1

10

100

Average fragment weight (LB)

Fig. 6.11 Bin weight as a function of average weight of a single fragment

Cumulative Weight vs. Fragment Weight

2500

Cumulative Weight (Ib)

2000

1500 arena data M ott’s Equations

1000

500

0 0

2

4

6 8 Fragment Weight (Ib)

Fig. 6.12 Cumulative weight as a function of fragment weight

10

12

14

80

6 Cased Explosives

6.3.5

The Modified Fano Equation

The fragmenting case carries away a significant fraction of the energy released by the detonation. For moderate to heavy cases this energy in the form of fragment kinetic energy carried away by the case fragments, as a general rule of thumb, reduces the available energy for air blast by about a factor of two. The original Fano equation first appeared in a Navy report [3] in 1953. In its original form, the effective charge weight producing blast is calculated as: Wb ¼ Wt ð0:2 þ :8=ð1 þ 2ðM=CÞÞÞ;

(6.10)

where Wt is the total charge weight, Wb is the energy available to blast, M is the case mass and C is the charge mass. This original form indicated that for large case mass to charge mass ratios, the effective blast yield approaches 20% of the total explosive weight. The Fano equation has been modified several times over the years and is currently in common use. The modified Fano equation is a commonly used equation to determine the fraction of energy available to generate air blast. The data used to find this relationship is based on TNT detonations in steel cases, although it is often applied to conditions outside of this data base. The relationship is given by: Wb ¼ Wt ð0:6 þ :4=ð1 þ 2ðM=CÞÞÞ:

(6.9)

We note that this ratio approaches a value of 0.6 as the case mass ratio increases. The results for this equation are plotted in Fig. 6.13. The Fano equation should be used to determine an approximate value for the effective yield. The coefficients are functions of the type of explosive and the Energy Fraction available for blast as a function of case to charge mass ratio

1 fraction of energy available

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0

1

2

3

4 5 6 7 Case to charge mass ratio

8

9

Fig. 6.13 Energy Fraction available for blast as a function of the case to charge mass ratio

10

6.4 First Principles Calculation of Blast from Cased Charges

81

material properties of the case and fall between the limits of the original and modified versions shown here. As with most “simple” formulae, there are several limitations to the applicability of this relation. At some point the case will become heavy enough to contain the explosive products completely. Then there is no blast, and no blast energy. For nonideal explosives, energy continues to be added to the detonation products which then continue to expand. The expanding gasses further accelerate the fragments after case break-up, resulting in a kinetic energy which may be as much as 70% of the detonation energy but less than 50% of the total energy released.

6.4

First Principles Calculation of Blast from Cased Charges

Treating the complex phenomena associated with the detonation of a cased charge can be accomplished with modern computational fluid dynamics (CFD) codes. The detonation can be calculated using a number of algorithms which propagate the detonation front through the explosive and deposit the energy released by the detonation in the fluid. The particular method that I favor is to calculate the local sound speed just behind the detonation front and advance the position of the detonation front at the sound speed [4]. This method satisfies the Chapman–Jouget conditions (if the sound speed is accurately represented), as well as providing a detonation propagation speed which is dependent on local conditions. We are able to make calculations of cased charges using a CFD code because the detonation pressure for almost all explosives is more than an order of magnitude greater than the strength of even the strongest steel. As one example, we will examine the early detonation process for a charge with a case mass approximately equal to the charge mass. The next series of figures show the calculated detonation propagation in such a device. Figure 6.14 shows the detonation sequence just after initiation. The detonator was cylindrical and positioned at the top center of the device. The left hand figure shows the detonation wave just outside the detonator. The imaging routine changes the color of the case material from white to blue and purple when the case obtains a velocity. This allows tracking of the shock wave through the steel case. In this instance, the detonation velocity is about twice the shock velocity in the case material. In the second frame, the detonation front has just reached the inner radius of the case. The end cap of the cylinder is starting to move. In the third frame the detonation wave has reached the

Fig. 6.14 Density plots showing early progression of the detonation wave

82

6 Cased Explosives

inner case radius and has progressed about one charge diameter down the tube. The detonation wave reflects from the case as a shock wave which is converging on the axis of symmetry. Note that the detonation front is curved. In the fourth frame, the detonation front has progressed to about two diameters. The reflected shocks have nearly converged on the axis. The case material is thinning at the corners and is about to break open. The detonation front remains curved at the edges. Note that the case is expanding linearly. This means that the case expansion velocity is a constant fraction of the detonation velocity. The air blast wave is initiated by the expansion of the case material. The case velocity can be determined from Fig. 6.14 by measuring the angle of the expanding case when the detonation velocity is known. If you don’t trust the calculation, another method of finding the case velocity is to use the Gurney equation shown earlier in this section. In this example, the charge and case were about the same mass, so the C/M is 1. Using Fig. 6.10 we find that the case fragment velocity is just over 6,000 ft/s or 2 km/s. The velocity is very close to the average case expansion velocity, but remember, this is an approximation. We can then use (3.9) from Chap. 3 to find the air blast pressure in the shock created by the expanding case. If we let ambient pressure be one bar and the ambient sound speed be 333 m/s we can solve a simple quadratic for DP as a function of the material velocity. Here the material velocity is the expansion velocity of the case or 2,000 m/s. This results in a pressure of about 20 bars. While this is not an insignificant pressure, the pressure of the detonation products in the case exceeds several kilobars. When the case begins to fragment, the internal pressure will be released and the initial 20 bar shock will rapidly be caught by the expanding detonation products and an air blast wave of about a kilo-bar will be formed. After many years of making calculations of the air blast from cased munitions, I have found that the better the case behavior is modeled, the better the air blast is modeled. The case significantly complicates the physics of the expansion of the detonation products. It confines the products for some time after the detonation. It may allow further chemical reactions to take place within the detonation products, depending on the explosive. It acts as a temporary interface between the detonation products and the air, thus reducing the initial tendency to form instabilities. Probably the greatest effect of a case is that it absorbs about half the detonation energy in the form of fragment kinetic energy. This energy results in a blast wave with half the effective yield as for a bare charge. Case effects may become even more important for non-ideal explosives. This is addressed in Chap. 18.

6.5

Active Cases

Because the case material takes up so much energy in the form of kinetic energy and reduces the amount of energy available for air blast, it seems reasonable to attempt to make the case from materials that release energy during or immediately after the

6.5 Active Cases

83

Fig. 6.15 Equivalent charge mass ratio as a function of case to charge mass ratio

detonation. In the early 1960s, Dr. Jane Dewey conducted a series of experiments at the Army Ballistic Research Laboratory in Aberdeen, Maryland [5] in which the air blast from cased charges of TNT was measured. Some of this data is summarized in Fig. 6.15 and shows that for some case materials, the air blast was enhanced by as much as a factor of 2 over a bare charge. The steel and cast cases were full metal casings; all other cases consisted of plastic bonded metal particulates. The solid line on this figure is the original FANO equation discussed in Sect. 6.3.5. We note that it tends to reasonably represent the trend shown by the steel case data. At a minimal case mass of only 0.1, the effective charge mass is reduced by over 10%. If we look at the steel case results, we see a large scatter in the data for case mass ratios between 0.2 and 0.06. There are two points that fall below 0.4, but there are also two points that are above 1.0. The FANO equation shows a decrease in the effective yield for all case mass ratios and is consistent with the effects of the steel case data. The explanation here is that the fragment kinetic energy is subtracted from the detonation energy. This argument is self consistent, logical and readily understood. There are a number of theories that have been proposed to explain the enhancement measured for the various materials. If we look first at the cast aluminum case and the plastic bonded aluminum particulate case we see that there is a measured enhancement when the case mass ratio is less than about 1. This enhancement factor reaches 2 at a case mass ratio of just over 1, which means that twice as much blast is generated when the case mass equals the explosive mass. This energy is in addition

84

6 Cased Explosives

to the loss of energy in the aluminum fragments kinetic energy. Thus over three times the energy must be generated in order to accelerate the fragments and double the air blast. One possible and reasonable theory is that the energy is coming from burning the aluminum case immediately upon detonation. This is easier to understand for the aluminum particulate case than it is for the solid cast aluminum case. The small particulates will be heated rapidly and burn in the atmospheric oxygen as they move through the air, away from the detonation. Each gram of aluminum, when burned, produces more than seven times as much energy as a gram of TNT when detonated. Thus for a case mass equal to the charge mass, burning only about 30% of the aluminum in the case would generate twice the energy of the explosive and account for the total blast enhancement. For the cast aluminum case, which showed enhancement of a factor of 1.8 over a bare charge, it is difficult to imagine the case breaking into so many small particles. If the case breaks into millimeter sized or larger fragments, the heating time will be far greater and the particles will never reach the ignition temperature before the gasses expand and cool. Yet we have the data indicating a significant enhancement in energy release. Another theory is that the aluminum case reflects infrared photon energy back into the detonation products, thus stimulating further chemical reactions which deposit photon energy in the back of the air shock when the case breaks. This mechanism is currently being studied experimentally. The magnesium case data, which is nearly indistinguishable from the aluminum case data, also shows enhancement, even when the case mass is more than twice the explosive mass. The same argument can be made here as for the aluminum. Magnesium burns readily in atmospheric oxygen and burning only a fraction of the case mass explains the enhanced energy release necessary to develop the measured blast enhancement. Magnesium and aluminum have very nearly the same reflectivity in the IR and the photon theory is consistent. If we look at the tungsten and lead cases, we see that all but two of the tungsten results are greater than 1.0 and all of the lead data is 1.0 or greater. Because lead and tungsten are essentially inert at the temperatures of detonating explosives, the energy cannot be explained in terms of energy added by burning the case metal. One plausible explanation is that the momentum of the high density case holds the detonation products together for a longer period of time due to inertial confinement and allows the chemical reactions to release greater energy. It also happens that the infrared reflectivity of tungsten and lead are only slightly smaller than that of aluminum. The IR theory may still be applicable. The silicon carbide case showed no consistent enhancement. It does not readily oxidize and its reflectivity is much smaller than the other materials mentioned. Just as a side note, the IR reflectivity of steel is the lowest of all materials tested. Another class of reacting case materials is those that will fragment upon detonation and will react with the target material upon impact. Some materials that may be used are aluminum, titanium, and uranium as well as a number of exotic mixes. When such case materials are accelerated to several thousand feet per second by the

References

85

detonation, the impact velocities approach that of the initial fragmentation velocity. When the fragments are suddenly stopped, their kinetic energy is converted to internal energy and raises the temperature to the point that significant chemical reactions can take place with the target material.

References 1. Gurney, R.W.: The Initial Velocities of Fragments from Bombs, Shells, Grenades, Ballistic Research Laboratories, report number 403, September, (1943) 2. Mott, R.I.: A Theoretical Formula for the Distribution of Weights of Fragments, AC-3642 (British), March (1943) 3. Fisher, E.M.: The effect of the steel case on the air blast from high explosives, NAVORD report 2753, (1953) 4. Needham, C. E.: A Code Method for Calculating Hydrodynamic Motion in HE Detonations, Air Force Weapons Laboratory, Kirtland Air Force Base, New Mexico, pp. 487, (1970) 5. Dewey, J.M., Johnson, O.T., and Patterson, J.D.: Some Effects of Light surrounds and Casings on the Blast from explosives, BRL Report No. 1218, (AD 346965), September, (1963)

Chapter 7

Blast Wave Propagation

In the previous sections I have addressed several methods of generation of blast waves. The propagation of the blast wave away from the source is a function of the geometry in which the blast wave is moving. A distinction needs to be made between the geometric representation of the blast wave and the number of degrees of freedom the expansion is permitted. A linear expansion, such as a shock tube, a cylindrical expansion such as generated by a long cylindrical charge and a spherical expansion and decay can all be accurately represented in one dimension. For linear propagation the cross section into which the blast wave is propagating remains constant. A cylindrical expansion may be accurately represented by increasing the cross section into which the blast is propagating proportional to the range to which it propagates. Similarly, a spherical expansion can be accurately represented by increasing the cross section proportional to the square of the range. This may be thought of as treating a unit length for the cylindrical case or a constant solid angle for the spherical expansion. Perhaps a thought experiment will help to visualize the differences between linear, cylindrical and spherical expansion. A shock wave traveling in a one dimensional tube of constant cross sectional area has no way of expanding, but propagates forward at constant velocity. The pressure behind the shock, in fact, all hydrodynamic parameters behind the shock remain constant, so long as information from the finite source does not reach the shock front. In the case of a cylindrical expansion, imagine a tall cylinder of high pressure gas that is suddenly released. If we look at a region near the center (in the long dimension) of this cylinder shortly after the gas has been released, the gas is expanding radially away from the source. The gas cannot move in the direction parallel to the axis of the cylinder because the gas above and below has the same pressure as our central sample. The gas can expand to the left and right because the volume it is flowing into is increasing as it travels radially from the source. The volume can be thought of as a wedge with a closed top and bottom with the source at the apex of the wedge. Energy is expanding from the wave front and the pressure falls as the wave progresses radially. All hydrodynamic parameters decay behind the front as the values at the front decline. The expansion has two degrees of freedom. C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_7, # Springer-Verlag Berlin Heidelberg 2010

87

88

7 Blast Wave Propagation

Divergence in Cartesian, Cylindrical and Spherical coordinates Divergence rA Cartesian @Ax @Ay @Az þ þ @x @y @z where x, y and z are three orthogonal space coordinates. Cylindrical 1 @ðsAs Þ 1 @Af @Az þ ; þ @z s @s s @f where s is the radius, f is the angle about the z axis and z is the axial coordinate Spherical 1 @ðr 2 Ar Þ 1 1 @Af ; þ ðAy sin yÞ þ 2 r @r r sin y r sin y @f where r is the radius vector and y is the angle between the z axis and the radius vector connecting the origin to the point in question. f is the angle between the projection of the radius vector onto the x-y plane and the x axis. For a spherical expansion, the gas expands radially and is not constrained above below or to the side. The energy expands into an increasing volume. This volume can be pictured as the wedge in the cylindrical case but the distance between the floor and ceiling are also increasing. Because the volume increases more rapidly than in the cylindrical case the peak values at the shock front decay more rapidly than in the cylindrical case and the decay behind the front is more rapid. There are also methods of representing flows in pipes by treating the flow “quasione-dimensionally”. This numerical approximation allows the cross section to vary as a function of the range, but the velocity is allowed only a radial component. Similarly, three dimensional flows can be represented by restricting the degrees of freedom by allowing only one or two velocity components. This is common practice in Computational Fluid Dynamics (CFD) codes. In two dimensions a sphere is represented as a circle in a cylindrically symmetric coordinate system. The usual representation uses radial and axial coordinates. The axial direction maintains a constant cross section while the radial cross section increases with the radius. In a shock tube with constant cross sectional area, the propagation is linear and one dimensional. Some blast wave properties may change, but the total energy, above ambient, remains constant. If a constant cross section shock tube changes to a variable cross section, the flow will take on two dimensional characteristics which

7.1 One Dimensional Propagation

89

may never be overcome. Reverberations perpendicular to the primary motion will continue at decreasing amplitude as the wave propagates. When the source of the blast wave is long compared to its diameter, the blast propagation perpendicular to the axis of symmetry is initially cylindrical and can be represented in one or two dimensions. In a free field or open air spherical detonation, the initial expansion is spherical. This expansion can be represented using one, two or three degrees of freedom. When the spherically expanding wave strikes the ground, the propagation may be accurately represented using two or three velocity components. When the blast wave strikes another object with a surface perpendicular to the ground, three dimensions are required to describe the behavior of the blast wave. Many applications of blast waves require combinations of geometrical descriptions of their propagation. A free air detonation generates a spherically expanding blast wave a portion of which may enter a long tube. The divergence of the blast wave suddenly changes to none. This sudden change in divergence generates secondary shock waves in an attempt to satisfy the new boundary conditions for propagation. The rate of decay of the blast parameters behind the blast front will be decreased and the rate of decay of the peak pressure will be decreased.

7.1

One Dimensional Propagation

The simplest geometry for blast propagation is one dimensional. The Riemann problem shown in Chap. 4 is a simple example of a one dimensional blast wave. If we make the driver section of a shock tube short compared to its length, the rarefaction wave from the back of the driver section will catch the shock front and cause a decrease in the shock parameters behind the shock front, thus forming a blast wave. Many of the worlds largest blast wave generating “shock tubes” use either a driver cross section which is smaller than the driven section of the tube or multiple drivers. The Large Blast and Thermal Simulator (LBTS) located at White Sands New Mexico (Fig. 7.1) was inspired by the large shock tube at Gramat, France. Both of these tubes use (or used) multiple compressed gas drivers to generate a decaying blast wave. In the case of the LBTS, the driven tube is 20 m wide, 11 m tall, with a semi-circular cross section, a flat bottom and is over 200 m long. This is the largest shock tube in the world. There are nine driver tubes, each having a nozzle opening of about 1 m in diameter and are spaced approximately symmetrically in the back wall of the driven section. The driver tubes can be filled to a maximum pressure of 100–200 bars. Flexibility in the operation of the facility is quite good because any number of the drivers can be used and they can be “fired” simultaneously or in any sequence. All of these combinations generate good approximations to decaying blast waves. They are only approximations to blast waves because the early expanding shocks from the drivers reflect from the walls of the shock tube. These reflections create secondary shocks within the decaying part

90


Fig. 7.1 Aerial view of the LB/TS located at White Sands Missile Range in New Mexico

of the main blast wave and do not clean-up before the blast wave reaches the test section near the end of the tube. One characteristic of a blast wave propagating in a confined one dimensional geometry with constant cross section is that the total energy, above ambient, remains constant. This means that the impulse of the blast wave remains unchanged as the blast wave propagates and decays. I find this point easy to understand because the impulse is a measure of the energy in the blast wave. A little more difficult to accept is the fact that the overpressure impulse remains constant and the dynamic pressure impulse remains constant, independent of the pressure level of the peak value in the blast wave. A blast wave decays as it travels the length of a shock tube. The Rankine–Hugoniot relations apply and the dynamic pressure decays at a faster rate than the overpressure, yet the overpressure and dynamic pressure impulses remain constant. The energy is redistributed behind the front, extending the positive duration and therefore the impulse. Many years ago, the Defense Atomic Support Agency (DASA) funded and built a conical shock tube at Dahlgren, Virginia, which was designed to eliminate the reflections caused by sudden changes in the divergence. The shock tube, designated the DASACON or DASA conical shock tube, represented a solid angle of a spherically expanding shock. Thus a true spherically diverging shock could be generated by detonating a small charge at the apex of the cone. Another large conical tube was funded by the Department of Energy and constructed by Sandia Corporation at Kirtland Air Force Base in New Mexico. This has been designated as the Sandia Thunder Pipe. In this instance the blast wave is generated by a gun at the apex of the cone. Whereas the DASACON had a continuously increasing cross section, the thunder pipe used several steps to increase the cross section. These steps created discontinuities which generated secondary shocks and detracted from the clean decay that was desired, but was successfully approximated.

7.1 One Dimensional Propagation

7.1.1

91

Numerical Representations of One Dimensional Flows

The region of interest is divided into zones which represent small increments in the direction of primary motion. The conservation equations for mass, momentum and energy with an equation of state are solved on this grid of zones. The conservation equations to be solved are give below. These are expressed in vector differential form in full three dimensions. The symbol definitions are as follows: t is the time U is the velocity r is the mass density P is the pressure F is any external field such as gravity k is the turbulence energy E is the total energy, internal plus kinetic H is the enthalpy Q is an energy source or sink The equation of state provides closure for the system. l

Mass:

@ ! ! þ U r r þ rr U ¼ 0 @t

l

Momentum: r

l

Energy: r

l

@ ! ! ! þ U r U þ rP þ rrF kr2 U ¼ 0 @t

@! ! ! U r E þ r P U þ r U rF kr2 H rQ ¼ 0 @t

Equation of State: P ¼ f ðr; I Þ

Numerical representations of one dimensional flows are restricted to three possible geometries: linear, the cross section is constant with range; cylindrical, the cross section is proportional to the range; and spherical, the cross section

92


Fig. 7.2 Sample geometry that may be represented in quasi-one dimension

is proportional to the square of the range. In all cases the flow is accurately represented using a single velocity. Flow fields can be numerically represented as “quasi-one dimensional” or 1½ dimensional. These numerical methods can be used to represent a flow whose primary motion is in a single direction but may have locally varying cross section. The cross sectional area at each zone boundary is varied according to the geometry of the object being represented. The flow then encounters larger or smaller masses and volumes as the cross section changes, but the flow velocity remains one dimensional (Fig. 7.2).

7.2

Two Dimensional Propagation

Two dimensional propagation of a blast wave is best exemplified by the expansion of a blast wave from a cylindrical source which is long compared to its radius. There are several such sources, for example, the blast generated by a lightening bolt. In this case the length is hundreds to thousands of feet and the radius is a few inches. The strength of the blast wave decays with the distance from the source in the radial direction. The UK has a munition called the Giant Viper which is an explosive charge a few inches in diameter and over 100 ft in length. When this munition is stretched out linearly and detonated, the expansion near the center (50 ft) of the charge is very nearly pure cylindrical until the rarefaction waves from the ends of the charge reach the center. In this case, the rarefaction waves don’t reach the center until the shock has expanded radially to a distance of nearly half the length of the charge. The advantage to this configuration is that the energy is spread more evenly over a wider area than a single charge having the same total explosive yield. For example, at a range of 100 charge radii, the energy is spread over a volume of about 10,000 times the initial volume, whereas the volume expansion ratio at the same distance from a sphere is one million and the pressure (energy per unit volume) is proportionately lower. The propagation of a blast wave in the two examples above can be well approximated using a one dimensional representation of the flow in which the volume increases proportional to the distance from the axis of the cylinder. Thus the restrictive geometry determines the rate of decay of the peak parameters in the blast wave and characterizes the rate of decay behind the shock front.

7.2 Two Dimensional Propagation

7.2.1

93

Numerical Representations of Two Dimensional Flows

Unlike one dimensional calculations, two dimensional numerical calculations can be carried out in a wide range of coordinate systems. In planar geometry, representing a region of fluid of unit thickness, a grid of zones can be established using any system of orthogonal coordinates. The simplest of these is an (x, y) or Cartesian coordinate system (Fig. 7.3) of rectangular zones. Each zone is defined as the area bounded by two consecutive values of x and y. This is a useful coordinate system for calculating generalized flow in two dimensions. Polar coordinates (r, y) are another popular and convenient method of representing a fluid (Fig. 7.4). In this case each zone is defined by the area between consecutive values of r (the radius) and y (the polar angle). This representation is especially useful for calculating cylindrical expansions when perturbations are expected in the y direction. Numerical schemes can be constructed using any other system of orthogonal coordinates such as parabolic or elliptic for special flow cases. Two dimensional flows can also be represented using axially symmetric coordinate systems. If we start with the (x, y) system as the computational plane and y

2D Carteslan Fig. 7.3 A two dimensional Cartesian coordinate system

x

θ

Fig. 7.4 Two dimensional polar coordinate system

r

94


Fig. 7.5 Cylindrically symmetric x,y grid

y

2D Cylindrical

x

invoke an axis of symmetry at x ¼ 0, we have a cylindrically symmetric system (Fig. 7.5). With this coordinate system, three dimensional flows can be calculated so long as the flow is axially symmetric. A sphere is represented as a circle in the computational plane and its expansion is defined with two velocity components. Cylindrical expansions with end effects can be calculated by representing the cylinder as a rectangle in the computational plane. For near spherical expansions an axi-symmetric grid can be formed by rotating a polar or (r, y) computational plane about the y ¼ 0 axis. Again a sphere is represented as a circle in the computational plane. Quasi-two dimensional flows can be represented by using “2½” dimensional grids. I have used such a 2½ D grid to represent the motion of a slowly rotating variable star. The grid was generated by rotating an (r, y) grid about the y ¼ 0 axis and assigning a third velocity component in the f or rotation direction. The f velocity is assumed to be symmetric about the rotational axis but can change with variations in the other two coordinates.

7.3

Three Dimensional Propagation

In three dimensions the blast wave expands freely in space. The volume into which the wave propagates is proportional to the cube of the radius and the cross section into which the front is propagating increases as the square of the radius. This divergence causes the most rapid decay of the shock front parameters and the corresponding decay of the blast wave behind the front.

7.3.1

Numerical Representations of Three Dimensional Flows

Three dimensional grid representations can be generated by any set of orthogonal functions. The simplest of these is the (x, y, z) or Cartesian grid. The flow is

7.3 Three Dimensional Propagation

95

represented with all three components of velocity. The Cartesian representation is shown in Fig. 7.6. It is also possible to represent a three dimensional flow field using an (x, y, f) grid as shown in Fig. 7.7. This grid might be useful for cylindrical flows that have a rotational component. Another useful representational grid for three dimensional flows is the polar or (r, y, f) orthogonal system. This system is especially useful for systems having a nearly spherical shape and is convenient for calculation of self gravitation. All of the mass interior to a given r coordinate contributes to the radial acceleration of the mass located outside of the given r. This system is used for describing the motion of convection within rotating stars. By setting an inner boundary at a fixed non-zero radius, fluid calculations can be made on the surface of near spherical geometries such as weather over the surface of the earth. Mountains can be constructed by using fine resolution to define the reflecting surface in all three coordinates.

z

3D Cartesian Fig. 7.6 An (x, y, z) or Cartesian three dimensional grid

x y

Φ

y

Fig. 7.7 An (x, y, f) grid for three dimensional flows

x

96

7.4


Low Overpressure Propagation

When the peak pressure of a blast wave decays to the level of a few tenths of a bar, the propagation becomes sensitive to the ambient conditions in which it is propagating. The propagation at any point in space and time can be obtained from the Rankine–Hugoniot conditions at the shock front; however, the overall geometry of the energy distribution can be influenced by temperature changes within the atmosphere. Remember that the propagation velocity of a shock at low pressures is strongly controlled by the ambient sound speed. The ambient sound speed is proportional to the square root of the absolute temperature. From the R–H relations, the equation for the shock velocity in low overpressure air is given by: 6DP 1=2 : U ¼ C0 1 þ 7P0 For example, if the peak shock pressure is 0.2 bars (3 PSI), the shock velocity is only 8% greater than ambient sound speed and at .1 bars (1.5 PSI) the shock propagation velocity is only 4% above ambient sound speed. When there are temperature gradients in the atmosphere, the low pressure shock will propagate at a velocity dependent almost entirely on the local ambient sound speed. Temperature inversions are often found under normal weather conditions. This condition is characterized by an increase in temperature with increasing altitude. If a temperature inversion exists in the ambient atmosphere, the blast wave will propagate faster in the higher temperature air. The portion of the blast wave at a higher altitude will outrun the blast wave following a lower and cooler path. Because the higher altitude shock is outrunning the lower altitude portion, the energy following the higher trajectory will begin to propagate downward. At some relatively large distance from the burst point, the energy following these multiple paths may converge and cause a significant increase in overpressure. Low overpressure blast waves are also influenced by wind velocities and shear velocity gradients within the atmosphere. The propagation velocity due to differences in sound speed can be enhanced (or diminished) by the addition of wind velocity. The wind has the effect of changing the shock front velocity through simple vector addition. The wind can have a pronounced effect on blast propagation even at moderate overpressures. Imagine an experiment with a 500 ton TNT charge, detonated midway between two structures. A near constant wind of 45 mph (20 m/s) is blowing from one structure toward the other. For a 3 PSI incident blast wave the distance to each structure is 2,000 ft or 600 m. The arrival time under no wind conditions is about 1.6 s. The arrival time at the upwind structure is delayed because it is traveling into a wind and has traveled effectively further by over 30 m (1.6 s times 20 m/s) than the ideal. In the opposite direction the shock is traveling with the wind and arrives earlier and has traveled effectively 30 m less than the ideal. The

7.4 Low Overpressure Propagation

97

arrival time difference at the structures is over 180 ms and the peak incident pressures differ by over 10%. A number of computer programs have been written to attempt to predict the behavior of low pressure shock trajectories using ray tracing methods. These programs use atmospheric soundings to determine the temperature and wind velocity as a function of altitude in the vicinity of a detonation. Rays are then propagated from the burst point, through the atmosphere and calculate the regions of convergence of the various possible paths. These programs are relatively simple, once the atmosphere has been described, and run in a matter of minutes on a modern personal computer. Such codes are used as standard procedure when determining the feasibility of conducting explosive tests anywhere near structures or populations. One such code is BLASTO, developed by J.W. Reed while at Sandia Corporation in Albuquerque, NM [1]. Some window breakage can occur at overpressures of only 0.01 bars. Under temperature inversion conditions or with strong velocity gradients, the blast wave can be ducted and enhanced pressures can occur at unexpectedly large ranges. The ray tracing codes are used to determine if a detonation can take place without causing damage to surrounding structures or alarming people. In several experiments with large amounts of TNT (500 tons or more), the blast wave broke windows at distant locations but was not heard at intermediate locations. A quote from [2]: “One of the first (actually the fourth) atmospheric tests (Operation Ranger, February 1951) broke large store windows on Fremont Street in downtown Las Vegas, Nevada, over 60 miles away. A similar 8-kt (kilotons) device had been fired the week before and a smaller, 1-kt device the day before, without being heard.”

7.4.1

Acoustic Wave Propagation

As a blast wave decays, it asymptotically approaches the behavior of a sound wave. In this sense, it never quite becomes a sound wave. Even at microbarograph measurement levels, a blast wave exhibits a faster rise to the peak than the decay after the peak and a higher positive overpressure than negative overpressure. The propagation of low overpressure blast waves can accurately be treated with the same methods as propagation of sound waves. If we assume that a sound wave is propagating in a constant atmosphere (no pressure or temperature gradients) without losses, the energy in the wave front is expanding spherically. The area of the wave front is given by 4pr2, where r is the radius of the front. The energy density in a sound wave is proportional to the square of the amplitude. It therefore follows that in a spherically expanding sound wave the amplitude (overpressure) varies as 1/r. For low overpressure blast wave propagation, the amplitude of the peak pressure falls somewhat more rapidly than 1/r. Referring to Fig. 4.17, the pressure decay coefficient from the blast standard has a value of 1.23 at a pressure of .25 PSI

98


(.017 bars) and a value of 1.19 at .1 PSI (.0068 bars). One example of the features of low overpressure blast waves at these pressure levels is given in Figs. 7.8 and 7.9. The first figure is a reproduction of the waveform resulting from the detonation of a 500 ton sphere of TNT that was placed on the surface. This waveform was MIXED COMPANY 1 LO 9350.

PRESSURE PSI

0.300

Range = 9150 ft. (2789 m)

0.200

0.100

0.00

–0.100

–0.200 7.00

9.00

11.0 TIME (SEC)

13.0

15.0

Fig. 7.8 Pressure waveform with 24 millibar peak pressure MIXED COMPANY 6 MBI 87200.

0.015

Range = 87,340 ft (26,621 m)

PRESSURE PSI

0.010

0.005 81.2 0.00 79.76 82.75

–0.005

–0.010

–0.015 79.0

81.5

84.0 TIME (SEC)

Fig. 7.9 Pressure waveform with 0.88 millibar peak pressure

86.5

89.0

References

99

measured approximately 2.5 km from the detonation. Note that the rise to the peak is very shock like, that there is a single peak and the decay is smooth. The negative phase pressure is about 1/3 of the peak positive pressure and is followed by a few minor oscillations about ambient. In Fig. 7.9, at a distance of 26 km, the rise time is a few tenths of a second. The peak indicates four or five peaks as a result of the shock having traveled over several different paths through the atmosphere to arrive at this location. The decay time from the peak is about the same as the rise time. The peak positive pressure is only 20% greater than the peak negative phase pressure. The waveform shown in Fig. 7.9 is approaching a sound wave with a frequency of about 0.4 Hz. This first pulse is followed by a damped sine wave with about the same frequency.

7.4.2

Non-Linear Acoustic Wave Propagation

A numerical method of propagating low pressure blast waves through an atmosphere is to solve the equations for acoustic wave propagation. The input parameters are the peak overpressure at the shock front, the positive duration assuming a triangular waveform, the radial distance to a target point and the geometry of the expansion. The solution method is posed such that a choice of geometry (cylindrical or spherical) may be chosen by specifying two and three dimensional expansion. The input waveform is then propagated through a specified atmosphere (either constant or exponential) with the desired expansion geometry. The overpressure waveform at the target point is calculated and characterized by the peak overpressure and the positive duration. The program numerically integrates the path of the wave through the specified atmosphere in less than one second on a modern PC and provides a very efficient means of approximating the propagation of low overpressure blast waves through atmospheres without inversions or velocities. This method provides a mean value for the strength of the blast wave propagated to that point through an unperturbed, quiescent atmosphere. Jack Reed’s program BLASTO uses insight and experience gained from many years of weather observations and blast experiments to estimate the enhancement or diminishing of the pressure, based on atmospheric conditions between the burst and the target point. The BLASTO code also runs in about a minute.

References 1. Reed, J.W.: BLASTO, a PC Program for Predicting Positive Phase Overpressure at Distance From an Explosion. JWR Inc. Albuquerque, NM (1990) 2. Cox, E.W., Plagge, H.J., Reed, J.W.: Meteorology Directs Where Blast Will Strike, Bulletin of the American Meteorological Society, 35, 3, March, 1954

Chapter 8

Boundary Layers

8.1

General Description

A boundary layer forms when a fluid flows over a solid surface. The fluid velocity goes to zero at the surface because of the roughness of a real surface. A general definition for a boundary layer is “a region in which the velocity gradient and related shear stresses become large enough that they cannot be neglected” [1]. Thus the consideration of the effects of a boundary layer is left to the user. Even very highly polished surfaces are rough on the scale of gas molecule separation distances. From Chap. 2.1 on the discussion of sound propagation we showed that the intermolecular distance was approximately 2.e-7 cm for sea level air. The surface would need to be smooth to a few times this distance for the surface to not form a boundary layer. For most applications a real surface may be considered “hydrodynamically smooth”. When the roughness of the surface must be considered for a particular application, a description of the roughness is required. For flow over a flat plate, the roughness can be characterized by ridges oriented perpendicular to the flow direction. These ridges may be circular, triangular or rectangular in cross section and are described by their height, shape and spacing. One common method of describing general surface roughness is to characterize it in terms of sandpaper roughness. This is accomplished by specifying a sandpaper grit number or, more precisely, by specifying the size and spacing of hemispherical roughness elements. Care must be used in specifying the size and spacing for such a representation. For a given size of hemispherical element, the spacing may range from zero to infinity. At both these spacing limits the roughness goes to zero. For zero spacing, the surface is covered by an infinite number of roughness elements and the surface is simply changed in position by the height of a roughness element. In the case of infinite spacing, there are no roughness elements and the surface is smooth. The greatest roughness effect occurs when the spacing is equal to twice the roughness height; the hemispheres are just touching at the surface.


101

102

8 Boundary Layers

A boundary layer is characterized by a reduced momentum and kinetic energy (velocity) near the surface, going to zero at the surface and approaching the free stream values of the blast wave at some height above the surface. It is the description of this height as a function of time or distance and how the velocity varies between the surface and the free stream which constitutes the greatest effort in the study of boundary layers associated with transient flows, such as blast waves. Boundary layers are divided into two major categories: laminar and turbulent. Laminar boundary layers form when the Reynolds number of the flow is low (Ernst Mach InstitutErnst Mach Institut< Freiburg, Germany, August, (1990) 3. Wisotski, J.: Sequential Analysis of Mighty Mach 80-6 and -7 Events from Photographic Records, DRI -5-31505, University of Denver, Denver Research Institute, May, (1981) 4. Henny, R.W.: Trinity- The Nuclear Crater, Proceedings of the 18th Symposium on Military Applications of Blast and Shock (MABS-18), (2006)

246

14 Height of Burst Effects

5. Martinez E.J.: editor, Hurricane Lamp- Volume 4 – Calculational and Data Analysis Reports, POR 7390-4, Defense Nuclear Agency, February, (1993) 6. Edited by Houwing, Proceedings of the 21st International Symposium on Shock Waves (ISSW), Great Keppel Island, Australia, (1997) 7. Needham, C.E., Crepeau, J.E.: A Revisit to Trinity, 2004, Applied Research Associates Topical Report, February, (2005) 8. Needham, C.E., Crepeau, J.E.: A Flux Dependent Thermal Layer Model (FDOT), DNA 5538-T, Defense Nuclear Agency, October (1980) 9. Miller R, Ortley, D.J., Needham, C.: NSWET – SMOKY Calculations, Contract No. DTRA01-03-D-0014, Defense Threat Reduction Agency, June, (2005)

Chapter 15

Structure Interactions

The study of blast waves, their generation, propagation and interactions with objects is more than an academic exercise. The importance of the study of blast waves is to understand how blast waves interact with objects, how the objects are loaded by the blast wave and how the blast wave is modified by these interactions. In this chapter I will discuss the roles of blast wave overpressure and dynamic pressure in generating loads on structures and vehicles. The damage caused by these loads is beyond the scope of this text and is the subject of an entire field of study. In general the overpressure manifests as a crushing force on the exterior of a structure and the dynamic pressure acts to accelerate drag sensitive objects. A structure which is flush with the ground surface will be loaded by the overpressure only. The vertical component of the dynamic pressure is stagnated and is included in the overpressure. The horizontal component of dynamic pressure simply passes over the flush target and the load is independent of the horizontal dynamic pressure. The overpressure, on the ground, is readily obtained from the height of burst curves described in Chap. 14. An object oriented parallel to the direction of flow will only experience the overpressure or side on pressure of a blast wave. This is the reason that overpressure gauges are placed in the center of large discs and the discs oriented with the minimum cross section in the direction of flow. This orientation keeps stagnation to a minimum and allows the gauge to record the true overpressure as the blast wave passes. If a blunt housing is used for the gauge, the flow is partially stagnated, secondary shocks may be formed, and the gauge records the complex waveform generated by the presence of the gauge mount rather than a free field value. If the mounting disc is oriented such that the flow is not parallel to its face, the recorded pressure will be higher if the face is oriented toward the flow because a partial stagnation of the dynamic pressure occurs and lower if the face is oriented away from the incident blast wave because the dynamic pressure causes a partial vacuum on the downwind side of the disc. For three dimensional objects the load descriptions are not so simple. The reflected pressure on the surface facing the blast wave causes modification of the C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_15, # Springer-Verlag Berlin Heidelberg 2010

247

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15 Structure Interactions

flow near the edges of the building. The high pressure on the face of the structure causes the gas to be accelerated parallel to the reflecting surface. This flow partially diverts the flow away from the sides and top of the structure, thus reducing, at least temporarily, the loads on these surfaces. The outward flow induces development of vortices on the edges of the structure. These vortices are relatively stable, do not dissipate rapidly and cause low overpressure regions on the sides and top of the structure. In some cases the vortices are sufficiently strong that the side walls of a closed rectangular building may fail by being pushed outward by the internal pressure when the loads on the side walls are reduced in the vortex region.

15.1

Pressure Loads

Once the parameters of a simple incident blast wave have been defined, the loads on a simple structure facing the detonation can be defined in terms of reflection factors which were given in Chap. 13. For flat faced structures, the reflection factors as a function of incident angle of Chap. 13 provide an excellent method of predicting the peak reflected pressure on the structure. In the case where the flat face is oriented perpendicular to the incoming wave the HOB curves of Chap. 14 may be used to find the peak pressure distribution across the face of the structure. These methods only provide information on the first peak overpressure. Remember that the reflection factors only apply to the peak pressure load. As the shock which is reflected from the structure moves away from the surface of the structure, the load is reduced to the stagnation pressure. The stagnation pressure is the sum of the overpressure and the stagnated dynamic pressure. I will use some experiments from the Ernst Mach Institute (EMI) to demonstrate these effects. A rigid block was placed in a shock tube with a gauge placed near the center of the upstream face of the block. A 1.4 bar shock struck the block. Figure 15.1 shows the measured pressure on the upstream face of the block as a function of time. The peak reflected pressure is 4.2 bars, in agreement with the Rankine–Hugoniot relations. The pictures in the lower part of the figure show the shock configuration at specific times during the shock interaction. The red curve which overlays the experimental pressure measurement is from a two dimensional CFD calculation made by Dr. Werner Heilig of EMI. At a time of 120 ms, the shock reflected from the front of the block can clearly be seen curving above the front of the block and joining the incident shock at about two block heights above the block. Early vortex formation can be seen at the top leading edge of the block. By this time, the pressure at the middle of the upstream face has dropped by about 1/3. At 400 ms, the shock front is well beyond the block, the pressure on the front face has reached the stagnation pressure and a vortex has formed at the back of the block. The shocks arriving near 500–600 ms are reflected from the roof of the shock tube. The block was reversed in the shock tube so the gauge was on the downstream end of the block and the experiment was repeated. Figure 15.2 shows the pressure measurement from this experiment. The first shock overpressure is only about 1/3

249

1 bar

15.1 Pressure Loads

200 µs

0

120

18

400

245

520

600

545

470

1150

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1725

Fig. 15.1 Upstream pressure measurement of a 1.4 bar shock interacting with a rigid rectangular block

0

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95

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445

1495

Fig. 15.2 Downstream pressure measurement of a 1.4 bar shock interacting with a rigid rectangular block

of the overpressure of the incident shock. The second shock (95 ms) is the shock reflected from the floor of the tube. This shock interacts with the vortex formed at the top edge of the block and temporarily changes the vortex motion.

250


The overpressure rises to a peak of just under the incident peak overpressure before smoothly decaying with some small perturbations caused by internal shock tube reflections. Again, the red line shows the results of a CFD calculation by Dr. Werner Heilig. If the structure is struck by a Mach shock, the loading can be more complex. At pressures above seven bars or so, a surface flush structure will be loaded by the Mach shock front followed by a decay followed by the second peak caused by the passage of the stagnation region of the complex or double Mach structure. The timing of this double loading may be important to the structure response. If the two peaks are separated by a time near the natural response frequency of the structure, the response to the second peak may be greater than to the first peak even when the second peak is lower than the first. In either case the impulse of the double peaked waveform is greater than for a simple decaying wave with the same peak pressure. For structures that extend above the ground and into the flow region, a Mach shock also complicates the structure load. The blast wave loading depends on the height of the structure above the surface and the relative height of the stagnation region behind the Mach stem. If the structure extends above the triple point when the blast wave passes, the lower part of the structure will be loaded by a single shock and a strong compressive wave. The upper part of the structure, above the triple point will be hit by two distinct shocks; the incident shock coming directly from the detonation and the reflected shock oriented from an image detonation below the ground. In general, both of the shocks above the triple point are weaker than the Mach shock and are typically half the overpressure of the Mach stem. The precise pressures and relative magnitudes of the shocks depend on the height of the point of interest above the surface, the height of the triple point and the distance of the point of interest above the triple point. The device used at Hiroshima was unique. The device was never tested before or after Hiroshima and initial estimates of the yield varied by nearly a factor of two. It is important to know the yield of the device because most of the data on radiation exposure is based on the population exposed during the Hiroshima detonation. Also a number of structural damage estimates are based on assumptions of the yield of that device. One of the methods of estimating the yield of the Hiroshima device was to examine the bending of steel utility poles in Hiroshima. The utility poles were in the region of the Mach stem. Pressures and impulses from the height of burst curves were used to estimate the total loading assuming uniform loading over the entire height of the poles. This estimate lead to a yield which was significantly lower than that obtained by other methods. As a part of the effort to obtain a better estimate of the yield of the Hiroshima device, a test was conducted at the Suffield Experimental Station in Alberta Canada. This test consisted of a 1,000 pound spherical charge detonated at a height of burst of about 70 ft. This corresponds reasonably well to the estimated geometry for the Hiroshima detonation. Gauges were placed at the ranges corresponding to the utility poles. Gauges were placed on the ground and at several heights above the surface corresponding to the heights of the utility poles. The test showed that the triple point passed below the tops of the poles. Thus the lower parts of the poles

15.2 Impulse Loads

251

were subjected to the environment of the high pressure Mach stem, but the tops of the poles were in a much lower pressure and impulse environment. The calculations of pole bending with the assumption of uniform loading overestimated the forces and the torque applied to the poles and therefore underestimated the yield of the device. Using the new understanding of the effects of triple point path on structure loads allowed the calculation of a new estimate of the yield of Hiroshima which was more in line with the estimate from other methods. I cannot confirm the source of the story, but I have been told that in Hiroshima, people survived on the upper floors of certain apartment buildings when nearly everyone on the lower floors were killed. Further examination showed that the lower floors of those apartment buildings were within the Mach stem region of the blast wave, while those on the upper floors were above the triple point. Those on the upper floors were subjected to two weaker shocks and those on the lower floors to a single shock of about twice the overpressure.

15.2

Impulse Loads

The load on a structure is usually expressed as a combination of the peak overpressure and the impulse delivered to the exposed area of the structure. Because the load on the surface of a structure may vary dramatically depending on the position of the measurement, a typical method of defining the load is to divide the surface area of the building into a number of panels which, if small enough, can be considered to be uniformly loaded. As a thought experiment, imagine a structure with a vertical wall facing directly into a blast wave. The width of the wall is twice its height. Suppose that the blast was initiated at a distance much larger than the linear dimension of the wall. The peak overpressure which loads the wall will be nearly uniform over the entire surface and will be equal to the reflected pressure of the incident blast wave. The peak overpressure load will be the reflected pressure of the incident blast wave. From the Rankine–Hugoniot relations, the peak reflected overpressure is given by: OPR ¼ 2OPI þ ðg þ 1ÞQI , where the subscript I refers to the incident blast wave value. The reflected pressure rapidly decays as the reflected shock moves away from the surface of the building and approaches the stagnation overpressure. The stagnation overpressure is given by: OPS ¼ OPI þ QI As the incident blast wave decays, the stagnation overpressure also decays. Because the wall is finite, rarefaction waves are generated at each edge of the wall. In the case of our free standing wall, there are three edges to be considered; the top and the left and right sides. The rarefaction waves move at sound speed from

252 Fig. 15.3 Regions showing relative importance of edge rarefaction waves

15 Structure Interactions Regions where overpressure is affected at a time when the rarefaction wave has reached ½ the height of the wall Red – 2 edges, Blue – 1 edge, Green - unaffected 2H

H

the edge toward the center of the wall, causing further decay of the stagnation overpressure. The impulse is the integral over time of the overpressure waveform. The impulse will be smallest and nearly equal, near the edges with the maximum impulse occurring at the ground level center of the wall. Because the upper left and right corners are affected by rarefaction waves from both the top and side walls, the pressure loads in these regions decay more rapidly than regions affected by only one edge. In regions where the rarefaction wave has not arrived, the overpressure load is the stagnation pressure of the incident blast wave. Figure 15.3 is a cartoon showing the regions affected by rarefaction waves when the rarefaction wave has reached a position equal to ½ the height of the structure from each edge. The red region has been affected by rarefactions from 2 edges, the blue by 1 and the green is as yet unaffected. Figure 15.4 shows the calculated overpressure distribution on a 45 ft tall by 80 ft wide wall subjected to the blast wave from a 5,000 pound detonation at a distance of 200 m (656 ft). The difference in distance from the detonation to the ground level center and the upper corner of the wall is less than 1%. The peak overpressure load on the wall of 2.6 psi was essentially uniform in both overpressure and arrival time. The decay of the incident blast wave was also essentially uniform over the entire face of the structure. The time is chosen such that the rarefaction wave from the top has reached approximately ½ the building height. Remember that a rarefaction wave is not a shock and is not discontinuous. The regions which have been affected by rarefactions from two edges are clearly shown in the upper left and right corners. The overpressure in center region near the ground has decayed to about half of the maximum, but the pressures in the upper corners are reduced by a factor of 5. The overpressures in the regions which have been affected by only a single rarefaction have decayed by a factor of approximately 3. This reduction in overpressure directly affects the impulse associated with the corresponding regions. The highest impulse is in the region least affected by rarefaction waves and the lowest impulse in the regions most affected.

15.2 Impulse Loads

253

Fig. 15.4 Calculated overpressure distribution on a nearly uniformly loaded wall at a time of 0.5 s

A more realistic situation is shown in Fig. 15.5. The contours represent the calculated peak overpressure at any time in the plane of the front face of the structure. Here the overpressure blast load was not uniform but is caused by the blast wave from 1,000 pounds of high explosive at a distance of only 4 m in front of the center of the structure. This structure has the same dimensions as that of Fig. 15.4, but has a number of open windows on the face toward the blast. The peak overpressure load is in excess of 7,000 psi near the center of the structure at ground level. The peak overpressure load at the upper corners is about 200 psi. Note that the peak overpressures in the openings are not uniform, but are affected by the reflected pressures on the structure surface near the openings. The high overpressure shocks generated by the reflection of the incident blast wave are propagated into the openings. At each edge of each opening the blast waves propagate into the openings. If we go back to the block shown in Fig. 15.1, the face of the structure corresponds to the leading edge of the block and the opening corresponds to the region above the block. In the case of the charge being close to the building, the reflected pressure changes rapidly with the position on the face of the structure. The reflected shocks do not reflect uniformly into the openings. The shocks from the sides of the openings are generally moving horizontally while the shocks from above and below are moving vertically. The interactions of the reflected shocks with different

254


Maximum Overpressure (PSI)

50 45

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6000

35 5000

Z (ft.)

30 25

4000

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15

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5 0 –40

0 –30

–20

–10 Range (ft)

Fig. 15.5 Peak overpressure loads on a structure with openings

shock strengths and different flow directions and the incident blast wave form a very complex three dimensional flow in the vicinity of every opening. The presence of the openings also initiates a rarefaction wave from each edge of each opening which travels over the surface of the structure. The rarefaction waves have a direct effect on the impulse load on the surface of the structure. The impulse load is the integral of the overpressure as a function of time and is evaluated at a few thousand points in the plane of the surface of the structure. The contours of Fig. 15.6 represent the integrated overpressure time histories from a three dimensional CFD calculation. The impulse values range from nearly 2,000 psi*ms at ground level to less than 200 at the upper corners of the structure. As was noted in the discussion of the overpressures, the impulse is a complex function of the incident blast wave and the geometry of the openings in the structure surface.

15.3

Non Ideal Blast Wave Loads

Structure loads resulting from non-ideal blast waves are not readily calculated from simplified techniques. If we take the example of the thermally precursed blast wave discussed in Chap. 14, the first arrival is not a shock for pressures above about 10 psi. This means that the Rankine–Hugoniot relations are not applicable. Because the incident wave is not a shock, the reflection factor curves and height of burst curves are not applicable. Because the rise in pressure load takes a finite amount of time, the pressure begins to relieve even before the peak is reached. The dynamic pressure also has a finite rise time and gradually stagnates as the pressure rises. The gradual increase in stagnating dynamic pressure and resultant pressure loading

15.3 Non Ideal Blast Wave Loads

255

Overpressure Impulse (PSI*ms) 1800 1600 1400 1200 1000 800 600 400 200 –40

–30

–20

– 10

0

10

20

30

40

Distance (ft)

Fig. 15.6 Impulse load on a flat face with openings

allows a flow around the object being loaded to be established. The loading of a structure then becomes a strong function of the dimensions of the structure as well as the parameters of the incoming blast wave. If we look at the precursor pressure and dynamic pressure waveforms of Figs. 14.25–14.27, we see that the duration of the non-ideal blast waves is the order of half a second or more and the rise to the peak takes a 100 ms or more at many ground ranges. As the loads on a structure are increasing with the slowly rising incident wave, relief waves and flow can be established even over relatively large buildings. In half a second, a rarefaction wave will move over 500 ft in ambient sound speed environments and upwards of twice that far in a non-ideal loading situation. Thus for structures with dimensions of 100 ft or so, the relief from structure loads occurs on the same time scale as the loading. The loads are more closely associated with the blast parameters of the incident blast wave. In conjunction with some of the high explosive height of burst with thermal layer experiments conducted at the Defence Research Establishment at Suffield (DRES) Alberta Canada, some rectangular blocks with pressure gauges were placed behind simple wedges in the thermal precursor region. The results showed that the pressure loads on the blocks were less than expected from the incident overpressure and that no significant loading was observed from the high dynamic pressures that were measured in the free field. Detailed CFD calculations showed that the initial loading, caused by the compressive wave, did not have a significant enhancement as expected from a shock of the same pressure level. More importantly, the vortex behind the precursor front, was deflected upward and over the rectangular block. Thus the high dynamic pressure at the bottom of the vortex passed over the block and resulted in essentially no significant enhancement of the load.

256

15.4


Negative Phase Effects on Structure Loads

A story which I have heard but cannot confirm the source, says that a major university designed a new shock tube that could generate a peak overpressure of several tens of bars. The tube was reinforced with external rings every few feet to ensure that the internal pressure would not blow out the tube. When the first high pressure shot was made in the new tube, the tube collapsed from external atmospheric pressure when the negative phase of the shock formed. Even if the story is not completely true, it provides a good lesson. The negative phase of the blast wave can be destructive also. Much of the damage caused by tornadoes has been shown to be the result of the sudden onset of low overpressure at the center of the tornado. The internal pressure (1 bar) in a structure cannot be relieved on the time scale of the passage of the tornado. The internal pressure simply blows out the windows, doors and walls of standard frame construction buildings. The high dynamic pressure winds then translate the “debris” to large distances. The dynamic pressure associated with a 200 mph wind (90 m/s) is only about 0.7 psi, whereas the overpressure in the center of a strong tornado is the order of minus 1.5 psi, more than twice the dynamic pressure. In Chap. 9 I discussed the entrainment of particulates into the flow behind a blast wave caused by the sudden decrease of pressure in the negative phase. The low overpressure above the surface generates upward velocities in the gas in small cavities in the soil which carries particulates into the flow. In a similar fashion, the negative phase of a blast wave can have dramatic effects on structures. While the negative phase is never as strong as the positive phase of a blast wave, the duration is longer. For low overpressure blast waves the positive and negative impulse are nearly equal. In most free field experiments, the negative phase impulse is greater than that of the positive phase. This is caused by the rising fireball. The rising fireball creates a partial vacuum near the surface and pulls air from large ranges into the stem of the mushroom cloud near ground zero. This affects the negative phase velocities and, to some extent, the density of the gas in the negative phase. The pressure remains below ambient for an extended period (seconds for large nuclear detonations) thus creating a large negative overpressure impulse. In some buildings hit by air blast, the glass from the windows is largely found outside the building. The blast wave breaks the window glass, but the negative phase of the blast arrives before the glass shards have gone very far and the negative phase pulls the glass outward. In high rise buildings exposed to high winds, the windows occasionally are “blown out”. This is caused by a combination of the bending of the building due to the stagnation pressure forces on the building and the low overpressure in the vortex formed on the sides and back of the building. The internal pressure of the building forces the windows out of their warped frames and the glass falls to the ground.

15.5 Effects of Structures on Propagation

15.5

257

Effects of Structures on Propagation

Just as rolling terrain had a significant effect on the propagation of a blast wave, man made structures also effect the propagation of blast waves as they encounter and pass over structures. A typical rule of thumb for the distance that a blast wave travels after encountering a single structure before the perturbation is “healed” is 4–5 times the dimension of the structure in the direction perpendicular to the flow. Thus a 4 in. diameter pole of any height, struck from the side requires less than 2 ft before the blast wave returns to its normal propagation. This is truly a rule of thumb and has many exceptions. In high velocity flows, the vortices that are shed from the object may travel large distances downstream. In precursor flows, which are dominated by dynamic pressure, it may take 40 or more structure dimensions before the blast wave returns to its undisturbed flow. Again, these are only approximations because, when an object is struck by a blast wave, the energy of the blast wave is redistributed and never returns to its unperturbed state. A reflected shock stagnates a portion of the flow and sends energy back upstream at locally supersonic velocity into a decaying blast environment. After the blast wave passes the object, energy is transferred from the higher pressure regions near the shock front, but the transfer of energy perpendicular to the flow direction is very inefficient. The perturbed shock front equilibrates with the neighboring regions of the shock front, but a weak pressure gradient remains in the direction perpendicular to the flow. In the decaying part of the blast wave the gradients are even smaller than near the shock front. The decaying region takes even longer to overcome the perturbation. In addition vortex formation may occur as a result of the interaction with the object. The vortex converts energy from flow in the direction of the blast to rotational flow. Because the vortices are stable, the rotational kinetic energy is slow to be converted back into “normal” blast wave flow. The shears induced in the flow by the object may also trigger Kelvin–Helmholtz instabilities and the energy associated with the turbulence will be returned to the flow through the cascade of ever decreasing size of turbulent vortices. While the energy is eventually returned to the flow, the energy has been displaced in time and space behind the blast wave. Let us examine the perturbation of a blast wave when it strikes a simple rectangular box structure. Figure 15.7 gives the pressure distribution on the ground from a blast wave as it engulfs a rectangular three dimensional structure. The detonation of 500 pounds of TNT took place at the origin in this figure. The blast wave reflects from both the long and short faces of the structure oriented toward the blast. The reflected shocks move away from the surface of the structure and interact with the incident blast wave. The reflected shock from the short side dissipates more rapidly than that from the long side of the structure because less energy is diverted by the reflection. The incident shock front is refracted around the corner of the structure and weakens as the energy expands into a larger volume. The refracted wave on the right side of the structure has decayed to about half the strength of the wave on the far side of the structure. A low overpressure region has formed at

258


Fig. 15.7 Blast Wave interacting with a rectangular block structure

the far corner of the long face of the building. The negative overpressure in this region has nearly the same magnitude as the positive overpressure in the incident wave. The incident blast wave is also proceeding over the top of the structure but cannot be seen in this view of the ground plane. The blast wave passes the structure and begins to “heal” on the backside. The shocks that have propagated around and over the structure combine on the far side of the structure and form a relatively high pressure region at the back corner of the building (Fig. 15.8). The blast wave interaction with two rectangular block structures is the next step in examining the effects of structures on the propagation of blast waves. In addition to the parameters of structure dimensions, the separation distance becomes an additional variable. Figure 15.9 is a cartoon of the geometric variables. We further

15.5 Effects of Structures on Propagation

9.660E+05

9.940E+05

259

1.022E+06

1.050E+06

1.07BE+06 DYNEA / Sq–CM

PRESSURE ZPLANE AT Z = 1.26E+01 CM

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RANGE (Y) M

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RANGE (X) M 500–LB. HEMI SPHERE OF TNT SHOCK DIFFRACTION OVER A BLOCK TIME 275.000 MSEC

CYCLE 1231.

PROBLEM 30703.080

Fig. 15.8 Blast wave after interaction with a rectangular block structure

S

H

L W

Fig. 15.9 Geometry for a simple blast wave interaction with 2 rectangular blocks

SHOCK

260


assume that the blast wave is incident along a line between the two structures, thus eliminating the angular dependency of the interaction. Even for this relatively simple case of a blast wave interacting with two identical structures, there are at least six variables that need to be examined in order to explore the entire range of interactions. The six variables that come to mind immediately are: the three dimensions of the block (height, width and depth), the separation distance between the blocks, the overpressure of the incident blast wave and the yield or energy of the source. The yield determines the positive duration of the incident wave. The calculated peak overpressure on the ground resulting from just one set of parameters is shown in Fig. 15.10. In this case the buildings were 50 m wide, 50 m high and 100 m long with a separation of 20 m. The blast was 5 kt with an incident peak overpressure of 3.4e4 Pa (5 psi). The figure clearly shows the decay of the

4.000E+04

2.000E+04

6.000E+04

Pa

Peak Overpressure

zplane at z = 5.00E–0l m 0.60 390 480

580 670 760 850 940 1030 1120 1210 1300 1390 1433

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range (y) km

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0.82

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1.06

1.18 1.30 1.42 Range (x) km

1.54

1.66

Fig. 15.10 Peak overpressure in the Ground Plane L ¼ 100, H ¼ 50, S ¼ 20

1.78

1.90

15.6 The Influence of Rigid and Responding Structures

261

incident wave with distance from the source. The reflected pressure on the face of the buildings for a 34 kPa incident wave is 78 kPa. The figure shows that the peak pressure just in front of the structures was indeed just under 80 kPa. The reflected shocks from the fronts of the two buildings interact in the region between the structures and the peak overpressure between them is greater than that of the unperturbed blast wave. There is a strong shadow region behind the structures where the shocks coming over the top of the structures interacts with the shock propagated between the structures. This interaction causes an interference pattern between the waves as they propagate downstream. A region of higher than incident overpressure extends for several hundred meters behind the structures. The low overpressure shadow extends from the back corners of the structures at an angle of about 20 for more than a kilometer. This is at least ten times the dimensions of the structure. The overpressure impulse is similarly affected because the incident blast wave dominates the positive duration of the overall flow for this simple case.

15.6

The Influence of Rigid and Responding Structures

In Chap. 14 the influence of the mass of the Mylar balloons was shown in thermal precursor experiments. The density of the Mylar is about 1,000 times the density of ambient air, thus a millimeter of Mylar has about the mass of a meter of air. The air responds much more rapidly than the Mylar. A blast wave interacting with a solid object behaves very similarly whether the object is rigid or responding. In this section I will site several examples of such behavior to illustrate the inaccuracy of the commonly held view that structure response influences blast wave propagation. In general, if the response time of the structure is greater than half the positive duration of the blast wave, or the propagation time of the blast wave over the dimensions of the structure, the rigid response approximation is valid. An experiment was conducted at White Sands Missile Range in 1999. A full scale (80 by 45 ft) structure was exposed to a large detonation which loaded the front face of the structure, nearly uniformly at 40 psi. The front face of the structure was solid and nearly planar. Glass windows which extended across most of the width of the structure, were installed on the fourth floor. The glass in the windows was 6 mm thick. There were short concrete stub wing walls at the sides of the structure which extended about 6 ft back from the front face. Pressure gauges were installed in the floor of the fourth floor, one near the center line and 10 ft back from the front face. Two predictive calculations were made. In the first calculation it was assumed that the glass would break immediately, thus allowing the air blast to enter the structure through the window openings. A second calculation was made with the windows closed and rigid. Figure 15.11 is a cartoon of the geometry of the fourth floor showing the two possible paths for the blast wave to reach the gauge. Other pressure gauges located on the floor confirmed the path of the blast waves.

262

15 Structure Interactions Air Pressure Gauge

windows

10 Feet

80 Feet

BLAST WAVE

Fig. 15.11 Top view of fourth floor of the test structure

When the experiment was conducted and the gauge record examined, we found that the first arrival at the gauge near the center line came from around the ends of the front wall. We could track the arrival of the shock front on other gauges placed on the floor across the width of the building. Several milliseconds later a very weak signal occurred which we attributed to the energy coming through the window openings. In addition, most of the glass from the windows was outside the structure on the ground in front of the wall. The conclusion is that the reflected shock moved 12 m in the time the glass moved through its thickness of 6 mm. Before any significant cracks could open in the windows, the shock front had traveled the 12 or 13 m around the end of the wall and to the gauge. It took several milliseconds more for any significant energy to get through the windows. Two more examples come from the Ernst Mach Institute (EMI) in Freiburg Germany. Dr. Reichenbach and his group were making shadowgraphs of the shock diffraction over a two dimensional version of a simple “house” in a shock tube. The house model had a front wall with a window opening, a solid back wall and a pitched roof which extended slightly beyond the front and back walls. The model was carefully machined out of mild steel and placed in the shock tube. Many good shadowgraphs were obtained. Another model was constructed of balsa wood and had the same dimensions as the steel model. The idea was that they could photograph the difference between the shock diffraction over the rigid model and that over the responding balsa model. When the photographs were examined, there was no difference in the shock geometry or any measurable difference in the position of the structures during the entire diffraction loading phase. The first noticeable motion of the balsa model did not occur until the shock wave had passed out of the test section of the shock tube. Numerical calculations were conducted by Dr. Georg Heilig [1] also at EMI. He examined the response of a 1 mm thick, 8 cm radius, aluminum shell to an incident shock with an overpressure of just over 160 kPa. The incident shock for these calculations was a square topped wave with long duration. He made two


263

AUTODYN–2D (K1530m) & SHARC–2D (15300-20): Fixed Targets (EULER) at the front of the original Shelter 600 SHARC Target: 2.5 deg. AUTODYN Target: 2.5 deg.

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2.4

Fig. 15.12 Rigid and deformable response of a thin cylindrical shell

calculations, one with the structure responding and one treating the structure as rigid. He then compared the calculated overpressure traces at a number of locations on the surface of the shell. Figure 15.12 shows a comparison of the overpressure waveforms just 2 above the surface of the shock tube. No significant difference is seen until nearly 500 ms after the shock strikes the leading surface of the shell. The second peak is from a reflection from the top boundary. In this time, the shock has traveled nearly 8 cylinder radii (26 cm) beyond the shell. Note that the pressure load is decreased at this location when the shell is responding. Figure 15.13 shows a comparison of the waveforms in the same experiment at the 90 location on the shell. The series of shocks near the peak are caused by reflections from the top boundary and from the cylindrical shell. There is essentially no difference in the overpressures until a time of over 400 ms. At that time the pressure load from the responding structure rises above that for the rigid nonresponding approximation. In a separate study, a series of calculations was carried out in which the loads on one building in an urban setting were calculated with a variety of structure response models ranging from rigid to dense fluid (no strength) representations of the building. The blast wave source was a 5 kt nuclear detonation. The comparison of the overpressure load waveforms is shown in Fig. 15.14. In all cases only minor differences were noted during the entire load time of the structure, independent of

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15 Structure Interactions AUTODYN–2D (K1530m) & SHARC–2D (15300–20): Fixed Targets (EULER) at the front of the original Shelter 600 Defromed Shelter at time 2.26 ms Undeformed Shelter at time 0 ms Fixed Targets

Y-Location [mm]

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40

gas pressure [kPa]

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300 250 200

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150 100 50

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1.6

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2.4

Fig. 15.13 Comparison of overpressure waveforms at 90 position for rigid and responding shell

Overpressure Versus Time Building 385, Top

40

Rigid Walls

30

Frangible Walls 1

Pressure (KPa)

Frangible Walls 2 Frangible Walls 3

20

Frangible Walls 3 with Dust

10

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–10 1

2

3 Time (s)

4

Fig. 15.14 Waveform comparisons for blast loads using several response models

5


265

the effective strength of the construction materials. The mass and density of the structure was sufficient to reflect the blast wave. The structure response velocity was small compared to the shock velocity and the shock traveled many building dimensions before any significant motion of the structure occurred. A realistic question to ask is “what is the overall effect on the blast wave of a large number of buildings in close proximity?”. A few scaled experiments have been conducted in the US, the UK and Canada, but the data is limited and applies to only one set of incoming blast parameters. With the current CFD capabilities, the more economical approach is to use large scale three dimensional calculations to answer this question both for specific cases and to examine the general behavior of blast propagation in urban terrain. To examine the effects of multiple structures on blast propagation, an artificial urban environment was constructed with taller buildings in the center and building height decreasing with distance from the center. Figure 15.15 shows the numerical model used in the CFD calculations. A 1 kt detonation was placed at street level between the two tallest buildings. All buildings were treated as rigid and nonresponding because it had clearly been demonstrated experimentally and with calculations, that this was an excellent approximation for blast waves. The results of the CFD calculation are summarized in Fig. 15.16. This figure shows the peak ground level overpressure as a function of location within the urban terrain. Note that the blast wave was channeled down the streets and exposed structures to higher overpressures at greater distances than in other directions. Another related phenomenon, not shown here, is that the fireball vertical radius at a time of one second was greater than the horizontal diameter. Of course the fireball was perturbed by the buildings and was not a simple geometric figure. The vertical diversion of energy also affected the blast wave propagation and loads on distant structures.

Fig. 15.15 Artificial numerical model of an urban terrain

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1010

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1

100 190 280

.48 .36

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.24 .12 .00

400

600 560 520 460 440 400 360 320 260 240 200 160 120 60 40 1

–.12 –.24 –.36 –.48 –.60 –.60

–.48

TIME

2.000

–.36

–.24

–.12 .00 .12 .24 .36 .48 .60 RANGE (X) KM 3D NIAE CITY CALCULATION S CYCLE 1846. PROBLEM 20308.209

Fig. 15.16 Peak overpressure at ground level for artificial urban terrain

Figure 15.17 is a comparison of calculated peak pressures on the ground as a function of distance from the detonation along various radials from the burst. The unobstructed radial, the one down the vertical street of Fig. 15.16, is shown as square symbols, other positions are marked with diamond symbols. The solid line is the free air blast peak overpressure scaled to 2 kt to represent a 1 kt surface burst. The points on the unobstructed radial are about a factor of 3 higher than the free field curve for all pressures above one bar. The majority of the diamond points are also above the free air curve. This is because the peak overpressures are the result of reflections from nearby buildings and the interactions of shocks coming over and around buildings causing partial stagnation of the dynamic pressure. Only in a few locations was the peak overpressure smaller than that from a free air detonation at the same ground range.


267

SHAMRC Non-Responding Building Calculation Peak Overpressure versus Radius 1000 1KT Standard Scaled to 2KT Unobstructed Radial Overpressure (bar)

100

Obstructed Radials

10

1

0.1 10

100 Radius (meters)

1000

Fig. 15.17 Comparison of free field overpressures with calculated peaks in an artificial urban terrain

For those locations where the increase in overpressure was caused by stagnation of the dynamic pressure, the pressure loads on a building at that distance will not be the reflected pressure if one used the R-H relations. If we take the extreme example of a wall positioned at a given distance and oriented perpendicular to the blast wave motion, the peak pressure load will be the reflected pressure. This is also the overpressure that would be reported for that location in Fig. 15.17. Thus the overpressure, in this case, is the reflected pressure and is not subject to further enhancement. Another three dimensional calculation was made by Applied Research Associates, Inc. in 2000, following the Oklahoma City bombing. The calculation started with the detonation of the ammonium nitrate fuel oil mixture in the back of a truck. The truck and many of the buildings and vehicles within about 2,000 ft of the detonation were included in the calculation. Three dimensional building geometries, foot prints, architectural features and heights were carefully modeled. Even vehicles in the parking lot opposite the detonation were included. Fig. 15.18 shows the resultant distribution of the peak overpressure on the ground. The light blue color corresponds to an overpressure of about one psi or 7,000 Pa. Such a pressure level will easily break most standard window glazing. Note the shape of the outline of this pressure level. The shadowing of the buildings and the channeling of the blast down the streets can be readily seen. Prior to these calculations a standard free air curve was used to estimate the pressure levels at various locations. The free air curve for the 1 psi level would have been a circle with a radius of about

268


102

101

100

10 –1

Fig. 15.18 Peak Overpressure distribution at ground level for Oklahoma city

100

102

101

Fig. 15.19 Overpressure impulse distribution at ground level for OKC

References

269

130 m. Such a distance did not explain such things as window breakage nearly 30 blocks from the detonation in a direction away from the Murrah building (the building that was destroyed in the bombing) and only five blocks in the opposite direction. The results of this calculations demonstrated how the blast wave propagated and was shadowed or reinforced by reflections and channeling caused by the structures. The overpressure impulse distribution at ground level is shown in Fig. 15.19. Note the extreme variations in impulse. The scale to the right is in psi *s. The high impulse found between the two buildings across the parking lot from the detonation is caused by the reflection and partial stagnation of the flow in that region.

References 1. Heilig, G.A.: Belastung einer nachgiebigen aluminiumschale durch eine Luftstosswelle. Ernst Mach Institue, Freiburg, Germany (1997)

Chapter 16

External Detonations

Previous chapters have dealt with blast loads on walls and exterior surfaces of buildings or structures. In this section I will briefly discuss how blast wave energy enters a building through windows and doors and the internal loads caused by external detonations. In general the walls floors and roof of a structure are much more substantial than the doors and windows. For most of the experiments that I will be using, the walls were reinforced concrete at least several inches thick. In several of the experiments the doors and windows were simply openings in the structure walls with doorways between rooms. In the first example a 775 pound cylindrical explosive charge with a very light weight aluminum case was detonated 25 ft in front of a three story reinforced concrete structure. The case diameter to thickness ratio was described as about the same ratio as an aluminum “coke” can. Each floor of the structure was divided into four symmetric rooms which were connected by door ways. There were window openings and door ways in the external walls on the ground floor. On the second and third floors there were 2.6 m2 window openings to each room. There was no roof on the structure and the back wall to one room on the ground floor had been removed by previous experiments. Figure 16.1 shows the geometry of the structure and the blast wave front just as the blast wave reaches the top of the structure. The figure was cut by a vertical plane through the center of the charge. The top bulge on the blast wave is caused by the cylindrical charge being detonated on top, thus causing an upward moving jet that accelerates more rapidly than the initial radial expansion. The extremely light case took only 8% of the detonation energy from the blast wave. The blast wave has just entered the windows of the upper floor and has not quite reached the upper outside corners of the structure. On the second floor, the blast wave is just entering the side window and the blast wave entering the front window has not yet reached the side opening. At ground level the shock front has passed the window opening and the interior shock and exterior shock have merged. The loading of a structure depends strongly on the architectural design and geometry of the exterior of the building. Figure 16.2 shows a portion of a building


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16 External Detonations

Fig. 16.1 Blast wave engulfing a three story building

Fig. 16.2 External geometry of a structure with four stories and complex architecture

having complex exterior design. The lower floor has a covered portico and the windows are recessed from the outer surface. Blast loading on this structure from a near surface detonation results in a complex overpressure and impulse distribution.


273

1e+07 8e+06 6e+06 4e+06 2e+06 0

Fig. 16.3 Overpressure loads from a near surface detonation

The blast wave reflects and refracts from the many corners and edges of such a structure. Pressure loads will be enhanced near reflecting surfaces, especially in corners such as those of each window. Figure 16.3 shows the complex peak overpressure loading on the exterior of this design. The units of pressure are dynes per square centimeter. Note the higher loads in the upper corners of each window. The lower floor pillars provide some shadowing, however, note that the shocks coming around the pillars collide on the side of the pillar opposite the incident blast wave and enhance the overpressure. The red and rust colors indicate lower pressures and show the effects of the blast wave turning the corner of the structure. The effects of the external geometry are enhanced when the overpressure impulse is examined. Figure 16.4 shows the impulse distribution on the surface of this structure. The units are cgs. The blast wave entering the covered portico has no place to expand and the overpressure remains higher for a longer time, thus enhancing the impulse on the lower walls. To a lesser extent the same is true for the recessed windows. The reflected shock is contained in the recessed volume and the impulse remains high on the windows. The impulse is significantly reduced as the blast wave rounds the corners of the buildings. This reduced impulse is caused by the formation of a low overpressure vortex at each edge. Note that the impulse is reduced near the top of the front face of the building. This is caused by the rarefaction wave that comes from the top edge of the structure thus reducing the pressure and impulse.

274


24000 19200 14400 9600 4800 0

Fig. 16.4 Impulse loading on a complex geometry structure

Figure 16.5 shows the overpressure distribution at a time of 12 ms on the surfaces of a four story building resulting from a surface detonation 25 ft in front of the center of the building. Only one half of the building is shown. The blast wave is approaching the upper corner of the structure. The red region indicates pressures below ambient. The detonation took place at the lower right of the figure. The low pressure region at the lower left side of the structure is caused by the vortex formed at the side of the building as the blast wave is diffracted around the corner. The interior overpressure loading can be seen through the window on the lower left of the structure. Because the detonation took place at ground level, the initial load on the interior ceiling of the first floor included the stagnation of the dynamic pressure and arrived prior to the blast wave entering the second story window. Thus the load on the floor of the second story was initially upward and no blast wave load in the downward direction occurred until many milliseconds later. Most multiple story buildings are constructed so they will take vertical downward loads on the floors of each level. The upward forces on the floors of the upper stories will initially cause the floors to rise from their supports with a sudden reversal of the forces caused by the blast loading entering through the windows of the next level. This dynamic loading and sudden change of direction may cause significant structural loads and damage. The next figure (Fig. 16.6) shows the interior loads at the same time as shown in Fig. 16.5. The pressure levels are shown in colors with the purple and dark blue


Fig. 16.5 Blast loads on a four story structure

Fig. 16.6 Pressure loading on the interior of a structure from an exterior detonation

275

276


Fig. 16.7 Side view of exterior blast propagating through a structure

Fig. 16.8 Blast wave approaching a three story structure with four rooms per floor; lower floor view, t ¼ 5 ms


277

being high overpressure and the yellow and red being low. The near half of the building has been stripped away so the interior can be viewed. The detonation took place at the centerline of the building; therefore the loads on the interior of the far wall include the stagnation of the dynamic pressure. This figure also clearly shows the delay of loading between the upper and lower floors which causing upward forces on the interior structure. Reflections from the interior columns can be seen in the loading on the ceiling near the columns. About 9 ms later the blast wave has nearly filled the lower floor. The peak overpressure in the blast wave does not decay as rapidly on the interior as the outside free air pressure because the interior expansion is restricted to two dimensions. Figure 16.7 is taken in a plane 10 ft from the centerline of the building. This plane passes through the windows nearest the centerline. The blast wave on the lower floor has nearly twice the pressure as the free filed wave on top of the structure. This figure clearly shows the distance that the shock on the lower level is ahead of the shock on the second level. This results in an upward force on the floors of each of the upper stories. This upward force is enhanced on the roof because a low pressure vortex forms on the roof just behind the front edge of the structure.

105

106

107 CM / S

OVERPRESSURE ZPLANE AT Z = 7.62E+00 CM

16.0 1

20

40

60

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100 120 140

160 170 161

12.8

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140 130 120 110 100 90 80 70 60 50 40 30

RANGE (Y) M

6.4 3.2 0.0 – 3.2 – 6.4 –9.6

20 10

–12.8 –16.0 –10.0 –6.8 –3.6 –0.4

1 2.8 6.0 9.2 RANGE (X) M

12.4 15.6 18.8 22.0

Fig. 16.9 Blast wave interacting with four rooms on the ground floor, t ¼ 8 ms

278


Fig. 16.10 Blast wave interacting with four rooms on the ground floor, t ¼ 13 ms

The next series of figures shows the propagation of the blast wave as it engulfs the first floor of the structure shown in Fig. 16.1 and illustrates the interior reflections from internal walls, the propagation through the door openings and the interaction between the internal and external propagating shocks. At each opening in the front wall the energy transmitted through the opening expands to fill the volume. This rapid expansion reduces the peak overpressure at the shock front and distributes the energy preferentially along a line between the source and the opening. The kinetic energy (dynamic pressure) tends to carry the energy in the direction of flow but the overpressure, a scalar, tends to equally distribute that fraction of the energy evenly into the room. Figure 16.8 shows the blast wave just as it reaches the front wall of the structure. The incident blast wave reflects from the front surface of the structure and enters the structure through the openings in the front surface. Only the lower floor is shown in Fig. 16.9 but this illustrates the complexity of the interacting waves. Note the low overpressure regions at the outside corners of the structure and on either side of the interior of the doorways. These are the result of vortex formation and rapid rotational flow induced by diffraction of the blast wave at each sharp corner. The blast wave propagated through the interior of the structure reaches the windows before the blast wave on the exterior, causing the initial flow to be out of the


279

openings. The interior shocks are reflecting from the side walls producing an outward load of more than 1 bar per cm2. The wave reflected from the front has caught the incident blast wave and enhanced the strength of the outer wave. The blast wave proceeds through the building and reflects from the interior walls. The exterior blast wave continues to decay and weaken. When the interior blast wave reflects from the interior middle wall, a load of over 3 bars per cm2 is generated and pushes outward on the exterior walls. Figure 16.10 clearly illustrates this interaction. Only a small amount of energy from the initial blast wave on the interior of the structure gets through the doors of the interior walls and reaches the back rooms. The interior wall reflects most of the energy into the front room of the structure. The exterior blast wave reaches the exterior openings (Fig. 16.11) of the back rooms before the blast wave propagated through the interior doors can expand to fill the rooms. The flow is inward through the windows. The blast wave propagating through the interior doors is weak and expands nearly spherically from the openings. Note also that the blast wave reflected from the interior center wall has reached the front wall of the structure and provides a load of about 2 bars per cm2 in the front corners.

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OVERPRESSURE ZPLANE AT Z = 7.62E+00 CM

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Fig. 16.11 Blast wave interacting with four rooms on the ground floor, t ¼ 21.5 ms

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2.8

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9.2

12.4 15.6 18.8 22.0

RANGE (X) M

Fig. 16.12 Blast wave interacting with four rooms on the ground floor, t ¼ 31.6 ms

The exterior blast wave overpressure peak has decayed to about 0.2 bars by the time the structure has been completely engulfed. This compares to a peak overpressure in excess of 4 bars when the incident blast wave struck the front wall. Figure 16.12 shows that the overpressure within the structure is generally higher than on the exterior of the structure. The back room on one side of the structure did not have a back wall and the blast wave exits that room while it reflects from the back wall of the adjoining room. The vortices formed at the corners of the front face of the structure have traveled down the sides of the structure and are now near the side window openings. A vortex is forming at the rear corner on the side of the structure with the rear wall intact. Remember, vortices generally reduce the pressure loading on the structure.

Chapter 17

Internal Detonations

Detonations inside structures present a number of complicating factors. Multiple reflections from walls, floors and ceilings interact and enhance the overpressure. In a structure which has few internal walls or partitions such as a parking garage, the blast wave reflects from the floor and ceiling. Mach stems form on both surfaces. As the triple points grow away from each surface, the two Mach stems will combine. At that time the expansion of the blast wave is essentially cylindrical and the effective yield of the blast wave is nearly four times that of the original detonation. Figure 17.1 is taken at a time when the blast wave has propagated more than five effective heights of burst. The burst took place at R ¼ 0 at a height of burst of zero between a floor and ceiling separated by 3 m. The Mach stems shown at the shock front are the second Mach stems caused by reflection of the reflected shocks. The first Mach stems combined before the front had traveled three heights of burst. The enhancement in effective yield is relative to a free field detonation. We will examine the behavior of the blast wave as it encounters a single opening in a wall. Think of this as a doorway with the door open. Figure 17.2 shows the results of a three dimensional CFD calculation at a time just prior to the blast front reaching the opening. The shock strength is about 3 bars (2 bars overpressure). Note that there is a strong negative phase about 8 m behind the front and the pressure returns to near ambient at the burst point. The geometry shown is about the simplest possible for the study of a blast through an internal opening. The burst point is 16 m from the wall and is aligned with the opening, the blast wave has separated from the detonation products and the expansion on the far side of the wall will be symmetric. The dots in Fig. 17.2 are monitoring points at which the blast wave parameters will be recorded as a function of time. The behavior of the blast wave on the far side of the wall is strongly dependent on the strength of the shock front. For incident pressures above about 4 bars, the dynamic pressure is greater than the overpressure. The blast wave momentum is aligned with the direction of the radial from the detonation point and the shock remains strongest in that direction. The overpressure is a scalar and the internal energy of the blast wave expands uniformly from the opening. The combination of the dynamic pressure with a vector for the momentum


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17 Internal Detonations

Fig. 17.1 Detonation between floor and ceiling PRESSURE ZPLANE AT Z – 1.60E+02 CM 30 1

10

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3 5 7 9 11 13 15 17

1.050E+06 1.350E+06 1.650E+06 1.850E+06 2.250E+06 2.550E+06 2.850E+06 3.150E+06

190 180 160 120 DX1 = –2.330E+03 80 MIN = 8.707E+05 40 X = –7.840E+02 30 Y = 1.926E+02 20 MAX = 3.429E+06 X = –1.750E+01 Y = 4.250E+01 10

6 0 –6 –12 –18 –24 –30 –30

1 –24

–18

–12

–6 0 6 12 18 24 30 RANGE (X) M MEA RUN9 3D (R = 16, AREA = 8, THETA = 90) TIME 17.000 MSEC CYCLE 0. PROBLEM 9704.1748

Fig. 17.2 Blast wave approaching an opening in a wall

and the overpressure interact to redistribute the blast wave energy on the far side of the wall. Figure 17.3 shows the pressure distribution after a strong shock (50 bars) has traveled through a hole in the wall. The reflected pressure on the wall is 380 bars but the transmitted pressure through the opening is near the incident value on the line from the detonation point through the center of the doorway. The pressure decays rapidly on either side of the center line. The energy of this expansion is drawn from


283 PRESSURE ZPLANE AT Z – 1.61E+02 CM

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4.000E+07 6.000E+07 1.200E+08 1.600E+08 2.000E+08 2.400E+08 2.800E+08 3.200E+08 3.600E+08

DX1 = –3.000E+02 MIN = 6.478E–02 X = 1.250E+00 Y = 2.876E+01 MAX = 3.806E+08 X = –8.750E+00 Y = –4.375E+01

20 10 1 1.5

RANGE (X) M MEA RUN1 3D (R = 1, AREA = 1, THETA = 90) TIME 500.000 USEC

CYCLE

79.

PROBLEM 9704.1738

Fig. 17.3 Blast wave propagation through a wall opening

the higher pressure region near the center line, causing the pressure along the center line to decay more rapidly than in the free air. In order to examine the behavior of shocks propagating through openings, a series of careful experiments were conducted at the Ernst Mach Institute in Freiburg, Germany. A Mach 1.31 shock was photographed as it propagated through a series of baffles in much the same way that a blast wave would propagate from room to room if all the doors were aligned. Dr. Heinz Reichenbach of EMI, graciously gave me a set of shadowgrams from these experiments when I visited his laboratory. These very detailed photographs were also used to evaluate the accuracy of first principles code calculations for this very complex flow. A series of calculations were conducted by several agencies and compared to the experimental results. The agreement between the calculated shock positions and the experiments provided confidence in the numerical results. The numerical calculations could then be used to determine the pressure and dynamic pressure distributions in these very complex flows. Figure 17.4 is the shadowgraph showing the shock positions at a time of 114.3 ms after the interaction with the first baffle. Vortices have formed on the leading and trailing edges of the first opening. The shock has just reached the opening in the second baffle. Note that the transmitted shock is nearly cylindrical and centered on the center of the opening.

284


Fig. 17.4 Experimental shadowgram of a shock wave traversing a baffle system, t ¼ 114.3 ms after interaction with the first baffle

Fig. 17.5 Mach 1.31 shock through a baffle system, t ¼ 174.3 ms

In the next photo, the leading shock has advanced to the middle of the region between the second and third baffle. The shock has expanded equally from the sides of the opening. This is a strong indication that the strength of the shock has fallen to the point that the dynamic pressure is nearly negligible. The vortices from the first baffle have shed from their original position and the shocks reflected from the second baffle have reflected from the top and bottom of the second section (Fig. 17.5). By a time of 234 ms, the transmitted wave has reached the third baffle. Figure 17.6 shows the shock configuration at this time. The leading shock wave has weakened significantly. Much of the energy has been trapped in the inter baffle regions in the form of reflecting shocks and vortices. If we skip ahead to a time of 354 ms, Fig. 17.7 shows the complex shock and vortex interactions after the leading shock has passed the final baffle. The vortices formed at the first baffle opening have reached the second baffle and the vortices formed by the second baffle opening are near the center of the region between the second and third baffles. Multiple shocks have interacted with the vortices and have been diffracted by this interaction. The leading shock has weakened to the point that it is just barely discernable. The reflected shocks to the left of the last baffle are not visible near the center of the last baffle.


285



Fig. 17.8 Pressure distribution from a cased explosive detonation

The reflected shock patterns in a single room can become very complex, even with little internal structure to perturb the blast wave. Figure 17.8 shows the calculated shock pattern at a time of 25 ms on the walls of a single large room with a box in one corner and a vertical cylinder placed asymmetrically within the room. A cased explosive was detonated near mid-height in the room at an off center location. The black dots are case fragments. All of the walls and the cylinder are non-responding. The pressure distribution of the shocks is plotted on the walls of

286


Fig. 17.9 Calculated results for a structure with responding walls

the room. The Mach reflections from the ceiling and floor are clearly seen. The incident shock is traveling radially from the burst point. The reflected shocks from the floor and ceiling have passed through one another and intersect about a room height behind the incident shock. The blast wave has not reached the back of the cylinder. A detonation within a structure having frangible partitions or internal dividing walls makes the prediction of the blast environment more difficult than for a structure with non-responding walls. In Fig. 17.9, the detonation of a cased munition containing more than 500 pounds of explosive takes place in the center hallway of a four room structure. The interior rooms were surrounded by a hallway and a hallway ran down the middle of the structure from upper left to lower right. The exterior walls were reinforced concrete and were supported by exterior earth berms and were treated as non-responding. A series of reinforced concrete pillars supplied support for the roof and end support for the frangible walls. A detailed three dimensional CFD calculation was made for this configuration and a full scale experiment was conducted. For walls that were within the direct line of sight of the explosion, fragment damage from the case of the device was an important part of the frangible wall failure. The fragments were only effective against the first wall that was struck because the fragment kinetic energy was significantly reduced by the interaction with the first wall. Fragments that reached a second wall after having passed through a frangible wall had lost about half of their momentum and 75% of their kinetic energy. Because the walls were relatively easily perforated by the fragments, there were no fragment reflections from the frangible walls. Experimentally it was noted that room contents that were not in direct line of sight with the explosion did not receive fragment damage.

17.1 Blast Propagation in Tunnels

287

Blast reflection and interaction with the frangible walls, even those struck by fragments, initially behave as non-responding walls. Reflection factors for blast waves can be applied with good accuracy for first interactions. During the time of interaction, the walls move a very small distance compared to the distance the air blast moves in the same time. The same argument cannot be made for the secondary shocks or reflections. The reflection is no longer simple. Breaches in the wall may be caused by the first interactions, so the “wall” for the second reflection may be curved, have holes and may be moving in a complex way. The results of the calculation indicated that the contents of the rooms remained within the walls of the rooms, however the walls of the rooms were translated to the far corners of the outer structure. In general, this was found to be the case experimentally as well. Most of the room contents were found within the walls of the rooms but were buried in debris from the walls and ceiling (which collapsed after being lifted vertically).

17.1

Blast Propagation in Tunnels

In this section is a brief discussion of blast waves propagating in tunnels. Some general principles, based on energy conservation and blast wave characteristics, are given. As with buildings and similar structures there is a significant difference between energy entering from an external detonation and propagation of a blast wave from a detonation within the structure or tunnel system. The first rule of thumb that is sometimes used for blast waves in tunnels is that the pressure, which has units of energy per unit volume, can be calculated as proportional to the volume of the tunnel into which the energy has expanded at a given time. Thus, by taking the volume of the tunnel behind the shock front, the pressure in the tunnel can be approximated. If the pressure distribution at any one time after detonation is known, then the pressure distribution at another time can be calculated by knowing the volume which the energy occupies at the other time. This rule of thumb works reasonably well so long as the energy in the tunnel is fixed. If energy vents in or out of an opening, the expectation of constant energy in the volume is lost. If the energy in the tunnel comes from a source external to the tunnel entrance, the amount of energy in the tunnel will vary as a function of time after the detonation. For detonations exterior to but near the tunnel entrance, locations near the tunnel entrance will be directly affected by the free field blast parameters. Energy will enter the tunnel opening during the free field positive duration at the opening. The negative phase of the free field blast will draw energy out of the tunnel. The energy exiting the tunnel will be greater than might be expected using the free field parameters because the pressure decay in the tunnel, (a one dimensional flow) decays less rapidly than the free field. Thus, when the pressure at the entrance drops due to formation of the free field negative phase, the higher pressure in the tunnel accelerates the mass and energy from inside the tunnel to the exterior. As the

288


blast wave propagates into the tunnel, a negative phase will form inside the tunnel. At this point, the shock is propagating independently of the external source. The rule of thumb assumption of fixed energy holds beyond this point and the simple rule gives very good agreement with data. When a detonation occurs just inside a tunnel entrance, initially half the energy is directed into the tunnel and half toward the exit. The rapid expansion of the blast wave as it exits the tunnel causes a rarefaction wave to travel into the tunnel at the local speed of sound. The high sound speed in the fireball inside the tunnel accelerates the rarefaction wave and may reverse the flow direction well inside the tunnel. This flow reversal takes additional energy out of the tunnel system leaving less than half the energy to propagate into the tunnel. The dynamic pressure in a blast wave in a tunnel is rapidly oriented along the direction of the tunnel. Radial reflections damp out rapidly and the primary flow is parallel to the tunnel axis. At the shock front, the Rankine–Hugoniot relations hold and they can be used to find all other shock front parameters if one parameter and the ambient conditions are known. For pressures above about 4.5 bars, the dynamic pressure exceeds the overpressure. The flow at high overpressure is dominated by the momentum of the flow, whereas at low pressures, the scalar overpressure will dominate. The partitioning of the blast wave energy between kinetic and internal will thus have a dominant influence on the propagation in a tunnel system. If tunnel walls are rough, such as may be found in blast and muck construction, the roughness tends to stagnate the flow near the walls. Large protuberances from the walls may cause reflections. The reflections have the effect of redistributing the energy by sending shock waves upstream against the incoming flow. In tunnels where the tunnel radius is only a few times the perturbation heights, the reflected shocks may provide a significant blockage of the flow through the tunnel. For smooth walled tunnels, the boundary layer effects on blast waves are usually minimal. Remember from Chap. 8 that the growth of a boundary layer is proportional to the shear gradient in the flow at the wall. This has a peak at the shock front and decays rapidly as the blast wave passes. Over sidewalk smooth concrete for large yield blast waves of hundreds of kilotons, the boundary layer has been measured at less than 3 in. in height. For any reasonable blast wave in a tunnel of a few meters in diameter the positive phase will be much smaller than that of a large nuclear detonation and the height of the boundary layer will not exceed a centimeter or so. Let us assume that we have a smooth walled tunnel and that boundary layers can be ignored. A strong blast wave (greater than 5 bars) is propagating along a straight smooth tunnel. A side drift emanates from the main tunnel perpendicular to the main tunnel. Assuming the side drift has the same diameter as the main tunnel, we can use a simple thought experiment to envision the blast wave behavior at the intersection. At the most basic level, only the energy associated with the overpressure will easily change direction. Thus a first approximation to the energy turning and going down the side drift will be about half of the internal energy of the blast wave in the main tunnel. If the side drift has a different diameter than the main tunnel, the fraction of the internal energy that turns the corner will be proportional

17.1 Blast Propagation in Tunnels

289

to the ratio of the cross sectional areas of the tunnels. The energy continuing along the main tunnel will be the fraction of the energy that is kinetic plus the remainder of the internal energy. These relatively simple ways of looking at flow in tunnels must be remembered as just rules of thumb. In the actual case of blast waves in tunnels, the dynamic component of the flow in the main tunnel will partially stagnate on the far side of the drift tunnel wall and send a shock back upstream. This reflected shock partially blocks the flow into the side drift and partially stagnates the flow in the main tunnel. Figure 17.10 shows the three simple tunnel intersection configurations that are considered here. When a tunnel turns a 90 corner, an L tunnel, the flow down the main tunnel stagnates at the end of the tunnel, the flow is essentially stopped, the energy is converted to internal energy and pressure. The flow must be re-established from the stagnation region. Because overpressure is a scalar, the pressure will act equally in the directions of the incoming flow and in the direction of the L tunnel. More of the energy will be propagated into the portion of the tunnel having ambient conditions than will move against the incoming flow. The flow will rapidly, within a few tunnel diameters, re-establish in the L tunnel, but because some of the energy has been reflected back up the incident tunnel, the blast wave will be weakened by such a corner turning. When a tunnel dead ends into a cross tunnel, a T tunnel, the flow is stagnated against the wall of the cross tunnel. The stagnated energy is divided equally into the three possible flow directions. The energy directed back against the incoming flow is partially stagnated by the incoming flow and is redirected along the two other channels. Thus the shock strength of the turned blast wave on each side of the cross tunnel is less than half of that of the incident blast wave. As the tunnel intersection become more complex, there are no simple rules of thumb for determining energy partitioning for blast wave propagation. When tunnel Side Drift

L Tunnel

Fig. 17.10 Simple tunnel configurations

T Tunnel

290


diameter varies along the length of the tunnel, the local variations of the cross section influence the propagation of the blast wave. For a tunnel system such as shown in Fig. 17.11, [1, 2] the blast wave flow can be very complex. The charge detonated in chamber A sends a blast wave toward the main tunnel entrance. At the same time a shock is reflected from the end of chamber A and also is directed toward the main entrance. At the intersection of the cross tunnel a portion of the blast wave energy is diverted into the cross tunnel. A reflection occurs at the end of the short part of the cross tunnel and the shock interacts with the flow at the main tunnel. Some of the reflected energy crosses the main tunnel toward chamber B. The blast wave expands in chamber B, reflects from the back wall and is directed back toward the main tunnel. In the mean time, the primary blast wave and the reflected wave from the back of chamber A are exiting the tunnel. All of the timing of reflected shocks is dependent on the size of the detonation. This flow is further complicated when the walls of the tunnels are very rough. Many tunnel systems have specialized regions to prevent debris such as case fragments, rocks or pieces of concrete, trucks or fork lifts from becoming sources of 60 m

15.5 m

25 m

12 m 2.5 m

4m Chamber A Charge

Tunnel Entrance

14.5 m Chamber B 17 m

Fig. 17.11 A simple cross tunnel with chambers

Fig. 17.12 A simple tunnel debris trap

References

291

damage as they may be accelerated by the flow in the tunnels system. One such mechanism is a simple debris trap. In its simplest form a debris trap is an extension of a tunnel at an L or T section. In Fig. 17.12, the momentum of the debris entrained in the blast wave flow carries the debris past the intersecting tunnel and is caught in the stagnated flow at the end of the tunnel. The initial gas flow of the blast wave is stagnated in the debris trap and the following flow continues around the corner and down the intersecting tunnel, while the debris remains in the stagnated flow of the debris trap. A succession of such debris traps may be constructed throughout a tunnel system in order to protect other regions of the tunnel from damage caused by debris impact.

References 1. Kennedy, L.W., Schneider, K., Crepeau, J.: Predictive calculations for Klotz Club tests in Sweden, SSS-TR-89-11049. In: S-Cubed, Dec. 1989 2. Vretblad, B.: Klotz Club tests in Sweden. In: 23rd Explosive safety seminar, Atlanta Georgia, vol. 1, pp. 855, August 1988

Chapter 18

Simulation Techniques

Air blast phenomena scale over many orders of magnitude. The scaling laws described in Chap. 12 are limited by the type of explosive source, not by the scale of the phenomena being studied. A spherical blast wave reflecting from a flat plane can be scaled over more than 12 orders of magnitude. Blast wave reflection phenomena are independent of the scale at which they are studied. At the Ernst Mach Institute in Germany, tests are often conducted in the laboratory using 0.5 g charges of PETN. Special care must be taken to ensure accurate geometry and instrumentation dimensions because a small deviation at this scale may be significant at full scale. For example, a 1 m diameter boulder at the 8 kt scale becomes a 0.4 mm grain of sand at the half gram scale. Many of the advances in the understanding of blast waves can be directly attributed to the nearly infinite scalability of air blast phenomena. A number of methods have been developed which permit the study of blast wave phenomena at laboratory or at least at manageable scales.

18.1

Blast Waves in Shock Tubes

A basic shock tube consists of a driver section, a run up region and a test section. The driver section may use a number of methods to generate the energy to produce the driven shock. To produce a blast wave, the driver section volume is small compared to that of the run up and test sections of the tube. This allows a rarefaction wave from the end of the driver tube to catch the shock front before the shock reaches the test section. The shock in the test section then decays as it propagates through the test or measurement section as a blast wave. Some examples of drivers for shock tubes that produce blast waves include: A high explosive charge detonated in a driver section, sudden electrical energy release (spark), compressed gas released by either a diaphragm or fast acting valve, gas compression by a piston or high explosive shaped charge. C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_18, # Springer-Verlag Berlin Heidelberg 2010

293

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18 Simulation Techniques

Another method of producing a blast wave in a shock tube is to change the volume of the shock tube as a function of distance from the driver. Conical shock tubes produce a realistic free field blast wave decay because they represent the spherical divergence of a free air detonation. Although I have not seen one, it would be possible to build a run up and test section in the form of a wedge. Such a tube would produce a blast wave with the decay of a cylindrical expansion. Shock tubes are in wide use throughout the world. Most mechanical engineering departments at any university has at least one shock tube. Practical shock tubes vary in size from an inch or two in diameter and a few meters long to the LB/TS at White Sands Missile Range in New Mexico. The LB/TS was designed to generate blast waves to simulate a full scale nuclear detonation. As mentioned previously, the LB/ TS is 11 m in radius and over 200 m in length. The energy source is a group of nine high pressure steel tubes about 2 m in diameter and with varying fixed lengths. The diaphragm on each of the driver tubes is 1 m in diameter. The diaphragms may be released simultaneously or in sequence, thus varying the duration of the blast wave. Shock tubes may be designed so that the test section and run up sections may be evacuated, thus allowing the study of high strength blast waves without exceeding the maximum pressure allowed by the construction of the tube. The behavior of blast wave phenomena can be studied to examine the effect of the gamma (ratio of specific heats) of the gas by filling the run up and test section with different gasses. The gamma can thus be varied continuously between 5/3 for a monatomic gas and a gamma of 1.065 for uranium hexafluoride or slightly greater than 1.08 for sulfur hexafluoride. Shock tubes can thus be used to study a very wide range of phenomena over a wide range of shock strengths from M 1.01 to M > 10.

18.2

High Explosive Charges

As was mentioned in Chap. 12, all blast wave phenomena can be scaled by the cube root of the charge size. Thus laboratory investigations can be conducted using whatever charge size is convenient. The restriction here is dominated by the minimum detonable charge size and the size and accuracy of the measurement systems. Because most explosives have a critical diameter, below which a detonation cannot be sustained, only a few explosives can be used at small scale. Nearly all of the explosives with small critical diameters are sensitive to handling and must be treated carefully. In order to study blast wave propagation and interactions, the initial detonation must be symmetric, whether cylindrical or spherical. The detonation of small charges requires special techniques. Most commercial detonators are larger than the gram sized laboratory charges and cannot be used. A carefully controlled electric discharge is the usual technique. The use of too low a discharge and the explosive burns but does not detonate; use of too large a discharge and the explosive may breakup and not detonate. Table 18.1 lists a few explosives and their critical diameters [1].

18.2 High Explosive Charges Table 18.1 Critical diameters for selected high explosives

295 Explosive PETN PBX-9404 RDX TNT, Pressed Octol Pentolite

Critical diameter (cm) 0.02 0.118 0.2 0.26 0.64 0.67

At larger scale, high explosive charges may be used to produce blast waves which simulate even larger detonations. Detonations of explosive charge weights of a few thousand pounds are common for field experiments which are conducted on a regular basis at test sites around the world. Simulation of a nuclear blast wave may be accomplished using almost any scale. The largest experiment for a nuclear blast simulation with which I have been associated was a hemispherical charge containing 4,800 tons of AN/FO. On such a test, full scale structures and equipment can be tested to validate their response to a nuclear blast. The use of a hemispherical charge has the good property of providing an easily characterized, smoothly decaying blast wave. It also has the undesirable effect of having a large area in contact with the ground surface. Such a large area of the surface exposed to the full detonation pressure of the explosive, creates a very large crater. The crater formation is accompanied by large amounts of crater ejecta which may fall on test articles at large distances from the detonation. In order to balance the size of the crater with the air blast, tangent spheres were used at large scale (up to 500 tons) for simulation of nuclear air blast. This configuration provided a good ratio between air blast and crater size, but the jet of detonation products which forms near the surface, perturbs the air blast and induces large vertical components of velocity to the flow that was desired to be parallel to the ground. Refer back to Fig. 14.7 to see an example of such a tangent sphere configuration and the resultant air blast. A good compromise was developed by using a cylindrical charge with a hemispherical cap. This geometry reduced the surface area exposed to the detonation pressure and had the added advantage of producing a cylindrically decaying shock, at least initially. The blast wave thus produced had all of the desired characteristics with the added bonus that the relative crater size could be controlled by adjusting the length to diameter ratio of the cylinder of explosives. In order to provide the desired blast wave moving parallel to the ground and oriented perpendicular to the surface, the cylindrical portion of the charge required multiple detonators. The detonators were placed on the vertical axis of symmetry of the cylinder. One detonator was placed at the ground surface and another at the center of the base of the hemispherical cap. Additional detonators were placed on the axis of the charge spaced evenly with a separation of less than one half charge radius. When the detonation took place, the detonation waves interacted when they reached a radius of about one quarter the charge radius. The detonation waves formed Mach stems before they reached half the charge radius. The Mach stems combined into a

296


very nearly cylindrical detonation wave before it reached the outer charge radius. This technique produced a very clean cylindrical blast wave for testing the response of structures.

18.3

Charge Arrays

A technique which has been used to simulate a blast wave from a large yield but requires only a small fraction of the simulated charge weight, is to use vertical arrays of individual charges or multiple strands of detonating cord. A sample of such an array is shown in the cartoon of Fig. 18.1. The blast wave generated by each individual charge coalesces with that of its neighbors. These then combine with those of the others and a plane blast wave is formed which decays inversely proportional to the overall array size. The overpressure for the generated combined wave can be adjusted by increasing the charge density. This can be accomplished by increasing the charge size or decreasing the separation distance. The duration and impulse can be adjusted by increasing the size of the array. The volume that contains a reasonably representative waveform for the total simulated yield is restricted to the regions on either side of the array, along the center line, perpendicular to the array and within about two array heights but at a distance greater than about half the array height. Larger arrays will have a larger usable test volume as well as a larger impulse. Rarefaction waves move in from the edges of the array and reduce the impulse as the blast wave propagates away from the array. Such arrays have been successfully used inside the LB/TS at White Sands. When the charge array is detonated inside the shock tube, the walls of the tube reflect the shocks and provide a much more efficiently generated blast wave by eliminating the rarefaction waves. The blast wave remains planar as it travels the length of the shock tube and radial waves dampen and coalesce into a relatively clean blast wave. The use of detonating cord in arrays has similar characteristics for the generation of blast waves. Figure 18.2 shows a typical array as used in the LB/TS at White Sands missile range NM. The cords in the array may be detonated simultaneously or in a sequence. Some success has been found by detonating alternating cords from

Fig. 18.1 Vertical charge array for simulating long duration blast waves with reduced total explosive weight

18.3 Charge Arrays

297

18° FROM CEILING

LB / TS Tunnel Cross Section

18° FROM FLOOR

Ignition Point

15 STRANDS, 200 GR. DET CORD-403’ 25 GR. DET CORD-58’ 11.7 LBS PETN 56’ END VIEW

Fig. 18.2 Detonating cord array used in a semi cylindrical shock tube

PETN Detonation at 2 msec

Jacket Afterburn at 9 msec

Fig. 18.3 Use of detonating cord as a driver in the LB/TS

the top and bottom, thus reducing the influence of the detonation direction on the formation of the blast wave. One drawback to using detonating cord is the fact that a large fraction of the mass of the cord is the jacketing material surrounding the explosive core. This mass must be accounted for when calculating the amount of explosive to be used in the simulations because the mass of the jacketing material initially detracts from the energy of the explosive that can generate a blast wave but then burns and adds energy to the tail of the blast wave. Figure 18.3 illustrates the detonation of the array and the afterburn of the jacketing material at three times the detonation time. As an example of the efficiency of such arrays, the array contained only 11.5 pounds of explosive but generated a planar blast wave with an impulse equivalent to more than 50 pounds. In a different facility, a very high operating pressure blast tube, about 5 tons of explosive could be used to generate the full impulse of a 2 megaton detonation at the 125 psi level. The tube is 20 ft in diameter with a 300 ft long high strength steel driver section built to withstand the pressure generated by a solid explosive driver. The driver section is designed to handle about 400 psi on the walls of the facility, therefore the solid driver charge must be distributed near the axis of the 300 ft long driver section. The maximum loading is thus just over 50 kg/m in the driver section. The remainder of the 825 ft long tube has a thickness of 1.5 in. of steel and is rated at 150 psi. This tube was built by the U.S. air force at Kirtland AFB NM.

298

18.4


Use of Exit Jets to Simulate Nuclear Thermal Precursor Blast Environments

Because of the difficulties of generating full scale thermal precursor environments using conventional helium layer techniques, an alternative was suggested by the army research laboratory. This technique made use of the fact that the blast wave exiting a shock tube expands rapidly, thus decreasing the overpressure. The dynamic pressure of the blast wave decays much less rapidly. In addition the decay of the overpressure on the exterior of the tube enhances the dynamic pressure behind the shock because the flow is further accelerated by the pressure gradient. The blast wave generated in the exit jet has many of the characteristics of a thermally perturbed blast wave. The overpressure is significantly decreased within a diameter or two outside the tube and decreases rapidly with increasing distance. The dynamic pressure peak occurs during a minimum phase of the overpressure and is caused by the acceleration of the flow behind the shock front. The peak dynamic pressure is several times that for an ideal wave based of the same overpressure and the dynamic pressure impulse is several times that of an ideal wave with the same peak overpressure. Figure 18.4 is a view of the LB/TS from the exit jet test region. Figure 18.5 shows a comparison of the precursed dynamic pressure impulse measured during a nuclear test. The yield was just under 40 kt. The data is not scaled or adjusted for altitude. The solid curves were generated using the SHAMRC CFD code described earlier. The only source of ideal information is from CFD codes or scaled high explosive tests. Note that at a range of 700 m the precursed

Fig. 18.4 View into the LB/TS from the instrumented earth berm. The exit jet is used to simulate thermally precursed blast waves

18.4 Use of Exit Jets to Simulate Nuclear Thermal Precursor Blast Environments

299

Fig. 18.5 Measured and calculated dynamic pressure impulse vs. ground range for ideal and precursed blast waves (40 kt)

dynamic pressure impulse is seven times that of the ideal and at 800 m is 8.5 times the ideal value. For reference, the ideal peak overpressures are called out at various ranges. The ideal and precursed impulse values converge at an overpressure level between 8 and 10 psi. Figure 18.6 is a comparison of the waveforms for dynamic pressure as a function of time for an ideal and thermally precursed wave at a ground range of 914 m. The curve labeled “Priscilla” is a fit to the average of several measurements. The peak dynamic pressure is about three times that of the ideal. The precursor arrival is 150 ms before that of the ideal and the impulse is more than 7 times that of the ideal. (The measured impulse points on Fig. 18.6 are the result of gauge failure prior to the completion of the positive phase.) The simulation of the Priscilla dynamic pressure waveforms is well represented by the exit jet method. Figure 18.7 compares the dynamic waveforms, measured and calculated for the LB/TS exit jet with the comparable smoothed waveform from the Priscilla event. While the waveforms differ in some details, the peak dynamic pressures and impulses provide a good match to the nuclear data. Remember that the dynamic pressure impulse can be directly correlated to the motion of vehicles exposed to the dynamic pressure environment. This is shown dramatically in Fig. 11.8, “Jeep displacement as a function of dynamic pressure impulse.”

18 Simulation Techniques Dynamic Pressure at 3000 ft Range

16

40

Dynamic Pressure [psi]

14

35

Ideal

12

PRISCILLA

30

10

25

8

20

6

15

4

10

2

5

0 1.0

1.1

1.2

1.3

1.4

1.5 1.6 Time [sec]

1.7

1.8

1.9

Dynamic Pressure Impulse [Kpa-s]

300

0 2.0

Fig. 18.6 Comparison of ideal and precursed dynamic pressure waveforms

Test Data SHARC 3D Calculation PRISCILLA

Dynamic Pressure (psi)

30

80 70

25

60

20

50

15

40

10

30

5

20

0 200 –5

400

600 800 Time (msec)

1000

Dynamic Pressure Impulse (KPa - sec)

35

Dynamic Pressure Comparison LB / TS Exit Jet Test, 30 m from Tunnel Exit Gauge Height = 3 m

10 1200 0

Fig. 18.7 Comparison of calculated and experimental exit jet dynamic pressure waveforms with smoothed Priscilla fit

18.4 Use of Exit Jets to Simulate Nuclear Thermal Precursor Blast Environments

301

As a result of six full scale exit jet tests, the exit jet method has been selected as the best feasible method of providing realistic thermally precursed loads on test articles. Because a dirt berm had been constructed outside the LB/TS, a realistic amount of dust was swept up by the blast wave and influenced both the overpressure and dynamic pressure waveforms. The dust is an important part of the contribution to the structure loads. Figure 18.8 is a comparison of the experimental measured waveform with the calculated waveforms with and without the inclusion of dust sweep-up. The measurements were made 40 m outside the tube and 1 ft above the surface. When dust is included in the calculation, the waveform is much closer to the measurement and has a similar impulse. The momentum of the dust acts as a damping mechanism to the oscillations in the flow. The dust also plays an important role in the timing of secondary shocks and the momentum of the flow around structures and the loads imposed on structures in the flow. Figure 18.9 is a comparison of the measured and calculated loads on a military vehicle. This figure is typical of the agreement obtained between calculated and experimental waveforms when dust is included in the calculations.

EJ214-4B 99 - CT- A - 004 34.dat, CL 40m 1ft 03 - 17 - 1999 Cal val = 25.30

Experiment SHAMRC 276 SHAMRC 276 w / dust

40

Static Overpressure, KPa

6

4

20

2

0

0

–20

–2

–40

–4

–60

0

250

500

750 Time, msec

1000

1250

–6 1500

Fig. 18.8 Comparisons of calculated and measured overpressure waveforms in an exit jet

Impulse, KPa-s

60

302

18 Simulation Techniques CHANNEL 2 P2

12

0.32

Down Stream Loads Measured Pressure Measured Impulse ARA Calculated Pressure ARA Calculated Impulse

10

0.24 0.16

6

0.08

4

0

2

–0.08

0

–0.16

–2

–0.24

–4

–0.32

PSI

8

–6 0.05

0.1

0.15

0.2

0.25 0.3 0.35 TIME-SEC

0.4

0.45

0.5

–0.4 0.55

Fig. 18.9 Comparison of calculated and measured loads on a military vehicle using the exit jet method

This measurement was made on the downstream side of the vehicle and includes the effects of vortex flow over the top of the vehicle and blast propagation under and around the ends of the vehicle. The overall agreement is excellent. When dust was not included in the flow, the total impulse of the load differed not only in magnitude, but in sign as well. This is because of the excessive negative phase formed in the overpressure when dust is not included. (see Fig. 18.8 for example).

References 1. Hall Thomas, N., Holden, James R.: Navy Explosive Handbook, Explosion Effects and Properties Part III, Naval Surface Warfare Center, Research and Technology Department, October, (1988)

Chapter 19

Some Notes on Non-ideal Explosives

My definition of a non-ideal explosive is: an explosive or detonable mixture of chemicals that releases some of its energy after the passage of the detonation front. Under this definition, many common solid explosives are non-ideal. The energy released can be divided into the heat of detonation and the heat of combustion, where the heat of combustion is generated by burning of or taking place in the products created by the detonation. As a classic example, TNT releases about 1,600 calories per cc upon detonation. Nearly 20% of the detonation products are carbon in the form of soot. This carbon has the potential to release, upon combustion, an additional 3,200 calories per cc or twice the detonation energy. The key here is the word potential. This means that only under special conditions can even a fraction of that potential be realized. Included in this non-ideal class are all explosives containing TNT and all plastic bonded explosives as well as many more. A sub-class of non-ideal explosives are those which have been labeled as “thermobaric”. A good working definition of a thermobaric explosive is: an explosive or detonable mixture of chemicals which includes active metal particulates. The metal particulates are commonly aluminum, magnesium, titanium, boron, zirconium or mixtures or alloys of these metals. The above list is not intended to be complete, but to serve as an example of the wide variety of possible particulates that may be used. The particulates may be spheroids or flakes with sizes ranging from nanometers to millimeters. The metal particles may be coated with Viton or Teflon, both of which release fluorine upon heating, at a temperature lower than the metal particles ignition temperature. The fluorine can react with the oxide coatings of the metal particulates before the oxide melts thus reducing the effective ignition temperature. A sub-set of the thermobaric mixtures is Solid Fuel Air Explosives (SFAE). In solid fuel air explosives, the metal particulates surround a central high explosive charge which disperses and initiates the burn of the particulates.


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19.1

19 Some Notes on Non-ideal Explosives

Properties of Non-ideal Explosives

Non-ideal explosives that do not contain metal particulates generate their combustion energy by mixing with atmospheric oxygen. Detonation products such as soot, carbon monoxide or hydrocarbons such as methane and ethane may burn when mixed with oxygen. The temperature of the mixture must be greater than the ignition temperature of the fuel to be burned. Because the detonation produces these unreacted species, non-ideal explosives are usually less sensitive than more ideal explosives may be. The temperatures immediately after detonation of nonideal explosives tend to be lower than from ideal explosives because the energy released is shared with the combustible detonation products.

19.2

Combustion or Afterburning Dependency of Non-ideal Explosives

The performance of non-ideal explosives is affected by several variables including: charge size, charge casing, proximity to reflecting surfaces, venting from the test structure, and oxygen availability.

19.2.1

Charge Size

Charge size is important because the detonation products cool more rapidly for small charges. All of the mixing takes place at the unstable interface between the detonation products and air. The mixing for larger charges continues while the temperature remains above the initiation temperature of the detonation products for a longer time, thus allowing a larger fraction of the detonation products to mix with the air. For TNT charges of a hundred tons or more, the fireball remains above the initiation temperature for carbon for several seconds. The afterburn of carbon takes place in the rising fireball as it forms the classic toroidal mushroom cloud. The energy released during this combustion process is released much too late to contribute to the blast wave but may have a significant effect on the fireball rise rate and stabilization altitude because the energy is added to that of the fireball.

19.2.2

Casing Effects

Casing material and weight also affect non-ideal explosive performance. Moderate to heavily cased charges (Chap. 6.3), convert 50–70% of the detonation energy into kinetic energy of the case fragments. The source of the kinetic energy is the heat of

19.2 Combustion or Afterburning Dependency of Non-ideal Explosives

305

the fireball detonation products. The sound speed in the detonation products is sufficiently high that near equilibrium for pressure and temperature is maintained in the detonation products during the early expansion. About 90% of the energy conversion takes place in the time that it takes to double the radius of the case. Case fracture takes place at about this time. This means that the detonation products have cooled by more than a factor of two before the case breaks and permits any mixing with the atmosphere. The detonation products may have cooled below the ignition temperature of the combustible products before any mixing can take place. Thermobaric and solid fuel air explosives on the other hand, initially are prevented from expanding rapidly and cooling at early times. This provides a slightly longer time for metallic particulates to heat in the elevated temperature of the detonation products. In almost all cases, the overall effect of the case is to reduce the combustion energy generated. As an illustration of this effect, Fig. 19.1 is taken from a paper by Kibong Kim [1] with the author’s permission. The figure shows that 6% of the aluminum burns in less than 200 ms in the cased charge. It takes twice that long for the bare charge to burn 6% of the aluminum, however by a millisecond, the cased charge has only burned 7% of its aluminum but the bare charge has burned 17% and continues to rapidly burn aluminum.

20% 18% Uncased 16%

Cased

Burned Al (%)

14% 12% 10% 8% 6% 4% 2% 0% 0

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 Time (s)

Fig. 19.1 Burned aluminum percentage vs. time for cased and bare charges of PBXN-109

306

19.2.3


Proximity of Reflecting Surfaces

The proximity of reflecting surfaces affects the performance of non-ideal explosives. The blast wave reflected from the nearby surfaces propagates back through the detonation products. The blast wave reheats the fireball material by compressive heating and fresh ambient air follows in the flow behind the front. Thus the reflected blast waves provide additional heating time and bring fresh oxygen to mix with the detonation products. The additional heating time and gas temperature increase are especially important to improving the metal particulate heating in thermobarics and SFAE. In a tunnel system, the availability of fresh air is very restricted. The detonation products quickly fill the diameter of the tunnel in the vicinity of the detonation. As the detonation products expand, the only mixing of fresh air is at the interface of the detonation products which is restricted to the cross sectional area of the tunnel. After the blast wave separates from the detonation product interface, the mixing slows and combustion slows accordingly.

19.2.4

Effects of Venting From the Structure

Two tests were conducted in a two room test structure at Kirtland AFB New Mexico in which the same sized and type of thermobaric explosive was detonated in the same location in the test structure. The only difference between the two tests was that in one test the doors and windows were left as openings, while in the second test the doors were covered with plywood and the windows with standard ¼ in. glass. The experiment with the closed window and doors showed an enhancement in the measured overpressure impulse of nearly 10%. Figure 19.2 is a comparison of the overpressure waveforms from the two tests from a gauge in the detonation room. The door blew out at a time of about 40 ms. The comparison shows that the waveforms are essentially overlays until a time of about 7 ms. The impulse from the test with the closed door then increases above that of the test with openings. After the door and window have been blown away, the room can then vent and the difference in impulse after that time is constant. SHAMRC calculations were conducted for the same test conditions. The calculations were completed before the experiments were conducted. These calculations were truly predictive. Because the agreement between calculation and experiment is excellent, the calculations can be used to understand the complex chemistry and the interactions between shock heating and mixing caused by instabilities. Using the calculations of these experiments, it is possible to determine not only the amount of aluminum burned as a function of time, but also to determine the oxidizer used as a function of time. Figure 19.3 shows the aluminum combustion results for the open window and doors. About 22% of the aluminum burned with 19% burning in the detonation products and only about 3% burning aerobically.

307

180

0.9

160

0.8

140

0.7

120

0.6

100

0.5 Test 1 Test 2

80

0.4

60

0.3

40

0.2

20

0.1

0 –20

Impulse (psi-sec)

Pressure (psi)


0 0

0.01

0.02 0.03 Time (sec)

– 0.1 0.05

0.04

Fig. 19.2 Comparison of blast waves from a thermobaric charge in a structure with open doors and windows and with closed doors and windows 30%

AL Mass Burned

25% 20% 15% % AI Mass Burned % brnd with det. O2 % brnd with amb. O2

10%

5% 0% 0.00

0.01

0.01

0.02

0.02 0.03 0.03 Time (s)

0.04

0.04

0.05

0.05

Fig. 19.3 Aluminum combustion for Test 1 (unrestricted openings)

For the case with the doors and windows covered with frangible materials, Fig. 19.4 shows that 24% of the aluminium burned, with 21% burning in the detonation products and just over 3% burning aerobically. This demonstrates that only a minimal change in confinement can increase the impulse of the blast wave from a thermobaric mixture by 10%.

308

19 Some Notes on Non-ideal Explosives 30%

AL Mass Burned

25%

20%

15% % AI Mass Burned % brnd with det. O2 % brnd with amb. O2

10%

5%

0% 0.00

0.01

0.01

0.02

0.02

0.03 0.03 Time (s)

0.04

0.04

0.05

0.05

Fig. 19.4 Aluminum combustion for Test 2 (doors and window in place)

19.2.5

Oxygen Availability

The burning efficiency of non-ideal explosives is dependent on the availability of ambient atmospheric oxygen. For explosive mixes that do not contain metal particulates, essentially no anaerobic afterburn energy generation is possible. The detonation product species are formed immediately behind the detonation front and very quickly come to chemical equilibrium. When the explosive mixture contains metallic particulates, the particulates may react with the detonation products (anaerobic reactions) without the presence of any other source of oxygen. The explosive mixture PBXN-109 contains approximately 20% aluminum particulates. The detonation products include water and carbon dioxide, as well as carbon, and a few other combustible species. Hot aluminum will burn in water or carbon dioxide. When aluminum burns in water, hydrogen is released and when aluminum burns in carbon dioxide, carbon monoxide is released. The hydrogen and carbon monoxide cannot react with oxygen in the detonation products because the strong reaction with aluminum has absorbed all the available oxygen. When atmospheric oxygen mixes with the detonation products, the hot aluminum competes with the hydrogen, carbon and carbon monoxide for the available oxygen. When the aluminum cools below its initiation temperature, the remaining species compete for the available oxygen. A pair of experiments was conducted in the same two room structure mentioned above. In the first experiment, the detonation took place in an ambient atmosphere (the baseline), in the other test the detonation room was filled with 99% nitrogen. PBXN-109 was used as the explosive source in both cases. Predictive SHAMRC


309

calculations were made for both the atmospheric oxygen content and the nearly pure nitrogen atmosphere. The same afterburn model was used as was used in the venting experiments mentioned above. Figure 19.5 compares experimental and calculated (SHAMRC) waveforms for a pressure gauge in the detonation room. The impulse for the baseline case is 1.5 times the impulse from the nitrogen fill experiment. This is a good indication that the aerobic combustion accounts for the majority of the energy for this explosive. The calculated results agree very well with the experimental data. Because the calculations agree so well with experiment, we can use the results of the calculation to determine the amounts of aluminum burned in the two experiments. Figure 19.6 compares the mass of aluminum burned in each case. In addition, the calculations provide the mass of aluminum burned in the water or carbon dioxide of the detonation products and in atmospheric oxygen. For the nitrogen filled detonation room, essentially all the aluminum that combusts, burns in the first millisecond and nearly all of that burns in the water of the detonation products. As would be expected, the same amount of aluminum burns in the ambient atmospheric case in the first millisecond, but after the case breaks nearly 50% more aluminum burns in the detonation product water. An additional 10 g burned aerobically in the first 30 ms. It is the aerobic burning that provided the additional energy to keep the aluminum particulates hot enough to continue to burn in the detonation products.

8-lb steel cased PBXN-109 4w12-30

1.05

175 Test II - 4a - Baseline SHAMRC - PBXN-109 Baseline SHAMRC - PBXN-109 Nitrogen Test IX-100a - Nitrogen

Pressure (psi)

125

0.9 0.75

100

0.6

75

0.45

50

0.3

25

0.15

0 –25

Impulse (psi - sec)

150

0 0

0.016

0.032 0.048 Time (sec)

0.064

– 0.15 0.08

Fig. 19.5 Blast wave comparisons, experimental and calculated, in normal and nitrogen atmospheres

310

19 Some Notes on Non-ideal Explosives Mass of Aluminum Burned

90

Standard Atmosphere - Solid lines Nitrogen Atmosphere - Dotted lines

80 70

Mass (g)

60 50 40

Total Burned In Atmosphere In H2O In Carbon Dioxide

30 20 10 0 0

0.005

0.01

0.015 Time (s)

0.02

0.025

0.03

Fig. 19.6 Comparison of aluminum combustion in standard and nitrogen atmospheres

19.2.6

Importance of Particle Size Distribution in Thermobarics

The previous sections discussed the importance of several parameters on the burning of aluminum particulates. For thermobaric explosives, including SFAEs, the particle size distribution (PSD) has a very strong influence on the efficiency of aluminum particulate burn. Specifying a mean particle size does not provide sufficient information to determine the efficiency of the aluminum combustion. Figure 19.7 shows a typical particle size distribution with a stated mean of 20 microns. The sizes vary from about 2 microns to more than 200. Only about 10% of the aluminum mass has a particle size between 20 and 30 microns. This figure also compares the PSD used in the calculation for this explosive with the measured size distribution. We have found that it is necessary to faithfully model the PSD in order to reach good agreement with experimental blast data. The importance of the PSD is shown in Fig. 19.8, the heating time required as a function of particulate diameter. The heating times were calculated assuming that the particles are soaked in a constant temperature bath and are in velocity equilibrium with the gas (no slip). The particles were assumed to be spherical and the time plotted is the time required to reach 2,050 K. The heating time goes as the square of the diameter; a one order of magnitude increase in diameter requires two orders of magnitude longer time to heat. A 1 mm particle takes 1 s to heat to 2,050 K in a 4,000 K bath. It takes a microsecond for a one micron particle to reach ignition temperature and a detonation wave travels about 0.7 cm in that time, there is no way that such a “large” particle could participate in the detonation process. Metal particulates can participate in the detonation process, but they must be much smaller than a micron.


311

12

Measured Distribution Generated Distribution

10

Percent Mass

8

6

4

2

0 1

10 100 Particle Diameter (micron)

1000

Fig. 19.7 Typical aluminum particle size distribution used in explosives Aluminum Particulate Heat Time vs Diameter for Different Soak Temperatures no Slip 1.0E+00 2500 K 3000 K 4000 K

1.0E – 01

Time (sec)

1.0E– 02 1.0E– 03 1.0E– 04 1.0E– 05 1.0E– 06 1.0E– 07 1

10

100

1000

Diameter (microns)

Fig. 19.8 Aluminum particle heating time as a function of particle diameter

If they do participate in the detonation, then the explosive has lost the advantage of not being required to carry the oxidizer with the explosive. One of the measures of the efficiency of thermobaric explosives is the energy obtained per unit mass of

312


the explosive mixture. This is the reason that SFAE charges are preferred in some applications because no oxidizer is carried, but is supplied by the surrounding atmosphere.

References 1. Kim, K., et al.: Performance of Small Cased and Bare PBXN-109 Charges, Proceedings of the International Symposium on the Interaction of the Effects of Munitions with Structures, Orlando, Florida, September 17–21, (2007)

Chapter 20

Modeling Blast Waves

In Chaps. 4 and 5 the nuclear blast standard and the high explosive or TNT blast standard were described. Each of these standards provide a full description of a free field blast wave in a sea level constant atmosphere. All blast parameters are given (or calculable from the provided parameters) as a function of range at a given time.

20.1

Non-linear Shock Addition Rules

Using one of these standards and a set of non-linear addition rules it is possible to construct the waveform at a given point which sees the effects of two or more blast waves. This is very useful for defining blast wave time histories that are generated by the combination of two or more detonations or by a blast wave reflection from a planar surface. The detonations need not be simultaneous nor do they need to be of the same or similar yields. The addition rules have been labeled as the “LAMB” addition rules in part in recognition of the work of Sir Horace Lamb which contributed to hydrodynamics and because the rules are used extensively in the Low Altitude Multiple Burst model. The addition rules are based loosely on the conservation laws of mass, momentum and energy. They are only as good as the free field models used to describe a single blast wave in the free field. Because the TNT and nuclear standards (Chaps 4 and 5) do provide a very close approximation to a free field blast wave and are very nearly conservative, application of the conservation laws provides a physically meaningful and consistent description of the interaction of multiple blast waves. Figure 20.1 gives the approximations used in the LAMB addition rules for “conservation” of mass, momentum and energy. The first equation states that the density at a point in space is equal to the ambient density plus the sum, over the number of detonations, of the over densities of each of the contributing blast waves. For momentum the total momentum at the point of interest is the vector sum of the momenta of each contributing blast wave at that point. The vector velocity is


313

314

20 Modeling Blast Waves

NB

P = P0 +

∑ ΔP

i

i=1

+

1 2

NB

∑ 1.2 r

∗ i

i=1

2

Vi

1

−2

2

r V

Fig. 20.1 The LAMB addition rules

obtained by dividing the summed momenta by the density calculated in the first equation. Conservation of energy is used to calculate the pressure at the point of interest. The pressure is the sum of the ambient pressure plus the sum of the overpressures from each contributing blast wave plus 1.2 times the total specific dynamic pressure minus the dynamic pressure of the combined waves as determined from the first two equations. This procedure is indeed non linear, it does preserve the vector nature of the velocity and momentum and in some sense conserves energy. Experience has shown that a modification to the above rules must be added. In some instances, multiple waveforms may be overlaid at a time and position such that several negative phases are coincident. In rare cases, the sum of the over densities may be negative and greater in magnitude than the ambient density. This leads to a non-physical negative density. In such cases the minimum density allowed should be set to a fraction (perhaps 10%) of ambient density and the calculations continued.

20.2

Image Bursts

To see how well this simple set of rules works, we can use the example of a blast wave reflecting from a smooth, flat, perfectly reflecting surface. In this case we can use the concept of image bursts in which an ideal planar reflecting surface can be represented by an image burst of the same yield at the same distance on the opposite side of the reflecting plane. Figure 20.2 is a cartoon demonstrating the concept of an image burst.

20.2 Image Bursts

315 Reflecting Surface Distance to Burst

Distance to Image

Burst Yield = Y1

Image Burst Yield = Y1

Fig. 20.2 Image burst representing a perfectly reflecting plane

A detonation at a distance H from a reflecting plane produces a blast wave with an incident overpressure of DP. The image burst produces a blast wave with the same pressure but moving in the opposite direction. Using the LAMB addition rules we can find the peak pressure at the plane by combining the blast wave parameters of the two incident shocks. Because the velocities of the two shocks have equal magnitude but opposite sign, the momentum rule results in a zero velocity at the plane. The pressure is found by summing the two overpressures and 1.2 times the two dynamic pressures. Because the resultant velocity is zero the resultant pressure is twice the over pressure plus 2.4 times the dynamic pressure of the incident blast wave. If we refer back to the Rankine–Hugoniot relations for the reflected pressure (3.13): DPr ¼ 2DP þ ðg þ 1Þq, we see that the reflected over pressure from the LAMB addition rules is the same as that from the R-H relations for a value of gamma of 1.4. Another example of the application of the LAMB addition rules is a comparison of overpressure waveforms from an experiment in which explosive charges were simultaneously detonated at heights of burst of 45 and 135 ft over the same ground zero. The test was conducted to provide a means of measuring the difference between the properties of blast waves reflected from the ground and from an ideal reflecting plane. The lower burst was 45 ft above the ground and the plane mid-way between the two charges was 45 ft from each charge. Overpressure measurements were taken at a large number of positions at various heights ranging from ground level to 50 ft. The name of the series of tests was Dipole West, conducted in Alberta Canada in the mid 1970s and was sponsored by the Defense Nuclear Agency and the Army Ballistics Research Labs [1]. Figure 20.3 is a comparison of the overpressure and impulse waveforms from the experiment and those obtained using the TNT standard and the LAMB addition rules for a gauge at ground zero. The experimental data are shown by the solid line and the model is given by the dashed line. Note the excellent agreement in the first peak and the entire positive duration of the first blast wave. The shock from the upper burst arrives at the ground 27 ms after that of the lower burst. In the

316


PRESSURE (PSI)

10

DIPOLE WEST VI ST 0.0 SYS 2 CH 3

6 2

IMPULSE (PSI –MSECS)

–2 DIPOLE WEST VI ST 0.0 SYS 2 CH 3

90 60 30 0 –0

10

20

30

40

50 60 70 MILLISECONDS

80

90

100

110

120

Fig. 20.3 Comparison of experimental and LAMB rule overpressure waveforms, DW-shot VI ground zero

waveform constructed using the LAMB addition rules the second shock is nearly 10 ms later than the data because shock from the upper burst was accelerated by the high sound speed of the lower fireball, thus arriving sooner. The LAMB methodology does not account for the time difference caused by passage through a high sound speed region. At a range of 40 ft from ground zero and 10 ft in the air, the incident and reflected shocks of both bursts can be seen. Figure 20.4 is a comparison of the experimental and LAMB generated overpressure waveforms and their impulses. The experimental data are again the solid curve and the model is the dashed curve. This gauge is above the triple point of the Mach reflection, therefore the incident and reflected waves of the lower charge are the first two blast waves to reach this gauge. The agreement here is quite good. The blast wave from the upper charge was significantly influenced by its passage through the high sound speed fireball of the lower burst. The shock from the upper detonation and its reflection from the ground are about 10 ms later in the model. At a range of 60 ft, (Fig. 20.5) the triple point of the lower blast wave passes below the gauge located 20 ft above the surface. For the upper burst, the 60 ft range is also in the regular reflection region and both the incident and reflected shocks are recorded. The experimental data show that the shock front from the upper burst arrives before the ground reflected shock from the lower burst. The model provides the correct arrival time for the ground reflected shock from the lower detonation but is about 10 ms slow on the incident shock from the upper detonation. This time delay in the model reverses the order of arrival in this instance. The calculation was not taken to a sufficiently late time that the ground reflected shock from the upper burst arrived at this position. The concept of image bursts is a useful method of modeling shock reflections, not only from a single plane but from walls, floors or ceilings of rooms or buildings.

20.2 Image Bursts

317

Fig. 20.4 Comparison of experimental and LAMB rule overpressure waveforms, DW-shot VI (40 ft range, 10 ft height) DIPOLE WEST VI ST 80.20 SYS 1 CH 11

PRESSURE (PSI)

6 3

0

IMPULSE (PSI – MSWECS)

–3 DIPOLE WEST VI ST 60.20 SYS 1 CH 11

60

40

20 0 –0

10

20

30

40

50

60

70

80

90

100

110

120

MILLISECONDS

Fig. 20.5 Comparison of experimental and LAMB rule overpressure waveforms, DW-shot VI (60 ft range, 20 ft height)

Suppose that a detonation takes place between two planes, not necessarily at the midpoint. Image bursts can be used to represent both planes simply by placing the image bursts at the appropriate distance on the far side of each plane. Figure 20.6 is a cartoon of the placement for one such configuration.

318


Fig. 20.6 Image burst configuration for two reflecting walls

Wall 1

H1

Wall 2

H1

Image Burst = Y1

Burst = Y1 Image Burst = Y1 H2

H2

This logic can be further extended to include multiple image bursts. The reflected shock from wall 1 will reflect from wall 2. In order to account for that reflection, an image of the image burst to the left of wall1 would be placed a distance 2H1 þ H2 to the right of wall 2. In the case of a three dimensional box, there are six image bursts that represent the reflections from the six walls of the structure. This method can also be extended to account for the reflections of shocks from the image bursts by adding additional images of the images. In three dimensions the number of secondary images to represent the reflections of the primary images is 26. This can further be extended to as many levels as are desired. The NB in the summation terms of the LAMB addition rules must be set to the total number of image bursts plus one for the original burst.

20.3

Modeling the Mach Stem

The formation of a Mach stem was described in Chap. 13. A model is presented here which provides a reasonable approximation to formation of the triple point as a function of height of burst (HOB) and ground range. The equations in Table 20.1 use the height of burst, scaled to 1 kt, to determine the scaled ground range at which the Mach stem first appears for any height of burst. These relations are attributed to Dr. Harold Brode [2]. For scaled HOB less than 99.25 m the ground range (scaled meters) is simply 0.825 times the HOB. For higher heights of burst, the second equation is used. Care must be taken to ensure that the units are converted to scaled meters and that the results are also in scaled meters. The path of the triple point is described by a cubic polynomial passing through the ground plane at the ground range described by the equations of Table 20.1. The triple point determines the height of the Mach stem as a function of time. The procedure is described in [3] Fig. 20.7 shows the results of this triple point path fit compared to interpolated experimental data points. The vertical and horizontal scales on Fig. 20.7 are not the same. The horizontal scale is exaggerated by a factor of 4. Note the good agreement over this wide range of scaled heights of burst. The height of the triple point can be used to define the geometry of the blast wave fronts for any detonation at a height of burst. A cartoon of this is shown in Fig. 20.8.

20.3 Modeling the Mach Stem Table 20.1 Equations for the ground range for initial Mach reflection

319 r0 = 0.825*HOB HOB < 99.25 m/kt1/3 For Higher HOB use: r0 ¼

170 HOB (1 þ 25:505 HOB0:25 þ 1:7176e 7 HOB2:5 Þ

250 Burst Height = 50 ft

Mach - Stem Hieght (ft)

200

100 ft 150

200 ft 300 ft

100 400 ft 500 ft 50

600 ft 700 ft

0

800 ft

0

200

400

600 800 1000 Ground Distance (ft)

1200

1400

1600

Fit to Triple Point DATA

Fig. 20.7 Triple point path for 1 kt detonations Incident Shock (TNT STD)

Burst

Triple Point Reflected Shock (LAMB Shock Addition)

Triple Point Path (Polynomial Fit) Mach Stem

Fig. 20.8 Shock geometry for evaluating the LAMB addition rules for a height of burst

The procedure for evaluating the LAMB addition rules are slightly modified although the addition rules remain unchanged. The radius of the blast wave from the image burst is stretched so that it passes through the triple point. The blast wave parameters are not modified. An arc with the radius of the distance from ground

320


zero to the triple point is drawn from the triple point to the ground. This gives a curvature to the Mach stem and ensures that the Mach stem is perpendicular to the ground at ground level. Below the triple point, both the incident and image waves are stretched to coincide with the position of the Mach stem.

20.4

Loads from External Sources

The modeling of loads on a structure resulting from an external detonation has been accomplished at the most accurate level by utilizing three dimensional CFD codes. This process is very expensive and requires a separate calculation for each change in blast yield or position. Various manuals have been written which provide graphs and rules to approximate the loads on structures. What follows is a description of recent models in which the accuracy falls between these two methods, require minimal computer resources and run in a matter of seconds on a laptop computer.

20.4.1

A Model for Propagating Blast Waves Around Corners

Several calculations were made using the three dimensional SHAMRC CFD code (AMR version of SHARC) to describe the loading on a single building. The pressure time histories were recorded on all sides of the building. The effects modeled in the first principles code included the reflection of the shock on the near surface, the refraction of the shock at building corners, formation of vortex fields at each corner and the rarefaction waves from the corners, including the roof. In order to gain some understanding of the behavior of the shock as it engulfs the building the pressures at a number of points on the various walls and in the near field as a function of time were monitored. The calculation was for a 2,000 pound TNT charge detonated approximately 70 ft from the front face of the building. In Fig. 20.9, the points on the light line labeled SHAMRC results were taken along a line from the detonation point to the corner of the building, along the side of the building and around the back side of the building. The curve labeled TNT standard is the free field peak overpressure as a function of range for the 2,000 pound charge. For comparison, the peak overpressure at twice and four times the distance for the free field overpressures was plotted. It was noted that the pressure at the shock front dropped as it rounded the corner of the structure and the decay fell parallel to the overpressure curve at twice the distance. Further, the peak overpressure dropped to correspond to the pressure at four times the distance when it rounded the second corner at 117 ft to the backside of the building. The curve labeled “ECD” was an earlier attempt to model this phenomenon. This observation provided the idea of using a simple geometric interpretation of the shock as it engulfed the building [4]. Figure 20.10 is a cartoon of the geometry of the blast wave and the building dimensions. The burst is not symmetrically

20.4 Loads from External Sources

321

Fig. 20.9 Overpressure vs. Range for 2,000 pound TNT detonation

Φ Burst

Rc

Rw θ Point of interest

Fig. 20.10 Treatment for points that are not in the line of sight

located, therefore, the angles F and Y are not equal. The point of interest is outside of the line of site from the burst point and the blast wave must turn a corner in order to reach this point. To find the overpressure at the point of interest we calculate the total distance from the burst point to the point of interest by summing the distance from the burst to the corner of the building, Rc, plus the distance from the corner of the building to

322


Fig. 20.11 Illustration of the diffracted blast wave engulfing a building

the point of interest, Rw. This is used to find the radius of the shock when it reaches the point of interest. Rt ¼ Rc þ Rw In fact, the shock did travel that distance to get to the point of interest. The resulting shock geometry is shown in Fig. 20.11. Note the curvature and “delay” of the shock as it travels around the building. When we evaluate the pressure from the model at that range, we find it is higher than what was calculated by the first principles code. Using the observation from Fig. 20.9 and the results of the first principles calculation, a relation was developed that the pressure at the point of interest is the pressure at the radius equal to Rp ¼ Rt ð1 þ sin yÞ Thus we have a two step procedure for determining the refracted shock geometry and the refracted shock pressure. We have found that this procedure can be used to describe not only the peak overpressure, but provides a good approximation to the time history of the overpressure. Note that this procedure accounts for the discontinuous drop in overpressure as the shock reaches the corner of the structure. This procedure works equally well for the shock being refracted around a second corner. Figure 20.11 illustrates the geometry for the evaluation of the pressure after the shock turns a second corner. The radius for the shock is measured as the sum of the radius from the charge to the first corner plus the radius along the length of the building plus the radius from the second corner to the point of interest: R t ¼ Rc þ Rw þ Rs

20.4 Loads from External Sources

323

The pressure at the point of interest is found by evaluating the pressure from the TNT standard at a distance of: Rp ¼ Rt ð1 þ sin yÞ ð1 þ sin aÞ Again, the total time history can be constructed by calling the TNT standard at a sequence of times for the same point. One of the complications with combining the shocks that have followed various paths is that only the minimum path length should be used for each surface of the building. Referring to Fig. 20.12, the path following the lower route is the minimum around the lower side, the route through angle F on the upper side can be readily calculated and a path over the top (out of this plane) of the building would provide a third shock path. Thus algorithms have been developed to find the shortest path over each surface. This can be accomplished by randomly choosing a large number of possible paths and finding the minimum for each of the sides/top of the building. When a blast wave strikes a finite planar object (such as a building), the image burst model can be used to describe the reflected shocks from the building surface and in the volume surrounding the structure. The image bursts are combined with the diffracted primary blast wave described above. Figure 20.13 is a cartoon showing the locations of the image bursts for an arbitrary burst location near a rectangular structure. This method also permits the use of the LAMB addition rules for the combination of shocks that come from the different sides or over the top of the structure. This method can then account for the interaction and stagnation of the shocks on the backside of a building. For a finite target, the addition rules are restricted to the region that is within the shadow region of the image burst. In the case shown in Fig. 20.13, the blast wave from image burst 1 is used only in the region below and to the right of the structure. This region is defined by the extension of the vectors from the image burst location to the lower corners of the structure. The region in which the effects of image burst 2 are included is to the left of the structure in the region defined by the extension of

Φ Burst

Rc

Rs α

Rw θ

Point of interest

Fig. 20.12 Geometry for turning a second corner

324


Fig. 20.13 Image burst locations for arbitrary structure orientation

Fig. 20.14 Modeled blast wave interaction with a structure

the vectors from the image burst 2 locations to the corners on the left side of the building. Figure 20.14 shows the combined blast wave diffraction and reflection when modeled using the above described procedures.

20.5 Blast Propagation Through an Opening in a Wall

325

Fig. 20.15 First principles calculated results of a blast wave interaction with a structure

As a check for the accuracy of this model, the modeled blast wave configuration of Fig. 20.14 can be compared with the results of the first principles code shown in Fig. 20.15. Note the geometry of the modeled refracted shock is nearly identical to that of the CFD result. The modeled reflected shocks are in the proper location, but they have abrupt terminations on both sides of the shock. The first principles CFD results show rapidly varying but continuous shock geometry.

20.5

Blast Propagation Through an Opening in a Wall

The assumption here is that the wall is infinite in extent and has a single opening of area A. The model [5] makes no assumptions about the shape of the opening because this would require specific information on the design of the building. The problem is

326


Fig. 20.16 Three dimensional results for a 15 m standoff, charge in line with opening

to define the distribution of pressure on the far side of the wall as a function of range and incident angle to the opening on the detonation side, the opening area, the range, and the angle from the opening on the far side of the wall. Figure 20.16 shows a CFD result just as the shock approaches the opening in the wall. We use this example problem with the angle between the opening and the detonation perpendicular (90 ) to the wall. The energy going through the opening is the fraction of the energy contained in the solid angle between the detonation point and the opening area. Thus the energy fraction through the opening is Ef ¼ 2A/3 R0 =ð4=3 p R3o Þ ¼ A/ð2 p R0 2 Þ where Ef is less than or equal to 1. At the door opening the effective yield is the original yield Y0, but the energy passing through the door is Y1 ¼ Y0 A/(2 p R02). The yield therefore transitions as (R0/R)2 between the limits of 1 at the opening and A/(2 p R02), where R is the total distance from the burst and R0 is the radius from the burst to the opening. As the shock progresses through the door, this fraction of energy is redistributed, but not uniformly. The angular distribution of the energy on the far side of the wall is proportional to the ratio of the dynamic pressure to the overpressure. Thus, at very low overpressures the opening will behave like a source of the reduced yield


327

located at the center of the opening. At high overpressures, the source will be directional, with a preferential direction aligned with the radius vector to the charge. Any point in alignment with the door opening will see the original yield for a greater distance than those points in the shadow region of the wall. The dynamic pressure at high overpressure is 2.5 times the overpressure. The overpressure, being a scalar, attempts to redistribute the energy equally in all directions. The dynamic pressure is directed and attempts to continue carrying the momentum and energy in the direction of the vector from the charge. When the detonation is not aligned with the opening, the effective yield, Y1, is further reduced by the effective size of the opening.

20.5.1

Angular Dependence of Transmitted Wave

Let F be the angle between the radial from the charge to the edge of the opening and the radial from the edge of the opening to the target point. When F is plus or minus 90 , the energy is proportional to DP (the overpressure, a scalar). We define the angle a to be the angular width of the opening. When F is outside the angular opening defined by a, the energy distribution is proportional to the ratio of the component of the dynamic pressure to the overpressure in the direction of the target point. Figure 20.17 illustrates the geometry and the angle relationships. For an ideal gas (g ¼ 1.4) Q¼

5ðDPÞ2 : 2ð7P þ DPÞ

Q 5DP Therefore, DP ¼ 2ð7þDPÞ , if we let P ambient ¼ 1. cos F and the The proportion of energy in the direction F is thus defined as DPþQ DPþQ effective yield is calculated accordingly. Each target point has an effective yield

Detonation

α

Φ3 Φ1 Target Point 2 (within line of sight) Target Point 1

Fig. 20.17 Geometry for general orientation of a burst with the opening

Target Point 3

328


associated with its location. Because the effective yield is specified at each target point, the model produces not only the arrival time and peak overpressure, but complete waveforms of all blast parameters. These waveforms may be integrated to provide impulses directly.

20.5.2

Blast Wave Propagation Through a Second Opening

Whether the source is in another room, or is in the open on the other side of the wall, the effective yield at the opening is modified by the same function of the opening area as described above. For the case of a second opening in the non-blast room, the effective yield at the center of the second opening becomes the effective yield as adjusted by the distance and angular position relative to the first opening. As the blast propagates through the second opening, the effective yield is further reduced by the opening area ratio and the angular adjustment, just as the yield was changed by passage through the first opening. Figure 20.18 shows the geometry for a second opening. The energy through the second opening is calculated in the same manner as the first and the blast environment in the second room is partitioned based on the angular distribution and the ratio of the overpressure and dynamic pressure at the second opening. The procedure is the same as described for the first opening except that the initial yield is now Y1 rather than Y0. The effective yield at the second opening Y2 is defined in terms of Y1 and the geometry. The model was exercised against a large number of three dimensional first principles (SHARC) hydrodynamic calculations. Figure 20.19 shows the results

TARGET TARGET

Detonation Room

Room 2

R2 Y0 0 R1 R1

R0 α? ? Θ

? Φ Y1

Room 1

Fig. 20.18 Geometry for propagation through a second opening

Y Y22


329

Fig. 20.19 Pressure distribution in second room, model vs. SHARC CFD code

of the pressure distribution in the second room for the case of a 100 kg TNT detonation placed 1 m in front of a 1 m2 opening into the second room. For perfect agreement, the data would fall on the diagonal line. Points above the line indicate that the model is higher than the CFD results and points below the line indicate that the model gives lower pressures than CFD. The model is on the low side at low values, but is consistently within a factor of two of the CFD results. The deviation at low overpressures is not considered to be a serious problem because the low overpressures are less important for most structure loads and response. Figure 20.20 shows the overpressure comparison for the case of the detonation being 4 m from a 4 m2 opening at a 60 angle. For this larger distance, and therefore less divergent flowfield, the model consistently tracks the SHAMRC results at all pressure levels and there is no falloff at the lower pressures. The largest differences occur when the position in the second room is on a line perpendicular to the line of sight at the opening and is minimal when the points fall along the line of sight. The algorithm presented here provides a very fast and efficient method of defining the air blast propagated into a second room through a relatively small opening. This method provides not only the peak overpressure waveforms as a function of time, but the dynamic pressures as well. These waveforms may also be integrated to provide the overpressure and dynamic pressure impulses at any location in the second room. The model is readily extended to the propagation of a shock through a second opening into a third room. This is accomplished by redundantly applying the same rules to the second opening as were applied to the first opening.

330


Fig. 20.20 Comparison of model and first principles calculations for the charge at 60 from the opening

The accuracy of the model (less than a factor of 2) is sufficient for most applications and is well within the known frangibility limits of most structures. Most overpressure points in the second room fall within 25% of the first principles calculations. One of the advantages of this method is that it requires no image bursts or shock addition logic. Improvements to the model which could be easily implemented include varying the yield in the second room using a similar algorithm to what is used in the detonation room to account for the reflections from the floor and ceiling. The effects of reflections from the walls of the detonation room and the second room could be included by adding image bursts and including the LAMB addition rules.

References 1. Keefer, J.H., Reisler, R.E.: Multiburst Environment- Simultaneous Detonations, Project Dipole West, BRL-1766. Ballistic Research Labs, Aberdeen, MD (1975) 2. Brode, H.L.: Height of Burst Effects at High Overpressures, DASA 2506, Defense Atomic Support Agency, July, (1970)

References

331

3. Needham, C.E., Hikida, S.: LAMB: Single Burst Model, S-Cubed 84-6402, October, 1983 4. Needham, C.E.: Blast Loads and Propagation around and over a Building. Proceedings of the 26th International symposium on shock waves. October, 2006 5. Needham, C.E.: Blast Propagation through Windows and Doors, Proceedings of the 26th International symposium on shock waves. October, 2006

Index

A Acceleration, 6, 7, 42, 43, 66, 115, 118, 120 drag, 118, 119 gravity, 42, 165 pressure, 71, 115, 119, 216, 231 radial, 95 shock, 119 Active cases, 82 Active gauge(s), 146 Adiabatic, 172 Adiabatically, 5 Algorithm, 28, 81, 323, 329, 330 Aluminum, 84, 153, 262, 303 burning, 84, 121, 305–307, 309, 310 case, 68, 83, 84, 271 foil, 144 fragments, 84 heating, 61, 311 particles, 61, 62, 83, 84, 308, 310, 311 Amplitude, 5, 6, 33, 89, 97, 135, 137, 149 Anemometer, 149 Arena test, 78, 79 Arrival, 5, 17, 18, 52, 67, 106, 145, 147, 180, 188, 189, 214, 216, 223, 232–234, 237, 238, 245, 254, 262, 299, 316 time, 48, 96, 97, 141–143, 166, 168, 169, 212, 234, 235, 238, 252, 316, 328

B Backdrops, 142, 143 Baffle(s), 283–285

Ballistic Lab Army, 83, 213, 315 pendulum, 154 Blast, 1, 75, 142 generator, 92 interaction, 48, 260–264, 313–320 loading, 245, 250, 253–256, 271–280, 301, 320 measurement, 48, 122, 144, 146, 211, 218, 233 parameter, 26, 32, 48, 141, 157–161, 212, 221, 222, 238 pressure, 10, 30, 48, 82, 122, 125, 146, 208, 242, 266, 280 propagation, 89, 96, 99, 102, 166, 226, 257, 265–269, 281–292, 302, 320–330 standard, 23, 28, 48, 97 Boundary layer, 101–113, 115, 116, 122, 123, 139, 153, 175, 192, 213, 214, 222, 288 Breakaway, 23, 34, 35

C Calculation, 4–6, 28, 40, 45, 48, 52, 53, 65, 68, 81, 82, 93, 95, 105, 106, 128–136, 146, 164, 166, 171, 179–198, 208, 219–240, 250–269, 281–287, 306–309, 314, 316, 320, 322, 328–330 Cantilever gauge, 105, 153 Cased explosive, 65–83, 283, 286, 304, 305, 309 heavily cased, 78

333

334

Casing, 65–83, 304 light, 65–68 CGS, 3, 4, 11, 30, 123, 273 Charge, 33, 39–50, 59–63, 80, 157, 210 array(s), 296, 297 bare, 65–67, 75, 82–84, 305 cylindrical, 69–71, 76, 87, 92, 271, 295 hemispherical, 209, 295 spherical, 65, 125, 157, 206, 242, 250 TNT, 23, 37, 83, 96, 304, 320 Collision(s), 5, 6, 189 Combustion, 51, 52, 303–310 Compression, 5–7, 34, 43, 71, 72, 104, 115, 158, 159, 184, 218, 220, 223, 231, 293 Computational Fluid Dynamics (CFD), 18, 38–40, 52, 68, 72 Conservation, 1, 9, 10, 14, 24, 34, 37, 41, 91, 116, 180, 216, 223, 230, 232, 287, 313, 314 Cubes, 105, 153

D Decay(s), 3, 5, 6, 17, 24, 26, 30, 32, 34, 40, 57–63, 87–99, 103, 105, 112, 116, 157, 160, 164–169, 177, 187, 201, 214, 225, 232–237, 242, 250–252, 260, 277–294, 320 Decomposition, 6 Decursor, 245 Density, 3, 4, 11–15, 17, 21 ambient, 9, 10, 12, 13, 18, 34, 38, 40, 42, 45, 50, 52, 72, 128, 223, 313 atmospheric, 4, 6, 40, 53, 189 loading, 40, 158, 159 over density, 4, 17, 24–26, 34, 35, 40, 53, 55 Deposition, 7, 18, 23, 163, 233 Detonable, 57, 294 gasses, 7, 51, 57 limits, 52, 58, 303 Detonation, 7, 29, 81 front, 37, 39, 59, 73 internal, 281 nuclear, 7, 17, 23–27, 31–33, 48, 50, 51, 116, 139, 140, 146, 152, 159–165, 194, 203–206, 212, 216

Index

TNT, 27, 37, 39–43, 48, 51–53, 55, 127, 168, 321 wave, 23, 37–43, 51, 70, 127, 295, 310 Diaphragm, 20–22, 145, 293, 294 Diffusion, 34, 243 Dimension(s), 1, 261, 265, 293, 320 one, 9, 30, 35, 39, 40, 87–92, 127, 224, 230 three, 3, 6, 7, 88, 93–95, 99, 131, 134, 226, 318 two, 88, 92, 93, 230 Dissociation, 10 oxygen, 11, 74, 189 nitrogen, 11, 74, 189 Distant Plain, 40, 241, 242 Drag, 68, 116–118, 247 coefficient, 118–120 force, 118–120, 149 gauge, 149 Duration, 108, 144, 255, 256, 296 positive, 17, 26, 34, 63, 90, 99, 110, 113, 116, 117, 120, 124, 152–154, 222, 225, 232, 236, 260–262, 287, 294, 315 precursor, 235, 240, 245 pressure, 17, 180, 212 Dust, 115, 116, 123, 150, 151, 222, 223, 232, 233, 240, 242, 264 acceleration, 117 entrainment, 116, 122, 147, 205, 222, 224, 226, 301 momentum, 116, 123, 223

E Energy, 4–7, 20, 23, 24, 37 conservation, 1, 9, 37, 41 internal, 3, 19, 38, 41, 43, 49, 73, 75, 106, 122, 159, 238, 281, 288, 289 kinetic, 15, 44, 49, 69, 71, 75, 80–85, 102, 115, 116, 118, 122, 154, 195, 203, 230, 257, 278, 286, 288, 304 rotational, 10, 74, 257 vibrational, 10, 74 total, 18, 30, 70, 88, 91, 159, 161 Equation of state (EOS), 38–40, 73, 74 Eulerian, 41, 48, 130 Evaporation, 121, 223 Exit jet, 298–302

Index

Expansion, 5–7, 10, 23, 24, 37, 39, 42–49, 55 cylindrical, 72, 87, 88, 93, 94, 294 free air, 89 spherical, 7, 29, 87–90, 94, 97, 207, 279 Explosive Fuel Air Explosive (FAE), 57, 60 Solid fuel air Explosive(SFAE), 60, 303, 306, 310, 312 External detonation, 271, 287, 320

F Fano equation, 80, 83 Fireball, 23–30, 34, 35, 46, 47, 50, 55, 116, 117, 122, 124, 129, 140, 159, 160, 164, 208, 217, 224, 226, 256, 265, 288, 304–306, 316 Flux, 224, 233, 239, 240 radiation, 160, 234 thermal, 160–164, 234 Foam, 124, 125, 221, 222, 245 Foil meter, 144 Fragment, 68–85, 154, 232, 285–287, 290, 304 Frequency, 4–6, 99, 146–151, 154, 161, 165, 180, 189, 211, 250

335

I Ideal surface, 201–216, 222, 340 Image burst, 314–319, 323, 324, 330 Impulse, 48, 50, 90, 157, 158, 221, 222, 232, 238, 245, 250, 296, 309, 328 dynamic pressure, 90, 105–112, 124, 152, 153, 212, 230, 232, 240, 298–310, 329 loads, 251–256, 272–274, 301 over pressure, 63, 90, 105–112, 212, 230, 261, 268, 269, 273, 306, 307, 315–317, 329 total impulse, 154, 302 Infrared (IR), 84, 159 Instabilities, 50, 82, 127–137, 208, 232, 306 Kelvin–Helmholtz, 132–135, 177, 257 Raleigh–Taylor, 45, 127–132, 207 Richtmeyer–Meshkov, 135–137 Instrumentation, 104, 105, 144, 146, 293 Interferogram, 147, 174, 179, 190–193, 198 Interior loads, 274 Ionization, 10, 11, 74, 165

J G Gamma, 9–11, 18, 21, 29, 30, 37, 39, 74, 75, 230, 245, 294, 315 Gauge electronic, 105, 143–146, 183, 235 greg gauge, 123, 150–153 passive, 105, 144, 145, 153 snob, 123, 150–153

H Heating, 5, 6, 52, 58, 61, 72, 84, 115, 118, 128, 149, 205, 223, 224, 234, 241, 303, 306, 310, 311 Height of burst (HOB), 167, 204, 205, 209–220, 248, 318, 319 Helicopter, 7 High explosive, 7, 24, 37, 39, 48, 127–130, 146, 161, 205–211, 241, 253, 255, 293–295, 298, 303, 313 Hiroshima, 250, 251

Jeep, 152, 153, 299 JWL, 38, 39, 74

L Lagrangian, 39–41, 45, 48, 128 Lamb, 313 addition rules, 313–316, 318, 319, 323, 330 Landau, Stanyukovich, Zeldovich and Kampaneets (LSZK), 38–40, 73, 74 Large Blast and Thermal Simulator (LB/TS), 89, 90, 134, 135, 243, 244, 294, 296–299, 301 Laser, 7, 23, 128, 129, 147, 179, 190, 191, 198, 213, 214, 231 Liquid Natural gas (LNG), 58 Loads, 122, 148, 154, 247–256, 263–265, 267, 271, 273–275, 277, 279, 301, 302, 320–325, 329

336

Index

M Mach, 3, 7, 43, 135, 146, 177, 179–181, 184, 187–198, 204, 209, 213, 214, 218, 224, 226, 227, 234, 242, 250, 251, 281, 284, 285, 295 Complex Mach reflection (CMR), 175–177, 181, 216 Double Mach reflection (DMR), 175–181, 184, 187, 189, 190, 193, 194, 213–216 number, 6, 15, 29, 166, 180–184, 187, 192, 193, 195 reflection (MR), 172–182, 184, 186, 188, 192, 195, 198, 202–205, 207, 209, 212, 213, 216, 217, 222, 226, 286, 316, 319 stem, 172–174, 176, 177, 179, 180, 187–190, 192, 195, 196, 198, 203, 209, 213, 218, 224, 226, 227, 234, 242, 250, 251, 281, 295, 318–320 transition, 175, 181, 196, 203, 204, 212, 213, 216–218 Mean free path, 24, 150, 160, 189–192 Measurement, 15, 48, 49, 52, 66, 97, 105, 120, 123, 129, 131, 139–154, 162, 164, 165, 179, 183, 187, 189, 205, 218, 220, 230, 232–234, 245, 248, 249, 251, 293, 294, 299, 301, 302, 315 Methane, 52–58, 74, 304 MKS, 4 Model, 28, 29, 48, 61, 62, 115, 116, 127, 128, 224, 226, 240, 262–265, 309, 310, 313, 315, 316, 318, 320–325, 328–330 Modeling, 313–330 Motion, 3–7, 10, 15, 24, 39, 40, 42, 57, 73, 89, 91, 92, 94, 95, 105, 118, 123, 134, 139–142, 145, 149, 152–154, 219, 232, 249, 262, 265, 267, 299 Mott’s Distribution, 77–79

N Negative phase, 17, 18, 26, 27, 32, 36, 47, 55, 99, 103, 115, 256, 281, 287, 288, 302, 314 Non-ideal explosive, 82, 303–312 Normal reflection, 172 Nuclear, 17, 23–29, 48, 50, 51, 122, 139, 143–145, 150, 159, 205, 206, 209, 212, 216, 217, 223, 224, 226, 230, 233, 236, 238, 240, 241, 298, 299, 313

blast wave, 144, 164, 295 detonation, 7, 17, 23–28, 31, 32, 48, 50, 51, 116, 139, 140, 146, 152, 159–161, 163, 165, 194, 203–206, 212, 216, 223, 233, 238, 241, 256, 263, 288, 294 scaling, 159, 206, 212

P Particle(s), 6, 24, 61, 62, 68, 72, 84, 115–121, 123, 139, 148, 151, 159, 223, 303, 310–312 Particulates, 61, 74, 84, 115–118, 121–123, 148, 154, 214, 223, 233, 234, 256 aluminum, 61, 83, 84, 308–311 metal, 83, 303–306, 308, 310 Photography, 24, 65, 66, 131, 139, 148, 223, 224 Photon, 84, 150 Piston, 7, 8, 145, 165, 231, 293 Point source, 17, 19, 206 Positive duration, 17, 26, 34, 63, 90, 99, 108, 110, 113, 116, 117, 120, 124, 152, 153, 222, 225, 232, 235, 260, 261, 287, 315 Positive phase, 17, 18, 26, 55, 63, 67, 103, 117, 154, 212, 220, 256, 288, 299 Power law, 30, 33 Precursor, 104, 224, 227–245, 255, 257, 261, 298–302 Pressure, 3–5, 9, 10, 13, 18–22, 24, 26, 27, 29–32, 34–36, 45, 46, 53–57, 65, 73, 74, 81, 97–99, 104–106, 186, 201, 231, 233, 242, 254–256, 261, 267, 285, 287, 294, 297, 309, 315, 320, 322, 323, 329 ambient, 5, 9, 12, 14, 18, 34, 40, 42, 53, 57, 82, 163, 183, 190, 191, 230, 233, 314 atmospheric, 4, 35, 44, 163, 165, 169 dynamic pressure, 3, 5, 7, 14, 15, 17, 29–32, 90, 102–112, 118–120, 122–124, 150–153, 203, 212–216, 218, 222, 225, 226, 230, 232, 236, 238–240, 242, 243, 247, 248, 254–257, 266, 267, 274, 277, 278, 281, 283, 284, 288, 298–301, 314, 315, 326–329

Index

over pressure, 4, 5, 12–15, 17, 25–28, 30–36, 40, 45, 47–51, 55, 56, 59, 63, 90, 96–99, 102, 105, 106, 108, 110, 112, 116, 119, 120, 123–125, 142–147, 150, 152, 159, 161, 168, 169, 171–173, 175, 177, 181, 182, 184, 186, 187, 189, 194, 196, 197, 202–207, 212–218, 220, 222, 225, 230–243, 245, 247–258, 260–269, 272–274, 277–282, 288, 289, 296, 298, 299, 301, 302, 306, 314–317, 320–322, 326–330 reflected, 4, 14, 48, 119, 120, 171, 183, 185, 195, 196, 198, 209, 217, 218, 247, 248, 251, 253, 261, 267, 282, 315 stagnation pressure, 5, 14, 15, 104, 105, 123, 150, 152–154, 180, 242, 243, 248, 252, 256 total, 5, 150, 152, 243 Priscilla, 233–238, 245, 299, 300 Propagate, 5, 20, 59, 81, 87, 89, 90, 94, 96, 103, 104, 165, 166, 193, 253, 261, 283, 288, 293, 296, 306, 328 Propagation, 1, 5, 6, 20, 27, 28, 30, 33, 37, 45, 65, 76, 77, 81, 87–99, 101–103, 105, 118, 135, 142, 157, 163–165, 172, 201, 202, 206, 216, 221, 226, 227, 230, 238, 239, 241, 247, 257–261, 265, 278, 283, 287–291, 294, 302, 325–330 Propane, 52, 57, 58

R Radiation, 24, 28, 31, 75, 117, 122, 159, 160, 223, 224, 226, 233, 234, 238, 239, 241, 250 Rankine–Hugoniot, 230, 234 Rarefaction, 20–23, 42–44, 53, 89, 92, 221, 251, 252, 254, 255, 273, 288, 293, 296, 320 Real, 1, 10, 34, 35, 45, 50, 127, 159, 162, 165, 194, 229, 232–241 air, 10–11, 229 surface, 101, 106–111, 115, 218–227 Reflection, 1, 4, 15, 67, 89, 90, 146, 172, 173, 175–178, 180–182, 222, 253, 288, 316, 318, 319, 330 factor, 171, 172, 180, 181, 186, 195, 196, 202, 203, 205, 226, 248, 254, 287

337

regular reflection (RR), 171–174, 181, 184, 186, 195, 198, 202, 204, 205, 209, 212, 213, 316 shock, 1, 70, 150, 180, 181, 185, 192, 197 wedge, 182–195 Riemann problem, 20, 89 Rotation, 94, 177, 196

S Scaling, 30, 50, 58, 113, 162–164, 240, 293 atmospheric, 161–167, 169 cube root, 161, 218 yield, 157–163, 218 Sedov solution, 18–19 Self recording, 52, 143, 145, 146, 235 Self similar, 18, 39, 157, 158, 189 Shadowgram, 176, 177, 192, 213, 214, 231, 283, 284 Shock, 1, 4, 6–10, 12–15, 17, 32–36, 45–47, 53–55, 68, 88, 97, 99, 104, 112, 150, 182, 191–195, 201–203, 218–220, 225–227, 234, 257, 273, 283, 293–295, 316, 323, 325, 326, 329, 330 Mach number, 4, 6, 180–184, 187, 192, 193, 195 tube, 20, 21, 87–90, 103, 134, 151, 172, 189, 196, 201, 230–232, 243, 244, 248, 250, 256, 262, 263, 293–294, 296, 298 wave, 1, 3–9, 12, 14, 18, 20, 24, 30, 33, 35, 52, 81, 82, 87, 89, 102–104, 116, 127, 132, 135, 136, 139, 150, 182, 189, 221, 222, 241, 284, 288 Signal, 5, 17, 33, 104–106, 116, 139, 143, 146, 154, 175, 176, 192, 193, 211, 230–235, 239, 243, 262 Simulation, 231, 241–245, 293–302 Slip line, 173–177, 179, 180, 187, 189, 190, 192, 193, 195, 198, 203, 213, 214, 216, 218, 243 Smoke, 139–141, 148, 208, 226, 241 puff, 140–142, 147 smoke rocket, 139–140 smoke trail, 105, 139, 140, 148 SMOKY, 226–229, 238–240 Snow, 118, 219–221, 245

338

Sound, 3–8, 13, 15, 20, 26, 29, 35, 37–40, 42, 43, 81, 82, 96, 101, 104, 150, 154, 158, 162, 167, 169, 201, 217, 219, 220, 225, 227–231, 233, 234, 236, 238, 239, 241–243, 245, 251, 255, 288, 305, 316 Sound wave, 5, 6, 17, 30, 97, 99, 104, 150, 231, 232 Specific heat, 4, 5, 10, 38, 39, 74, 122, 123, 294 Spectral analysis, 149 Speed, 3–8, 13, 15, 20, 24, 26, 29, 35, 40, 66, 72, 82, 96, 158, 162, 229–231, 234, 243, 255, 288, 305, 316 material, 20, 29, 37, 219 shock, 5–7, 13, 20, 37, 96, 104, 150, 169, 201, 207, 217, 220, 230, 231, 234 Steel can, 144 Structure, 96, 122, 148–150, 154, 190 interaction, 48, 247–269 responding structure, 261–269 rigid structure, 261–269 Supersonic, 6, 8, 257 Surface, 4, 14, 23, 30, 32, 33, 41–44, 48, 50, 52, 57, 58, 61, 65, 66, 71, 89, 102, 113, 123, 154, 177, 187, 201, 207, 214, 218, 224–226, 239, 254, 274, 278, 295, 316, 320, 323 rough, 192–194, 222 smooth, 101, 192, 194, 201, 209, 222, 314 snow, 219, 220 Sweep up, 115–116, 224, 301

T Taylor Wave, 17–18 Temperature, 4–6, 9–12, 14, 15, 23, 24, 29, 35, 39, 50, 52, 57, 58, 61, 75, 84, 85, 96, 97, 117, 118, 122, 124, 132, 134, 149–150, 159, 160, 163, 165, 169, 171, 206, 223, 224, 229, 230, 233, 238, 240, 241, 303–306, 308, 310, 311 Terrain, 50, 57, 105, 117, 134, 201, 224–229, 232–241, 257, 265–267 Thermal flux, 160–164, 234 Thermal radiation, 117, 122, 159, 223, 224, 226, 233, 234, 238, 239, 241

Index

Thermobaric(s), 303, 305–307, 310–312 Time, 5, 6, 9, 17, 18, 21, 24–28, 33–36, 39–42, 44–47, 53–55, 58, 59, 61, 62, 67–69, 72, 82, 84, 91, 98, 99, 102, 116–121, 123, 127–131, 134, 136, 137, 144–150, 177, 180, 194, 207, 237–240, 252, 263, 308–310, 314–316, 323, 327, 329 arrival, 48, 52, 67, 96, 97, 141–143, 166, 168, 169, 180, 212, 234, 235, 238, 252, 316, 328 duration, 103 Train(s), 6–8, 150 Transmitted shock, 55, 283 Triple point, 172–176, 187, 189, 190, 193, 195, 196, 203, 212–214, 216–218, 222, 224–227, 239, 242, 250, 251, 281, 316, 318–320 Tube, 5, 7, 20, 82, 87, 89, 90, 103, 104, 134, 140, 150, 151, 153, 218, 231, 243, 244, 249, 256, 293, 294, 296–298, 301 Tunnel, 7, 8, 124, 287–291, 297, 300, 306 Turbulence, 91, 257 Turbulent, 7, 102, 115, 177, 214, 224, 257 Two phase flow, 117, 118

U Urban terrain, 265–267

V Vector, 3, 4, 14, 17, 88, 91, 96, 171, 172, 281, 313, 314, 323, 324, 327 Velocity, 3–7, 9, 13, 14, 17–21, 23–30, 34–45, 47, 48, 50, 53–55, 57, 67, 69, 70, 72, 75, 76, 81, 82, 85, 87–89, 91, 92, 94–97, 101–106, 108, 110, 112, 115–123, 132, 134, 135, 137, 139, 141, 143, 148–150, 152, 157–159, 162, 163, 165, 169, 171–174, 177, 189, 190, 196, 198, 202, 204, 216–218, 220, 222–226, 229–232, 234, 238, 240, 257, 265, 295, 310, 313–315 Vibration, 3, 7, 10, 74 Vortex, 117, 148, 153, 192, 207, 214, 216, 218, 226, 232, 236, 243, 248, 249, 255–257, 273, 274, 277, 278, 280, 284, 302, 320

Index

W Water, 118, 123–125, 127, 134, 165, 219, 221, 223, 224, 240, 308, 309 Wave, 5, 6, 20–23, 29, 37–63, 67, 69–71, 81, 82, 87, 89, 92, 94, 97–99, 104, 127, 150, 165, 172, 175, 176, 180, 183, 187, 192, 195, 201, 206, 214, 221, 222, 230–232, 239, 243, 248, 250–252, 254, 255, 257, 258, 260–262, 273, 277–279, 284, 288, 290, 293, 295, 296, 298, 299, 310, 314, 316, 320, 327–328 Waveform, 34–36, 98, 99, 146, 150, 184, 188, 189, 197, 214–217, 220–222, 232,

339

234–238, 242, 247, 250, 252, 255, 263, 264, 296, 299–301, 306, 309, 313–317, 328, 329 Window, 97, 148, 232, 253, 256, 261, 262, 267, 269, 271–274, 277–280, 306–308 Wolfe–Anderson, 121 Work, 5, 10, 73, 75, 77, 102, 159, 161, 182, 192, 287, 313, 314, 322

X X-rays, 24, 159, 160, 163

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