Blast Waves

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. Suc´eska : 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

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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

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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

2 Some Basic Air Blast Definitions

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

2 Some Basic Air Blast Definitions

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.

C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_3, # Springer-Verlag Berlin Heidelberg 2010

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

3 The Rankine–Hugoniot Relations

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

3 The Rankine–Hugoniot Relations

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

17

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

19

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.”

20

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

21

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]

23

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

4 Formation of Blast Waves

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

4.4 Nuclear Detonation Blast Standard [7]

25

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

RADIUS CM × 102 SAP 1KT STANDARD 50 CM

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

RADIUS CM × 102 SAP 1KT STANDARD 50 CM

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

4.4 Nuclear Detonation Blast Standard [7]

27

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.

28

4 Formation of Blast Waves

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

29

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.

30

4 Formation of Blast Waves

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

4.5 The 1 KT Nuclear Blast Standard

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

4 Formation of Blast Waves

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

4.5 The 1 KT Nuclear Blast Standard

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

4 Formation of Blast Waves

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

4.5 The 1 KT Nuclear Blast Standard

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

4 Formation of Blast Waves

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

5.2 Solid Explosive Detonation

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

5 Ideal High Explosive Detonation Waves

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

5.2 Solid Explosive Detonation

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

Fig. 5.5 TNT hydrodynamic parameters at 4.5 radial expansion factor

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

Fig. 5.6 TNT hydrodynamic parameters at 11.7 radial expansion factor

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

5.2 Solid Explosive Detonation

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

5 Ideal High Explosive Detonation Waves

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

5 Ideal High Explosive Detonation Waves

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

5 Ideal High Explosive Detonation Waves

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

5 Ideal High Explosive Detonation Waves

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

5 Ideal High Explosive Detonation Waves

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

5 Ideal High Explosive Detonation Waves

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

5 Ideal High Explosive Detonation Waves

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

5 Ideal High Explosive Detonation Waves

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.

6.3 Moderate to Heavily Cased Charges

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

6.3 Moderate to Heavily Cased Charges

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.

6.3 Moderate to Heavily Cased Charges

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)

78

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.

6.3 Moderate to Heavily Cased Charges

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

7 Blast Wave Propagation

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

7 Blast Wave Propagation

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

7 Blast Wave Propagation

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

7 Blast Wave Propagation

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

7 Blast Wave Propagation

(.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.

C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_8, # Springer-Verlag Berlin Heidelberg 2010

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 (<
500). Laminar boundary layers are much thinner than turbulent layers, in general, by more than an order of magnitude. For all blast propagation applications, the boundary layers are turbulent and all further discussion here deals with turbulent boundary layers. The study of boundary layers and boundary layer behavior in steady or quasi-steady flows is a much broader subject and will not be considered here. The boundary layer affects only the velocity. It is important to note that the overpressure in the blast wave is not affected by the presence or size of the boundary layer, while the dynamic pressure is directly affected.

8.2

Boundary Layer Formation and Growth

There have been numerous studies of the boundary layer generated by a shock wave traveling over a flat plate. Harold Mirels [2, 3] has done extensive work on boundary layers behind shocks beginning in 1958 and has extended that work to boundary layers behind blast waves and behind reflecting blast waves. These works concentrate on the growth of the boundary layer in simple shock wave and blast wave flows. An approximate solution for the growth of the boundary layer behind a shock was given by [4]. They define the boundary layer thickness d as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Um t  x; d¼ Um  a where a ¼ 74U/189 and b ¼ 40n/3 and U is the material velocity behind the shock, Um is the shock velocity and n is the coefficient of kinematic viscosity of the fluid. The value x is the position of the evaluation relative to the shock front. At x ¼ Umt, the thickness is zero and grows as the square root of the distance behind the shock front. Because this thickness is defined as the height at which the velocity asymptotically approaches the free stream value, we choose, somewhat arbitrarily, to interpret this thickness as the point at which the velocity reaches 99% of the free stream value. If we examine the first term in the thickness equation we see that the boundary layer thickness is proportional to the square root of the viscosity of the fluid. For weak shocks, the material velocity approaches zero and the thickness grows as the

8.3 Termination of a Boundary Layer

103

inverse square root of the shock velocity. As the shock strength gets large, the material velocity becomes a large fraction of the shock velocity and the layer thickness grows as the inverse square root of the difference between the shock velocity and a fraction of the material velocity. For a given shock strength in a square wave shock, the first term is a constant. The second term indicates that the thickness grows as the square root of the distance behind the shock. If we apply this relation to shock tubes, care must be taken to ensure that the boundary layer does not grow to become a significant fraction of the tube radius. If the boundary layer becomes a large fraction of the tube radius before the test time is complete or before the duration of the shock wave is complete, the flow will be stagnated across the entire tube, thus eliminating a free stream velocity and destroying the flow behind the propagating shock.

8.3

Termination of a Boundary Layer

For blast waves, the growth of the boundary layer varies as a function of the shear velocity at a surface. Because the material velocity decays as a function of distance and time behind the blast wave front, the shear velocity at the surface also decreases as a function of time. The rate of growth of the boundary layer slows as the free stream velocity goes to zero. For blast waves with a negative phase, the velocity reverses at the end of the positive phase of the dynamic pressure. At the end of the positive phase, the flow velocity in the free stream goes to zero and no boundary layer can exist because the zero velocity at the surface is the same as the zero velocity in the free stream. When the velocity begins to flow in the opposite direction in the negative phase, a boundary layer forms and grows in the negative phase. Boundary layers can be terminated in other ways. For flows with secondary shocks, the boundary layer can be compressed by the secondary shock. If the shock is sufficiently strong, the boundary layer may be pushed back to the surface and a new boundary layer initiated by the reflected shock. Depending on the boundary layer thickness and the strength and direction of the secondary shock, the shock wave may be diffracted by the low velocity conditions found in the boundary layer. If a secondary shock is moving in the same direction as the shock which generated the boundary layer, the second shock will not propagate at the same velocity in the free stream as in the reduced velocity of the boundary layer. The Rankine–Hugoniot relations, discussed in Chap. 3, can be used to calculate the magnitude of the change in shock propagation velocity as a function of the velocity in the boundary layer. Thus the second shock will be delayed near the surface and curve upward to join the free stream shock front. If the second shock is moving in the opposite direction of the shock which generated the boundary layer, the second shock will propagate faster in the boundary layer because the free stream second shock is traveling against the flow behind the first shock and there is reduced opposing momentum near the wall for the

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second shock. This situation creates a new set of possibilities for the fluid flow. The part of the shock propagating faster in the boundary layer may create an upward flow ahead of the second shock, thus creating a kind of “precursor” shock on the second shock near the free stream values. If the conditions are right, the second shock traveling in the opposite direction may just cancel the free stream flow behind the first shock and the boundary layer will no longer be defined (exist). The second shock traveling in the opposite direction may overcome the free field flow of the first shock and reverse the direction of the flow in the free stream. In this case the boundary layer will be eliminated and will initiate again behind the second shock. Another possibility is that the secondary shock pressure is not sufficiently strong to sustain a shock in the boundary layer because the secondary shock velocity is less than the relative sound speed in the boundary layer. The shock will then propagate as a sound wave or compression wave in the boundary layer but will out run the shock in the free stream. This situation also leads to a type of “precursor” signal propagating upward from the boundary layer ahead of the shock front. In this case the “precursor” is a compressive wave which connects to the shock wave at some point above the boundary layer. Within the boundary layer the velocity varies from zero at the surface to the free stream velocity at the top of the boundary layer. The variation of the velocity with height varies as the logarithm of the ratio of height above the surface to the boundary layer thickness. Thus the momentum density varies as the log of the height and the dynamic pressure varies as the square of the log of the height above the surface.

8.4

Calculated and Experimental Boundary Layer Comparisons

There are two large scale surface burst detonations on which experiments were conducted to measure the parameters within the boundary layer as well as the extent of the boundary layer as the shock propagated over a natural soil surface. The first of these was an event named Mixed Company. A 500 ton sphere of TNT was detonated on a ranch outside of Grand Junction Colorado in 1973 [5]. Instrumentation was placed at several ground ranges to measure the stagnation pressure as a function of height above the ground. The instrumentation consisted of pitot tubes mounted at heights of ½ inch, 1.5 in. and several other heights gradually increasing the separation to a height of about 4 ft. The second large scale detonation was for the Minor Uncle shot at White Sands New Mexico. This was a 2,400 ton hemisphere of an ammonium nitrate fuel oil (AN/FO) mixture. Again the instrumentation was placed at several ground ranges between distances corresponding to the 150 psi and 5 psi predicted peak pressure. Several methods of instrumentation were fielded in attempts to measure the

8.4 Calculated and Experimental Boundary Layer Comparisons

105

boundary layer parameters as a function of height. Instrumentation included a standard set of electronic gauges for overpressure and stagnation pressure, a series of metal and wood cubes which had been calibrated for motion induced by dynamic pressure, smoke trail generators and cantilever gauges. The wooden cubes and cantilever gauges require no active recording and are referred to as passive measurements. The smoke trails required electronic triggering and high speed photography. The photographs would then be calibrated and measured in a laboratory in the days following the experiment. A more detailed description of each of these measurement techniques can be found in Chap. 11. The cantilever or lollipop gauges were used to measure the dynamic pressure impulse as a function of height above the surface. The lollipop gauges consist of a cylindrical head mounted on a ductile metal rod. The rods are then attached horizontally to a vertical mount at a number of heights above the ground with the gauges spaced more closely near the surface and extending to a height of 6 ft. The cantilever gauges supplemented several electronic stagnation pressure gauges, cubes and smoke trails fielded nearby. Overpressures were also measured electronically at several heights above the surface at the same ground ranges. Results from the several methods of measurement were then compared and evaluated against the calculated predictions. Large scale computational fluid dynamics (CFD) calculations were conducted for the Minor Uncle event with sufficient resolution that the boundary layer could be resolved. The calculation included the initiation of the charge, the detonation of the explosive, the formation of the blast wave, and the propagation of the blast wave over the desert surface. One of the input parameters needed for the calculation was the actual surface roughness at the site. A few days before the test, I traveled to White Sands Missile Range and walked the half mile or so from the charge to the 5 psi overpressure range. I took measurements of the surface roughness every few feet. Surface roughness included truck and tractor tire tread marks, clods of dirt from the excavation of the emplacements holes for the various gauge mounts, and occasional rocks and clumps of vegetation. There was no practical way to smooth over the ground over such a wide area, besides we were looking for the effect of realistic terrain on the propagation of a blast wave. The next series of figures illustrate the results of the CFD calculations at three of the ground ranges of interest. Figure 8.1 shows the overpressure impulse as a function of height above the surface at a range of 664 ft. Note that there is very little variation with height above the surface. This is in agreement with the statement that the overpressure is not changed by the boundary layer. The departure from a constant value is caused by the difference in momentum density behind the blast wave front. Signals from the surface boundary layer can catch the shock front and because the velocity near the surface is lower than above the surface, reinforcing pressure signals reach the front above the surface before they reach the front near the surface. As the shock front pressure decays with distance, the pressure near the surface decays slightly faster than that above the surface.

106

8 Boundary Layers 2.4 KT ANFO REAL SURFACE OVERPRESSURE IMPULSE VERTICAL PROFILE 664 FEET 60 PSI 80.0 72.0 64.0

HEIGHT (INCHES)

56.0 48.0 40.0 32.0 24.0 16.0 8.0 0.0 1.9

2.0 2.1 2.2 2.3 OVERPRESSURE IMPULSE (PSI–S)

2.4

Fig. 8.1 Overpressure impulse vs. height at 664 ft range

Figure 8.2 shows the peak dynamic pressure as a function of height above the ground at the 60 psi range. The two curves show the actual calculated values and the values when corrected for numerical overshoot found in the calculations from over 30 years ago. Note that the peak values are only affected in the lower 3 or 4 in. above the surface. This effect may be interpreted as a boundary layer; however, the reduction in dynamic pressure peak is at the shock front. The boundary layer has not had any time or distance to grow. The reduction is therefore the result of the interaction of the shock front with the actual roughness elements when the shock was at a higher pressure level. The surface level velocity was stagnated and the resultant internal energy (pressure) was directed backwards and upward from the surface. Because this is a blast wave, the pressure and velocity behind the shock front are decaying, thus allowing these surface signals to catch the shock front just above the surface. We see a slight enhancement in the dynamic pressure just above the surface. This is the result of the redistribution of the stagnated energy at the shock front at a higher pressure level being carried downstream as the free stream shock decays. In Fig. 8.3 we see the integrated effect of the boundary layer on the dynamic pressure. The dynamic pressure impulse deficit near the surface is caused by a combination of the reduced dynamic pressure in the boundary layer and the changing height of the boundary layer as a function of time after shock arrival.

8.4 Calculated and Experimental Boundary Layer Comparisons 2.4 KT ANFO REAL SURFACE DYNAMIC PRESSURE VERTICAL PROFILE 664 FEET 60 PSI 80.0 72.0 64.0

MAXIMUM AS CALCULATED CORRECTED FOR OVERSHOOT

HEIGHT (INCHES)

56.0 48.0 40.0 32.0 24.0 16.0 8.0 0.0 32.0 40.0 48.0 56.0 64.0 72.0 80.0 88.0 96.0 104.0112.0 DYNAMIC PRESSURE (PSI)

Fig. 8.2 Dynamic pressure vs. height at 664 ft range 2.4 KT ANFO REAL SURFACE DYNAMIC PRESSURE IMPULSE VERTICAL PROFILE 664 FEET 60 PSI 80.0 72.0 64.0

HEIGHT (INCHES)

56.0 48.0 40.0 32.0 24.0 16.0 8.0 0.0 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50 1.65 1.80 DYNAMIC PRESSURE IMPULSE (PSI–S)

Fig. 8.3 Dynamic pressure impulse vs. height at 664 ft range (60 psi)

107

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8 Boundary Layers

The dynamic pressure impulse at the surface must be zero, because the dynamic pressure is zero at the surface for a single decaying blast wave. The dynamic pressure impulse approaches the free stream value at a height of about 4 ft (1.22 m) at the range of 664 ft (202 m). The rate of approach is caused by the combination of reduced velocity in the boundary layer and the fact that the boundary layer has just reached this height at the end of the positive duration of the blast wave. The next three figures show the calculated effects at a range of 887 ft (270 m) and a pressure level of 30 psi (
2 bars). The first Fig. 8.4 shows the overpressure impulse as a function of the height above the surface. The impulse is constant within a few percent from ground level to 6 ft (2 m). The peak dynamic pressure Fig. 8.5 is showing the same characteristics with height that it did at the higher pressure level. The only noticeable effects are in the bottom 8 in. (20 cm) at the shock front. This height is unchanged from that at the higher pressure level. The dynamic pressure impulse of Fig. 8.6 shows that the height to which the layer has influence has been reduced by about 20% from the height at the 60 psi level. The dynamic pressure impulse has reached the free stream value by a height of
40 in. (1.0 m). The reduction in the height of the effects of the boundary layer is caused by the reduction in the surface shear because the velocity associated with a 60 psi shock is over 1.5 times the material velocity behind a 30 psi shock. The 2.4 KT ANFO REAL SURFACE OVERPRESSURE IMPULSE VERTICAL PROFILE 887 FEET 30 PSI 80.0 72.0 64.0

HEIGHT (INCHES)

56.0 48.0 40.0 32.0 24.0 16.0 8.0 0.0 1.5

1.6 1.7 1.8 1.9 OVERPRESSURE IMPULSE (PSI–S)

Fig. 8.4 Overpressure impulse vs. height at 887 ft range

2.0

8.4 Calculated and Experimental Boundary Layer Comparisons 2.4 KT ANFO REAL SURFACE DYNAMIC PRESSURE VERTICAL PROFILE 887 FEET 30 PSI 80.0 72.0 64.0

MAXIMUM AS CALCULATED CORRECTED FOR OVERSHOOT

HEIGHT (INCHES)

56.0 48.0 40.0 32.0 24.0 16.0 8.0 0.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 31.5 33.0 34.5 DYNAMIC PRESSURE (PSI)

Fig. 8.5 Dynamic pressure vs. height at 887 ft range 2.4 KT ANFO REAL SURFACE DYNAMIC PRESSURE IMPULSE VERTICAL PROFILE 887 FEET 30 PSI 80.0 72.0 64.0

HEIGHT (INCHES)

56.0 48.0 40.0 32.0 24.0 16.0 8.0 0.0 0.16 0.24 0.32 0.40 0.48 1.56 1.64 0.72 0.80 0.88 0.96 DYNAMIC PRESSURE IMPULSE (PSI–S)

Fig. 8.6 Dynamic pressure impulse vs. height at 887 ft range (30 psi)

109

110

8 Boundary Layers

height of the boundary layer is reduced, even though the positive duration of the dynamic pressure at 887 ft is greater than the duration at 664 ft. As we advance to the 15 psi (1 bar) range of 1,210 ft (369 m) the material velocity at the shock front is about 1/3 of what it was at the 2 bar pressure level and the dynamic pressure is less than 9% of that at the 2 bar level. The overpressure impulse curve shown in Fig. 8.7 confirms that the overpressure is not influenced by the presence of a boundary layer. The impulse at ground level is within a few percent of the free stream value. Because of the weaker shear at the surface it is expected that the effects on the peak dynamic pressure at the shock front will be smaller than were observed at higher pressure levels. Figure 8.8 confirms these expectations. The dynamic pressure impulse at the 1,210 ft (369 m) range (Fig. 8.9) shows that the influence of the boundary layer is less extensive than at the higher pressure levels. The impulse reaches the value of the free stream at a height of just over 32 in. (0.8 m). The surface shear at this range is reduced by more than a factor of 2.5 from that at the 664 ft range. The boundary layer has, none the less, grown to a height of nearly 3 ft. This is because the positive duration is more than a factor of two greater at this range than at 664 ft. The computational results are in very good general agreement with the experimental data. Figure 8.10 summarizes the measured dynamic pressure impulse as

2.4 KT ANFO REAL SURFACE OVERPRESSURE IMPULSE VERTICAL PROFILE 1210 FEET 15 PSI 80.0 72.0 64.0

HEIGHT (INCHES)

56.0 48.0 40.0 32.0 24.0 16.0 8.0 0.0 1.2

1.3 1.4 1.5 1.6 OVERPRESSURE IMPULSE (PSI–S)

Fig. 8.7 Overpressure impulse vs. height at 1,210 ft range

1.7

8.4 Calculated and Experimental Boundary Layer Comparisons 2.4 KT ANFO REAL SURFACE DYNAMIC PRESSURE VERTICAL PROFILE 1210 FEET 15 PSI 80.0 72.0 MAXIMUM AS CALCULATED CORRECTED FOR OVERSHOOT

64.0

HEIGHT (INCHES)

56.0 48.0 40.0 32.0 24.0 16.0 8.0 0.0 6.4

6.8

7.2

7.6 8.0 8.4 8.8 9.2 9.6 10.0 10.4 DYNAMIC PRESSURE (PSI)

Fig. 8.8 Dynamic pressure vs. height at 1,210 ft range 2.4 KT ANFO REAL SURFACE DYNAMIC PRESSURE IMPULSE VERTICAL PROFILE 1210 FEET 15 PSI 80.0 72.0 64.0

HEIGHT (INCHES)

56.0 48.0 40.0 32.0 24.0 16.0 8.0 0.0 0.12 0.15 0.18 0.21 0.24 0.27 0.30 0.33 0.36 0.39 0.42 DYNAMIC PRESSURE IMPULSE (PSI–S)

Fig. 8.9 Dynamic pressure impulse vs. height at 1,210 ft range (15PSI)

111

112

8 Boundary Layers

Fig. 8.10 Minor uncle dynamic pressure impulse as a function of height above ground at several ranges

a function of height above the ground at several overpressure ranges. The data has been normalized to the free stream blast wave impulse. The curves in the expanded box clearly show the behavior of the impulse relative to the free stream value as a function of ground range. The low overpressure (5 psi) curve rises more slowly than at any other pressure level. The thickness then increases as the overpressure level increases with the maximum height measured at the 50 psi range. The thickness of the boundary layer then decreases as the overpressure continues to increase. This observation is in agreement with the calculated results. As one point of comparison we look at the 60 psi range where the height at 99% of ambient is 900 mm experimentally while the calculated height is about 35 in. or 860 mm, a difference of less than 5%. A few summarizing comments on boundary layers in blast waves resulting from free field surface detonations. Because this example used a surface burst, there were no reflected shocks. The fact that this was an AN/FO detonation rather than TNT makes only minor differences in the blast wave at the ranges used above. The second shock (“repeat” from Chap. 5.2) is trapped in the negative phase of the primary shock and is essentially unaffected by the boundary layer. The growth rate of the boundary layer is proportional to the shear velocity at the surface. This is initially directly related to the material velocity at the shock front. As the layer grows, the shear decreases near the surface; first from stagnation of the flow and second from the decay of the blast wave. Between the 664 ft range and the 1,210 ft range, the boundary layer affects the dynamic pressure impulse to heights of the order of a meter, even though the peak shear has dropped more than a factor

References

113

of 2. This growth is permitted because the positive duration has increased more than a factor of two, thus giving more time for the growth of the boundary layer. The example I have used here is for a very large charge (2,400 tons). This large size meant that the boundary layer would be large and could be readily detected and parameters within the layer measured using conventional techniques. The thickness of the boundary layer scales with the roughness of the surface and the size of the charge. Methods of scaling will be discussed in Chap. 12.

References 1. Prandtl, Tietjens: Applied hydro and aeromechanics, p. 59. Dover, New York (1934) 2. Mirels, H.: The wall boundary layer behind a moving shock wave, Grenzschichforschung Symposium, pp. 283–292. Springer-Verlag, Freiburg, Germany (1958) 3. Mirels, H.: Boundary layer growth behind Mach reflections, 10th Mach reflection symposium, pp. 20–23. Abstract Book, Denver, CO (1992) 4. Sakurai, H., Adachi, T., Kobayashi, S.: Effect of boundary layer on Mach reflection over a wedge. In: Proceedings of the 22nd International Symposium on Shock Waves, pp. 1,249– 1,252 (2000) 5. Mixed Company/Middle Gust Results Meeting 13–15 March 1973, vol. 1, Sessions 1, 2A, and 3A, Published by, General Electric Company-Tempo, 1 May (1973)

Chapter 9

Particulate Entrainment and Acceleration

We saw in the previous chapter that a blast wave traveling over a real surface can be modified significantly by even small surface roughness. In this chapter we will discuss the effects on the blast wave of a loose layer of particulates on the surface. The particulates share momentum and kinetic energy with the gasses in the flow. They exchange thermal energy with the gasses, either heating or cooling the particles. The particles that may be entrained in the flow have a density of the order 1–3 g/cm3, while the gasses have a density of 1–2
103 g/cm3. It takes a very small volume of particulate matter to have the same mass as the gas in which they are entrained. If we take a dust particle with a 100 mm diameter, only 100 such particles will have a mass of 10% of the mass of air in a volume of 1 cm3. With such a density of opaque material, the optical depth is about 30 m. This is then the equivalent of a very severe dust storm with a visibility of only a 100 ft or so. It is very difficult for a gas to hold a mass of particulates in suspension when the mass of the particulates approaches the mass of the gas. The particulates will simply fall out of suspension under gravity.

9.1

Particulate Sweep-up

There are two major mechanisms that have been identified with particulate sweepup, lofting and entrainment in the flow of a blast wave. The first, and probably most important, is the sweep up induced by the shear velocity of the gas flow near the surface. The second is the compression of gas within the particulate layer (soil) during the positive pressure loading of the blast wave and the sudden upward acceleration of the particulates by this compressed gas when the negative phase of the blast wave passes. In the first instance, the particulate sweep-up can be closely approximated by using the shear velocity in the boundary layer. The mass of dust lofted into the flow field is proportional to the turbulent shear velocity at the surface. Another model which provides very similar results when a boundary layer is not defined is to inject C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_9, # Springer-Verlag Berlin Heidelberg 2010

115

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particulate mass proportional to the momentum of the gas in the air just above the surface. Either of these models is self limiting. The momentum of the particulates is initially zero because the ground is at rest. When the blast wave passes, the initial shear is the velocity of the gas immediately above the surface which is the velocity of the air at the shock front. When the air exchanges momentum with the particles behind the shock front, the particles are accelerated by drag and the momentum of the air is reduced accordingly. The reduced velocity caused by momentum conservation is the new shear velocity and less dust is swept-up as the blast wave progresses. In addition the velocity is being reduced by the decay of the blast wave behind the shock front. The more dust that is entrained, the lower the velocity, the less dust that is entrained. Thus we make the statement that the momentum based models are self limiting.

9.2

Pressure and Insertion Velocity

In order for the particulates to be entrained in the flow, they must have a vertical velocity component. There are two possible sources for this vertical lofting velocity. In the discussion of the boundary layer in the previous section, the stagnation of the velocity in the boundary layer causes an increase in the local pressure near the surface. This pressure induces velocities upward into the flow. The upward velocity will carry particulates with the gasses. The second mechanism is that the particles are dragged horizontally by the flow. The particles bump into one another and are lofted through momentum transfer. In either case the lofting velocity can be closely matched by using a velocity of 10% of the horizontal shear velocity. The momentum density of the dust overwhelms the boundary layer and creates a moving, growing dust layer within the positive duration of the blast wave. The particulates partially stagnate the gas flow, converting kinetic energy into internal energy causing a vertical pressure gradient. The vertical gradient generates upward moving velocities which carry the dust to greater heights and involve greater volumes of the gas. At high overpressure levels the particulates may be accelerated to a large fraction of the shock velocity. As the blast wave decays with distance the high momentum density of the particulates prevents them from slowing as rapidly as the gas and the particles may punch through the shock front. The dust induced signal travels faster than the air shock above it. A graphic example of this behavior is shown in Fig. 9.1 below. Trinity was the first nuclear detonation. The yield was approximately the equivalent of 19,000 tons of TNT. The device was detonated 100 ft (30 m) above the ground. At the time this picture was taken, the shock wave is coincident with the edge of the radiating fireball. Near the ground surface the jetting of the dust leads the shock front by about 100 ft (30 m). The jetting is caused by the high velocity of the dust as it was swept into the flow at ground ranges between 100 and 400 ft (30–120 m). Early interpretation of this photo mistakenly identified the cause of this jet as a hot layer

9.3 Drag and Multi-Phase Flow

0.016 SEC. N

117

100 METERS

Fig. 9.1 Photograph of the Trinity detonation at a time of 16 ms

of air near the ground generated by the thermal radiation from the fireball into which the blast wave was accelerated. Later analysis proves that, at this early time, insufficient thermal radiation has escaped the fireball to cause such a heated layer. The jet is caused by dust accelerated to high velocities by the blast wave at smaller ground ranges which move ahead of the shock front. The “dust” density is highest near the ground and decreases rapidly with height above the ground. The dust layer at this time extends to a height of about 25 ft and varies as a function of the terrain over which the blast wave is propagating. The dust is clearly characterized by complex vortex flows and is not simply described.

9.3

Drag and Multi-Phase Flow

When the particles are small, the distance required to accelerate the particles to 99% (again an arbitrary number) of the flow velocity is small compared to the positive phase of the blast wave and the particles can be assumed to be in velocity equilibrium with the flow. For small particles, the surface area to volume ratio is large and the particles exchange heat rapidly with the gas. Thus “small” particles, where the definition of small depends on the parameters of the blast wave, (such as positive duration) heat transfer is also very rapid. The particles can then be assumed to be in temperature and velocity equilibrium with the gas. A general definition of two phase flow includes any combination of materials involved in a flow when the fluid velocities are not in equilibrium. These combinations include solids in gasses or liquids, gasses in solids and gasses in liquids. Some applications include solids of one density in a mixture with solids of another density in which the flow induces relative velocities. Blast waves may be propagated in all such combinations. In discussing air blast with solid particulates, we could just as easily talk about blast waves in porous soil or sand; these are nearly the same

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9 Particulate Entrainment and Acceleration

problem but the volume of the gas is a much smaller percentage of the total volume. In a similar vein, blast waves in air or other gasses may entrain water droplets or there could be blast waves in water with air bubbles or solid particulates. One interesting subject is the propagation of blast waves through snow; a solid with air spaces, such that the density is about 10% of water. This means that the water (ice) occupies only about 10% of the volume and air fills the remaining 90% of the volume. In the interest of placing limits on the subjects covered, I will discuss gaseous mixtures which contain less than about 15% liquid or solid by volume. When particles are sufficiently large, they will have a separate velocity from the gas. This is one example of two phase flow. The particles will be accelerated by drag of the flow and still exchange momentum and kinetic energy with the gas. Each particle will have, in general, a different velocity from all other particles. The particles have different sizes (masses) and have experienced different acceleration histories. In addition, each particle will have a different temperature because the heating rate is dependent on the surface to volume ratio and the temperature time history to which it has been exposed. Numerical methods that attempt to represent massive particles of different sizes in two phase flow by creating a few artificial bins of particles having the same velocity (and temperature) are missing important parts of the physics of two phase flow. The massive particles experience a drag force proportional to the dynamic pressure caused by the relative velocity between the particle and the fluid. The acceleration of the particle is given by: a ¼ F=m. The drag force is proportional to the cross sectional area of the particle, the dynamic pressure caused by the relative motion of the gas and particle, with proportionality constant which is by definition the drag coefficient, Cd. We can then write the equation for the acceleration of the particle as: ! ! d VR 1 rc Cd V R jVR jprp2 ¼ : dt 2 rp ð43 pr 3 Þ Remember from the definitions in Chap. 2 that the dynamic pressure is given by: ! ! Q ¼ r U jU j=2. We substitute the relative velocity for the velocity and use the density of the gas to determine the relative dynamic pressure. The mass of the particle is the density of the particle times its volume. Figure 9.2 shows the drag coefficient for a sphere as a function of the Reynolds number of the flow. The Reynolds number is a useful relation between the flow parameters and the dimensions of interest for a given problem. The dimension for our problem is determined by the diameter of the particle. Re ¼

rg Ug d mg

The Reynolds number is defined as the momentum density of the gas times the diameter of the particle divided by the viscosity of the gas.

9.3 Drag and Multi-Phase Flow

119

Fig. 9.2 Drag coefficient for spheres as a function of Reynolds number

If we evaluate the Reynolds number for a moderate blast wave of about 1 bar peak overpressure encountering a 100 micron particle. The value of the Reynolds number is: Re ¼ 2.e3*2e4*1e2/2e4 ¼ 2,000. (Reynolds number is dimensionless) As the particle equilibrates with the flow velocity, the Reynolds number will decrease because the relative velocity is decreasing, as is the gas density. When the particle has reached 90% of the material velocity behind the shock, the Reynolds number will be the order of 100 or so. The drag coefficient remains nearly constant over this range of Reynolds number. Depending on the rate of decay of the velocity behind the shock front, the particle may find itself traveling faster than the fluid because the particle has been accelerated by high velocity flow, but during the time the particle has been accelerating, the flow velocity is decreasing. When particles are accelerated by a blast wave, there are two very different forces which act to accelerate the particle. The drag force, as described above, is many times the only acceleration considered. The second is the acceleration caused by the reflected pressure differential as the shock engulfs the particle. This shock acceleration only lasts for the time that it takes for the shock to engulf the particle, but the forces may be much larger than the drag force. If the effect of reflected pressure of the shock on the particle is not separately resolved, the drag coefficient appears to be larger and is no longer independent of the particle size and is not a linear function of the dynamic pressure. The shock pressure acceleration is a function of the particle size because the time over which the pressure force is acting is a function of the particle size. The larger the particle, the greater the time for a given shock to engulf the particle and the greater the shock acceleration forces act. The reflected pressure at the point on the particle where the shock strikes is given by: DPr ¼ 2DP þ ðg þ 1Þq from the

120

9 Particulate Entrainment and Acceleration

Rankine–Hugoniot relations introduced in Chap. 3. The reflected pressure is equal to twice the incident overpressure plus more than twice the dynamic pressure. The particle acceleration increases as the shock front begins to engulf the particle because the area over which the pressure is acting is rapidly increasing. As the shock reaches the backside of the particle the pressure forces start to equilibrate. The shock from the circumference of the spherical particle converges at the center of the back of the particle and for a brief time the pressure forces on the back may exceed the combination of drag and pressure forces in the direction of the flow. When these forces are interpreted as being caused by drag alone, the drag coefficient is much greater than for a uniform flow when the shock first hits the particle and goes negative when the shock converges on the back side of the particle. An impulsively loaded sphere appears to have a different, higher drag coefficient than for a sphere in steady flow (for example [1]). Devals [2] compares the results for the experimental measurements primarily with results for impulsive acceleration of a sphere in Temkin and Mehta, [3]. Figure 9.3 compares the “drag coefficient” for standard and impulsive flows. The higher coefficient is caused by the interpretation of shock pressure reflection as a drag force. The particles accelerated by the flow may interact with the gaseous flow in other ways. The particles may change size as a result of the interaction with the flow. Solid particles may melt, the liquid particles may evaporate or react chemically with the flow. Liquid particles may condense to form solids, or the products of a chemical reaction may be solid, liquid or gaseous. Each change of phase absorbs or releases energy which must be shared with the flow. Chemical reactions may release significant amounts of energy either within the blast wave positive duration or, if the reaction is slow, in some later portion of the blast wave. When the particles are liquid, the relative velocity between the particle and the gas causes deformation of the particle. When the deformation becomes large, the Drag Coefficient Comparison Sphere in Steady vs. Unsteady Flow 5

CD

Standard Drag Coefficient Impulsive Drag Coefficient

1

0.5

0.2 200

500

1000

5000 10000 Reynolds Number

Fig. 9.3 Modified drag coefficient for blast waves

50000 100000

9.3 Drag and Multi-Phase Flow

121

particle may divide into multiple particles. The particle size into which a liquid particle will split in a flow is well approximated by a relationship developed by Wolfe and Anderson [4]. This relationship gives the largest particle diameter that can exist in a shear flow as a function of the particle material properties and the relative flow velocity.  1:5 :5
13 ms D Dm ¼ C 2 :5 4 ðg r U Þ The droplet particle properties needed to evaluate this equation include: m: The liquid viscosity, s: The surface tension, D: The current particle diameter, g: The gas density, r: The liquid density and U: The relative velocity between particle and gas. C is a fitting constant for a given material. The smaller particles, thus formed, will transfer energy more rapidly to the surrounding gas; they will be accelerated to the gas velocity in less time. In other words, they will come to equilibrium with the gas in a much shorter time and distance. Particulates can also change size through evaporation. As a liquid particle evaporates, the particle radius and mass change and the mass lost by the particle becomes part of the mass of the surrounding gas. Some solid materials go directly from the solid state to gas in a process called sublimation. The particulates may undergo size change through chemical reactions with the surrounding gas. The chemical reactions change the energy in the gas and may produce more gas or solid particulates. One example is the burning of aluminum in oxygen to form Al2O3. Every two moles of aluminum that burn take 3 moles of oxygen from the gas and forms solid particulates. The number of gas molecules is reduced. Particulates do not have a pressure of their own that contributes to the pressure of the gas, but can be thought of as having the same pressure as the surrounding gas. The particles contribute to the gas pressure by occupying local volume and by exchanging energy with the gas. In the example of burning aluminum in oxygen, large amounts of energy are transferred to the gas but the mass of gas is reduced as solid particulates are formed. It is therefore possible, at least in a thought experiment, to burn aluminum in stoichiometric oxygen in a closed chamber which would create a lower pressure than the initial pressure in the chamber even though it would be significantly hotter than it was prior to the burn. In many situations the volume occupied by the particulates can be ignored because the density of solids and liquids is much higher than that of gasses at normal conditions. For a region containing equal masses of air and sand, for instance, the sand will occupy only 0.05% of the volume of the air. Even when the mass of sand reaches ten times the mass of the air, the volume occupied by the

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sand is only 0.5% of the total volume. To have approximately the same volume, the sand would have 2,000 times the mass of the air. This corresponds to very loose sand characterized as having 50% porosity.

9.4

Particulate Effects on Dynamic Pressure

Determining the contribution of particulates to the dynamic pressure of a blast wave is of considerable interest because of the differences observed in loading effects on structures exposed to clean vs. dusty flows. The loads transferred to a structure can be about a factor of two greater in a heavily dust laden flow than in a clean flow with the same flow velocity. Higher than clean flow loads can be obtained when the flow velocity of the dusty flow is less than that of the clean flow. An enhancement of only a few percent in dynamic pressure can mean the difference between a structure being damaged but surviving and destruction. The damage caused by solid particulates entrained in a high speed flow behind a blast wave can be impressive. On some nuclear tests, several inches of concrete were “sand blasted” from the faces of concrete structures by the passage of a single blast wave in less than 1 s. In some cases the concrete was stripped down to the reinforcing steel. Thus, it is not just the enhancement to the dynamic pressure that may cause damage to structures, but the impact of individual solid particulates causing damage to the surface of a structure. The entrainment of particulates in high speed flows can also produce some interesting and unexpected phenomena. Again some observations at the nuclear test site in Nevada showed that the dust entrained in blast waves provides some of the characteristics of a boundary layer flow. Large diameter pipes (8–18 in in diameter) were used to support blast gauges on some nuclear tests. Near the ground surface, dust was piled to a height of a few inches in front of the pipes by the blast wave. There was little or no erosion to this height. Above the dust was a region where the pipe surface had been “sandblasted” and above that was a region where semi-liquid sand had coated the surface with granular material imbedded in coarse glass. The glass may have been formed by the thermal radiation from the fireball, but in this case it is more likely caused by the kinetic energy of the sand being converted to internal energy when the sand hit the pipe. The velocity of the sand when it stagnated against the pipe would need to be about 560 m/s in order to have sufficient energy density that the kinetic energy converted to internal would exceed the melting temperature of sand. This can be roughly calculated by assuming that the melting point is 1,600 C and the specific heat of sand is 0.2 J/g degree C (2.e6 ergs/gm degree C). Sand grain density is about 2 g/cc. Then 1 2 rU ¼ Cp DT: 2

9.5 Effects of Water

123

Using cgs units this becomes U 2 ¼ 2  2e6  1600=2 ¼ 3:2e9 or U ¼ 5.6e4 cm/s. Such a velocity is readily attained in a high pressure blast wave. We can use the Rankine–Hugoniot relations to find what the overpressure level is for such a blast wave. The reader is invited to check the result that a blast wave with a peak pressure of about 80 psi or just over 5 bars would have the required material velocity. The various states of the steel pipes are an indication of the effects of the dust induced boundary layer. The bottom few inches of the flow behind the shock front had very low velocities. The sand which hit the pipe was simply stopped by the pipe and fell to the ground. A little higher, the sand was moving sufficiently fast that the sand eroded the surface of the pipe and bounced away. Higher on the pipe, the velocity was sufficient to melt the sand on impact and coating the pipe with a glassy particle laden cover. Above the glassy layer either the dust density was low or the higher velocities wiped the molten glass from the surface of the pipe. A number of methods were used in attempts to measure the dust component of the dynamic pressure. Using the fact that the overpressure is only slightly affected by the presence of equilibrium dust, the Rankine–Hugoniot relations could be used to define the dynamic and therefore the stagnation pressure associated with a given blast wave. Force plates, large circular disks were placed in the flow. The combined stagnation of the gas and the particulates resulted in a total force against the face of the plate. The difference between the gas stagnation pressure times the area of the plate and the measured total force on the plate could be attributed to dust impact. Some success was found with these gauges, but the response time of the plate motion was slow and this method gave little information about the time history of the flow. Uncertainties in how much dust was deflected by the flow without striking the plate increased the error in the measurements. The measured forces were a function of the particle size distribution of the dust, but it was difficult to know what the particle size distribution was at the height of the gauge. Two classes of gauge designs were developed which were nick named Greg and Snob gauges. The idea was that Greg gauges would measure the stagnation of the gas plus all of the particulates, a gregarious gauge. The snob gauges attempted to measure only the gas stagnation pressure, ignoring the dust. Even recently designed Snob and Greg gauges have shown limited success on full scale tests in dusty environments. A discussion of these gauge designs is in Chap. 11.9.

9.5

Effects of Water

Water is one of the best substances for absorbing energy. First, its specific heat in the liquid form is 1 cal/g/ C. More important, its latent heat of vaporization at standard conditions is 539 g-cal/g. W.F. Penney (Lord Penney) [5] showed that the water in fog or rain can reduce the blast performance of conventional or atomic bombs. This was physically demonstrated in the U.S. detonation of Operation Castle [6]. The shot Koon device was detonated during a rainstorm, in which the

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9 Particulate Entrainment and Acceleration

peak overpressure was reduced by up to 20% at a given range compared to the pressures on the “dry” line. In order for the water to efficiently absorb energy from a blast wave it must be in the form of droplets so the large surface area will allow rapid absorption of the energy by the water. Some have suggested that when munitions are stored, walls filled with water surrounding the explosives would provide a method of reducing the pressure generated in case of an accidental detonation. Two tests were conducted in order to test this hypothesis. A large number (180) of 6 in (155 mm) shells were simultaneously detonated in a storage chamber at the end of a tunnel system. Pressures were measured in the chamber and along the tunnel as well as outside the tunnel entrance. The test was repeated with large water filled balloons surrounding four sides of the stack of shells. Again pressures were measured at a variety of distances. The results were surprising in that it had been expected that the water would significantly reduce the pressures near the tunnel entrance. Instead the data show that the pressures were enhanced by the presence of the water. The explanation is that the bulk water has a very small surface area and very little heat is absorbed in the water. The thin layer of water on the outer edges of the balloons that was exposed directly to the high temperatures of the fireball was evaporated, thus releasing more gas (water vapor) into the system and increasing the pressure because the gas density was increased more than the energy was decreased. The water vapor that was generated was hot. Remember, the gas pressure is the product of the gas density and its energy density. The water not only increased the overpressure near the tunnel exit, it increased the positive duration of the dynamic pressure impulse because more gas had to escape the tunnel. Pressure vs. Range Comparison Free Air and with Aqueous Foam

1.00E+04

Pressure (PSI)

1.00E+03 With Foam Free air 1.00E+02

1.00E+01

1.00E+00 0

5

10

15

20

Range (ft)

Fig. 9.4 Overpressure comparison for a blast wave in free air and in aqueous foam

25

References

125

One example of an efficient use of water to reduce the effects of blast pressure is from a series of tests conducted by Sandia National Laboratories. Figure 9.4 shows a comparison of the overpressure as a function of range from a 50 pound free air charge of C-4 explosive with the measured pressure as a function of range when the same charge was detonated when surrounded by aqueous foam. The foam supplied the matrix in which small water droplets were suspended. The overpressure reduction is more than a factor of ten. The pressure at 10 ft is reduced from 100 psi to less than 10 psi. At a range of 12 ft the pressure is reduced from 80 psi to 9 psi and at 18 ft from 22 psi to 2.5 psi. Such differences would mean the difference between severe damage or injury to a level which could be survivable.

References 1. Paterson, A.R.: A first course in fluid dynamics, pp. 236–239. Cambridge University Press, Cambridge (1997) 2. Devals, C., et al.: Shock Tube Spherical Particle Accelerating Study for Drag Coefficient Determination. Shock Waves 12, 325–331 (2003) 3. Temkin, S., Mehta, H.K.: Droplet drag in an accelerating and decelerating flow. J Fluid Mech 116, 297–313 (1982) 4. Wolfe and Anderson: PROCEEDINGS, 5th International Shock Tube Symposium, pp. 1145–1169 (1965) 5. Penney, F.W. FEW-43, The Physical Effects of Atomic Bombs, Part 5, Loss of Performance of H.E. Bombs and Atomic Bombs When Exploded in Fog or Rain, September 9, (1955) 6. WT 904, Operation Castle–Project 1.2a, Ground Level Pressures from Surface Bursts, October 30, (1957)

Chapter 10

Instabilities

The study of blast waves would be much simpler if high explosives behaved as consistently and smoothly as our one dimensional models tell us they do. The real world is not so simple or smooth. For example, in Chap. 5 we showed a number of illustrations of blast wave formation from a TNT detonation. Beginning with Fig. 5.8 of that chapter there is a significant spike in density at the interface between the high density expanding detonation products and the air. Such a discontinuity is not found in the real world. In fact it was so strongly believed that nature could not support a discontinuity that at one time it was thought that a shock wave could not exist. In this section I will give a brief introduction to three of the most common instabilities and the conditions for their development.

10.1

Raleigh-Taylor Instabilities

Raleigh-Taylor instabilities develop when the density gradient and the pressure gradient are in opposite directions (have opposite signs). The classic example of this is an experiment in which a container is partially filled with oil. Water is then placed in the container on top of the oil, separated by a thin membrane. The system is allowed to rest so there are minimal residual velocities in the fluids. When the membrane is removed, the water and oil exchange places. The interface moves slowly at first, then rapidly accelerates and the fluids reach a natural equilibrium with the lighter fluid on top. In this classic case, the oil is lighter than water so initially the density gradient is positive upward. The pressure gradient at the interface is due to gravity and is vertically downward. In the case of solid high explosives the pressure and density gradients are in the same direction when the detonation wave reaches the surface of the charge. The high pressure is in the detonation products and the detonation products are more dense than the air. As the detonation products expand and compress the air, the pressure gradient reverses direction while the density gradient retains its direction. Figure 10.1 is replicated from Chap. 5 and illustrates the calculated pressure and C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_10, # Springer-Verlag Berlin Heidelberg 2010

127

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10 Instabilities

Fig. 10.1 Calculated TNT hydrodynamic parameters at 4.5 radial expansion factor

density gradients at the interface when the detonation products have expanded by a factor of 4.5. The density of the detonation products, labeled D, has decreased but is still 30–40 times ambient air density near the interface. The air density has been increased to a factor of 5 or so times ambient. The pressure, labeled P, in the air has been increased by the increase in density but even more by the compressive heating and is now at a pressure of about 4,000 bars while the pressure in the detonation products has dropped due to expansion to less than ambient atmospheric near the interface. The conditions are very strong for the formation of R-T instabilities. The one dimensional Lagrangian model used for this calculation does not allow formation of instabilities. To illustrate this behavior I am including a series of photographs taken by a high speed camera using laser light with a reflective backdrop. The backdrop was about the same distance as the camera and laser from the charge but on the opposite side. In Fig. 10.2 the detonation products have expanded by about a factor of three. The air shock can be seen to have separated from the detonation products interface. More importantly, the detonation products interface is smooth except where the straps used to suspend the charge are present. The next Fig. 10.3 shows that the shock separation is only slightly greater at 4 diameters of expansion than it was at 3 diameters. The detonation products interface has a slight ripple, but is still generally smooth. By the time the products have expanded to 6 diameters, (Fig. 10.4) the separation between the air shock and the interface has disappeared. Surface instabilities have grown and fill the space all the way to the shock front.

10.1 Raleigh-Taylor Instabilities

129

Fig. 10.2 Solid high explosive detonation at about 3 diameters expansion

Fig. 10.3 Solid high explosive detonation at about 4 diameters expansion

By the time the expansion has reached 8 diameters, (Fig. 10.5) the instabilities have punched through the air shock front at a number of locations. No laser light can be seen behind the blast wave front. The air shock front remains indistinguishable from the detonation products from this time to a range of about 20 diameters expansion. The detonation products expansion then slows dramatically and the blast wave pulls away from the detonation products. A number of experimenters have made measurements of the shock and fireball expansion of TNT spheres. Figure 10.6 is a summary of a number of these measurements compared to results of recent calculations using the Sandia

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Fig. 10.4 Solid high explosive detonation at about 6 diameters expansion

Fig. 10.5 Solid high explosive detonation at about 8 diameters expansion

Laboratories code CTH and the U.S. government owned code SHARC (Second order Hydrodynamic Advanced Research Code). Both codes used the same equation of state for the detonation products. Both codes are Eulerian based and second order accurate and used similar resolution for this calculation. The primary difference for this example was that the detonation product material interface was tracked by CTH and was not allowed to form R-T instabilities, whereas SHARC used no restriction on the expansion or mixing of the detonation products. Two of the curves are the location of the primary blast wave and the location of the secondary shock described in Chap. 5. The codes are in reasonable agreement with the locations of these fronts for the entire time plotted here. The light blue

10.1 Raleigh-Taylor Instabilities

131

Fig. 10.6 Radii of various parameters as a function of time for a one pound sphere of TNT

squares are measurements of the radius of the detonation products as a function of time using high speed photography. Note that the CTH and SHARC results agree with the data to a range of just less than 2 ft. In the CTH calculation, the detonation products separate from the primary shock front at that range. The radius then oscillates about the 2 ft radius for the remainder of the plot. The experimental data indicate that the detonation products continue to expand and stabilize near a range of 3.5 ft. The SHARC results are plotted with a black triangle with an “error” bar. The triangle is the average radius of the detonation products and the extent of the vertical lines represents the range over which the instabilities vary as a function of polar angle at the specified time. To illustrate the extent of the instabilities, the SHARC result plots for a C-4 detonation are shown in Figs. 10.7 and 10.8. These are density contours of the detonation products at a time of 0.5 and 1 ms. These plots are two dimensional representations of a three dimensional phenomenon but give the overall extent and magnitude of the instability growth and mixing. The data points tend to be in agreement with the outer edge of the extent of the calculated instabilities. The data were taken from photographs which give the outermost edge of the instabilities as the surface. The photo cannot show the extent of the incursions of the mixing. This is illustrated in Figs. 6.2 and 6.3 of Chap. 6. With modern computers and high order numerical techniques, it is possible to resolve many of the R-T instabilities in three dimensions. One of the first calculations to accomplish this is shown in Fig. 10.9. This image is taken at a time of 60 ms when the detonation products have expanded to about three times

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Fig. 10.7 SHARC calculation showing the growth of instabilities at the detonation products-air interface at 0.5 ms

their initial diameter. The location of the shock wave is shown as a projection in the three planes. Note in the projection, the instabilities are small ripples on the surface of the detonation products. This is in agreement with the photographic data in Figs. 10.2 and 10.3 above.

10.2

Kelvin–Helmholtz Instabilities

This instability was studied independently by Lord William Thomson Kelvin and Hermann Ludwig Ferdinand von Helmholtz in the late 1860s. In [1, 2] are two of the initial reports on the subject. Kelvin–Helmholtz instabilities occur at interfaces in fluids when shear velocities exist at the interface. Any small perturbation to the flow can initiate growth of K-H instabilities. The growth rate is dependent upon the magnitude of the shear velocity gradient but is also a function of several fluid properties, including density, viscosity and surface tension. K-H instabilities may occur independent of other fluid interface parameter differences. The type of fluid, the density and the temperature of the fluid may be the same or different in the two flow regions. The two regions are otherwise in pressure equilibrium or in stable equilibrium under gravity. Such K-H instabilities are often observed in cloud

10.2 Kelvin–Helmholtz Instabilities

133

Fig. 10.8 SHARC calculation showing the growth of instabilities at the detonation products-air interface at t ¼ 1 ms

Fig. 10.9 One pound TNT detonation at 60 ms

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10 Instabilities

formations, especially near mountainous terrain, or in water near the convergence of two rivers having different flow velocities. The classic experiment for Kelvin–Helmholtz instability study is a thin tube with a fluid having constant flow velocity in pressure equilibrium with the ambient atmosphere. The thin tube terminates in ambient air. The flow soon reaches a “steady state”. In this case the “steady state” is not steady in the sense that the conditions are not changing with time. Vortices are continuously forming and growing as they move down stream and the two fluids mix. The mixing of the fluids also mixes the momenta of the regions. The velocity outside of the original tube radius increases and the velocity of the fluid within the original tube radius decreases. Axial velocities are observed with increasing radii as the distance from the end of the tube increases. The velocity varies continuously from the initial constant velocity at the inner edge of the mixing region to zero at the outer edge. When the two fluids are the same material at a constant temperature and pressure, the rate of growth of the mixing layer can be measured by placing gauges across the stream, perpendicular to the axis of the tube at several distances from the end of the tube. As the K-H instabilities grow, the momentum of the two fluids is mixed. Figure 10.10 is a cartoon of one such possible experimental set up. Figure 10.11 shows the results of a SHAMRC calculation of the after flow from a blast wave generated in the Large Blast and Thermal Simulator (LB/TS) at White Sands Missile Range in New Mexico. The LB/TS is 11 m in height and 20 m in diameter with a flat floor. This figure clearly shows the formation and growth of K-H instabilities at the exit of the shock tube. The calculation shown here is fully three dimensional and the plot is made at the midline of the shock tube. The blast wave is moving from right to left. The dimensions on the plot are in meters. The R-T and K-H instabilities are usually studied together. If we look carefully at the R-T instabilities, we see that they form small jets of one material moving into another. Along the edges of these small jets are shear velocity regions because the one gas is moving faster than the gas into which it is moving. These shear velocities lead to the formation and growth of K-H instabilities on the sides of the jets. The computational examples shown here make no distinction between types of instabilities, but are solving the equations of fluid motion. The computational results therefore contain a mixture of R-T and K-H instabilities.

Fig. 10.10 Configuration for measuring mixing region due to Kelvin–Helmholtz instability

10.3 Richtmyer–Meshkov Instabilities

135

Fig. 10.11 Kelvin–Helmholtz instabilities formed at the exit of the LB/TS

10.3

Richtmyer–Meshkov Instabilities

Richtmyer–Meshkov instabilities are generated when a shock passes through inhomogeneous media. Richtmyer describes these instabilities for compressible fluids in his paper “Taylor Instability in Shock Acceleration of Compressible Fluids” [3]. The shock induced instabilities are closely related to the Raleigh-Taylor instabilities discussed previously in this chapter. They are of importance to blast wave formation and propagation because shock waves accelerate and expand any existing instabilities with which they interact. As a demonstration of the R-M instabilities, a series of SHARC (Second order Hydrodynamic Advanced Research Code) calculations was run with a shock interacting with an interface between two different density fluids. The interface was perturbed by a simple sinusoidal geometry. The initial conditions are illustrated in Fig. 10.12. The shock velocity is Mach 1.25 and travels from the light gas into the more dense gas. Figure 10.13 is a sequence of times showing the density distribution as the shock passes through the gas density interface. This particular calculation used a grid of 200 by 300 zones of 0.5 cm in each dimension. The initial amplitude was 2 cm or 4 zones. The shock progresses from left to right. Because of the different densities on

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RIGID BOUNDARY

DENSITY INTERFACE

SHOCKED REGION

SHOCK WAVE

TRANSMISSIVE BOUNDARY

RIGID BOUNDARY

100 cm.

RIGID BOUNDARY

Fig. 10.12 Initial conditions for Richtmyer–Meshkov calculations

Fig. 10.13 Time sequence of Richtmyer–Meshkov instability growth, 200  300 zones

References

137

either side of the interface the shock strength and the shock velocity are modified as the higher density fluid is encountered. As the shock travels beyond the interface, it begins to “heal” and return to its initial planar geometry. The fluid interface deforms because of the differing velocities induced by the changing density. The amplitude of the oscillation at the interface increases by more than a factor of 4 in 700 ms. (the first plot in the upper left hand corner is at a time of 50 ms after the shock interaction). The interface is no longer sinusoidal. The shape of the low density incursions is much more rounded than the spikes at the leading edges of the high density fluid.

References 1. Helmholtz, H.L.: About the discontinuous motions of fluids, Monthly reports of the Prussian Academy of Science in Berlin, vol. 23, p. 215 ff., (1868) 2. Kelvin, W.T.: Hydrokinetic solutions and observation. Philos. Mag. 42, 362–377 (1871) 3. Richtmyer, R.D.: Taylor Instability in Shock Acceleration of Compressible Fuids, Communications on Pure and Applied Mathematics, Vol. XIII, pp 297–319, (1960)

Chapter 11

Measurement Techniques

In this chapter I will mention a few of the methods used to measure blast wave parameters. Some measurement techniques described are of an historic nature and demonstrate the ingenuity of the people asked to make quantifiable measurements of the various parameters associated with blast waves. This is meant to be a survey of methods and is intended to provide the reader with a general knowledge of measurement techniques.

11.1

Use of Smoke Rockets

One of the early methods of measuring the position of blast waves from a nuclear detonation as a function of time was the use of smoke rockets in conjunction with photography. This technique was used on most of the US above ground nuclear tests at the Nevada Test Site (NTS) during the 1950s. When I was very young, I had seen movies of a number of nuclear tests and had concluded that the smoke rockets must be a signal that a nuclear detonation was about to occur. I quickly learned that this was a measurement technique used in combination with other methods for visualizing the shock wave from a nuclear detonation. Small “rockets” or mortars were placed at evenly distributed distances along a radial from ground zero. These rockets or mortars were launched vertically and left a reasonably uniform trail of smoke, usually white. Using this array of vertical lines, photography was used to monitor the motion of the smoke trails as the blast wave accelerated the smoke particles. Because the smoke particles were very small, they were assumed to be moving at the velocity of the air in the blast wave. This worked especially well when the blast wave was moving perpendicular to the orientation of the smoke trails. It was such an array of vertical smoke trails that were used to measure the boundary layer velocity distribution on the Minor Uncle shot described in Chap. 8. Linear smoke arrays were developed such that crossing patterns of smoke lines could be generated, thus providing a rectangular array of smoke trails. Blast waves C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_11, # Springer-Verlag Berlin Heidelberg 2010

139

140

11 Measurement Techniques

Fig. 11.1 The Met detonation (
22 KT) showing smoke trails

moving in any direction could then be readily photographed and measured. These arrays were photographed with fixed photo markers within the field of view. The photo markers were of a known size and were placed at carefully surveyed locations. The motion of the air in the blast wave could then be precisely measured from the film. Figure 11.1 is a photograph of the Met shot, a 22 kt nuclear detonation on a 300 ft tower. The blast wave has passed out of the frame of the picture by this time. This picture clearly illustrates the early use of smoke rockets in a crossing pattern above the detonation. In addition to evaluating the blast wave parameters, these smoke trails were also used to measure the motion of the atmosphere outside the fireball. As the rising fireball formed the now familiar toroidal flow of the “mushroom” cloud, the smoke trails allowed measurements of the ambient air entrainment to the fireball. In one test, the smoke mortars were carefully prepared so that a rectangular array would be generated. The mortar tubes were aligned so that half would be inclined toward ground zero and half would be inclined at the same angle away from ground zero. At the appropriate time before detonation the mortars were launched simultaneously. The launchers had been too carefully prepared however and the smoke generating munitions collided in pairs, thus creating a chaotic background for the cameras. [1]

11.2

Smoke Puffs

A direct outgrowth of the smoke rockets but more sophisticated and much more difficult to field technique has been used on explosions with yields ranging from a few hundred pounds to a few thousand pounds. Small balloons are loaded with a

11.2 Smoke Puffs

141

small amount of smoke producing flash powder and an initiator. The balloons are then suspended in a vertical array along a radial line from ground zero. These arrays may have a balloon every foot (or other known distance) in the horizontal and vertical directions. An array of photo markers is placed in the overlapping fields of view of the cameras. At a time of a second or so before detonation, the initiators are triggered and a small puff of smoke is generated at the location of each balloon. An array of cameras is then used to photograph the motion of the smoke puffs. Using this technique, a number of blast parameters are readily measured from a single, consistent set of data. Measurements are made of the positions of each smoke puff for each photographic frame. Using a combination of light diffraction and early smoke puff motion, the arrival time can be measured using the framing rate of the cameras and the position of the smoke puffs. Knowing the ambient conditions and using the Rankine–Hugoniot relations all of the blast front parameters can be calculated. The velocity of the air behind the blast front can be measured directly from the puff motion between consecutive frames. Using the positions of four puffs at the corners of each initial square in the array, the density of the air can be determined by assuming that the air between smoke puffs remains in the volume defined by the quadrilateral defined by the smoke puff positions. The convergence or divergence of the flow can be measured in complex, multiple shock environments. After several experiments, it was noted that it was very difficult to pick out dark puffs against the ground below the horizon in the photographs. White puffs were tried, which worked very well against the dark background but were difficult or impossible to locate on a partly cloudy sky above the horizon. For more recent tests the balloons appearing below the horizon were filled with white smoke producing powder and the balloons above the horizon were filled with red smoke powder. This made distinguishing the puffs much easier when color film is used. Figure 11.2

Fig. 11.2 Colored smoke puff array prior to charge detonation

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11 Measurement Techniques

shows an array of colored smoke puffs after initiation and prior to the charge detonation. This method of data reduction is manpower intensive and tedious for the graduate students making the measurements. Modern methods may be used to teach a computer to recognize the puff patterns and automate the system, the measurements thus requiring minimal human intervention.

11.3

Painted Backdrops

For test yields of a few thousand pounds or less, a more reliable visualization backdrop was used. Large canvas “curtains” can be hung in an array, within the field of view of several cameras, on the opposite side of the blast from the cameras. The backdrops are positioned sufficiently far from the detonation that the blast wave would travel between the backdrops and the cameras before the blast wave struck either the cameras or the backdrops. These curtains are painted with black diagonal stripes of a known width. Photo markers of a known size and position are included in the field of view of the cameras. These large canvas curtains are difficult to field and take preparation time. The canvases are hung between telephone poles, 20–30 ft high and spaced every 10–15 ft. Ten to twenty canvases are needed to cover the distance of interest for blast wave propagation. Any unforeseen winds prior to the detonation can destroy the canvases and the experiment. The canvases can usually be used only once because the blast wave destroys the canvases. The diffraction of light by the shock front can be seen to traverse across the striped backdrops. Measurement of the positions of the intersection of the shock front with each stripe provides a nearly continuous history of the motion of the shock front. This technique also provides a method of measuring complex shock combinations and multiple shock geometries. Measurements were more reliable when the shock was traveling in the same direction as the stripe, but were difficult to make as the shock was crossing a stripe. To resolve this problem, the back drops were painted with “Polka Dots” so the relative orientation of the motion of the shocks did not matter. Figures 11.3 and 11.4 show two examples of the use of painted backdrops.

11.4

Overpressure Measurements

There is a long history of making pressure measurements. Two of these were mentioned in the previous sections of this chapter. Such measurements that require the use of Rankine–Hugoniot relations are referred to as indirect measurements, because some other parameter is actually measured and the overpressure is derived from that measurement. One example of an indirect overpressure measurement is to measure the arrival time of the shock at a series of closely spaced positions. Using

11.4 Overpressure Measurements

143

Fig. 11.3 Shock visualization using striped painted backdrops

Fig. 11.4 Shock visualization using “polka dot” painted backdrops

this arrival time data the shock velocity is calculated by dividing the difference in range of the points by the difference in arrival time. In order to smooth the data, sometimes a numerical fit to the arrival time data is found using a least squares fitting technique. The equation for the fit can then be differentiated to provide a continuous function for the shock velocity. The Rankine–Hugoniot relations are then used to find the overpressure. A number of gauges were developed that were self contained. The gauge, supporting electronics, power supply, triggering mechanism and recording were in a single portable package. They did not require timing and firing signals nor power lines nor cable runs. These self recording gauges were used to obtain most of the nuclear air blast data.

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11 Measurement Techniques

There are a number of techniques that are referred to as passive measurements. These techniques are characterized by their ability to make the measurement with no electronic, photographic or personnel interaction. These techniques are especially useful for regions that are difficult to reach, remote, or dangerous. The largest and most popular class of instrumentation is that of active instrumentation. This class requires electronic circuitry support or a remote triggering system or both. A fairly complete and detailed description of air blast instrumentation is given in [4]; I will mention here a number of techniques which I find interesting and useful in understanding the history of blast wave overpressure measurements.

11.4.1

Passive Techniques

Beginning with passive techniques, and remembering that the overpressure can be interpreted as the crushing force of a blast wave, the simplest and easiest to obtain during early nuclear testing in the Nevada desert was the 12 ounce beer can. The cans were emptied and resealed. The cans were then exposed to a nuclear blast wave and the degree of crushing was correlated with the overpressure. Two types of cans were used; aluminum for low overpressures (26–40 psi) and steel for higher overpressures (above 50 psi). Use of this technique was terminated after the first few tests. For overpressures ranging as low as 0.3 bars, another plentiful can was used; a 5 gallon “jerry” or gas can. For larger events, 55 gallon drums were sealed and exposed to blast waves. Overpressures were determined by measuring the change in volume of the container before and after exposure to the blast wave. The difference in volume was taken to be a measure of the peak blast wave overpressure taken within “a few” milliseconds of passage of the blast front. Because most of the nuclear blast waves had durations which were a substantial fraction of a second, this was indeed a good approximation to the peak overpressure. The consistency of these crushed can “gauges” was estimated to be between 5 and 10%. The accuracy, which depended on an outside calibration, ranged from about 10% for the five gallon gas cans to as much as 30% for the steel beer cans to 50% for the aluminum cans. A foil meter was developed at about the same time (1947) in the US and independently developed in Germany in 1957. The U.S. foil meter consisted of a circular brass plate, 16 in. in diameter and half an inch thick. A series of as many as 14 holes were then machined through the plate with rounded ridges at the edges of each hole in order to hold a smooth piece of aluminum foil to cover the hole. The hole sizes ranged from 3 in. in diameter to 0.166 in. Each hole was covered with aluminum foil which was held in place by a ring over the ridge at the edge of the hole. These gauges were then exposed to a nuclear blast wave. The gauges were calibrated to measure blast wave overpressures from 4.62 psi (3 significant figures?) for breaking the foil on the 3 in. hole to 113 psi for the 0.166 in. hole. Accuracy was quoted as 20% or so for overpressures above 15 psi.

11.4 Overpressure Measurements

145

A similar gauge was developed in France in about 1957 which used glass covers for the holes. The holes ranged in diameter from 100 to 52 mm. The pressure range capability was from 1 to 14 psi with a quoted accuracy of 10%. Several piston or diaphragm and spring gauges were designed. In general these gauges scratched or marked the deflection of the piston or diaphragm as it was moved by the blast wave pressure. One of these piston deflection gauges used a soft metal disk to record the deflection of the piston. The piston was mounted on a rod with a conical point on the end touching the softer metal disc. As the piston was deflected, the pressure was calibrated to be proportional to the diameter of the impression left in the metal disc. These gauges, with slightly different designs, could be used to measure pressures from 5 to 250 psi. The overall response time of the gauge was about 5–10 ms with an accuracy of about 10%. It was the piston and diaphragm gauges that led to the development of the active electronic gauges that are in common use today. Although electronic gauges had been developed and were fielded on the Trinity atomic test in 1945, they were not very successful because of the large electromagnetic pulse generated by the detonation of a nuclear device. This Electro-Magnetic Pulse (EMP) induced large potentials and currents in the cables connecting the gauges to the recording devices. Most of the early electronic gauges failed because of this phenomenon. The experimentors were aware that EMP might be a problem before the test but had underestimated the effects by orders of magnitude.

11.4.2

Self Recording Gauges

Because of the problems with long conducting cables and the fact that most of the passive gauges recorded only peak overpressure, a series of “self recording” gauges was developed. These devices contained a small motor either electric or spring operated that would move a recording drum, disc or spool while a pen or needle was moved by a deflecting diaphragm or piston. Some of the self recording gauges developed by Stanford Research Institute were triggered by a photo sensor on the gauge itself with a backup thermal link initiator. Other self recording gauges could be triggered by a sensor placed a few feet in front of the gauges which would, upon passage of the blast wave, start the motion of the recording device prior to arrival of the blast wave at the recording device. Several types of manometers were modified from their usual meteorological designs to handle the higher frequencies and much higher pressures of a blast wave. The manometers consist of a closed volume of gas which contracts when pressurized externally. The motion of the contraction is recorded by attaching a moving pen or needle to write on a revolving drum. Any of these mechanical systems required the motion of some mass and the response time of this mass, usually several milliseconds, was a significant limitation of the measurements.

146

11.4.3

11 Measurement Techniques

Active Electronic Gauges

At the risk of over simplifying the development of active electronic instrumentation, I refer the reader to [3, 4] for details and a broad discussion of instrumentation. I will briefly discuss the application of active gauges to the measurement of blast wave overpressure. Most of the self recording gauges had a fairly low frequency response of 500 Hertz or so. For nuclear detonations with yields of kilotons, this was sufficient for airblast measurements. Some people have argued that the observed long rise times on the overpressure waveforms of several nuclear detonations were the result of this low frequency response. This has since been shown by numerical calculations and small scale high explosive experiments not to be the case. This subject will be further discussed in Chap. 13. When blast waves from more conventional sized explosions of a few pounds to a thousand pounds are measured, it is necessary resolve the passage of the blast wave on a much smaller time scale. Instead of a few milliseconds, it is necessary to resolve the overpressure on a time scale of microseconds. It is this high frequency capability along with reliability and enhanced accuracy, which has made electronic gauges the technique of choice for measuring blast wave overpressure. It should be pointed out that the frequency response needed is not determined solely by the gauge characteristics but by the entire measurement system. The recording device must be able to record the signals at a rate commensurate with the gauge response. The system also includes the cables connecting the gauge to the recording device. Signal losses of both magnitude and frequency can be caused by long cables. In order to reduce these problems, two popular methods are used. One method is to employ line drivers or small signal amplifiers near the measurement point that increase the signal strength so that line losses are compensated. Another method is to place the recording devices in protective bunkers near the gauges, thus eliminating the need for long cables but emplacement costs can be significant. Modern electronic gauges have some physical limitations on their frequency response due to their design. The rise time of a blast wave is limited to the time it takes the blast wave to cross the surface of the sensitive part of the gauge. For a Mach 1.5 blast wave and a sensing element 1 cm in diameter, the rise time for the shock front will be 20 ms if the measurement system response frequency is infinite. To reduce the system response time, the sensing element size must be reduced. For a 1 mm sensing element and a Mach 2 shock, the rise time, (crossing time) is about 1.5 ms. Installation of modern electronic gauges is usually accomplished by placing the pressure sensing element in the middle of a large flat plate. The plate is then aligned with the anticipated direction of flow to permit the blast wave to pass over the gauge with no reflections. For ground level overpressure measurements, the plates are mounted in concrete or steel blocks flush and oriented with the ground surface. Figure 11.5 shows a typical overpressure gauge installation showing the plate and the sensor at the center.

11.5 Density Measurements

147

Fig. 11.5 Typical blast wave overpressure gauge mount

11.5

Density Measurements

In Sect. 11.2, I mentioned the use of smoke puffs to find the density of the blast wave as a function of time and space by measuring the positions of the individual smoke puff positions and calculating the convergence or divergence and thus the density of the fluid. In a laboratory, density can be measured directly using laser interferograms. The frequency of the laser light remains constant but the wavelength changes as a function of the density of the gas through which it passes. If the path length of the light is sufficient the arrival of the light traveling through the different densities will be shifted sufficiently that interference patterns can be photographed. The density distribution at a given time can be visualized and with knowledge of the ambient conditions, the shock strength, the index of refraction and the R-H equations, the interference fringes can be directly related to a density change. Such interferograms are a very accurate method for defining the entire density field in a controlled laboratory experiment on blast waves. Laser absorption has also been used to measure the density distribution within a blast wave. For gasses, this technique requires either a long path length or a gas with absorption near the frequency of the laser. A more common use of laser absorption is in the measurement of dust laden flows. The path lengths can be the order of a few centimeters for flows with high density dust entrainment. The laser and the receiver can be placed in an underground bunker, thus protecting them from damage by the blast wave. Mirrors are used to route the light up from the bunker, across a measuring region and back to the receiver in the bunker. Either photographic film or digital recording can be used with a laser pulsed every few microseconds or milliseconds depending on the application.

148

11 Measurement Techniques

Another technique uses a radioactive source and a counter. The source and counter are located at a fixed separation distance at some height above the ground. The source and counter are protected from the blast wave by a heavy mount with windows made of a material of known absorption between the source and counter. The counter readings are then recorded as a function of time as the blast wave passes. One problem with this technique is the frequency response of the measurement. Very highly radioactive sources must be used to ensure that the number of counts measured in a short time (milliseconds) is statistically significant.

11.6

Velocity Measurement

The most straight forward method of measuring fluid velocity is by photographic measurement of smoke trails or puffs. The displacement of a smoke trail or puff can be measured by finding the position of the smoke on consecutive frames and dividing by the interframe time on the film or digital recorder. If there are particulates in the flow, they may obscure the smoke positions after only a few frames. When particulates are present, velocities can be measured using either Doppler radar or Ladar. Ladar uses the same principle as radar but at a much higher frequency. Such techniques measure only the highest speed particle in the flow, and are not very useful for obtaining velocity as a function of time at a given position. They can measure the velocity as a function of time just behind the shock front as the blast wave moves toward the measuring device. To obtain a more continuous record of flow speed as a function of time at a point in space, a number of gauges have been developed to measure the vortex shedding frequency of the fluid as it passes over an obstruction inside a pipe. Vortex shedding is a function of the viscosity of the fluid and the Reynolds number of the flow. Performance of this gauge also depends on the fluid not having imbedded particulates. The vortex shedding frequency can be measured using photography of smoke trails or strain gauges can be used to measure the response of the obstruction as the loading changes with time as the vortices are shed. Pressure gauges can also be mounted in the side of the fixture to measure the variations as the vortices are shed. These gauges are commonly used for industrial applications of steady or nearly steady flow speeds. These measurements have been successful but have been found very difficult to interpret for decaying blast waves. The vortex shedding frequency is a function of the shape of the obstruction and the ratio of the area of the obstruction to the area of the pipe. Each design must be calibrated separately.

11.7

Angle of Flow Measurement

It is sometimes necessary to measure the direction of the flow as a function of time. This is especially important when a structure is exposed to a blast wave. If the flow behind the shock is not parallel to the ground, significant side loads can be induced.

11.8 Temperature Measurement

149

For targets above the ground, loading may come from several different directions in quick succession. The response of such a structure would be difficult to understand without the magnitude and timing of the flow velocity. Measurement of the angle of flow can be accomplished by two or three dimensional “drag gauges”. The two dimensional drag gauge is a cylindrical pipe fastened at both ends and oriented with the long axis perpendicular to the anticipated flow direction. The pipe is instrumented with strain gauges to monitor the strain on two perpendicular axes in the radial direction. The flow produces flexing of the pipe which is measured by the strain gauges. The three dimensional drag gauge consists of a sphere which is instrumented with three independent and orthogonal force gauges. The sphere is mounted in such a way that the actual motion is small compared to the diameter of the sphere. The drag force direction gives the direction of the flow velocity. Attempts have also been made to measure blast wave flow direction by using light weight wind vanes. There are two problems with this method. The reaction of the wind vane to a change in direction is slow unless the mass of the moving vane is small indeed. If the mass of the vane is small, a sudden change of direction can bend or break the vane.

11.8

Temperature Measurement

Temperature measurements of blast waves or temperature of gasses subjected to blast waves remains a particularly difficult problem. One method of measurement is the use of thermocouples using bimetal strips, constant current or constant voltage circuits across a small wire. Bimetal thermocouples are typically far too massive to have the millisecond response time that is needed. The heating time of even a thin metal strip is too great and the response to a temperature change is many milliseconds. Hot wire anemometers can be used to measure the gas temperature with millisecond response times. Constant current or constant voltage bridge circuits monitor the resistance of the wire as the gas temperature changes. The very fine wires that are needed to obtain the millisecond response times are very fragile and in general cannot survive the blast wave environment. Some success has been obtained by protecting the hot wires, but the protection increases their response time. Multi frequency spectral analysis can provide fast and accurate gas temperature measurements. High speed photographic recording provides the ability to focus on a single small region as the blast wave passes or can be expanded to cover a large area and measure the average temperature of the surface. Measurements of the amplitude at several frequencies are made within a given frequency range. By comparing the relative amplitudes as a function of time, the temperatures can be determined with an accuracy of a few percent.

150

11 Measurement Techniques

Different parts of the spectrum can be selected for accuracy at a variety of temperature levels. For moderate level blast waves, the most efficient frequencies are in the infra-red (IR). The drawback to this method is that when the gas is heated to the point that the mean free path for photons is short, only the surface temperature of a gas is measured. This may or may not be the maximum temperature and no information is found for the gas behind this surface. One class of temperature gauges designed to measure transient temperature variations, specifically for blast waves is called “whistle gauges”. These gauges take advantage of the fact that the sound speed of a gas is proportional to the square root of the absolute temperature. A standing sound wave is generated in an open ended tube of known length. The wavelength l is fixed by the length of the tube. By monitoring the frequency of the standing wave, the temperature of the gas can be found directly using the relationships: C ¼ l=f , and ðC=C0 Þ2 ¼ T=T0 where l is four times the length of the tube, f is the frequency and C0 and T0 are the ambient sound speed and temperature. Another method for temperature measurement in a closed structure is to use the timing of reverberating shock waves in the room. By careful analysis of the overpressure waveforms measured in a closed rectangular room, the transit times of shock waves can be determined. Given the shock strength and the transit time, the Rankine–Hugoniot relations can be used to find the average sound speed through which the shock has traveled and thereby the average temperature of the gas in the room. This method requires a good understanding of the shock reflections in a room and the ability to sort out the various waves in a complex train of shocks.

11.9

Dynamic Pressure Measurement

The dynamic pressure is defined as the density of the fluid multiplied by the square of the fluid velocity and has a direction commensurate with the velocity. The dynamic pressure is important because it is the primary damage mechanism for above ground targets exposed to a blast wave. From Chap. 3 and the Rankine– Hugoniot relations, the reflected pressure of a blast wave is twice the incident overpressure plus 2.4 times the dynamic pressure. Also for overpressures above about 50 psi the dynamic pressure is greater than the overpressure. In Sect. 11.3.4, reference was made to snob and greg gauges for measuring the dust component of dynamic pressure in a blast wave. Snob and greg gauges measure the stagnation pressure of the flow with snob gauges designed to measure just the gas component and greg gauges designed to measure the sum of the stagnation of gas and dust. The stagnation pressure or the total pressure is used with an overpressure measurement to derive the dynamic pressure. The total pressure is defined to be the overpressure plus the dynamic pressure. A variety of designs were created with an example shown in Figs. 11.6 and 11.7 of gauges used on nuclear tests. None of the designs were perfect. The Snob gauges measured some of the dust contribution and the Greg gauges didn’t measure all of the dust contribution. If the sensing area was too

11.9 Dynamic Pressure Measurement

151

Probe Details 8/1

Dust Cavity

Pressure Sensing Section

Pressure Line to Forward Transducer

Fig. 11.6 Schematic diagram of a snob gauge

Silicon Rubber Protector

Pressure Transducer (Head-on Pressure)

Fig. 11.7 Schematic diagram of a greg gauge

large the frequency response was degraded. If the tube in which the sensing element was placed was too small, the dust would plug the gauge and only partial waveforms would be recorded. The gauges had to be calibrated in a shock tube with a known gas pressure and a known particle density. Each gauge design was then assigned a dust registration coefficient. The results for each gauge were then adjusted by this registry coefficient. Significant improvements have been made in frequency response and recording technology since the original designs. In Fig. 11.6 a small opening in the bottom near the left entrance to the tube transfers the stagnated gas pressure to a pressure transducer, called the forward transducer. The dust laden flow passes directly into the dust cavity and stagnates. Only the gas pressure is supposed to reach the transducer. The greg gauge, Fig. 11.7, on the other hand, is designed to measure the sum of the pressures caused by the stagnation of the dust and the gas. The open cavity at the front of the gauge traps the dust and gas and the resultant pressure is transferred to the pressure transducer through the rubber protective cover. Even with this large open volume in front of the sensing element, heavily dust laden flows could jam the dust in the cavity and the sensing element would continue to measure high pressures after the blast wave had passed.

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11 Measurement Techniques

Both the greg and snob gauges require the simultaneous measurement of overpressure in the very near vicinity. Typically a side on or overpressure measurement was made in the side of the mount containing total pressure gauge or was mounted in the ground directly below the gauge. In many instances the overpressure gauge was mounted in the center of a flat plate and positioned at the same height and ground range but separated by a few feet from the total head gauge. Another limitation of the greg and snob gauges is that a true reading of the stagnation pressure, even in clean flows, can only be obtained when the flow velocity is within about 20 of the axis of the probe. In other words, alignment is very important and non-radial flows or flows with significant vertical components cannot be accurately measured. Dynamic pressure impulse (DPI) is the integral of the dynamic pressure as a function of time. It is the dynamic pressure impulse which generates the displacement of objects in the flow. One of the earliest correlations of dynamic pressure impulse was made with the displacement of jeeps exposed to nuclear detonations. Figure 11.8 shows the correlation of displacement in feet as a function of the dynamic pressure impulse in psi. Thus one of the earliest sets of units for DPI was the jeep foot. Figure 11.8 was generated from the collection of data on jeep motion from a number of nuclear detonations having relatively long positive durations and a number of different yields. The jeeps were all parked side on to the blast wave and the data was limited to jeeps that remained somewhat intact. As you might imagine, a jeep that has been rolled for over 1,000 ft is not in drivable condition. Engine blocks, sheet metal and frames tend to be widely separated.

Fig. 11.8 Jeep displacement as a function of dynamic pressure impulse

11.10 Stagnation Pressure Measurement

153

Some examination of this plot is worthwhile. The jeeps did not move for impulses less than about 0.07 psi-s. At these levels either the peak force is insufficient to move the jeep or the positive duration is so short that the motion is absorbed in the springs and shock absorbers. After all, a jeep does not move when the wind blows, even a 50 mph side wind will not displace a parked jeep. Between 0.07 and 0.2 psi-s the jeep displacements go from zero to 10 ft. Some of these jeeps were drivable and had sheet metal and windshield damage. The slope of displacement changes dramatically at the 10 ft displacement, but is nearly linear in this log-log plot. An order of magnitude increase in impulse gives an order of magnitude increase in displacement. Keeping this correlation in mind, the idea of placing cubes of known mass and cross section in blast waves was developed [2}. For complex multidimensional flows and for very long duration flows, the displacement of cubes gives a high degree of correlation with the dynamic pressure impulse. Cubes of several dimensions from 2 to 8 in. or more can be constructed of materials with densities ranging from steel (7.8 gm/cc) to aluminum (2.7 gm/cc) to oak and balsa wood. These cubes have been calibrated to measure impulse over a large range of peak dynamic pressures. One of the advantages of the cubes is that they are passive gauges and require only an initial survey and a post detonation survey to determine displacements. The cubes can be used to measure the variation of the dynamic pressure in a boundary layer by placing the cubes at different heights above the surface. The cubes exposed to higher dynamic pressure impulse will travel further. Another advantage of the cubes is that they will turn in vortex flows and provide an indication of the swirling motion which cannot be measured by a stationary gauge. Another passive measurement of dynamic pressure impulse can be made using cantilever or lollipop gauges. These gauges are usually constructed using a rigid cylindrical rod with a cylindrical disk on one end and a soft pliable attachment on the other. When the disk is moved by the dynamic pressure of a flow, the cantilever bends the pliable metal at the attachment point. The final angle between the mount and the rigid stick can be very accurately correlated with the dynamic pressure impulse. Gauges have been designed to measure a wide range of dynamic pressure impulse by stiffening or increasing the diameter of the pliable metal hinge. If the hinges are cylindrical and the loading mass on the top of the rod are in the shape of spheres, the direction of the resultant impulse can also be measured because the hinge will bend in the direction of the average total impulse.

11.10

Stagnation Pressure Measurement

The greg and snob gauges of the previous section measure the total head or stagnation pressure. A pitot tube, in common use on aircraft to measure air speed, is actually a type of stagnation pressure gauge. Several other interesting techniques have been used to measure stagnation pressure. A flat plate mounted on a spring with a simple displacement gauge can be used to measure the stagnation pressure as

154

11 Measurement Techniques

a function of time. These gauges are placed with the flat face toward the detonation. The measured displacement is a direct function of the force on the surface of the plate which is proportional to the stagnated pressure loading the plate. These gauges were labeled as force plates. The primary problem here is the frequency response of the system. A heavy plate responds slowly to the accelerating load and light plates may bend under the loads. A similar gauge called a load cell was adapted to measure stagnation pressure. In this case the plate is mounted to a heavy base with a gauge to measure the load being transferred from the plate to the mount. This design requires very little motion of the plate and the forces are transferred at sound speed in the metal to the gauge. This significantly reduces the response time of the gauge, but induces large ringing signals at or near the natural frequencies of the cymbal (plate) when it is struck by the blast wave.

11.11

Total Impulse Gauge

The easiest and most popular gauge for making measurements of the total impulse is the ballistic pendulum. These have been in wide use for a number of decades and are reliable and reasonably simple (in concept) to construct. The main advantage to a ballistic pendulum is that it responds to the total momentum delivered to the area of the surface of the movable mass. This momentum includes the air blast, any case fragments and any particulates in the flow. Care must be taken in the design of such a pendulum to assure that the response will be measurable. Too heavy a pendulum and the response will be too small to measure and too small or light a pendulum and the motion may be too large to be meaningful. Care must be taken to assure that the air blast does not engulf the pendulum prior to the completion of the loading on the front face. This is usually accomplished by placing the movable pendulum in a fixture that extends sufficiently far into the surrounding space that the distance the blast wave must travel to reach the back side of the pendulum is greater than the blast wave positive phase duration. The protective structure must be constructed with sufficient strength to withstand the total load of the blast wave with no significant response. If the fixture does move, the fulcrum of the pendulum will move and the interpretation of the data becomes very difficult. The ballistic pendulum records the total impulse by its responding motion. As the pendulum swings, the center of mass is raised and the kinetic energy of the pendulum is converted to gravitational potential energy. By measuring the maximum height to which the pendulum swings, the initial kinetic energy and therefore the initial momentum of the pendulum mass is measured.

References 1. Dewey, J.M., Johnson, O.T., Patterson, J.D.: Some Effects of Light surrounds and Casings on the Blast from Explosives, BRL Report No. 1218, (AD 346965), September, (1963)

References

155

2. Ethridge Noel: Use of Cube Displacements as a Measure of Airblast, MABS 13, 13th International Symposium on the Military Application of Blast Simulation, Proceedings, Vol. 1, September (1993) 3. Ethridge, N., Keefer, J.H., Reisler, R.E.: MABS Monograph Airblast Instrumentation 1943–1993, Vol. 1, Defense Nuclear Agency, August, (1995) 4. Reisler, R.E., Keefer, J.H., Ethridge, N.H.: MABS monograph, Air Blast Instrumentation 1943–1993, Defense Nuclear Agency, August (1995)

Chapter 12

Scaling Blast Parameters

It is often useful to be able to scale the results from one experiment to a similar experiment having a different yield. Such scaling allows the pressure or impulse from one experiment to be used to predict the pressure from another or to compare the results of two different experiments.

12.1

Yield Scaling

In Chap. 5 it was shown that the detonation pressure of a condensed high explosive was independent of the charge size. We can make use of this fact in developing scaling relations for different charge sizes. The pressure inside the charge, the detonation pressure, extends to the surface of the charge. The total energy released by the detonation is the detonation energy, measured in energy per unit mass, times the mass of the explosive. For a spherical charge, the energy released, also called the yield, is therefore proportional to the cube of the radius of the charge: 3 Y ¼ 4p 3 rEdet Rc , where Rc is the charge radius, r is the explosive density and Edet is the detonation energy of the explosive. The pressure at a distance measured in charge radii has the same value and is thus independent of the charge size. The development and propagation of the blast wave is self similar. The development of the blast wave in Chap. 5 made use of this fact and was described in terms of charge radii. Thus at one charge radius, the pressure is the detonation pressure independent of the charge size. The pressure decays proportional to the distance from the charge. At any distance R we can define the parameter D ¼ R/Rc, where the pressure, velocity and density of the blast wave will have the same value for all charges of the same explosive. The radius at which a given pressure occurs can then be expressed in terms of the ratio of the yields. Because the detonation energy and the mass density of an explosive are

C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_12, # Springer-Verlag Berlin Heidelberg 2010

157

158

12 Scaling Blast Parameters

properties of the explosive and not a function of charge size, we can write the ratio of the yields of two charges as: Y1 ¼ Y2




Rc1 Rc2

3 (12.1)

We can now write the radius at which a given blast wave parameter occurs in terms of the radius at which that same value occurs for a different yield. rffiffiffiffiffi 3 Y1 : R1 ðP1 Þ ¼ R2 ðP1 Þ Y2

(12.2)

Thus the radius at which a given parameter occurs for one charge size is equal to the radius at which that same parameter value occurs for a reference charge size times the cube root of the ratio of the yields of the two charges. The detonation velocity is a constant throughout the detonation process and therefore time can be expressed in terms of R/V. Because the process is self similar, this expression holds at any range. The time can also be scaled as the cube root of the yield ratio. The velocity does not scale, therefore, when the distance scales, the time to reach that distance scales in the same manner. This relationship holds very precisely for spherical charges of the same explosive. The values for the pressure, density, and velocity remain the same; only the radius and time are scaled. This relationship is still useful for comparing one explosive to another in an approximate manner. The ideas of pressure and impulse equivalency were introduced in Chap. 5. Equation (12.1) or (12.2) above can be used to establish an equivalent yield for different explosives for any blast parameter. The procedure is to find the radius at which the blast parameter of interest reaches the value of interest for two different explosives having the same mass. The blast equivalency for that blast parameter at that range is then found by taking the ratio of the cubes of the radii. Unfortunately, in general, this results in a different equivalency for each blast parameter at every value of that parameter. These differences are caused by differences in the explosive loading density, the detonation pressure, the detonation velocity and to some extent the species in the detonation products. The higher loading density gives the explosive a higher momentum density during the detonation. This higher momentum density carries into the expansion phase of the detonation products and the early compression of the surrounding air causing the pressure, for instance, to fall off less rapidly during the early expansion. Clearly a higher detonation pressure tends to generate a higher blast pressure in the surrounding medium. A higher detonation velocity is composed of either a higher sound speed in the detonation products, a higher material velocity at the detonation front or a combination of the two. The higher sound speed means that a higher expansion velocity is achievable and a higher material velocity enhances the momentum density at the charge surface. Both of these conditions lead to higher

12.1 Yield Scaling

159

early expansion rates and stronger compression of the surrounding gas which in turn generates higher pressures. The species concentration within the detonation products has a marked effect on the effective g of the detonation products. The effective g is a measure of the ability of the energy in the detonation products to do work on their surroundings. The higher the g (actually the g1), the larger the fraction of internal energy is available to do work on the surroundings. Differences in g1 manifest themselves when the detonation products have expanded by more than an order of magnitude in density and thus have an influence at larger radii than the explosive loading density or detonation velocity. To avoid the confusion produced by attempting to find an effective yield for various blast wave parameters at many different values, the concept of effective yield is introduced. The effective yield of an explosive can be determined in terms of the energy released. In Chap. 5, Fig. 5.10 we showed a comparison of the peak overpressure as a function of range between several explosives which had been cube root scaled to the same energy release. This plot demonstrates that the individual explosive detonation characteristics influence the pressure (as well as all other blast parameters) for significant radii, even when the energy scaling has been accounted for. It is not until the blast overpressure has fallen below about 50 bars that the energy equivalency or effective yield shows its real value. Below 50 bars, all of the explosives scaled to the same effective yield, give near overlays of the pressure as a function of range. This effective yield scaling is thus a powerful tool for comparing data and calculational results for pressures below about 600 psi.

12.1.1

Application to Nuclear Detonations

These relations hold for conventional explosives and are used for nuclear detonations as well. In the case of nuclear generated blast waves the scaling relations are not as well founded as for conventional explosives. The primary reason is the relative importance of thermal radiation from a nuclear fireball. In general about half of the energy generated by a nuclear detonation in the low atmosphere is radiated away. Thirty-five to forty-five percent of the energy is lost in the form of thermal radiation with an additional 5–15% lost in nuclear radiation in the form of a and b particles, g rays and X-rays. Thus only 40–60% of the energy released is available to generate blast. This percentage is dependent on a number of factors related to the design and yield of the device. Yield scaling or total energy scaling between nuclear detonations works reasonably well. The problem comes when comparisons are made at small ranges or early times. The timing of the thermal radiation loss does not cube root scale. This means that the energy left to produce air blast does not cube root scale. The radiation loss from a nuclear fireball is a function of the temperature of the radiating gas and the surface area over which it is radiating. The thermal radiation is characterized by wavelengths ranging from the far ultraviolet to the far infrared but

160

12 Scaling Blast Parameters Thermal Flux vs. Time (1 kt) 2.00E+20 1.80E+20 1.60E+20

Flux (ergs/sec)

1.40E+20 1.20E+20 1.00E+20 8.00E+19 6.00E+19 4.00E+19 2.00E+19 0.00E+00 1.00E – 05

1.00E–04

1.00E–03

1.00E–02

1.00E–01

1.00E+00

1.00E+01

Time (sec)

Fig. 12.1 Thermal flux as a function of time for 1 KT at sea level (log scale)

is dominated by the visible spectrum. This is primarily because the visible spectrum has a very long mean free path in air and the radiation which escapes the fireball reaches ranges well beyond the blast wave unless something other than air is encountered. The radiation rate or radiative flux as a function of time is shown in Fig. 12.1 for a 1 KT nuclear detonation in a sea level atmosphere. Note that there are two distinct peaks and that the time scale is logarithmic. The rise to the first peak is caused by the conversion of X-rays absorbed in the air and reradiated in the form of thermal reradiation. The drop in flux after the first peak is caused by the rapid cooling of the radiating gas. The temperature is decreasing at a decreasing rate as the fireball cools because the radiation rate is proportional to the temperature to the fourth power. The rise to the second peak is caused by the expanding fireball and the rapidly increasing surface area of the radiating fireball. The increase in surface area more than makes up for the effects of the decreasing temperature. The decay after the second peak is caused by the fireball reaching its maximum radius and the general cooling of the fireball. Some relevant numbers for this figure (1 KT) should be pointed out. The radiation rate drops by an order of magnitude by 1 ms. The fireball radius at minimum time, (the time the minimum flux is observed) is 15 m. The fireball grows to a radius of about 100 m at a time of 40 ms. The fireball continues to radiate for a full second. Only 1% of the radiated energy is radiated during the first pulse (between time zero and 2.56 ms). This is illustrated in Fig. 12.2, which is the same radiated flux plotted on a linear time scale. The timing of the thermal flux does not scale as the cube root of the yield although the total radiated energy comes very close to cube root behavior. The physical reason for this is primarily that radiation mean free paths are not a function

12.2 Atmospheric Scaling

161 Thermal Flux vs. Time (1 kt)

2.00E+20 1.80E+20 1.60E+20

Flux (ergs/sec)

1.40E+20 1.20E+20 1.00E+20 8.00E+19 6.00E+19 4.00E+19 2.00E+19 0.00E+00 0.00E+00 1.00E – 01 2.00E – 01 3.00E – 01 4.00E – 01 5.00E – 01 6.00E – 01 7.00E – 01 8.00E – 01 9.00E – 01 1.00E+00

Time (sec)

Fig. 12.2 Thermal flux as a function of time for 1 KT at sea level (linear scale)

of the yield of the detonation and are only a function of the radiating frequency. The minimum time, that is, the time at which the thermal flux reaches a minimum between the first and second peaks, scales as the yield to the 0.39 power. The maximum time, the time at which the second peak occurs, scales as the 0.44 power of the yield. The magnitude of the energy flux scales more like the square root of the yield (the 0.59 power). In the previous section we showed that the overpressure below about 30 bars scaled well with the total energy released for high explosives. A similar rule holds for nuclear detonations, however, cube root scaling works very well for pressures below about 10 bars. As an example of the thermal flux scaling, Fig. 12.3 shows the thermal flux as a function of time for a 100 KT nuclear detonation at sea level. The minimum time is 15.5 ms rather than 2.56 ms for 1 KT. The maximum time shifts from 38 ms for 1 KT to 288 ms for 100 KT. The peak flux is increased from 1.5e20 ergs/s to 2.25e21 ergs/s.

12.2

Atmospheric Scaling

The previous section dealt with scaling blast parameters as a function of the yield of the source. In this section we will discuss the effects of atmospheric scaling with altitude. As with yield scaling, there are accepted scaling laws, the most used are the modified Sachs scaling rules. The expected accuracy of the modified Sachs scaling rules is about 20% for a general application of blast waves propagating through

162

12 Scaling Blast Parameters Radout vs. Time (100 kt)

2.50E+21

Flux (ergs/sec)

2.00E+21

1.50E+21

1.00E+21

5.00E+20

0.00E+20 1.00E – 05

1.00E–04

1.00E–03

1.00E–02 Time (sec)

1.00E–01

1.00E+00

1.00E+01

Fig. 12.3 Thermal flux as a function of time for 100 KT at sea level (log scale)

a real atmosphere. A reference burst is defined with a set of known parameter definitions as a function of range for a given yield and altitude of detonation. The modified Sachs scaling relations can then be used to define the parameters for any other burst at any measurement position. In order to use the atmospheric scaling relations, certain values for the atmosphere must be known. These include the pressure, density and sound speed at the reference location (denoted by a numeral 1) and the pressure, density and sound speed at the point of interest (denoted by the numeral 3). In addition the yield of the reference burst (Y1) and the yield of the burst of interest (Y3) must be known. Notice that information about the altitude of the burst of interest is not needed, only the altitude of the measurement point. The basic scaling parameters given here include the yield scaling described in the previous section. The scale factor for time is: =

=

SFt ¼ C1=C3ðP3=P1Þ1 3 ðY3=Y1Þ1 3 ;

(12.3)

where C is the sound speed and P is the pressure. The scale factor for distance or range is given by: =

=

SFR ¼ ðP1=P3Þ1 3 ðY3=Y1Þ1 3

(12.4)

Note that the distance scaling is inversely proportional to the cube root of the pressure ratio. As before, the pressure, density and velocity are not functions of the yield. They are therefore scaled only as a function of the corresponding atmospheric parameter and are directly proportional to those values.

12.2 Atmospheric Scaling

163

The pressure scaling factor is: SFp ¼ P3=P1;

(12.5)

SFr ¼ r3=r1:

(12.6)

SFv ¼ C3=C1:

(12.7)

The density scales as:

and the velocity scales as:

While the above relations give the values of the scaling factors, some care must be taken to apply them properly. First calculate the combined yield and altitude scaling factors. Then calculate the scaled time at the target point by multiplying the reference time by the time scale factor (12.3). Evaluate the scaled distance for the reference burst. Use the scaled distance to evaluate the pressure at the scaled distance. Then use the pressure scaling factor to obtain the pressure at the target point. It is worth some discussion of the applicability of these atmospheric scaling relations. If we look at the atmospheric pressure scaling, the blast wave peak pressure is proportional to the ambient atmospheric pressure. At an altitude of 36,000 ft, the ambient pressure is only 22% of the pressure at sea level. The scaling relations therefore reduce the shock pressure by a factor of 4.5. Remember that the detonation pressure for a solid explosive is independent of the altitude at which it is detonated. Also remember from the discussion of air shock formation in Chap. 5 that the peak pressure in the blast wave may not be at the shock front but is in the expanding detonation products. The ambient atmospheric pressure has very little effect on the peak pressure or the velocity of the expanding detonation products until the products have expanded to at least ten charge radii. This is the radius at which the mass of the sea level air is about equal to the charge mass for TNT. The scaling relations do not apply well at high pressures for solid explosives. At the other end of the pressure scale, the discussion on shock propagation in Chap. 7 demonstrated that temperature and wind velocity gradients within the atmosphere can change the blast wave peak pressure at the same altitude and at the same distance by an order of magnitude, depending on the direction. The above atmospheric scaling laws do not apply well in these cases either. In the case of a nuclear detonation at high altitude, the phenomena associated with blast wave formation and propagation change dramatically. The radiative thermal flux from a 100 KT device detonated at an altitude of 150,000 ft is shown in Fig. 12.4. If we compare this curve with the curve of Fig. 12.3 we see a number of striking differences. First there is only a single peak at high altitude because there is little atmosphere to absorb the X-rays so the deposition radius is much larger. The time of the first peak is not much different between the two cases, but the value of the peak flux has increased by nearly two orders of magnitude.

164

12 Scaling Blast Parameters 100 kt@150 kft 1.80E+23 1.60E+23

Energy Flux (ergs/sec)

1.40E+23 1.20E+23 1.00E+23 8.00E+22 6.00E+22 4.00E+22 2.00E+22 0.00E+00 1.00E – 05

1.00E – 04

1.00E – 03

1.00E – 02

1.00E–01

1.00E+00

1.00E+01

Time (sec)

Fig. 12.4 Thermal flux from a 100 KT device detonated at 150,000 ft

The sea level fireball radiates for about 10 s, whereas the high altitude fireball has finished radiating in less than 100 ms. At a time when the sea level fireball has reached a minimum flux of about 1.e20, the high altitude fireball is radiating at 500 times the rate of the sea level detonation. The vast difference in the behavior of the radiated energy has a marked effect on the formation and propagation of the blast wave which is not accounted for by the modified Sachs scaling rules. For small to moderate changes in the altitude of burst, (perhaps less than a few kilometers) and for small variations in yield (one order of magnitude), the scaling relations provide a good approximation to the behavior at intermediate (30–0.1 bars) pressure levels. I will make one more observation about nuclear blast waves. When comparisons are made between the scaled pressure vs. range curves from the Nevada Test Site (NTS) and the Pacific Proving Ground (PPG) tests a systematic difference is found between the two sets of data. The pressure vs. range curve for the PPG tests falls well below the NTS data with only a few exceptions. Closer examination of the comparisons shows the majority of the PPG tests being compared were for yields of half a megaton or more, whereas the yields of the NTS test were all less than about 70 KT. The lower yield PPG tests show good agreement with the NTS test data. Numerical calculations of the large yield PPG tests show that the radius of the shock at intermediate level pressures (30–0.1 bars) is the order of or greater than the scale height of the atmosphere. This means that the changing pressure of the atmosphere in the upward moving shock can be communicated to the horizontally propagating shock and energy is diverted to the vertical direction. The pressure therefore decays more rapidly in the horizontal direction for megaton yields than for kiloton yields. Nearly all pressure measurements in the Pacific were taken at ground level or within

12.2 Atmospheric Scaling

165

3 km above the ground. The 3 km altitude data falls along the same line as the sea level measurements when scaled back to sea level. The pressure scale height, the distance over which the pressure falls by a factor of e, the base of the natural logarithms (2.72), is the order of 8 km. A blast wave generated by a surface or near surface nuclear detonation propagates vertically through the atmosphere. The energy in that blast wave decays at a rate between 1/R2 and 1/R depending on the yield of the detonation and the distance traveled. The atmospheric pressure drops exponentially with altitude with a scale height of 8 km. Thus a strong shock at sea level decays rapidly during its early vertical propagation. The rate of decay of the blast wave soon becomes smaller than the rate of decay in the ambient atmosphere. The shock strength, relative to ambient atmospheric pressure, becomes larger. At an altitude of 90–100 km or so, a very strong shock may interact with the ionosphere. The ionosphere is characterized by the presence of a sudden change in temperature gradient. The temperature rapidly increases by more than two orders of magnitude and continues to increase with increasing altitude. This temperature increase is accompanied by the appearance of partially ionized atoms of primarily oxygen and nitrogen atoms. This change in temperature also creates a rapid decrease in fluid density. The electron density of the ionosphere increases with increasing altitude. Radio frequency waves are reflected from the lower portion of the ionosphere. As the frequency increases, higher levels of ionization are required to cause their reflection. Thus higher frequency waves are reflected from higher altitudes within the ionosphere. The phenomenon of enhancement of blast waves from near surface bursts as they propagate vertically through the atmosphere provides an interesting technique for detecting large scale detonations in the atmosphere. Strong blast waves perturb the ionosphere and because of the rapid decrease in density in the ionosphere, can induce gravity waves. These gravity waves then propagate from the perturbed region in exactly the same way that waves move on the surface of a quiet pond when a stone is thrown into the water. By observing the movement and the magnitude of the waves with multi-frequency radar, it is possible to not only detect a near surface detonation, but to obtain information on the approximate yield of the device. In 1970 a magnitude 7.8 earthquake struck northern China. The ground was lifted about 1 m over an area of several square kilometers. The local villagers reported that the sky glowed green for several minutes following the earthquake but no explanation was given. The author made a one dimensional numerical calculation of the pressure wave generated by a piston with an impulsive velocity that would decay to zero under gravitational acceleration in a distance of 1 m and propagated the wave vertically through a real decaying atmosphere. When the pressure wave, now a strong blast wave, reached an altitude of 100 km it had sufficient strength to further excite the ionosphere well above ambient temperature causing further ionization and could have produced the observed green glow over a wide area. The low densities at that altitude would not allow the induced ionization to equilibrate rapidly and the glow could have lasted many minutes.

166

12 Scaling Blast Parameters 50.0 LEGEND Mod Sachs 500 MT Mod Sachs 27 MT CFD 500 MT CFD 27 MT Sachs 27 MT Sachs 500 MT

45.0 40.0

ALTITUDE (KM)

35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0 0

5

10

15

20 25 30 Arrival Time (sec)

35

40

45

50

Fig. 12.5 Calculated and scaled altitude vs. time from sea level detonations

Accurately scaling the arrival time as the blast wave propagates vertically upward through the atmosphere presents some special problems. When the blast wave is propagating downward from a detonation, the scaling parameters defined in (12.1)–(12.7) have proven perfectly adequate. However, when the blast wave propagates upward from the detonation, some modification of the time and distance scale factors can improve the results significantly. Figure 12.5 illustrates a comparison of the arrival time as a function of altitude for two, admittedly large, surface bursts. The computational fluid dynamics (CFD) results used first principles calculations to propagate the blast wave through the atmosphere. The results of the CFD calculations are taken to be accurate and the scaling results are evaluated against them. The x and diamond shaped symbols represent the results using the standard Sachs scaling rules. The vertical velocities of the shocks accelerate to unrealistic values for altitudes above about 10 km. The shock front for the 500 Mt burst moves 20 km in the 2 s between 10 and 12 s for an approximate Mach number of 30. The CFD calculation indicates that the distance should be about 1 km over the same time interval, for a Mach number of 1.5 (a much more believable value). To achieve the much better agreement shown for the modified Sachs scaling rules, the scaling factors for both range and time were modified. We assumed the

12.2 Atmospheric Scaling

167

atmosphere to be exponentially varying with altitude. With this assumption we can use the burst point and the field point altitudes to evaluate constants in the exponential equations for pressure and density. PðzÞ ¼ Ap expðBp  zÞ rðzÞ ¼ Ad expðBd  zÞ

(12.8)

The four constants A and B can then be found by evaluating the pressure and density at the burst point and at the field point. The distance scale factor (12.4) can then be modified to use the averaged pressure between the burst point and the field point. The averaged pressure is calculated using the equation: Hfpt R

Pdif ¼

PðzÞdz

Hob

Hob  Hfpt

;

(12.9)

where Hob is the altitude of the burst and Hfpt is the altitude of the field point. The distance scale factor becomes:  1=3 = ðY3=Y1Þ1 3 : SFR ¼ P1=Pdif

(12.10)

Similarly the average density can be evaluated as: Hfpt R

rdif ¼

rðzÞdz

Hob

Hfpr  Hob

:

The average sound speed over the distance between burst and field point can now be evaluated as: sffiffiffiffiffiffiffiffiffiffi gPdif C¼ rdif The time scaling factor can be modified in a manner similar to that of the distance scale factor by using the averaged pressure and sound speed. The time scale factor becomes:  1=3 = SFt ¼ C1=C Pdif =P1 ðY3=Y1Þ1 3 : When the modified time and distance scale factors are used, the agreement shown in Fig. 12.5 is obtained.

168

12.3

12 Scaling Blast Parameters

Examples of Scaling

To demonstrate the use of the scaling relations, a few examples of their use are presented. We will start with some examples where the scaling relations are very accurate, that is, where the same explosive is scaled in yield at the same atmospheric conditions. The known source is a one pound TNT sphere in a sea level atmosphere. Table 12.1 provides the peak overpressure as a function of radius for some selected pressure levels. To find the radius at which a given pressure level will occur for any TNT yield, simply multiply the radius from our known source in Table 12.1 by the cube root of the ratio of the requested yield to the known yield. For example, to find the radius at which 60 psi will occur for a 1,000 pound charge we multiply 3.61 ft by 10 and get 36.1 ft. To find the range at which 110 kPa occurs for a 1 g charge, multiply the radius for 1 pound by (1/454)1/3 ¼ 0.130. The radius is then 0.2485 m or 0.841 ft. Suppose that we wish to find the arrival time of the blast wave at a distance of 22 m from a 1,000 pound TNT detonation. The procedure is to scale the distance from 1,000 pounds to our reference yield of 1 pound. R ¼ 22 (1/1,000)1/3 ¼ 2.2 m. Table 12.1 Pressure and arrival time as a function of range for 1 pound TNT Source ¼ 1 pound TNT at sea level Range (ft) R (m) Op (PSI) Op (Pa) 2.95E + 00 9.00E01 1.00E + 02 6.89E + 05 3.00E + 00 9.15E01 9.60E + 01 6.62E + 05 3.06E + 00 9.31E01 9.20E + 01 6.34E + 05 3.11E + 00 9.48E01 8.80E + 01 6.07E + 05 3.17E + 00 9.66E01 8.40E + 01 5.79E + 05 3.23E + 00 9.85E01 8.00E + 01 5.52E + 05 3.30E + 00 1.01E + 00 7.60E + 01 5.24E + 05 3.37E + 00 1.03E + 00 7.20E + 01 4.96E + 05 3.44E + 00 1.05E + 00 6.80E + 01 4.69E + 05 3.53E + 00 1.07E + 00 6.40E + 01 4.41E + 05 3.61E + 00 1.10E + 00 6.00E + 01 4.14E + 05 3.71E + 00 1.13E + 00 5.60E + 01 3.86E + 05 3.81E + 00 1.16E + 00 5.20E + 01 3.59E + 05 3.92E + 00 1.20E + 00 4.80E + 01 3.31E + 05 4.05E + 00 1.23E + 00 4.40E + 01 3.03E + 05 4.19E + 00 1.28E + 00 4.00E + 01 2.76E + 05 4.38E + 00 1.33E + 00 3.60E + 01 2.48E + 05 4.59E + 00 1.40E + 00 3.20E + 01 2.21E + 05 4.86E + 00 1.48E + 00 2.80E + 01 1.93E + 05 5.20E + 00 1.58E + 00 2.40E + 01 1.65E + 05 5.64E + 00 1.72E + 00 2.00E + 01 1.38E + 05 6.26E + 00 1.91E + 00 1.60E + 01 1.10E + 05 7.22E + 00 2.20E + 00 1.20E + 01 8.27E + 04 8.95E + 00 2.73E + 00 8.00E + 00 5.52E + 04 1.34E + 01 4.09E + 00 4.00E + 00 2.76E + 04

Toa (s) 5.26E04 5.43E04 5.62E04 5.82E04 6.03E04 6.27E04 6.52E04 6.80E04 7.11E04 7.44E04 7.82E04 8.23E04 8.69E04 9.21E04 9.80E04 1.05E03 1.14E03 1.26E03 1.40E03 1.59E03 1.86E03 2.25E03 2.89E03 4.12E03 7.63E03

12.3 Examples of Scaling

169

Find the arrival time at the reference yield for this distance (2.89 ms) and scale the time to the desired yield. The arrival time at 22 m for 1,000 pounds of TNT detonated at sea level is 28.9 ms. As a reasonableness check for this answer we can calculate the average velocity of the blast wave to be 22m/0.0289s ¼ 761 m/s or about twice ambient sound speed. The shock velocity is greater than sound speed and is just a factor of two or so greater than sound speed. This is quite reasonable for the pressure level of 82 kPa associated with the 22 m range. To demonstrate the method of application for atmospheric scaling we will find the distance and arrival time at which 12 psi occurs for a 1 pound TNT detonation at 6,500 ft above sea level. The pressure ratio at 6,500 ft is approximately 0.78. We find the overpressure at sea level by using the ambient atmospheric pressure ratio and find that this corresponds to 15.4 psi. The distance to 15.4 psi at sea level is 6.4 ft, which is the distance to 12 psi at 6,500 ft. To find the arrival time for the blast wave at this distance and altitude we find the arrival time for the reference burst at 6.4 ft (2.32 ms) and scale it to altitude using the combination of sound speed ratios and the cube root of the pressure ratio. To find the sound speed ratio we use the square root of the temperature ratio. A good approximation to the decay of temperature with height in the standard temperate atmosphere is 6.4929e05 degrees per centimeter of altitude. Thus at 2 km (6,500 ft) the temperature is reduced about 13 . The ratio of the sound speeds is then  (275/288)1/2 ¼ 0.977. Multiplying this by the cube root of the pressure ratio we get 0.901. The arrival time is 2.2 ms.

Chapter 13

Blast Wave Reflections

13.1

Regular Reflections

The Rankine–Hugoniot equations presented in Chap. 3 include the equations for the calculation of the reflected pressure, given the incident overpressure and the ambient conditions. The reflected overpressure divided by the incident overpressure is the reflection factor. The R–H reflected overpressure equation can only be used when the incident shock strikes a plane surface at an incident angle of 90
, that is, when the vector for the shock velocity is perpendicular to the surface. Even in this case, the reflection factor is a function of the g of the gas. Brode [1] gives a curve for the reflection factor for air as a function of the incident overpressure. Figure 13.1 gives the reflection factor as a function of incident overpressure for constant g gasses and for air as the g changes with overpressure. The dashed lines represent gasses from monatomic with g of 5/3 to partially ionized gasses with g near 1.17. This can be confirmed by referring to Fig. 3.3 which shows that the value of g for sea level density air approaches a value of 1.17 at an internal energy density of 2.e11 ergs/g. The temperature at this energy density is approximately 9,000 K. For low to moderate overpressure shocks (below 250 psi) the value of g for air is well approximated by a value of 1.4. If g remained constant at a value of 1.4, the maximum reflection factor would reach eight at an incident overpressure of about 10,000 psi and would never exceed that factor. For noble gasses such as helium, neon, and argon, g retains a value of 5/3 to very high pressures and the maximum reflection factor is six. As the overpressure increases in air, the value of g decreases as shown in Fig. 3.3. This decrease in g affects the reflection factor as shown in Fig. 13.1. The reflection factor for air reaches a maximum of 14.3 at an incident overpressure of approximately 18,000 psi and decreases at higher pressure levels. This is because the g begins to increase at higher pressure (energy/unit volume). At very high energy densities and temperatures the air becomes a dissociated, monatomic gas with a g of 5/3 and the reflection factor approaches a value of six at incident pressures above 20 kilobars.

C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_13, # Springer-Verlag Berlin Heidelberg 2010

171

172

13 Blast Wave Reflections 16 g =1.17 14 Reflection factor ΔPr / ΔP1

g =1.2 12 10

g =1.3 g =1.4

8

g =5/3

6 4

2

1

10

100 1000 Incident overpressure (psi) (ΔP1)

104

105

Fig. 13.1 Normal reflection factor for adiabatic shocks in sea level air as a function of incident overpressure

13.1.1

Regular Reflection at Non-perpendicular Incidence

When a shock impinges a planar surface at any angle other than 90
, the reflection factor changes. There are two regions for the reflection. In the first region the incident and reflected shocks meet at the reflecting plane. This is called “regular reflection”. When the reflected shock is able to catch the incident shock, a single combined shock is formed near the reflecting plane. This is called the Mach reflection region and the merged shock is called the Mach stem. The point at which the incident, reflected and Mach shocks intersect is called the triple point. Figure 13.2 is a cartoon of the shock geometry for a regular reflection. The incident shock (SI) strikes a plane inclined at an angle a relative to the velocity vector of the incident shock. A reflected shock (R) forms behind the incident shock but cannot catch the incident shock because the phase velocity of the incident shock along the reflecting plane is greater than the propagation velocity of the reflected shock. The phase velocity (VP) is the incident shock velocity divided by the cosine of the wedge angle a, in the direction parallel to the reflecting plane. The region denoted as (I) contains ambient conditions. Region II contains gas that has been processed by the incident shock and region III gas has been processed by both the incident and reflected shocks. In region II the gas properties remain constant because the incident shock is a square wave for most shock tube experiments. It is important to note that the gas in region III varies in pressure, velocity, energy density and density as a function of position.

13.2 Mach Reflection

173

Fig. 13.2 Cartoon of Mach reflection shock configuration

I

SI II

Vp

R III α

Because the velocity of the reflected shock is a function of the incident shock strength, the wedge angle at which regular reflection occurs is a function of the incident shock strength. The angle at which the reflection changes from regular to Mach reflection is called the transition angle.

13.2

Mach Reflection

When the reflected shock is able to catch the incident shock, a single combined shock is formed. This phenomenon is called Mach reflection and the merged shock is called the Mach stem. The point at which the incident, reflected and Mach shocks intersect is called the triple point. Shock theory does not permit the intersection of three discontinuities. The “triple” point is always accompanied by a slip line from the triple point, extending between the reflected shock and the Mach stem.

13.2.1

Simple or Single Mach Reflection

Figure 13.3 is a cartoon of the shock configuration for a “single” or “simple” Mach reflection. Region I contains ambient gas. SI is the incident shock moving from left to right and reflecting from a flat plane placed at an angle a relative to the incident shock velocity. Region II contains gas that has been processed by a single shock, the incident shock. Region III contains gas that has been processed by a single shock at higher overpressure, the Mach shock. For planar shocks and flat reflecting planes, the Mach stem is perpendicular to the reflecting plane at the surface, although the Mach stem is generally not straight as depicted here. R is the reflected shock and is curved.

174

13 Blast Wave Reflections

Fig. 13.3 Shock geometry for a regular reflection

SI

II

I

TP M

R IV

III α

Interferogram Courtesy of K. Takayama Instititute of High Speed Mechanics Tohoku, University, Sendai. Japan

Fig. 13.4 Interferogram of Mach reflection showing density distribution

Region IV contains gas that has been processed by the incident shock and the reflected shock and is therefore at a different state than the gas in region III. The curve separating the two states of the gas in regions III and IV represents the slip line. The pressure is in equilibrium across the slip line but the velocity, energy density and density are different. The velocities on either side of the slip line are parallel to the slip line but have different magnitudes. Because the velocities have different magnitudes at the slip line interface, the slip line may be Kelvin– Helmholtz unstable. In several of the experiments shown in this section, the slip lines show this instability as the slip line approaches the surface. The instability occurs earlier for high pressure shocks because the differential velocity across the slip line is greater at high pressures. Although the incident shock may be flat topped (non-decaying), the pressure, density and velocity vary behind the reflected shock. Figure 13.4 is an interferogram of a Mach reflection. The interference fringes mark constant density lines, or isopycnics. The density variations behind the reflected shock and the Mach stem are clearly shown. The discontinuity in density across the slip line is also clearly marked. When the wedge angle is near the transition angle, the growth of the Mach stem is very slow. The line from the beginning of the wedge, through the triple point is at

13.2 Mach Reflection

175

a very small angle relative to the wedge. Many discussions, some rather heated, have been carried out over the decision that a reflection is regular rather than Mach. Some of these discussions have been settled by using larger wedges so the triple point can be distinguished above the wedge surface. This decision is also influenced by the presence of a boundary layer which exists behind the shock front and may hide the triple point and keep it from being detected or delay its appearance above the boundary layer.

13.2.2

Complex Mach Reflection

Mach reflections are further divided into classes of reflections “simple” or single, “irregular” or “complex” or “transitional”, and “double”. There have been reports of the observation of “triple” Mach reflections at very high pressures. For a discussion of 13 different classifications of reflections see [2]. I will discuss the four most commonly studied reflections. Each of these types of Mach reflections is dependent on the incident overpressure and the angle of reflection. Careful experiments have been conducted by numerous experimenters to define the boundaries of these different types of Mach reflection. Figure 13.5 illustrates the observed geometry of a complex or transitional Mach reflection. The letter K indicates a kink in the reflected shock. This marks the point at which the flow transitions between supersonic relative to the triple point (TP) but subsonic relative to the intersection point of the slip line and the wedge. The reflected shock is straight between TP and K because the signal from the slip line intersection with the wedge has just reached K. The dotted line from K to the slip line indicates a weak wave which is the location of the leading edge of the signal from the intersection of the slip line with the wedge. Figure 13.6 illustrates the shock configuration for a double Mach reflection. This is labeled a weak double Mach configuration but keep in mind that double Mach reflections only occur at pressures above several hundred PSI. Keep in mind also that the shock geometry changes very nearly continuously. When looking at complex Mach reflections near the transition from simple Mach reflection, the questions to be answered are “is there a kink in the reflected shock and how close is the kink to the triple point?” SI

II

I

K TP M

R

Fig. 13.5 Observed Mach reflection geometry for complex reflection

IV

III α

176

13 Blast Wave Reflections

Fig. 13.6 Weak double Mach reflection geometry

SI II

I

K

R

TP

M2 IV

M

V III

VI α

When deciding if the reflection is a complex Mach or a double Mach reflection the decision must be made by the experimenter or the calculator as to whether the compressive wave is a shock by answering the question “how sharp is the discontinuity?” in the photograph or the plot.

13.2.3

Double Mach Reflection

The major difference between the complex Mach reflection and the weak double Mach reflection is the existence of a second triple point in addition to a kink. This second triple point joins the straight part of the reflected shock, the curved part of the reflected shock and a second Mach stem. A second slip line emanates from the second triple point and terminates on the surface. The second Mach stem extends from the second triple point (K) and usually terminates at the first slip line. For “weak” double Mach reflections the second Mach stem is curved. The presence of the second Mach stem also generates two more regions of gaseous states within this geometry. In the simple Mach reflection, region IV represents gas processed by the reflected shock. This region is divided into two additional regions by the second Mach stem and the slip line associated with the second triple point. The gas in region IV has been processed by the incident and the reflected shocks. Region V has been processed by the incident and the reflected shock ahead of the kink. The presence of the kink indicates that the reflected shock strength differs from that of the curved part of the reflected shock. The gas in region VI has been processed by the incident shock, the reflected shock ahead of the kink and the second Mach stem. Figure 13.7 is a shadowgram of a Mach 2.7 shock on a 47
wedge. This photo was labeled by the experimenter [3] as a complex Mach reflection. The primary reason for this label is the absence of a second slip line at the second triple point. Note the curvature of the signal below the second triple point. Note also that this compressive wave terminates in the slip line from the first triple point. The slip line

13.2 Mach Reflection

177

Fig. 13.7 Shadowgram of complex Mach reflection

in this case, is unstable and generates a turbulent region due to Kelvin–Helmholtz instabilities induced by the velocity shear at the slip line between regions III and IV as shown in Fig. 13.5. As the shock strength increases, the strong double Mach reflection is observed. Figure 13.8 illustrates the geometry for a strong double Mach reflection. Here the primary distinction is that the second Mach stem is straight rather than curved. The second Mach stem extends closer to the surface, near the tail of the first slip line. One of the important aspects of complex and double Mach reflections is the overpressure time history recorded on the surface as the shock passes. The Mach shock has a discontinuous rise to a peak; the overpressure then smoothly decays to a minimum and rises smoothly to a second peak. The second peak may be higher or lower than the pressure in the Mach shock. The timing of the second peak corresponds to the passage of the intersection of the slip line with the reflecting surface. The velocity of the gas above the slip line has a component perpendicular to the reflecting plane. When the gas above the slip line strikes the reflecting plane, the perpendicular component of the flow velocity is stagnated and causes a rise in pressure over a small region near the termination of the slip line. In double Mach reflections, there are two slip lines and two secondary peaks. The partial stagnation of the flow parallel to the slip line causes a rise in pressure in a small region near the reflecting surface. The increased pressure near the surface accelerates the flow and may cause the slip line to curl forward. For high pressure flows the slip line may form a counter clockwise rotation behind the Mach stem with the highest velocities near the surface. This is illustrated in the cartoon of Fig. 13.9.

178

13 Blast Wave Reflections

Fig. 13.8 Shock geometry for a strong double Mach reflection

SI II

I

K

TP M

R

V III IV

M2 VI α

gm cm3

Po = 0.769 × 10–5

MS I T

M

R T1 R1

M1

S 70

n

P /Po

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1.000 5.530 6.685 6.830 7.806 8.872 9.756 10.734 11.710 12.685 13.631 14.637 15.613 16.589 17.564

S1

60

50

40

30

20

10

0

BEN-DOR 26.56 DEGREE WEDGE EXPERIMENT (M = 8.05)

ZONES

318

280

240

200

160

120

80

40

1

10.1

80.0

70.0

60.0

50.0

40.0

30.0

Fig. 13.9 Density fields from experiment and calculation

20.0

10.0

0.0

13.2 Mach Reflection

179

Calculations as early as 1980 [4] were able to resolve the double Mach reflection flow field. Comparisons of the experimental and CFD results of the density field are shown in Fig. 13.9 for a Mach 8.06 shock on a 26.56
wedge. The slight bulge in the calculated Mach stem is the result of assuming a constant g of 1.32 for the gas. The calculation was repeated with a constant g of 1.30. Figure 13.10 compares the density profile along the reflecting surface for the experiment and the two CFD calculations. The calculation with g ¼ 1.30 shows a significantly higher density in the vicinity of the stagnation region at the base of the slip line. The two calculations span the data in this region. The experimental density measurements were made using a laser interferogram technique and counting fringe numbers.

17

Ms = 8.06 θw = 26.56

16 15

RELATIVE DENSITY ALONG WEDGE (ρ / ρo)

14 13 12 11 10 9 8 7 6 5 4 3

Code results for γ = 1.30 Code results for γ = 1.32

2

Experimental Values Ben-Dor, 1978

1 70

60 50 40 30 20 DISTANCE FROM CORNER OF WEDGE (mm)

Fig. 13.10 Comparison of the density distribution at the reflecting surface

10

0

180

13 Blast Wave Reflections

The second peak in the pressure time history is caused by the passage of the stagnation region and closely follows the profile of the density. The second peak pressure rises to a maximum over a distance which is finite. The second peak is a compressive wave and not a shock. The time between the arrival of the Mach shock and the stagnation region is small when the Mach stem has just begun to form. When the stagnation region is close behind the Mach shock is also when the stagnation peak pressure is greater than that of the Mach shock. In this case the duration of the rounded stagnation pressure is short because the distance over which the stagnation is taking place is small. Unless very high frequency response pressure gauges are used, the separation of the Mach shock pressure and the stagnation region cannot be resolved. Because the peak pressure is not a shock, the reflection characteristics do not obey the Rankine–Hugoniot relations. The R–H relations are based on one dimensional conservation laws, however the second peak in a Mach reflection is a two dimensional phenomenon and is not properly represented in one dimension. Because the second peak is the result of a partial stagnation of the flow and is not a shock, the reflection factors do not apply when the second peak is greater than that at the Mach shock front. As the Mach stem grows, the distance (and time) separation increases, but the relative magnitude of the second peak pressure decreases. The pressure at the base of the slip line decreases as the separation increases and the magnitude of the rounded peak falls below the peak at the Mach shock (Fig. 13.11). Figure 13.12 comes from Ben-Dor [2]. This plot defines the latest understanding and labeling of shock reflection regions in terms of incident shock Mach number and wedge angle. Shock Mach number is the horizontal axis and wedge angle is the vertical axis. The various symbols represent experimental observations by a number of investigators. The symbols indicate the observed type of reflection and the lines represent our current best theoretical understanding of the reflection regions. Where some symbols may encroach into a neighboring region, the secondary shocks or slip lines could not be discerned in the experimental photographs. Keep in mind that the

SI II I

TP

M

R III IV

Fig. 13.11 Shock geometry illustrating stagnation of flow near termination of the slip line

α

13.2 Mach Reflection

181

Fig. 13.12 Regions of reflection types as a function of wedge angle and shock Mach number

weakest shock occurs as the Mach number approaches one. The region above 50
is labeled “RR” for regular reflection. For all incident Mach numbers, regular reflection occurs for all wedge angles greater than 60
and is therefore not shown on this plot. The region generally to the right and just below the regular reflection region is labeled “DMR” for double Mach reflection. Ben-Dor uses the term “transitional Mach reflection” or TMR for the region below and to the left of the DMR. I have called this reflection the complex Mach reflection. The remainder of the plot represents the region where Single or Simple Mach Reflection or SMR occurs. Ben-Dor actually divides shock reflections into 13 different classes. For details I refer the reader to his book [2]. For the weakest shocks, regular reflection is observed to wedge angles approaching 25
. It is a complement to the experimenters that such low shock strengths can be studied. For example, at 0.005 psi (3,447 dynes/cm2) or Mach 1.001, the density increase at the shock front is only 0.2% above ambient. With such a small density discontinuity, it is very difficult to observe such shocks, let alone distinguish transition to Mach reflection. For stronger shocks, above Mach 2, the transition wedge angle from regular reflection remains very nearly constant at a value of just over 50
. If we look at a Mach 3 shock, for example, as the angle decreases, it is more difficult to concentrate the stagnated energy behind the shock front and the reflection geometry transitions from regular to double Mach to complex Mach to simple Mach reflection. Each of these reflection types occurs for the same strength shock. The reflected peak overpressure, not necessarily a shock pressure, changes dramatically. A Mach 3 shock in ambient sea level air has an incident overpressure of 9.5 bars or 137 psi. The reflection factor varies from nearly five at a wedge angle of 50
(double Mach reflection) to a value of 1.5 at a wedge angle of 15
(single Mach reflection).

182

13.3

13 Blast Wave Reflections

Planar Reflections

The peak pressure enhancement associated with Mach reflection has been under study by a specialized group of experts with expertise in experimental, analytic and numerical methods for several decades. The International Mach Reflection Symposium (recently changed to the International Symposium on the Interaction of Shock Waves) was established to provide a forum for the exchange of information amongst active researchers in the field. Experts from many countries have participated in this symposium on a regular basis. The countries typically represented include: US, UK, Canada, Japan, Israel, Australia, Russia, China, Norway, Sweden, South Africa, India, Italy and Germany. Proceedings, other than abstracts, are not regularly published because this is a working level meeting at which frank and critical evaluations of ongoing work is encouraged. The 14th Mach Reflection Symposium was an exception in that papers were published. Dewey et al. [5] comes from that symposium.

13.3.1

Single Wedge Reflections

Figure 13.13 comes from a paper presented at the 14th symposium by John Dewey from the University of Victoria in British Columbia, Canada, [5]. The solid curves were calculated using von Neumann’s two- and three-shock theories [2, 3, 15] for a shock with Mach number 1.36 (incident shock overpressure of just over 1 bar). Note that the shock theories have a discontinuity at Mach reflection. (For a detailed description of the two and three shock theories, see [2]) This is an analytic singularity that is not supported by experiment or numerical calculation. Theory

PRESSURE

Experiment

Ms = 1.36

MR

RR

ANGLE OF INCIDENCE

Fig. 13.13 Two and three shock theory and experimental data, pressure as a function of wedge angle (Dewey)

13.3 Planar Reflections

183

Fig. 13.14 Measured pressures as a function of angle for a cylindrical reflector

9

M = 1.65

8

Pressure

7 6 5

M = 1.62

4 3 2

M = 1.06

1 10

20

30

40 50 Degree

60

70

80

90

The experimental points are from electronic transducers flush mounted with the reflecting surface. “Measurements such as these have been made in several laboratories and clearly showed an increase of pressure in the transition region, but not of the magnitude predicted by theory [5].” Figure 13.14 compares the reflected pressure as a function of angle for a plane wave on a cylindrical reflector. Experimental data for shocks of three different Mach numbers are shown. The original experimental data comes from [6]. Note that there is no measured enhancement as a function of angle for these cases. An argument was made that the radius of the cylinder was too small and the gauges were too large for the enhancement to be seen. In response to this, Dr. John Dewey, at the University of Victoria undertook a series of experiments in his laboratory. The data shown in Fig. 13.15 was collected by Dewey et al. [5]. The incident shock had a Mach number of 1.2 and reflected from a 1.42 m radius cylinder. The large radius of the cylinder provided higher resolution for both photographic and electronic measurements. The vertical dotted line marks the transition angle based on detachment criteria, whereas the dot–dashed line at 50
marks the experimentally observed transition. The data points are scattered because each point comes from a different experiment with slightly different shock properties and ambient conditions. The dashed lines show the results of numerical calculations at several levels of resolution. This experimental data shows an enhancement near the observed transition angle. The calculational results show a trend toward higher reflected pressures as the resolution is increased and the angle at which the peak is calculated is approaching the observed transition angle. Clearly the shock geometry is more easily observed than is the measurement of the pressure as a function of time and angle on the cylindrical reflecting surface. Figure 13.16 is taken from [6]. The incident shock Mach number for Fig. 13.16 was 3.36. These experiments were conducted in a partial vacuum with an ambient pressure of 0.839 bars. The solid lines are from von Neumann theory [7]. For this

184

13 Blast Wave Reflections

Fig. 13.15 Peak overpressure as a function of incident angle for Mach 1.2 shock

shock strength at an incident angle of 38
, a regular reflection is observed, while at an incident angle of 38.5 a double Mach reflection forms. The two sets of measured and calculated data correspond to the two peaks in the double Mach reflection waveform. The von Neumann theory predicts only the lower pressure level in the Mach reflection region. As was stated earlier, the second peak is a compression and not a shock and therefore should not be expected to be predicted by the von Neumann shock theory. Figure 13.17 is also taken from [6]. Here the incident shock Mach number is 1.26 in an ambient atmosphere of 0.987 bars. This is considered to be a weak shock and only a simple Mach reflection occurs. The von Neumann theory fails in the Mach reflection region at these low overpressures. Each of the numerical calculation points, labeled SHARC-Code, represents the results of an entire two dimensional computational fluid dynamics calculation and each experimental point represents at least one experiment. The analysis, calculations and experiments were conducted at the Ernst Mach Institute in Freiburg im Breisgau, Germany.

13.3 Planar Reflections

185

Fig. 13.16 Measured, calculated and theoretical points for reflected pressure ratio as a function of angle (Heilig)

Fig. 13.17 Comparison of theoretical, numerical and experimental results for a weak shock reflection from a wedge

186

13 Blast Wave Reflections 10 Pso

α

Pro

LIMIT OF REGULAR REFLECTION

7 MACH REFLECTION REGION

6

Pso

10 70 0

70 50 30 20 10 5

30

3 2

2

50° 47.5° 45° 42.5° 41.25°

1

55°

0.5 60°

0 85° 80°

1

600 400 300

100

50

4

0 100 900

200 150

20 15 0 0

5

100 900 0 30 400 600 0

8

70°

REFLECTION FACTOR, RF = Pro / Pso

9

1.5

2 Z- col α

2.5

3.0

3.5

Fig. 13.18 Reflection factors as a function of the cotangent of the incident angle for several pressure levels

Figure 13.18 shows the behavior of the reflection factor as a function of the cotangent of the incident angle for shocks on a planar wedge. The incident overpressures are in PSI. This is a compilation of data from several sources and the compilation has been attributed to Werner Heilig of the Ernst Mach Institute in Freiburg, Germany. This figure illustrates the general behavior of planar shocks on planar wedges. The reflection factors at normal incidence agree with the Rankine–Hugoniot calculated values. As the angle a increases, the reflection factor drops smoothly until the onset of Mach reflection. Near the transition angle, the reflection factor is greater than at normal incidence for all incident overpressures less than 100 psi. Note that the transition angle is near a minimum in the vicinity of 100 psi. For weak shocks, the reflected and incident shock velocities are nearly equal and the reflected shock requires a greater angle to catch the incident shock. At 2 psi the transition angle approaches 60
. As the overpressures increase, the reflection factor is larger and the reflected shock more readily catches the incident shock.

13.3.1.1

Pressure Above the Reflecting Surface

The entire set of discussions, plots and descriptions of the pressure behavior of reflected shocks, refers only to the pressures at the reflecting surface. Even at small distances above the surface, the pressure time histories are very different. If we look

13.3 Planar Reflections

187

at Fig. 13.2 and imagine a point above the reflecting surface, the overpressure experienced will be the passage of the incident shock, followed by the reflected shock. The relative timing of the two shocks is a function of the height above the reflecting plane and the incident shock strength. For the same incident shock strength, the strength of the reflected shock is a function of the distance above the reflecting plane because the reflected shock is a decaying wave. If we now look at Fig. 13.3 and imagine a point above the surface, there are a number of possibilities. If the point is below the triple point, only a single shock, the Mach shock, passes. The pressure will decay uniformly behind the shock until the slip line passes. The pressure is continuous across the slip line, but the rate of decay after the slip line passes will be different. There will be a discontinuous change in slope of the overpressure. If the point is above the triple point, the passage of two distinct shocks will be measured. The first shock is the incident, followed by the reflected shock. The reflected shock may be stronger, equal to or weaker than the incident shock depending on the incident shock strength and the distance of the point above the triple point. For a double Mach reflection, even more variations are possible, again depending on the height of the point above the wedge. Using the cartoon in Fig. 13.4 we can construct a qualitative representation of the overpressure as the double Mach reflection passes. At the surface, the Mach shock pressure discontinuously rises to a peak, followed by a decay and a rounded rise to a second peak with a smooth decay from there. If we move the point to just above the surface, the Mach shock is the first shock, followed by a smooth decay until the slope of the decay changes when the slip line passes. If the point is sufficiently close to the reflecting surface, it may see some of the pressure increase in the stagnation region near the base of the slip line and then a further smooth decay. If the point is below the triple point, but above the region affected by the stagnation at the base of the slip line, the overpressure time history is as follows. The first peak is the Mach shock with a smooth pressure decay. There is an inflection point on the decay as the slip line crosses the point of interest, followed by a sharp rise as the second Mach stem passes. This shock is followed by a decay with an inflection point as the second slip line passes and continued decay. A series of experiments were conducted and reported by [8], in which three gauges were placed on the reflecting plane and three gauges placed on a wall oriented parallel to the flow and above the surface of the plane. Figure 13.19 shows the gauge positions relative to the reflecting plane. A number of experiments were conducted with different angles for the reflecting plane and measurements made at each of the gauge positions. Calculations were made at the Air Force Weapons Laboratory of one of those experiments and reported in [9]. For the experiment shown here, the incident shock had a Mach number of 4.94 (90 psi) into a 0.2275 bar ambient atmosphere with a wedge angle of 40
. According to Fig. 13.12, this combination of shock strength and wedge angle forms a double Mach reflection. Figure 13.20 shows four of the gauge records from Bertrand’s experiment. Station 2 is flush with the wedge surface about half way up the wedge. The passage of the Mach shock is shown with a discontinuous rise to

13 Blast Wave Reflections

In ci d

en

ts ho

ck

188

gauges

α

2 inches

5

4 1

Reflecting plane

3 inches

2

½ in

6

3 inches

3

gauges

Fig. 13.19 Gauge locations for Bertrands Mach reflection experiments

91 415 188 0

0 STATION 6

STATION 5

478

492

198

201

0

0 STATION 2 SHOT 10 / 18 / 71–88 P21 = 27.3 P1 – 3.30 PSIA a = 50˚

STATION 3 Ps(EQUIV) = 387 PSI SWEEP: 10m SEC / DIV

Fig. 13.20 Waveforms measured by Bertrand for a Mach 4.94 shock on a 40
wedge. (Bertrand used the complement of the wedge angle)

198 psi followed in about 50 ms by the stagnation region with a peak of 478 psi and a smooth decay. Station 3 is on the wedge flush with the surface and 13 cm from the beginning of the wedge. The discontinuous rise marks the arrival of the Mach stem. The peak

13.3 Planar Reflections

189

pressures in the Mach stem should be the same and the measurements are within 1.5% of one another. The second peak arrives 90 ms after the Mach shock and should have the same peak pressure as gauge 2. The longer time between the Mach arrival and the peak pressure than in gauge 2 is caused by the growth of the self similar geometry of the double Mach reflection. The magnitude of the peak pressures are within 3% of one another. Such agreement between independent pressure measurements is excellent. Gauge 5 is 2 in. above gauge 2. Gauge 5 is above the triple point and we see only the incident shock wave with a non-decaying waveform which confirms the stated 90 psi incident pressure within a 1% error. Gauge 8 is 0.5 in. above the reflecting plane and below the triple point. This gauge measured the arrival of the Mach shock at 188 psi, which is 6% less than the Mach stem overpressure at the surface, and the arrival of the second Mach shock with a pressure of 415 psi. The peak pressure 0.5 in. above the surface is lower than at the surface by over 14%.

13.3.1.2

Mean Free Path Effects

In Chap. 2.1 a calculation was made for the collision frequency of nitrogen molecules at standard sea level conditions. For sea level conditions, very high strength and high velocity shock waves cause molecular dissociation of the nitrogen and oxygen in the air. These chemical reactions can be assumed to take place instantaneously for typical atmospheric densities. The reverse reactions, i.e., formation of molecular nitrogen and oxygen (relaxation) also take place rapidly when the energy density decreases. The time for the reactions to occur increases exponentially with decreasing density. At high altitude or in evacuated shock tubes the dissociated molecules do not readily recombine into molecules after passage of a blast wave. When the time that it takes for molecular formation behind a strong blast wave is of the order of or greater than the time for the passage of the shock wave, the phenomena behind the shock front may be significantly modified. As one example, a series of experiments were conducted at the university of Toronto Institute for aerospace studies by Deschambault. A shock tube was evacuated to a pressure of 2 Torr and a Mach 14.55 shock was propagated over a plane wedge with a 10
angle relative to the flow. The incident shock front and the Mach stem were very sharp; however, the details of the reflected shock and the slip line could not be clearly discerned. This particular combination of shock velocity and wedge angle would produce a strong double Mach reflection in sea level air. To help explain the observed diffused slip line and loss of the second Mach stem, two CFD calculations [4] were run using the Second order Hydrodynamic Advanced Research Code (SHARC). In the first calculation, an equilibrium air equation of state was used and in the second the air was treated with non-equilibrium chemistry. The air in the second calculation was represented as eleven constituent species using 58 reaction rate equations for the equation of state. The calculations captured the sharp rise at the shock front for the incident and Mach shocks.

190

13 Blast Wave Reflections Energy 6.0 601 630 660 690 720 750 780

810 840

870

Contour Scale ergs / gm

5.4 4.8

240

Altitude cm

230 220 210 4.2 200 190 180 3.6 170 160 150 3.0 140 130 120 2.4 110 100 90 1.8 80 70 1.2 60 50 40 0.6 30 20 10 1 0.0 12.0 12.6 13.2 13.8 14.4 15.0 15.6 16.2 16.8 17.4 18.0 range cm Planar Shock Reflecting from a Wedge Time 32.000 usec Cycle 1931 Problem 1.230

3 5 7 9 11 13 15 17 19

8.400E+10 9.000E+10 9.600E+10 1.020E+11 1.080E+11 1.140E+11 1.200E+11 1.260E+11 1.320E+11

dx1 = 2.000E–02 min = 2.044E+09 x = 1.701E+01 y = 4.930E+00 max = 1.334E+11 x = 1.675E+01 y = 3.850E+00

Fig. 13.21 Mach 14.5 shock on a 10
wedge 2 Torr ambient pressure with equilibrium chemistry

The slip line in the equilibrium calculation (Fig. 13.21) was distinct but unstable due to the high shear velocity across the slip line. The structure of the second Mach stem was more like a weak double Mach reflection and the second slip line was not resolved. In the non-equilibrium calculation, the reflected shock in the linear region (see Fig. 13.8) behind the triple point, was indistinct at best. Figure 13.22 shows the energy distribution in the vicinity of the reflection region. The slip line was about 1.5 cm thick with a continuous density gradient. There were no distinct regions below the triple point that could be labeled. The oxygen and nitrogen are dissociated very rapidly at the shock front. The atomic nitrogen and oxygen do not react on a time scale commensurate with the passage of the shock system. The distinct regions pointed out in Chap. 13.2 do not exist because the state of the gas is a continuum behind the Mach stem. The gasses have been dissociated and relax over a period of 10–20 ms or so, depending on the specific reaction path being followed (Fig. 13.22). The experimental laser interferogram is difficult to interpret because the density of the gas is so small that little interference can be generated. Figure 13.23 was generously supplied by Dr. James Gottlieb from the Institute for Aerospace Studies at the University of Toronto. This is a copy of the original photo used in [10].

13.3 Planar Reflections

191

Fig. 13.22 Mach 14.5 shock on a 10
wedge 2 Torr ambient pressure with non-equilibrium chemistry

Fig. 13.23 Laser interferogram of Mach 14.55 shock on 10
wedge in 2 Torr ambient pressure air

192

13 Blast Wave Reflections

The incident and Mach shocks are clearly defined. The reflected shock and the kink are clearly distinguishable; however the subtle gradients in the vicinity of the slip line and second Mach shock demonstrate that the sharp gradients found in higher density gasses do not exist. The continuous density gradients generated by the continually reacting species prevent the formation of sharp shocks or slip lines.

13.3.2

Rough Wedge Reflections

There are several reasons to examine the effects of roughness on shock reflection. The effects of a boundary layer could be examined by starting with a very rough surface and decreasing the roughness in steps to very fine roughness and extrapolating to a smooth surface. Regular rough surfaces have been used to visualize the path of waves in the flow following shock passage. This was done to help understand the way that signals catch the incident shock during Mach reflection. Rough wedges have also been used to examine the effects of roughness on Mach stem growth and the geometry of waves behind the Mach stem. Ben-Dor et al. [11], used triangular elements to represent roughness on the reflecting plane. Each element had a base twice as wide as its height. Roughness heights ranged from less than 0.1 to 2 mm. Reichenbach [12], used rectangular crosssection and cubical roughness elements on his wedges. The rectangular elements varied from 0.5 to 5 mm in height. In both cases, smooth wedges were also used for comparison. The experiments provided information helpful to quantifying most of the anticipated effects. One conclusion from these experiments that CFD calculators were very glad to see is that as the roughness elements became small, the shock reflection and geometry uniformly approached the smooth wedge results. I include here just three examples from Reichenbach’s work as examples of the results obtained. The reader is referred to the [11, 12] references for more details of their results. Figure 13.24, is an interferogram of a Mach 1.803 shock passing over a rough surface with 5 mm deep rectangular elements. The incident shock is smoothly curved beginning at the intersection of the wave emanating from the initial interaction with the wedge with the incident shock. This figure illustrates the variation of gas density behind the incident shock and the path of the signals from each of the roughness elements. Note that a small vortex forms and sheds from the top of each roughness element as the shock progresses (Fig. 13.24). The Reichenbach experiments continued with variations of (1) the incident Mach number of the shock, (2) the wedge angle and (3) the roughness element height. Figure 13.25 shows the wave reflections for a Mach 1.352 shock reflecting from the same wedge as in Fig. 13.24, but rotated by 16
. Referring back to Fig. 13.12, we see that this combination of Mach number and wedge angle results in a single or simple Mach reflection. There are two sets of signals shown in this shadowgram. The first set comes from the reflection of the incident shock from the

13.3 Planar Reflections

193

Fig. 13.24 Mach 1.803 over a flat rough surface (courtesy of H. Reichenbach)

Fig. 13.25 Mach 1.352 shock on a 16
rough wedge

top of the roughness elements and the second from the bottom of the surface between the elements. The individual signals clearly show the convergence of the signals as they near the triple point and the divergence of the same signals as they propagate away from the wedge and the incident shock (Fig. 13.25). Figure 13.26 is an interferogram for a much stronger shock reflecting from the same wedge oriented 37
from the horizontal. This shock has a Mach number of 3.065. Referring to Fig. 13.12 we see that this combination of shock strength and wedge angle yields a double Mach reflection. The signals reflected from the tops of the roughness elements have formed the double Mach reflection and the first slip line is clearly seen. The reflections from the bottom of the slots fall several element

194

13 Blast Wave Reflections

Fig. 13.26 Mach 3.065 shock on a 37
rough wedge

spacings behind and below those from the tops. The double Mach reflection from the bottom of the slots has not yet emerged above the slots. These reflection phenomena are important for the reflection of blast waves over rough surfaces. This roughness might represent clumps of grass or large gravel on a surface from a few pounds of explosive. For a roughness corresponding to the above wedge geometry and a nuclear detonation, the roughness elements might be multi story buildings with streets and alleys running between them. Several of these experiments were simulated using numerical computational fluid dynamic (CFD) codes [13]. With modern computing methods and real gas equations of state, the definition of the behavior and evolution of these complex flows is now competitive with the best experimental and photographic techniques. Studies of these flow fields are best approached using a combination of experimental and computational techniques.

13.4

Reflections from Curved Surfaces

Shocks reflecting from convex or concave surfaces may be thought of as reflecting from wedges with continuously changing angles. A number of investigators, including Ben-Dor, Reichenbach, Heilig, Heilig and Takayama, have investigated the behavior of shocks reflecting from smooth curved surfaces. While Ben-Dor and Takayama have examined the shock geometry of reflections from curved surfaces in great detail, Reichenbach and Heilig have attempted to measure overpressure– time histories as the shocks transit the surface of the cylinders.

13.4 Reflections from Curved Surfaces

195

For a planar shock striking a cylindrical surface, the initial wedge angle is 90
(shock angle relative to the wedge is 0
). The kinetic energy of the material at the shock front is fully stagnated and the full reflected pressure is attained. A regular reflection occurs as the shock progresses over the surface of the cylinder. For shock Mach numbers above two, regular reflection persists to a wedge angle of 50
(shock angle of 40
) where a Mach reflection is first generated. The Mach stem grows above the surface of the cylinder as the reflection angle continues to decrease. The peak pressure as a function of the reflection angle in the regular reflection region and the initial Mach reflection region is well represented by the curves given in Fig. 13.6. Note that for most pressure levels, the reflected pressure drops as the shock engulfs the cylinder until a wedge angle of about 50
(shock angle of 40
), where the reflected pressure then increases over a sector of 10
or so. For wedge angles less than 40
, the reflected pressure rapidly decays. In the case of reflection from a cylinder, the Mach stem forms and grows as the wedge angle decreases and it is the pressure in the Mach stem that persists at small wedge angles. Therefore, the peak reflected pressure is dominated by the Mach reflection and the planar wedge reflection factors are no longer applicable. Georg Heilig also performed a number of calculations with the commercial CFD code Autodyn and with the US government code SHARC. He then compared the results of his calculations with the results of his experiments [14]. The experiment used in the following example is a Mach 1.308 shock, incident on a non-responding 7.5 cm radius cylinder. For a Mach 1.3 shock, Mach reflection is expected to begin at a wedge angle of 47
. Figure 13.27 is an experimental Schlieren photo taken when the shock is at the top of the cylinder, 90
from the initial reflection point and at a wedge angle of 0
. The pressure in the Mach stem is a factor of two greater than in the incident wave. The pressure decays between the triple point and the cylinder’s surface, but is more than 50% greater than the incident pressure at the surface. Figure 13.28 shows the density distribution in the results of Heilig’s SHARC calculation at the same time (220 ms after striking the cylinder), as the shadowgraph in Fig. 13.27. The curved Mach stem and the slip line are clearly shown in both

Fig. 13.27 Schlieren picture of a Mach 1.308 shock reflecting from a 7.5 cm cylinder

196

13 Blast Wave Reflections

Fig. 13.28 SHARC calculated reflection of a Mach 1.308 shock from a 7.5 cm radius cylinder

figures. The noticeable wiggle in the calculation that starts near x ¼ 11.7 cm and follows the contour of the cylinder is a propagated discontinuity that was inadvertently introduced as part of the initial conditions in the calculation. The discontinuity corresponds to the location in the fluid of the initial position of the shock front and has propagated at the material velocity for 220 ms. The density distribution clearly shows the variation of the shock strength within the Mach stem. The density near the triple point is 90% greater than ambient and falls to about 65% above ambient near the cylindrical surface. Included in the experiment were pressure gauges mounted flush with the surface of the cylinder at angles of 10 and 40
relative to the lower shock tube wall. Figure 13.29 is a comparison of the experimental overpressure time histories with the results of the Autodyn and SHARC calculated results for the two gauge positions. The much sharper representation of the SHARC shock fronts relative to those of Autodyn was noted by Heilig. The agreement shown is quite remarkable and demonstrates the power of modern computing techniques. The SHARC results and experimental data are virtual overlays. The reflection remains regular at both gauge locations. Mach transition is expected at a wedge angle of 47
which corresponds to a rotation angle of 43
. For a Mach 1.308 shock in air at sea level, the incident overpressure is 0.84 bars. The reflection factor is 2.63. The peak reflected pressure (including ambient) is 3.21 bars (321 kPa). At a wedge angle of 80
, the reflection factor falls to about 2.55. The measured (and calculated) peak pressures at the 10
position are in good agreement with this value. As expected, the peak reflected pressure grows smaller

13.4 Reflections from Curved Surfaces

197

Fig. 13.29 Overpressure waveform comparisons of a Mach 1.308 shock reflection from a cylinder

198

13 Blast Wave Reflections

Fig. 13.30 Mach 1.8 shock in air reflecting from a concave cylindrical surface

as the wedge angle decreases and at the 40
position (50
wedge angle), the peak reflected pressure has fallen to about 2.8 bars. Similar experiments were conducted by Takayama and his students over concave cylindrical surfaces. Figure 13.30 is a laser interferogram showing the density distribution behind the reflection of a Mach 1.8 shock. The initial reflection forms a Mach stem. As the shock progresses, the slope increases. At some point the angle of the “wedge” reaches the critical angle for regular reflection. At that point the incident and reflected shock should intersect at the wedge surface, however, the flow field already established by the Mach reflection interferes with the regular reflection shock geometry. Transition to regular reflection is delayed as the phase velocity of the incident shock along the surface must overcome the separation produced by the growth of the Mach stem. Note that the slip line is not increasing in height with distance along the reflecting surface. It is difficult to see here, but the height of the slip line above the surface is beginning to decrease as it approaches the Mach stem. The Mach stem height is decreasing at this angle; the incident shock is approaching regular reflection and the Mach stem is decreasing in size.

References 1. Brode, H.L.: Height of burst effects at high overpressures, DASA 2506. Defense Atomic Support Agency, Washington, DC (1970) 2. Ben-Dor, G.: Shock wave reflection phenomena. Springer, New York (2007) 3. Takayama, K.: Private communication figures 13.4 and 13.7, July 1991 4. Needham, C.E.: Chemical non-equilibrium effects of Mach reflection. International workshop on strong shock waves, Chiba, Japan (1991)

References

199

5. Dewey, J., van Netten, A.A.: Particle velocity at transition and its relationship to height of burst curves, University of Victoria, B.C., Canada. Published in Proceedings of the 14th International Mach Reflection Symposium by the Shock Wave Research Center, Institute of Fluid Science, Tohoku University, Sendai, Japan, October 2000 6. Heilig, W.: The pseudo-steady shock reflection process as predicted by the von Neumann Theory and by the SHARC-Code, Ernst Mach Institute, Freiburg, Germany. August, 1990. Published in FESTSCHRIFT zum 65. Geburtstag von Dr. re. nat. Heinz Riechenbach by the Fraunhofer- Institute f€ ur Kurzzeitdynamik, >Ernst Mach Institut
Chapter 14

Height of Burst Effects

In the previous chapter we have discussed plane waves reflecting from plane and curved surfaces. In nearly all cases the incident shocks were constant, square topped waves rather than decaying waves which are characteristic of blast waves. All of the reflections and shock geometries mentioned can be observed behind blast waves as well. It is easier to illustrate the behavior of such reflections in planar two dimensional geometry than in the more complex three dimensional geometry which is more common for blast waves. We will use blast waves in shock tubes to illustrate some of the more complex interactions in the following discussions.

14.1

Ideal Surfaces

When a detonation occurs above a smooth planar reflecting surface, the angle of incidence of the blast wave, a spherical front, is continuously changing with distance from ground zero. Ground zero is defined here to be the point at the reflecting surface directly below the burst point. That is the point of intersection on the reflecting plane with a vertical line through the burst point to the plane. This definition will continue to be used in later sections where we will discuss height of burst effects over rolling or mountainous terrain. Note that this is not the point of closest approach nor is it necessarily perpendicular to the plane. The decay of the blast wave immediately behind the shock front influences the propagation of the reflected shock on a macroscopic scale. It has been argued that shock front reflection of a planar or spherical shock should be indistinguishable. On a microscopic scale this is certainly true. However, the decay of the blast wave parameters behind the leading front means that the propagation of the reflected wave is altered because the sound speed in the gas into which the reflected shock is moving is decreasing with distance behind the incident shock. Thus, everywhere except at the point of intersection of the incident shock front and the reflecting plane, the reflection of a blast wave is moving into a lower sound speed medium. In addition, the reflected wave is also decaying as it moves away from the surface and C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_14, # Springer-Verlag Berlin Heidelberg 2010

201

202

14 Height of Burst Effects

its propagation velocity would be decreasing even if it were moving into a nondecaying medium. Perhaps these differences help to explain the slightly different reflection factors that are measured for spherical blast waves over planar surfaces when compared to plane shocks on plane wedges. Figure 14.1 gives the reflection factors for spherical blast waves reflecting from a flat plane. These curves are very similar to those given in Fig. 13.18 and have subtle differences in the Mach reflection region. They have been assembled by Brode [1]. The assumptions here include a spherical blast wave reflecting from an ideal, non-responding, perfectly reflecting plane of infinite extent. For the ground burst (zero height of burst), the energy is contained in a hemisphere above the reflecting plane. This has the same effect on shock propagation as doubling the yield from a free air burst. The range to a given overpressure for a ground burst is thus increased by the cube root of 2 or 1.2599 with no reflection factor. The reflection factors can be used in conjunction with an equation for the peak blast wave over pressure as a function of radius in free air to generate a curve of the peak over pressure on the ground as a function of the distance from ground zero from any height of burst. This is done as follows: for a point on the ground, calculate the radius from the burst point. Calculate the angle of incidence or perhaps more readily (except at ground zero), the height of burst divided by the ground range. Note that this is the coordinate used in Fig. 14.1. Find the incident overpressure from the equation for overpressure as a function of radius. Using this overpressure and the reflection angle, evaluate the reflection factor using Fig. 14.1.

20,000

z = y/x

LIMIT OF REGULAR REFLECTION (THEORETICAL - RIGOROUS)

y

11

5000 3000

x

2000

9

1200

MAC HR (INA EFLECT DEQ IO UAT N REG ED ATA ION )

REFLECTION FACTOR R = ΔPr / ΔPi

13

7

5

600 300 200 100 70 30

3 2

1

0

1

2 z = y/ x

3

3.7

Fig. 14.1 Reflection factor as a function of the cotangent of the incident angle for several pressure levels

14.1 Ideal Surfaces

203

Multiply the incident overpressure by the reflection factor and this gives the ground level overpressure for the given ground range. This can be a tedious task and a different curve is found for each height of burst. To aid the user, a type of plot called a height of burst curve has been developed. A height of burst curve is plotted with the ground range on the horizontal axis and the height of burst on the vertical axis. They are usually scaled to a convenient yield such as 1 pound, 1 kg or 1 kt. Distances are then expressed in scaled units such as feet or meters per cube root of the yield. The curves provide the peak blast overpressure at ground level as a function of scaled ground range and height of burst. They are constructed by using a measured or calculated overpressure curve as a function of ground range for a single height of burst detonation. The ground range at which a particular overpressure occurs is then plotted at the given ground range for the given height of burst. The following examples are for nuclear detonations over ideal or “near-ideal” plane reflecting surfaces.

14.1.1

Nuclear Detonations

Figure 14.2 shows the height of burst curves for a few selected overpressures between 10,000 and 100 psi for a 1 kt detonation at heights of burst between ground level and 500 ft. The dashed line indicates the range at which a Mach stem forms for the given height of burst. This is the currently accepted set of height of burst curves for nuclear detonations. They are consistent with the currently accepted overpressure as a function of radius and the currently accepted reflection factors. As an example of the use of these curves, using Fig. 14.2, at a height of burst of 300 ft, the Mach stem forms at a ground range of 250 ft for a 1 kt nuclear detonation. At ground zero, the peak reflected overpressure from a burst at 300 scaled feet will be just under 500 psi. At a scaled ground range just prior to Mach reflection, the overpressure will have dropped below 200 psi for this 300 scaled foot height of burst. As another example, we follow the range at which 100 psi occurs as a function of the height of burst and see that the range actually increases with increasing height of burst to a maximum value near a height of 250 scaled feet. This is the direct result of applying the reflection factors as a function of incident shock overpressure. The ground range at which 100 psi occurs for a surface burst is 340 ft. For a height of burst of 300 ft, 100 psi occurs at a ground range in excess of 350 ft. The radius from the 300 ft high burst point is more than 480 ft. Energy is not being created to generate these results, but the increase in overpressure is caused by a partial stagnation of the dynamic pressure which converts kinetic energy into internal energy and therefore pressure. The volume into which this energy is confined near Mach transition is restricted to the small region between the triple point, the slip line and the reflecting surface. As the Mach stem grows, the volume increases and the stagnated energy is distributed over a larger volume. Remember that the

204

14 Height of Burst Effects 500 100 psi

400 REGULAR REFLECTION REGION

HEIGHT OF BURST (FEET)

200

300

MACH REFLECTION REGION

500 1000 200 2000

10,000

5000

100

0

0

100

200

300

400

DISTANCE FROM GROUND ZERO (FEET)

Fig. 14.2 Height of burst curves for 1 kt; high overpressures

highest rate of energy redistribution is restricted to the shock velocity of the reflected shock. At high overpressures the enhancement in ground range is rather modest. If we look at the height of burst (HOB) curves for intermediate overpressures in Fig. 14.3, we see somewhat more ground range enhancement with height of burst. This figure shows the height of burst curves for pressures between 200 and 10 psi. These are the currently accepted HOB curves. The dashed line marks the transition from regular to Mach reflection. Points to the right of the dashed curve are in the Mach region and to the left are in regular reflection. The ground range enhancement is quite impressive. The range to a 10 psi blast wave from a ground level burst is about 1,020 ft. If we elevate the burst to a height of 700 ft, the ground range increases to over 1,450 ft, an enhancement of over 40% in range or nearly a factor of two in area coverage for pressures above 10 psi. An interesting characteristic of the 10, 15 and 20 psi curves is that they are triple valued in range. For example at an HOB of 875 scaled feet, a pressure of 10 psi is found at a range of 1,000 ft. The pressure then falls below 10 psi to a range of 1,150 ft where the pressure rises above 10 psi and again falls below 10 psi at a range of 1,350 ft. It is not clear to me that this behavior has been observed on any nuclear

14.1 Ideal Surfaces

205

1,000 REGULAR REFLECTION REGION

HEIGHT OF BURST (FEET)

800

10 psi

20

600

MACH REFLECTION REGION

15

30 50

400 100 200

200

0

0

200

400

600

800

1,000

1,200

1,400

DISTANCE FROM GROUND ZERO (FEET)

Fig. 14.3 Height of burst curves for 1 kt; Intermediate Overpressures

test over an ideal or near ideal surface. Most nuclear tests, on which pressure measurements were made, were detonated over desert surfaces and were affected by thermal heating and dust scouring from the surface. Both of these phenomena tend to redistribute the blast wave energy in a manner that is consistent with the HOB curve interpretation. More will be discussed about this in Sect. 14.5.2 below. The curves in Fig. 14.4 are for the low overpressure region. The triple valued characteristic is continued to pressure levels as low as 4 psi. The position of the enhancements corresponds to the formation of Mach reflections. If we look at the 1 psi curve, we see that the surface burst pressure extends to 3,800 scaled feet. When the detonation is raised to a height of 1,500 scaled feet, the range for 1 psi increases to 7,000 scaled feet. This is an increase of over 80% in range and more than a factor of 3 in area coverage for pressures above 1 psi.

14.1.2

Solid High Explosive Detonations

When solid high explosives are used to generate a blast wave, the mass of the high density detonation products must be included in the reflection factors and the resulting height of burst curves. The primary difference in HOB curves for high explosives comes from the fact that high explosives are about twice as efficient at generating air blast as nuclear detonations. When a nuclear device is detonated in the atmosphere at an altitude of less than
20 km, approximately half of the energy is radiated away to large distances in the form of thermal and visible light. As was described in Sect. 12.1.1, the precise fraction of the energy radiated away varies between 40 and 60%. This energy is lost to the blast wave and does not contribute to

206

14 Height of Burst Effects

Fig. 14.4 Nuclear height of burst curves for 1 kt; low overpressures

the formation or propagation of the blast wave because very little visible light is absorbed in the ambient atmosphere. When a solid explosive is detonated, the temperatures are much lower than for a nuclear detonation and a very small percentage of the energy is radiated away. The HE energy radiated away can be ignored when considering blast wave generation. Thus solid high explosives are very nearly twice as efficient at producing air blast as are atmospheric nuclear detonations. When scaling solid high explosives to compare with nuclear detonations, this factor of
2 must be taken into account. Another major difference when compared to the nuclear curves occurs near zero height of burst. Because the high explosive has a finite size, the charges cannot be considered as point sources as in the nuclear case. The zero height of burst charge of HE is a hemisphere sitting flush on the ground with the detonation point on the reflecting surface at the center of the charge. For an ideal reflecting surface, the resulting distance to a given pressure is the same as for a sphere of twice the yield detonated in free air (no ground reflection). When the HE charge is raised one charge radius to a tangent above ground configuration, one half of the charge mass is between the detonation point and the reflecting surface. When the detonation shock reflects from the surface, the reflected wave cannot overcome the mass and momentum of the downward moving mass of the lower half of the sphere. Essentially half the energy released by the detonation is trapped in a layer with a height of the charge radius. Figure 14.5 shows the pressure distribution for a tangent sphere C-4 detonation when the upward moving shock has expanded by 2.5 charge diameters. The shock at ground level has expanded by more than 4 diameters. The upward moving reflected shock is trapped in the downward

14.1 Ideal Surfaces

207

Fig. 14.5 Pressure distribution for a 1 pound tangent sphere of C-4

moving detonation products at a height of less than one charge radius and the only direction that the pressure can be relieved is outward along the ground. The detonation products form a high speed, high density jet near the surface that initially produces a toe on the outward moving shock. This jet of high density detonation products over runs the Mach reflection and dominates the near surface flow field. The jet of detonation products eventually is forced up behind the ground level shock and forms a counterclockwise rotating vortex. Figure 14.6 shows the distribution of the detonation products when they have expanded to nearly seven charge diameters along the ground. The tip of the jet has moved up and back toward ground zero behind the blast wave. The spherically expanding region above the reflected shock shows the effects of Rayleigh–Taylor instabilities. The modified shock, caused by the high speed jet of detonation products, dominates the near surface blast wave to overpressures of less than two bars at distances greater than 250 charge diameters. The shock never forms a hemisphere and is influenced by the initial energy distribution for all time. When the overpressure has reached a level of 0.25 bars, the radial extent of the shock is 10% greater

208

14 Height of Burst Effects

Fig. 14.6 Detonation products density distribution from a 1 lb. C-4 tangent sphere detonation

than the vertical extent, even though the vertical shock is traveling through a decreasing density atmosphere. Figure 14.7 is a picture of a 500 ton TNT tangent sphere fireball showing the dominance of the near surface jet and the rollup of the detonation products behind the shock. The boxes in the foreground are three story buildings and the object at the right side of the fireball is a 50 ton armored tank. The position of the shock front can be seen along the horizon because the density change diffracts the light. Smoke mortar traces can be seen to the right above the ground level jet. Because the jet is expanding rapidly, it is more unstable than the upper part of the fireball. The near surface level jet is divided into individual jets in the azimuthal direction. These instabilities cannot be resolved by two dimensional calculations because they assume axial symmetry. The instabilities shown in Fig. 14.6 on the upper part of the fireball can be distinguished on the surface of the fireball in Fig. 14.7. This comparison is also an indication of the scalability of all of the blast phenomena from a 1 pound to a 500 ton charge. When the HE is raised so the charge center and detonation point are 1 diameter above the surface, the ground zero pressure is reduced by more than a factor of three. The formation of the detonation products jet is delayed and a strong Mach

14.1 Ideal Surfaces

209

Diffraction by Blast Wave

Three story buildings

50 ton tank

Fig. 14.7 500 ton TNT tangent sphere detonation

reflection occurs. At a ground range of about two charge diameters, the pressure in the Mach stem is greater than that for the tangent sphere. The detonation products jet then overtakes the Mach shock and replaces the flow field with relatively cooler, expanded detonation products. The pressure from the tangent sphere then exceeds that from the 1 diameter height of burst for all larger ground ranges. As the height of burst increases, the importance of the detonation products on the blast wave is reduced and the HOB curves become independent of the source of the blast wave and depend only on the energy in the blast wave. A set of experimentally based height of burst curves for solid high explosives were constructed by Kuhl and reported in 1990 [2]. Kuhl used 0.5 g charges of PETN in a laboratory chamber. The surface over which the reflected pressures were measured was “smoother than the paint on a new Porsche”. Multiple experiments were conducted for each height of burst. Figure 14.8 shows the resulting HOB curves. Each pressure contour is plotted with error bars. The data were cube root scaled from 0.5 g of PETN to 1 kg of TNT on the basis of the detonation energy ratio. The heavy dots on the contours are the mean interpolated ground range for that pressure level from the given height of burst. The lighter dots on either side of the mean represent the 2 sigma error bars on the interpolated values. The dotted lines that deviate from the solid curves represent the nuclear curves shown in Figs. 14.3 and 14.4. Agreement in the regular reflection region between the nuclear and HE curves is excellent. The effects of the charge mass can be seen at all pressure levels for heights of burst less than 50 scaled cm. The “wiggles” in the curves near the surface are caused by the changing influences of the Mach reflection and the detonation products jet. The change in pressure at a given ground range is not monotonic with height of burst. The laboratory experiments show no triple valued ranges for any pressure level, such as were shown in the nuclear curves. The enhancements in range are somewhat reduced from those of the nuclear curves. For example the nuclear 10 psi curve showed an enhancement in ground range of just over 40% whereas the HE curve shows an enhancement of approximately 25%. Just raising the HOB for the HE charge from a hemisphere to one charge radius has the effect of increasing the ground range for high pressures by about 20%. As the ground range increases, the range enhancement decreases but remains about

210

14 Height of Burst Effects

Fig. 14.8 Experimentally based HOB curves for high explosives from Kuhl and ScheklinskiGluck at the Ernst Mach Institute

10% greater than for the hemispherical detonation. Once the energy is directed along the ground, it is very difficult to change the direction of flow because the pressure gradients in the vertical direction are small. Figure 14.9 compares the suggested height of burst curves from Kuhl and Scheklinski-Gluck with experimental data collected independently by Carpenter and by Sauer. To give the reader an idea of the difficulty of obtaining high quality data, I provide a brief description of Carpenters experiments. Carpenter used carefully machined spherical 8 pound charges of high quality explosives. The detonation wells were machined to put the detonator at precisely the center of the charge. He suspended the charges with a nylon panty hose belt around the equator of the charge, thus exposing the bare charge below the equator. This ensured that there would be no material other than air between the source and the gauges on the ground. He used specially wound cables of Invar wire to suspend the charges at the precise height of burst. Using Invar cables was necessary because the normal steel cables changed length in the southern California sun and the charge

14.1 Ideal Surfaces

211

Fig. 14.9 Comparisons of carpenter and sauer solid high explosive height of burst curves

moved as the cool winds blew across the test pad. The test pad was circular high strength, steel reinforced, concrete 30 ft in radius. The depth varied from several feet at ground zero to 14 in. near the outer edge. The pad surface was 6 in. above the surrounding ground surface. The air blast gauges were mounted in steel plates that were recessed into the concrete and welded to the rebar. The entire surface was then ground flat, smooth and level. The resulting pad resembled a terrazzo floor and was at least as smooth. The gauges were mounted with shock isolating material to ensure that signals propagating through the concrete and steel would not interfere with the air blast signal. Pressure gauge signals were dual recorded on magnetic tape and on oscilloscopes with cameras. There was an oscilloscope and a camera for each gauge, and the oscilloscopes provided sub-microsecond frequency response. Each gauge was calibrated before and after each shot. Multiple experiments were conducted for each height of burst. Sauer took similar steps for his experiments to ensure high precision and quality. Carpenter and Sauer used different explosives and different sized charges. The data has been scaled using the detonation energy to kilograms of TNT.

212

14 Height of Burst Effects

Fig. 14.10 Arrival time Height of burst curve for 1 kt Nuclear

The open and closed symbols are the results of Carpenter’s and Sauer’s experiments respectively. There are differences between the results from the three experimenters which have not been resolved. In all cases, the enhancement in range to a given pressure due to HOB is much less than in the curves derived from nuclear data. None of the HE data suggests that the pressure as a function of range for any HOB increases with range. Height of burst curves can also be constructed for any blast parameter. Some examples include: arrival time, horizontal component of dynamic pressure, positive phase duration, and both overpressure and dynamic pressure impulse. Included here are examples of such HOB curves for 1 kt scaled nuclear detonations. Figure 14.10 is a set of HOB curves for arrival times. In the regular reflection region, the arrival times are those of a free air detonation arriving at a radius corresponding to the distance from the burst point to the point on the ground. In the Mach reflection region (below the dashed line), the arrival times account for the change in shock geometry. Figure 14.11 shows the HOB curves for the horizontal component of dynamic pressure. This is the component which is parallel to the ground at the surface for a perfect reflecting surface. At zero range the flow is purely vertical and there is no horizontal dynamic pressure. The horizontal dynamic pressure increases rapidly with a small change in ground range. Near the transition from regular to Mach reflection, the dynamic pressure reaches a peak. At distances where the Mach reflection is well established and the triple point is well above the surface,

14.1 Ideal Surfaces

213

Fig. 14.11 HOB curves for the horizontal component of dynamic pressure

the dynamic pressure can be obtained from the overpressure HOB curves and the Rankine–Hugoniot relations. At these ranges the Mach stem is a single blast wave and smooth horizontal flow has been established behind the shock front. For the range between the transition to Mach reflection and a well established Mach reflection, the flow is complex with vortices and possible secondary shocks. In this region, the dynamic pressure may exceed the R-H values associated with the shock front and is plotted with dashed lines. Most of the experimental data on which these dynamic pressure curves are based were measured at heights of 3 ft or more above the surface in order to avoid the boundary layer. A test series was designed by scientists from the Army Ballistics Research Laboratory, the Defense Nuclear Agency and their contractors and was conducted at the Defence Research Establishment in Suffield, Alberta Canada. The series was dubbed “Mighty Mach” and was intended to measure the blast wave parameters in the Mach reflection region during the transition from regular reflection. The following discussion refers to the results of the Mighty Mach test series. Figure 14.12 is a laser shadowgram showing the Mach reflection region of the blast wave from the detonation of 216 pounds of pentolite at a height of burst of 12 ft. [3] The þ sign near the triple point is 2 ft above the surface at a ground range of 16 ft. This photo clearly shows the incident shock, the first Mach stem the slip line, the kink in the reflected shock and the second Mach stem for this double Mach reflection. This photo was made by synchronizing a pulsed laser and a high speed framing camera. The laser and camera were located next to each other and pointed toward a reflecting backdrop on the opposite side of the detonation. Ground zero is about 8 ft to the right of this picture. The picture is taken through the expanding

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14 Height of Burst Effects

Photo courtesy of John Wisotski Denver Research Institute

MIGHTY MACH DNA-BRL-DRES-DRI

Fig. 14.12 Laser shadowgram of a double Mach reflection from a height of burst detonation

shock. The vortex that forms at the surface is swept forward by the stagnated pressure near the termination of the slip line at the surface. The vortex has a minor radius of approximately 6 in. and a major radius of about 14 ft at this time. The vortex appears black because the line of sight path through the turbulent vortex is very large and the light is refracted away from the camera. The light and dark, nearly horizontal, lines starting near the second triple point are caused by the refraction of the first and second triple points rotated around ground zero. The dark line that travels nearly vertically from the second triple point is the reflected shock behind the kink at the second triple point. These complex reflection phenomena result in complex blast wave time histories. Figure 14.13 is a waveform from the same Mighty Mach test series as Fig. 14.12. This waveform was measured at a ground range of 19 ft from a detonation at a height of burst of 13.7 ft. The gauge was positioned ½ in. above the reflecting surface. The concrete pad was swept and wet mopped just before the shot in an attempt to reduce the effects of particulates and boundary layers on the blast wave and to provide a clear line of sight for the cameras and lasers. The measured shock front peak overpressure is 3.4 MPa (493PSI). The overpressure then decays to a minimum of about 2.4 MPa at a time of 200 ms after shock arrival. The second peak is rounded, indicating a compressive wave, and has a peak value of 4.2 MPa (609 psi) at a time of 350 ms and then decays smoothly for more than a millisecond. The peak pressure reported for this ground range is the 4.2 MPa (609 psi) value even though this is not the shock front overpressure. If we use the Rankine–Hugoniot relations to obtain the dynamic pressure that corresponds to the peak overpressure, we find a value of 9.0 MPa. If we use the shock front overpressure, the corresponding dynamic pressure is 7.0 MPa. Figure 14.14 is the measured dynamic pressure waveform at the same position. The shock front dynamic pressure corresponds to the shock front overpressure, with the exception of an overshoot just at shock arrival. The maximum dynamic pressure of 13.5 MPa is the second peak and exceeds the R-H value when the peak overpressure is used (9 MPa). There are several reasons for this disagreement.

14.1 Ideal Surfaces

215

12 MIGHTY MACH III–3 N19.05

11

X = 5.79E + 000 M Y = 1.27E – 002 M MM TIME SPAN = ~1 ms

OVERPRESSURE (MPa)

10 9 8 7 6 5 4 3 2 1 0 0

0.1

0.2

0.3

0.4

0.5 0.6 TIME (ms)

0.7

0.8

0.9

Fig. 14.13 Overpressure waveform in the double Mach reflection region 24 22

DYNAMIC PRESSURE (MPa)

20 18 16

MIGHTY MACH III–3 N19.05 0

14

X = 5.79E + 000 M Y = 1.27E – 002 M MM TIME SPAN = ~1 ms

12 10 8 6 4 2 0 0

0.1

0.2

0.3

0.4

0.5 0.6 TIME (ms)

0.7

0.8

0.9

Fig. 14.14 Dynamic pressure waveform in the double Mach reflection region

Neither the peak overpressure nor the peak dynamic pressure is at a shock front and therefore application of the R-H relations is questionable. The primary reason for the discrepancy is that the peak overpressure and peak dynamic pressure do not occur at the same time. The peak overpressure occurs at a time of 350 ms after shock

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14 Height of Burst Effects

arrival, while the peak dynamic pressure occurs more than 100 ms earlier, at
220 ms. The conservation equations of the R-H relations can not be expected to hold if the location is not the same. The waveforms are out of phase because the peak overpressure, caused by the partial stagnation of the flow at the base of the slip line behind the vortex accelerates the near surface gas. The high pressure gradient is driving the acceleration of the fluid forward at the bottom of the vortex. The resulting higher velocity near the surface causes the higher dynamic pressure and is ahead of the stagnation region, thus arriving earlier. (You can refer to Fig. 13.9 for the flow geometry) As the shock expands, the pressure decreases, the separation between the first and second peaks increases and the pressure in the second peak decreases faster than at the shock front. The geometry of the double Mach reflection changes to a complex Mach reflection and eventually into a single Mach reflection.

14.2

Range for Mach Transition

Understanding of the reflection of blast waves must include the transition from regular to Mach reflection. An approximation to the ground range at which this transition occurs can be made by using the following two simple relations. For nuclear detonations the transition over ideal surfaces for HOB less than 99.25 m/kt1/3, the ground range is given by: r0 = 0.825 * HOB. For HOB greater than 99.25 m/kt1/3, the slightly more complex equation below can be used. r0 ¼

170  HOB ð1 þ 25:505  HOB0:25 þ 1:7176e  7  HOB2:5 Þ

These relations, although derived from nuclear data are applied to high explosives as well. When being used for HE, the caveats of remembering that the mass of the explosive must be taken into account for low HOB and that HE is twice as efficient at producing blast as a nuclear source must be invoked. The massive detonation products will prevent or delay the formation of a Mach reflection for heights of burst less than several charge diameters. In scaling the height of burst to a kiloton, the effective yield should be increased by a factor of 2 for solid high explosives. To illustrate the growth of the Mach reflection region and get a better understanding of the geometry of the reflected shocks, the path of the triple point can be plotted for various heights of burst. Again, the figure is derived from nuclear detonations and has been scaled to 1 kt. Figure 14.15 shows the triple point path for a number of scaled heights of burst between 50 and 800 ft/kt1/3. Note also that the vertical scale is exaggerated by approximately a factor of 4. The point at which the triple point first rises above the surface is given by the equation above. The triple point path is dominated by the shock geometry and the decaying shock propagation velocity.

14.2 Range for Mach Transition

217

250 Burst Height = 50 ft

Mach-Stem Height (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

S3 Fit to Triple Point Path Experimental Date

Fig. 14.15 Triple Point Path for 1 KT Nuclear

Note that for a 50 ft height of burst, the triple point reaches the burst height at a ground range of less than 180 ft and continues to rise sharply beyond that point. In contrast, the triple point from a 200 ft HOB reaches the burst height at a ground range of over 700 ft. A few observations related to Fig. 14.15. For a 50 ft height of burst, the reflected pressure at ground zero is in excess of 100,000 psi or 7 kbars. At these pressure levels the fireball plays a significant role. For heights of burst where the fireball nearly reaches the ground or is flattened on the bottom by the reflected shock, the reflected shock velocity is accelerated as it passes through the high sound speed gas in the nuclear fireball. The reflected shock may catch up to the initial shock at a point above the fireball before the triple point reaches the height of burst. At a 200 ft HOB, the incident shock pressure is just over 250 psi with a reflected pressure of about 1,700 psi. A strong Mach reflection occurs at a range of 165 ft. Remember that the HOB curves provide the overpressure at ground level only. The triple point path gives an indication of the type of blast pressure waveform that will be encountered at locations above the ground. If the location is above the triple point, two shocks will be encountered. The first will be the incident shock traveling radially from the burst point. The peak overpressure from this shock can be obtained from a free air blast curve such as the 1 kt standard from Chap. 4 or from the TNT standard in Chap. 5. The second shock encountered will be the reflected shock. This shock will be moving upward with an apparent origin below ground zero at a distance of approximately the HOB below the surface. The relative strength and timing of the second shock will be dependent on the proximity of the

218

14 Height of Burst Effects

location to the ground (the reflected pressure) and to the triple point. If the location is just above the triple point, the reflected shock will be very close behind the incident shock and will be stronger than the incident shock. If the location is near the ground, the incident shock will be followed closely by the reflected shock with the reflected shock having a higher pressure. If the location of interest is below the triple point and above the surface, a single overpressure peak will be encountered as the Mach stem passes. The dynamic pressure will initially be oriented perpendicular to the shock front, i.e., having the same direction as the shock. Behind the Mach stem, a vortex flow may be encountered with rapid changes to upward and then downward flow. If the location is above the vortex region, the slip line will cross this location. The density, magnitude of the velocity and the direction of the flow, change discontinuously across the slip line. The overpressure is continuous across the slip line and the passage of the slip line will not be observed on an overpressure gauge (except as a possible change in slope). The dynamic pressure will change discontinuously in both magnitude and direction as the slip line passes. Measurement of the dynamic pressure in this region cannot be made using standard stagnation or pitot tube techniques because they are direction sensitive.

14.3

Height of Burst Over Real Surfaces

The reflection of a blast wave from a surface is usually treated as an ideal reflection. There are many instances in which the properties of the reflecting surface should be taken into account. Historically, the effective blast yield of a surface burst of high explosive is reduced by as much as 20% to account for the energy partitioned into cratering and ground shock. Using cube root yield scaling, a reduction of 20% in yield corresponds to a reduction in range to a given overpressure of less than 7%. Results of large scale (over 100 kg) experiments do not support such a large decrease in the available blast wave energy. Over soft sandy soils the blast wave measurements are more consistent with a reduction of 10% in energy. This scales to a reduction in range of just over 3%, which falls within the scatter of many large scale blast measurements. From these observations, treating the surface, even a soft sandy one, as an ideal reflector, results in an accountable reduction in range of about 3%. For non-zero heights of burst the perfectly reflecting approximation is even better. For example, the Trinity shot (19 kt at 100 ft HOB), scales to an HOB of about 37 ft. The incident pressure was over 30 ksi onto a sandy alluvial surface. A shallow depression (compression) crater was formed with a diameter of 400–600 ft (some reports indicate 1,000 ft) [4] and a depth of 10 ft. There was no crater “ejecta” but there was significant scouring of the surface. Dr. Robert Henny [4] reported in the 18th MABS symposium that there is good evidence that secondary craters caused by “ejecta” from the primary crater were actually eroded and swept out by the scouring effects of airblast. He sites both experimental and high resolution computational

14.3 Height of Burst Over Real Surfaces

219

fluid dynamic calculations that indicate that about 70% of the Trinity crater volume was caused by air blast scouring of surface material. This poses a difficult question as to what fraction of the yield went into cratering and ground shock. If the air blast is responsible for excavating 70% of the crater volume, and the cratering and ground motion energy is 10% of the yield, then the fraction of the yield that went into air blast must be close to 100% (minus the radiated energy).

14.3.1

Surface Response

The effective reflection of a blast wave from a real surface can be approximated by comparing a parameter called the acoustic or mechanical impedance. This parameter is calculated as the density of the material times the square of the sound speed. It can be used as a measure of the transmitted to reflected energy of a blast wave traversing a material interface. Table 14.1 contains the typical density, sound speed and acoustic impedance value for a few materials of interest. The closer the impedance values match, the greater the energy transferred across the interface. The greater the difference, the smaller the fraction of blast energy transmitted. A few examples will be illustrated. For a blast wave in air striking a region of helium or sulfur hexafluoride, the impedance match is very close and the blast wave will be propagated into the second region with little or no reflected energy. On the other hand, an air blast wave striking a sand, concrete or steel interface will reflect nearly perfectly, with little energy going through the interface. Some interesting experiments have been carried out to examine the effects of a snow layer over a concrete surface. The sound speed in snow is only about half the Table 14.1 Density, sound speed and impedance for several materials Material Sorted by material name Density g/cc Sound speed cm/s Air 1.23E03 3.40E þ 04 Concrete 2.26 3.40E þ 05 Helium 1.79E04 1.02 þ 05 Sand 2 2.40E þ 05 Snow 0.35 2.00E þ 04 Steel 7.7 5.80E þ 05 Sulfer hexafluride 6.16E03 1.49E þ 04 Water 1 1.50E þ 05 Sorted by acoustic impedance Sulfer hexafluride 6.16E03 Air 1.23E03 Helium 1.79E04 Snow 0.35 Water 1 Sand 2 Concrete 2.26 Steel 7.7

1.49E 3.40E 1.02 2.00E 1.50E 2.40E 3.40E 5.80E

þ þ þ þ þ þ þ þ

04 05 05 04 05 05 05 05

Impedance 1.42E 2.61E 1.86E 1.15E 1.40E 2.59E 1.37E 2.25E

+ 06 + 11 þ 06 þ 11 þ 08 þ 12 þ 06 þ 10

1.37E 1.42E 1.86E 1.40E 2.25E 1.15E 2.61E 2.59E

þ þ þ þ þ þ þ þ

06 06 06 08 10 11 11 12

220

14 Height of Burst Effects

sound speed in air. The original thought was that a snow layer would cushion the impact of a strong shock striking a concrete surface. A 1,000 pound sphere of 85% HMX was detonated at an HOB of 5 ft above the concrete surface. The experiment was repeated for the same yield and HOB with a 3 in. thick snow layer on the concrete. The surprise was that at a ground range of 2 ft, the measured overpressure under the snow layer was nearly three times that on the exposed concrete surface (120,000 psi vs. 45,000 psi). Multiple measurements on different radials confirmed the findings. Calculations were made to assist with the understanding of the physics of such a blast wave reflection and showed good agreement with the data. Looking at the impedance mismatch at the air snow interface, we see a ratio of 100, indicating a relatively weak reflection compared to the other materials in the table. The weak reflected shock does not overcome the downward momentum of the air and HE detonation products in the positive phase of the incident blast wave. The reduced sound speed in the snow layer produces a shock which is traveling at a small fraction of the velocity of the shock front in the air above. The energy in the incident blast wave does not efficiently reflect from the surface and the energy is not rapidly transmitted through the snow layer. The result is a significant increase in the energy per unit volume within the snow layer. Remember that pressure has units of energy per unit volume. Thus the pressure of the blast wave is increased, in this case, by about a factor of 3 as measured and calculated. Figure 14.16 [5] shows the calculated energy density when the blast wave front in air has reached a ground range of 2 m. The incident shock front is radial from the burst point and intersects the top of the snow layer at 2 m. About 15 cm behind the blast wave front is the detonation products interface. The air that was trapped between the shock front and the detonation products is further compressed and heated by the interaction with the snow layer. The downward momentum of the high density detonation products and the slow moving snow interface provide a strong mechanism for this compression. The highest energy density in this figure occurs at a ground range of 1.25 m in the snow layer. The snow layer has been compressed to a height of about 4.5 cm at this ground range. Essentially all of the energy of the blast wave that was initially between the shock front and the detonation products is confined to this 5 cm thick region. Figure 14.16 also shows the unstable mixing in the vicinity of the snow, air and detonation products interface. The air is moving horizontally over the surface of the snow at a velocity nearly 8.5 times as fast as the snow velocity. The mixing layer is generated by Kelvin–Helmholtz instability. The calculation treated the concrete as a perfectly reflecting surface. If we look at the ratio of the impedance match between snow and concrete, a factor of more than three orders of magnitude is seen. This would correspond to a reasonable assumption of a perfectly reflecting interface. However, when the experimental data was examined in detail, there were some unexplained discrepancies in the measured and calculated pressure waveforms. The pressures were measured using steel mounted gauges which were imbedded in the concrete surface. The blast wave that was striking the concrete and steel

14.3 Height of Burst Over Real Surfaces

221 ENERGY

1.50 110 170 230 290 350 410

440

450

1.35

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1.20 1.05

ALTITUDE M

468

460

0.90 0.75 0.60

1.000E+10 2.000E+10 3.000E+10 4.000E+10 5.000E+10 6.000E+10 7.000E+10 8.000E+10 9.000E+10 1.000E+11 1.100E+11 1.200E+11 1.300E+11 1.400E+11 1.500E+11 1.600E+11

DX1 = 3.212E+00 MIN = 1.000E+06 X = 1.139E+02 Y = 1.370E+01 MAX = 1.646E+11 X = 1.245E+02 Y = 3.700E+00

0.45 0.30 0.15 0.00 1.00

179

CONTOUR SCALE ERGS/GM

1.15

1.30

1.45

1.60

1.75

1.90

2.05

2.20

2.35

170

160

150 140 130 120 100 70 40

1 2.50

RADIUS M

Fig. 14.16 Blast wave propagation in air interacting with a snow layer

interface was no longer in snow but in compressed and heated liquid water. The impedance mismatch from water to concrete is just one order of magnitude while the impedance mismatch from water to steel is two orders of magnitude. In order to obtain good agreement between the measured and calculated waveforms it was necessary to take into account the geometry of the gauges and the material properties of the concrete and steel. The calculations showed that the energy from the blast wave in the “water” was more efficiently coupled into the concrete than into the steel. This caused a rarefaction wave to originate at the edge of the steel gauge mount and move across the face of the gauge, thus reducing the impulse that was originally calculated with the assumption of an ideal reflector. The calculated peak pressures were only slightly reduced but the impulse was reduced by about 10%. Several experimenters have confirmed the general finding that foams and porous materials tend to enhance, rather than mitigate the peak pressure of a strong incident blast wave. As one example, J. Nerenberg and his colleagues at McGill University reported their findings with polymeric foams at the 21st international symposium on shock waves (ISSW) [6]. From the results section of their report, “. . . at high incident blast pressures, the stress at the rear surface of the foam may be amplified above that measured with no foam present”.

222

14 Height of Burst Effects

These findings have changed the design of blast protective gear for military and civilian organizations faced with defusing bombs. Rather than foams, solid, light weight Kevlar pads are more efficient blast wave reflectors and mitigators.

14.3.2

Surface Roughness Effects

The effects of surface roughness on shock waves over planer surfaces were discussed briefly in Sect. 13.3.2. The effects on blast waves from heights of burst have somewhat different consequences. A blast wave reflecting from a rough surface changes the character of both the geometry and the blast parameters of the reflected region of the wave. The incident wave front reflects from each roughness element. The near surface velocity is stagnated and enhances the overpressure. Because overpressure is a scalar and thus acts omnidirectionaly, the blast wave energy is directed both upward and backward from the wave front. The velocity of propagation of the blast wave near the surface is reduced, causing a curved wave front with a slight convex curvature near the surface. Unlike a boundary layer, a rough surface affects the overpressure as well as the vertical distribution of the dynamic pressure. A realistic rough surface provides many reflecting roughness elements that cause the reflections and are generally sufficiently numerous that the effects of an individual roughness element cannot be distinguished. The overall effect of a rough surface is to reduce the pressure at the front, thus slowing the velocity of the wave front near the surface. The energy is redistributed behind the wave front at the reflected shock velocity. Energy is conserved, so the impulse is not affected. Waveforms generally have lower peaks and longer durations, thus conserving the impulse. Because the energy of the reflected wave is redistributed by the rough surface, the formation of a Mach reflection is delayed to greater ground range than over an ideal surface. The reflected wave velocity is also reduced, thus changing the pressure waveform well above the surface. The path of the triple point is lower than for a smooth surface. All of these effects are dependent on the relative size of the roughness elements and the positive duration of the blast wave. Micron sized roughness may affect the blast wave from a gram sized explosive. On the other hand, a residential neighborhood acts like a rough surface on a megaton sized detonation.

14.3.3

Dust Scouring Effects

Nearly any surface subjected to a high pressure blast wave will have some dust and debris entrainment in the flow field. Smooth concrete pads that were swept and wet mopped just prior to detonation show the effects of swept up particulate material. The dust may have a minor effect on the blast parameters and show up on

14.3 Height of Burst Over Real Surfaces

223

photography as a light absorbing region or it may cause significant modifications to the blast wave and the associated flow parameters. The first approximation is that the dust particle size is sufficiently small that the dust can be considered to be in velocity equilibrium with the gas. Dust lofted into the flow will share momentum and thermal energy with the flow. Because the gas in a blast wave is generally traveling faster than the surface dust particles, momentum conservation will slow the gas flow and increase the particle velocity. The dust laden flow will have a higher density, a lower horizontal velocity and a higher momentum density than the cleaner gas flow above. This vertical shear velocity tends to mix the particulates above the surface. The density in such a layer is well represented by an exponential distribution with a scale height which depends on the magnitude of the shear gradient and the time since the passage of the shock front. The surface recedes due to the combined effects of the mass being lofted and the compression of the surface material by the blast pressure. The lofted particulate material is generally cooler than the compressed and heated gasses in the blast wave. The particulates absorb energy from the gas in the flow and reduce the thermal energy and thus the pressure near the surface. The reduction in near surface pressure causes a negative vertical pressure gradient above the surface and downward velocities are generated. The hot gasses from above are brought into the dust laden layer and contribute a vertical component of momentum within the dusty layer. This vertical component continues the scouring process. Dust densities several inches above the surface can reach values of ten times ambient air density. In the case of nuclear detonations over sandy desert soils at relatively low heights of burst, the dust can be lofted through a mechanism dubbed “popcorning” by the early nuclear experimenters. Popcorning is caused by high rates of thermal radiation incident on the ground prior to shock arrival. The radiated energy is absorbed in a thin (1 or 2 mm) layer of soil at the ground surface. The sudden temperature rise causes any water in the soil to explosively boil. These small steam explosions carry hot particulates into the air above the ground. The dust layer was observed to be as high as 10 ft with a dust density variation ranging from soil density at the surface to near zero at the 10 ft height. An exponential representation of the dust density with a scale height of a few inches is representative of the observed dust distribution. Most of the layer has a dust density of less than 1% of the air density. It was initially thought that the dust density was significantly higher because the pre-shock dust obscured targets such as busses and trucks from the view of cameras. A fairly simple calculation shows that the 105 gm/cc dust density is more than sufficient to obscure visible light over a path length of a few kilometers. In the case of popcorning, the dust is lofted ahead of the shock and becomes a part of the momentum of the blast wave. The fact that the popcorning is caused by sudden evaporation of water in the soil has been demonstrated in a number of experiments. Various soil samples have been exposed to intense thermal heating in solar furnaces. The experiments showed that soil which had been exposed to the radiation and exhibited popcorning, did not respond during a second exposure. Samples which were heated in an oven to more

224

14 Height of Burst Effects

than 400 K did not exhibit popcorning. Samples which were soaked in water also did not respond. The maximum response was found at water concentrations of 6–10% by weight. An extreme example of dust scouring is the TRINITY event conducted at White Sands New Mexico. A 20 kt device was detonated on a 100 ft tower. As was mentioned in Sect. 14.3, about 70% of the crater volume is attributed to “dust” scouring. A high resolution calculation was made of the event as part of the review process for issuing a new edition of “The effects of Nuclear Weapons” edited by Glasstone. The report on the calculation is entitled “A revisit to Trinity-2004” [7]. The calculation was made using the U.S. government owned SHAMRC CFD code. The problem was initialized using the results of a one dimensional coupled radiation transport- hydrodynamic code at a time just prior to the shock front reaching the ground. The ground elevation corresponded to that of ground zero for the test. A thermal radiative cooling model in the calculation removed energy from the fireball as a function of time. Another model called “FDOT” [8] calculates the air temperature caused by the thermal radiation striking the ground, as a function of the incident flux as a function of time. A dust sweep-up model which injects dust from the surface using a simple shear velocity model was also used. The rate of dust injection, mass per unit area per second is proportional to the surface shear velocity above a threshold. The dust was assumed to be small and was treated as being in velocity and temperature equilibrium with the gas. The calculation used a 16 million zone grid in two dimensional cylindrical symmetry to represent the computational domain. Comparisons of the results of the calculation were made with the photography from the event. The original film was digitized at the Los Alamos Laboratories and a copy of the digitized photos was sent to the author. The comparisons were used to verify the phenomena in the photographs and to validate the models used in the CFD calculation. Figure 14.17 is a comparison at a time of 34 ms after detonation. The upper position of the shock and fireball are in excellent agreement. The height of the triple point and angle of the Mach stem are in excellent agreement. The turbulent “toe” near the surface is composed primarily of high density, high velocity dust. This was labeled in the original Glasstone as a precursor caused by thermal radiative heating of the air ahead of the shock. The calculation showed that no appreciable heating of the pre-shock air occurred at this early time. The dust overtaking the shock front is caused by the high momentum obtained by the dust at smaller ranges and the continued scouring and entrainment of the dust in the flow. The detail in the center rectangle shows the very complex nature of the flow within the dust jet and the interaction with the Mach Stem.

14.3.4

Terrain Effects

So far the discussion of height of burst effects has included only flat, level reflecting surfaces. For large yield detonations, the surface over which the blast wave is propagating will most likely rise and fall at varying slopes. First approximations

14.3 Height of Burst Over Real Surfaces

225 Pressure

Density 180 160 140 120 100 80

Mach Triple Point

60 40 20 0

0

20 40

60 80 100 120 140 160 180 200

RADIUS M

Fig. 14.17 Comparison of the Trinity event with CFD results at 34 ms

to the effects of this rolling terrain on a blast wave can be made by examining the behavior of shocks on wedges, such as discussed in Chapter 13. The blast wave peak pressure will be modified by the slope of the terrain it encounters relative to the direction of the flow at the shock front. A gently upward sloping region will cause a partial stagnation of the flow at the shock front. The stagnation enhances the peak overpressure, decreases the dynamic pressure and changes the direction of flow to be parallel to the surface. The density near the surface is also increased. The blast wave is affected behind the shock front as well. An upward sloping terrain modifies the volume expansion of the gas behind the shock front. The restricted volume expansion causes the pressure behind the shock to decay more slowly than over a flat surface. The shock geometry will be compressed. Parameters such as the height of the triple point above the surface will be affected because the altitude of the ground is increasing as the triple point is moving away from the reflecting surface. The energy in the blast wave below the triple point is confined to a smaller volume and the pressure in the entire affected volume is increased. For terrain sloping down, the effects, not surprisingly, are just the opposite. The volume into which the blast wave is expanding is greater than over a flat reflecting surface. The peak overpressure is reduced as the shock expands into the larger volume. The near surface velocity is oriented to be parallel to the surface, resulting in a convex curvature at the shock front. The reduced pressure and changed flow direction near the surface, communicate these changes to the rest of the flow field at the speed of sound. The distances to which the changing slope has an effect on the flow field is thus a function of the shock strength, the positive duration of the flow and the slope change.

226

14 Height of Burst Effects

Suppose that a strong Mach reflection has formed over a flat, level reflecting surface. A strong vortex has formed near the surface behind the Mach stem. When a down slope is encountered, the Mach stem develops a convex curved front as the gas velocity turns to be parallel to the surface. The vortex flow behind the front is very stable; think of the stability of a smoke ring. When the vortex reaches the down slope, it is difficult for the established flow to change directions because the pressure gradients are small; unlike the gradients at the shock front. The vortex, as a whole, continues in the same direction. The center of the vortex continues along a horizontal trajectory as the surface falls away. The dynamic pressure near the surface, at the shock front and in the vortex region is significantly reduced. This is the reason that vehicles or structures more readily survive when they are on a slope facing away from a detonation. Reasonable approximations to the peak pressure can be made using reflection factors as a function of angle for the local terrain. However, the effects of rolling terrain are not just a function of the slope of the terrain at the shock front. Because the entire flow field is affected, the effect of another change in slope is a function of the history of the flow over the previous terrain encountered. To illustrate the effects of sloping terrain on blast wave propagation, I will use the SMOKY nuclear event as an example. The actual event took place over a variety of terrain and the measured blast waves included the effects of dust entrainment (Chap. 13) and thermal radiation from the fireball (next section). Using advanced numerical methods, a series of calculations were performed of the SMOKY event. [9] One of these calculations used the actual terrain of the SMOKY event but treated the surface as an ideal, non-responding reflecting surface. The results of this calculation thus isolate the effects of terrain alone. The calculation was made in three dimensions. Figure 14.18 is a plot showing the surface level contours and the five instrumented blast lines from the SMOKY event. Lines 1 and 3 are highlighted because they were heavily instrumented by SRI and BRL. Figure 14.19 gives the profile of the surface along each of the 5 blast lines. The vertical scale is only slightly exaggerated, about 30%. The terrain varies from a gentle down slope on line 1, to an undulating surface along line 2, to a steep up slope on line 4 and a sharp rise and fall along line 5. Figures 14.20 and 14.21 give close up views of the near surface pressure distribution at a time of 200 ms. The terrain slice is taken along line 4 on the up slope and extends to the south near line 1 for the down slope. On the down slope the curvature of the Mach stem is clearly visible. Because the shock pressure is reduced in the down slope direction, the propagation velocity is also reduced, whereas the propagation velocity in the up slope direction is increased. At this time there is about a 10 m difference in the ground range of the shock front along the two slopes. The up slope surface was also bumpy. The calculation attempted to model the terrain which was strewn with boulders some 3–6 ft in diameter using small discontinuous changes in slope. These small discontinuities resulted in reflected shocks in the up slope calculation. Note that the triple point height in the up slope direction is only 1 or 2 m due to the combination of slope and roughness. By a time of 0.7 s, the shocks have advanced to a ground range of about 680 m. The down slope shock remains slower and with lower pressure than the up slope shock.

14.4 Thermal Interactions

227

Fig. 14.18 Elevation contours for the SMOKY event showing the five instrumented blast lines

The up slope shock has a near surface pressure of just over 4 bars while the down slope shock peak pressure is about 3.2 bars. The triple point height above the surface is 150 m on the down slope and 120 m on the upslope. It is interesting to note the height of the triple point above ground zero is also different along the two slopes. The triple point has risen nearly 40 m higher in the upslope direction. This is caused by the generally higher pressures in the flowfield caused by the decrease of the expansion volume available in the upslope direction. Note also that the curvature of the Mach stem remains convex outward in the down slope direction but is very nearly straight on the upslope.

14.4

Thermal Interactions (precursors)

Propagation of a shock is significantly modified by the properties of the gas into which it is moving. In the case of a nuclear detonation, an intense thermal layer may be formed in a thin layer of air just above the surface. The sound speed in this layer

228

Fig. 14.19 Terrain profiles for the 5 SMOKY instrumented blast lines

14 Height of Burst Effects

14.4 Thermal Interactions

229

a

b 9.000E–01

4.500E+00

8.100E+00

1.170E+01

1.530E+01 BAR

9.000E–01

4.500E+00

1.170E+01

1.530E+01 BAR

XPLANE AT X = 1.00E+00 M

1440.

1460.

1430.

1450.

1420.

1440. 1430. ALTITUDE M

1410.

ALTITUDE M

8.100E+00

PRESSURE

PRESSURE XPLANE AT X = 1.00E+00 M

1400. 1390.

1420. 1410. 1400.

1380. 1390. 1370. 1380. 1360. 1370. 1350. 1360. 250.

1340. –350. –340. –330. –320. –310. –300. –290. –280. –270. –260. –250.

260.

270.

280.

290.

300.

310.

320.

330.

340.

350.

RANGE (Y) M SHAMRC: SMOKY CALCULATION WITH REAL AIR AND GRAVITY CYCLE 1628. PROBLEM 30608.220 TIME 200.000 MSEC

RANGE (Y) M SHAMRC: SMOKY CALCULATION WITH REAL AIR AND GRAVITY TIME 200.000 MSEC CYCLE 1628. PROBLEM 30608.220

Fig. 14.20 Close-ups of the effects of mild down and up slope terrain, 0.2 s

a

b 8.000E–01

2.000E+00

2.600E+00 3.200E+00 BAR PRESSURE XPLANE AT X = 1.00E+00 M

1660.

1660.

1620.

1620.

1580.

1580.

1540.

1540.

1500. 1460.

8.000E–01

1.600E+00

450.

530.

1700.

ALTITUDE M

ALTITUDE M

1700.

1.400E+00

3.200E+00 4.000E+00 BAR PRESSURE XPLANE AT X = 1.00E+00 M

1500. 1460.

1420.

1420.

1380.

1380.

1340.

1340.

1300. –800. –760. –720. –680. –640. –600. –560. –520. –460. –440. –400.

1300.

RANGE (Y) M SHAMRC: SMOKY CALCULATION WITH REAL AIR AND GRAVITY TIME 700.000 MSEC CYCLE 4537. PROBLEM 30608.220

2.400E+00

490.

570.

610.

650.

690.

730.

770.

810.

850.

RANGE (Y) M SHAMRC: SMOKY CALCULATION WITH REAL AIR AND GRAVITY TIME 700.000 MSEC CYCLE 4537. PROBLEM 30608.220

Fig. 14.21 Close-ups of the effects of mild down and up slope terrain, 0.7 s

may be several times ambient atmospheric sound speed with temperatures near or above the vaporization point of sand (2,250 K). When the blast wave encounters this layer the shock front accelerates rapidly causing an increase in the gas velocity and a corresponding drop in the pressure. If the sound speed in the layer is greater

230

14 Height of Burst Effects

than the shock velocity in the air above, the shock can no longer be supported and becomes a compressive wave traveling at approximately the air shock velocity. A weak signal travels ahead of the compressive wave at the sound speed in the layer. This occurs because the vast majority of the energy is in the region of the blast wave above the thermal layer and the overall flow field is controlled by the free field blast wave. The high velocity of the front in the high sound speed layer and the reduced pressure occurring ahead of the expected main blast wave have led to the phenomena being labeled as a “precursor”. Observations from nuclear tests show that the overpressure in a precursor may be reduced by about a factor of two and the dynamic pressure increased by a factor of 4 or more. The overpressure impulse remains nearly the same as for an ideal wave but the dynamic impulse may be increased by an order of magnitude. Let us examine these effects using energy conservation with the Rankine-Hugoniot relations as a guide.

14.4.1

Free Field Propagation in One Dimension

For a 30 psi overpressure blast wave, the shock velocity in sea level air is 564 m/s. The material velocity at the front is 299 m/s and the sound speed is 340 m/s. Helium at ambient pressure and temperature has a sound speed of just over 900 m/s. When the shock enters the helium layer the peak overpressure at the blast front drops to 14 psi and the material velocity increases to 400 m/s with a shock velocity of about 1.2 km/sec. A shock front is maintained in the helium layer. The Rankine Hugoniot relations hold at the blast wave front when the ambient conditions are changed from those of air to those of helium. The gamma also changes from 1.4 for air to 5/3 for helium. Using the parameters for the shock in helium, we note that the dynamic pressure actually decreases when the shock enters the helium in a one dimensional flow. This decrease in dynamic pressure is caused by the decrease in density at the shock front, now in helium. The kinetic energy density (¼1/2 U2) increases by about a factor of 2 and the overpressure drops by about a factor of 2. The measurements on the nuclear tests showed that the dynamic pressure increased much more than a factor of two. Both static and dynamic pressures have units of energy per unit volume. If the overpressure drops by a factor of 2, how does the dynamic pressure increase by more than a factor of 2? The observed enhanced dynamic pressure is not generated in a one dimensional flow. The answer is that the enhanced dynamic pressure impulse is caused by two dimensional phenomena.

14.4.2

Shock Tube Example

I will use an illustration of the dynamic pressure enhancement generated using a square topped shock in a shock tube interacting with a horizontally layered gas.

14.4 Thermal Interactions

231

Fig. 14.22 Precursor simulation using a helium bag in a 6 ft shock tube

Figure 14.22 is a reflected laser shadowgram of the shock structure of a planar shock interacting with a helium layer contained in a polyethylene balloon. This shock configuration was generated in a 6 ft diameter shock tube with a flat floor approximately 1.5 ft above the bottom of the tube. A polyethylene sheet was stretched across the floor of the tube, 6 in. above the floor and secured to the sides of the tube with duct tape. The region between the floor and the polyethylene was filled with helium gas and started about 2 ft to the left of the test section where this photo was taken. The sound speed in the helium layer is three times the sound speed in the air above the polyethylene. (Table 14.1) The overpressure in the incident shock was approximately two bars. Because the sound speed in the helium gas (1.e5 cm/s) exceeds the shock velocity in the air (5.64e4 cm/s), the shock is not supported in the helium and the front becomes a compressive wave which travels faster than the shock in the air above. The compressive wave has the effect of a piston in the helium layer and a sound wave travels ahead of the compressed region. This sound signal travels ahead of the compression region in the helium layer at the speed of sound. The overpressure in the sound wave is small compared to the rest of the flow field. As the compressive wave advances in the helium layer, an upward moving shock is generated ahead of the incident shock. This shock is seen ahead of the incident shock in Fig. 14.22 extending from the top of the helium layer to the incident shock front. In this case the gas velocity in the helium layer is greater than the shock velocity in the air above. The sudden expansion of the gas near the floor of the tube causes a local drop in the pressure. The air behind the incident shock is accelerated downward by the pressure gradient and the downward component of velocity is stagnated near the floor behind the low pressure region in the helium layer. The stagnated pressure causes acceleration of the flow horizontally near the surface. This accelerated flow near the surface induces a rotational flow in the region that the incident shock would be located if there were no helium layer.

232

14 Height of Burst Effects

In Fig. 14.22, the incident shock is nearly vertical just to the left of center at the top of the photo. The weak shock which joins the incident shock near the top of the photo and extends downward to the left is the remnant of the reflection of the incident shock from the upstream end of the polyethylene bag. The dark horizontal line that runs across the entire frame is the tape that held the polyethylene to the window. The heavy line that is below the precursor shock and above the tape is the shadow of the broken polyethylene sheet. The pieces of the polyethylene can be seen curving below the tape line behind the vortex near the bottom left of the photo. The pieces do not follow the flow because of their size and density but slip relative to the gas motion. The shock that starts from the intersection of the incident shock and the precursor shock and curves downward to the left is the precursor shock curved by the induced rotational flow behind the incident shock. An overpressure gauge mounted flush with the floor of the shock tube would record a pressure-time history that can be described as follows. The first signal would be very weak and marks the arrival of the sound wave. The first significant pressure marks the arrival of the compressive wave with a rounded rise at the front and a slow decay. This is followed by a stronger rounded rise to a higher pressure indicating the passage of the stagnation region. The rise and decay from the stagnation region may be noisy because of the instabilities generated by the vortex and later by the impact of the polyethylene fragments. A dynamic pressure measurement would not show the arrival of the sound wave, but a rounded first peak, marking the passage of the compressive wave, with a decay followed by a steep rise to a maximum. This maximum marks the arrival of the highest velocity region of the vortex and arrives before the maximum in the overpressure measurement. The peak then slowly decays as the stagnation region passes. This decay also contains noisy spikes and oscillations. The reason that the impulses are modified now becomes clear. The overpressure is reduced by a factor of two or so but the positive duration is increased by a similar amount. The integral of the waveform thus remains nearly constant. The dynamic pressure impulse near the surface is significantly enhanced by energy pulled from the region behind the incident shock above the helium layer. The enhanced velocities near the vortex region are associated with high density air from behind the incident shock. If the velocities are increased by a factor of two and the density is higher by a factor of two, the dynamic pressure is locally increased by a factor of eight. The higher dynamic pressures, combined with the increased positive duration easily accounts for the observed order of magnitude increase in dynamic impulse. Local conservation of energy is not violated because the increase in energy near the surface comes from a volume in the second dimension, above the helium layer.

14.4.3

Thermal Interactions Over Real Terrain

In Sect. 14.3.3, the phenomenon called “popcorning” was described with regard to dust loading of the air prior to shock arrival. The dust that is lofted by popcorning

14.4 Thermal Interactions

233

has been heated to temperatures above 100 C, and as high as 2,000 K. The hot dust conductively heats the gas in which it is embedded, resulting in a temperature distribution that decays exponentially from the surface with a scale height of a few centimeters. The energy is deposited very rapidly in the gas above the ground because the dust particulates are small.

14.4.3.1

Generating a Heated Layer

In Chap. 12 we showed that the majority of the thermal output from a 1 kt device is radiated prior to 100 ms, with the peak incident flux arriving at about 40 ms. This rapid deposition of energy creates a high pressure layer near the surface which expands upward at ambient sound speed. Because the pressure gradients are small in the radial direction, the vast majority of the expansion takes place in the vertical direction. The upward moving gas may have an overpressure of as much as a few PSI. The hot layer expands, thus reducing the density and the overpressure in the layer. Remember from the discussion on popcorning, the dust densities in the thermal layer are less than 10% of the air density. A small fraction of the thermal radiation is absorbed in the dust in the layer but most of the radiated energy reaches the surface and continues to heat the ground and the gas in the layer as it expands. The layer remains close to pressure equilibrium with ambient pressure. The sound speed in the layer increases as the square root of the temperature. The layer thus has a sound speed which decreases exponentially with height above the surface. The signal propagating through the heated layer travels fastest near the ground. It is this leading signal that initiates the upward moving precursor shock. One assumption that has been made historically, is that the sound speed within the layer is constant to a height of 3–10 ft. Photographs of nuclear precursors can only show the precursor shock above the thermal layer because the dust obscures the lower 10 ft or so of the shock structure. There is good evidence from pressure measurements within the thermal layer, that the precursor shock extends linearly to the surface. For most nuclear experiments, blast gauges were placed at ground level, and at 3 and 10 ft above the surface. The shock arrival at the surface is earlier than at 3 ft and at 3 ft the shock arrives earlier than at 10 ft. On some nuclear detonations that exhibited precursor formation, gauges were placed on the ground and on the front of structures as little as 6 in. above the surface. In all cases the shock arrived earlier at the surface than at the 6 in. height. This confirms the decrease in sound speed with the height above the surface. I will use the Priscilla event as a representative example of a thermally precursed blast wave from a nuclear detonation. Priscilla was a 36.6 kt detonation at a height of burst of 700 ft over a dry lakebed at the Nevada test site. The dry lake bed had a deep layer of fine dust several inches thick. This fine dust provided the means of generating the hot air layer near the surface, and provided an essentially unlimited amount of dust that could be swept up by the rotational flow behind the precursor blast wave.

234

14 Height of Burst Effects Priscilla Sound Speed at Shock Arrival

Speed of Sound (ft / s)

4500 4000 Data 3500 3000 2500 2000 1500 1000

0

500

1000

1500 2000 Range (feet)

2500

3000

3500

Fig. 14.23 Sound speed at shock arrival for the Priscilla event

The sound speed in the thermal layer was obtained by using the measured overpressures and arrival times of the precursor front with the Rankine-Hugoniot relations. The sound speeds thus obtained are plotted in Fig. 14.23. Note that the sound speed near ground zero is less than at the 600 ft ground range. This is because the shock arrives at ground zero before the maximum thermal flux has been reached. As the shock proceeds along the ground, the radiated flux continues to increase more rapidly than the 1/R2 geometric factor decreases the incident flux. At larger ranges the 1/R2 factor combines with the decreasing radiated flux and produces a cooler layer. Because these are “measured” sound speeds, the expansion of the pre-shock thermal layer, heating and cooling of the dust particulates and additional incident thermal radiation effects are included. Note also that the maximum sound speed is approximately four times that of ambient conditions. The precursor shock does not form until the sound speed in the thermal layer exceeds the phase velocity of the shock along the ground. Near ground zero the phase velocity is essentially infinite and decreases as the inverse of the cosine of the incident angle. When the Mach stem forms, the Mach shock moves parallel to the ground and the phase velocity is the shock velocity. The strength of a precursor can be estimated by using the ratio of the sound speed to the phase velocity of the ideal blast wave. The greater the ratio, the stronger the precursor. The extent of a precursor, the time between signal arrival and the peak overpressure, is a function of the time that the precursor has outrun the ideal shock. At early times the precursor is barely separated from the shock front. The extent grows to a maximum and decreases as the sound speed in the layer approaches ambient sound speed. For yields of kilotons to tens of kilotons, the range at which the precursor “cleans up” corresponds to about the 10 psi ideal overpressure level. The measurements made for Priscilla air blast included two different types of gauges fielded by two different laboratories. The measurements were taken along the same radial and were placed near to one another at several ground ranges. I include here a few examples of the waveforms measured by the two different

14.4 Thermal Interactions

235

organizations. The SRI gauges were electronic and provided an accurate time line. The BRL gauges were self recording and were triggered just prior to the shock arriving at the gauge. In some instances the recording mechanism took a fraction of a second to come up to speed and the time line is not linear. Figure 14.24 shows the overpressure waveforms measured at 850 ft from ground zero. The BRL data has been shifted to match the arrival time of the SRI gauge. The precursor arrives 30–50 ms prior to the peak pressure. The peak overpressure is about 250 psi and the positive duration is more than 200 ms. The precursor duration is relatively short at this range and has an overpressure of about 20 psi. As the blast wave progresses, the length of the precursor grows while the peak overpressure decays. Figure 14.25 shows the measured overpressure waveforms at the 1,350 ft ground range. The first signal arrives over 100 ms prior to the peak, Priscilla Overpressure at 850 Feet

300

Overpressure (psi)

250 SRI BRL

200 150 100 50 0 150

170

190

210

230

250 270 Time (ms)

290

310

330

350

Fig. 14.24 Overpressure waveforms from Priscilla at 850 ft ground range

Priscilla Overpressure at 1350 Feet Overpressure (PSI)

80 70

BRL SRI

60 50 40 30 20 10 0 250

300

350

400

450 500 Time (ms)

550

Fig. 14.25 Measured Overpressure waveforms from Priscilla at 1,350 ft

600

650

700

236

14 Height of Burst Effects

expanded from
30 ms at 850 ft, and the overpressure in the precursor region is nearly constant at about 15 psi. The peak overpressure has decayed from over 200 psi at 850 ft to about 60 psi. The dynamic pressure is significantly enhanced at these ground ranges. Figure 14.26 shows a measured dynamic pressure waveform at a range of 1,350 ft for the Priscilla event. The dynamic pressure in the low overpressure region of the precursor is between 40 and 60 psi. The overpressure in this region is, from Fig. 14.25, nearly constant at about 15 psi. The Rankine Hugoniot relations give a dynamic pressure of less than 5 psi for a 15 psi overpressure shock, so the dynamic pressure in the precursor region is ten times the “normal” value. For the peak overpressure of 60 psi, the R-H relations give a dynamic pressure of 55 psi but the measured peak in Fig. 14.26 is over 140 psi or nearly three times the “normal” value. Note also that the time of the peak dynamic pressure is about 30 ms prior to the peak in overpressure. This timing difference is in agreement with the explanation of the structure of the vortex formed in the precursor region and helps explain why the R-H relations should not be expected to apply here. The solid line in Fig. 14.26 is the result of a fit developed by the author using a collection of all nuclear dynamic pressure waveform data. Figure 14.27 shows the overpressure waveforms measured at a ground range of 1,650 ft. The precursor arrives 170–230 ms prior to the peak. This is about twice the separation seen at 1,350 ft and means that the thermal layer continues to cause the duration to increase. The sound speed in the thermal layer remains significantly above ambient. Figure 14.23 shows that the sound speed at 1,650 ft is more than

Estimated Non-Ideal Dynamic Pressure Waveform Priscilla at Ground Range = 1350 ft 160

Dynamic Pressure (psi)

140 120 100 80 60 40

Waveform Fit Experimental

20 0

0

50

100

150

200 250 Time (msec)

300

Fig. 14.26 Dynamic pressure waveform from Priscilla at ground range 1,350 ft

350

400

14.4 Thermal Interactions

237 Priscilla Overpressure at 1650 feet

40 Overpressure (psi)

35 SRI BRL

30 25 20 15 10 5 0 300

350

400

450

500

550 600 Time (ms)

650

700

750

800

1500

1600

Fig. 14.27 Overpressure waveforms from Priscilla at 1,650 ft ground range Priscilla Overpressure at 2500 feet

Overpressure (psi)

14 12

SRI BRL

10 8 6 4 2 0 600

700

800

900

1000

1100 1200 Time (ms)

1300

1400

Fig. 14.28 Overpressure waveforms from Priscilla at 2,500 ft ground range

twice that of ambient. The pressure behind the precursor is comparable to that measured at 1,350 ft, but the peak overpressure has decayed to about half of that measured at 1,350 ft. All of the waveforms from 850 ft to over 2,000 ft do not have shock fronts at the first arrival but a slow rise to a nearly constant overpressure. All of the peak overpressures also have a slow compressive rise to the peak with an immediate decay from the peak. As the pressure drops and the layer cools, the precursor begins to shrink. Figure 14.28 shows the overpressure waveforms measured at 2,500 ft from ground zero. There is no distinct precursor and the peak over pressure has dropped to less than 1 bar. There is a finite rise time at the front indicating that the precursor has not fully cleaned up. There are multiple peaks near the front indicating a complex flow in this region.

238

14 Height of Burst Effects

Table 14.2 Comparisons of blast parameters ideal vs. precursed for priscilla Air blast parameters at selected ideal overpressure levels Priscilla Mean values IDEAL NON-IDEAL Range OP OPPPD OPI DP DPI OP OPI DP DPI Ft. PSI SEC PSI-SEC PSI PSI-SEC PSI PSI-SEC PSI PSI-SEC 1,990 30 0.51 4.2 19.4 1.90 9 4.2 80 10 2,900 15 0.62 3.1 5.6 0.80 7 3.1 12 6 3,520 10 0.7 2.5 2.6 0.46 8 2.5 3.2 0.7 4,360 7 0.95 2.1 1.5 0.29 7 2.1 1.6 0.33

DT (s) TPK-AT 0.35 0.2 0.1 0.01

Table 14.2 contains a summary of a few blast parameters at a few ground ranges for the Priscilla event. The values on the left side of the table are for an ideal reflecting surface with no thermal layer. The values on the right side are mean values from the precursor blast lines. At a ground range of 1,990 ft, the overpressure is only 30% of the ideal value but the impulse is the same. The dynamic pressure peak is more than four times the ideal value and the impulse is more than five times the ideal dynamic impulse. The separation between precursor arrival and the arrival time of the peak are tabulated in the right hand column. At the 30 psi range the separation is 350 ms. The precursor slows as the ground range increases and by the time the 10 psi range is reached, there is only a 100 ms separation. By the 7 psi range, the blast parameters on the non-ideal side are comparable to the ideal values. The precursor has “cleaned up”. The Priscilla event took place over a large flat (and dusty) area where the thermal layer was continuous. Some recent CFD calculations have indicated that if the thermal layer is interrupted by small cool or ambient temperature regions, the development of the precursor is stopped and even reversed. This is because the low overpressure in the precursor flow cannot be sustained when ambient atmospheric conditions are encountered. For example, if the flow represented by the waveform in Fig. 14.24 suddenly encountered ambient sound speed, the velocity of the 20 psi or so precursor front would suddenly slow to a velocity less than that of the 250 psi peak region. Some of the stagnated dynamic pressure from this sudden slowing would convert to internal energy and thereby overpressure, but the rise to the peak would be much sharper. The extent of the precursor would be rapidly reduced as long as ambient conditions were encountered. If a new region of thermal enhancement were to be encountered, the precursor would have to start all over again. When a nuclear detonation occurs over rolling terrain or in a region where bushes or boulders provide shadowing from the incident thermal radiation, the thermal layer is interrupted. Regions of relatively cooler air are interspersed in the hot layer. Formation and growth of a precursor is delayed or slowed or prevented. The nuclear event SMOKY was designed to answer many of the questions about blast and precursor propagation over realistic terrain. Smoky was a 44 kt device detonated on a 700 ft tower. It was therefore very close to the Priscilla event in yield and height of burst. Some examples from the SMOKY calculation were used in the

14.4 Thermal Interactions

239

a

b 1.000E+00 9.000E– 01

8.100E+01

1.170E+01

1.530E+01 BAR

1450.

1420.

1440.

1410.

1430.

1390. 1380.

9.000E+00

1.300E+01

1.700E+01

BAR

XPLANE AT X = 1.00E+00 M

1460.

1430.

1400.

5.000E+00

PRESSURE

PRESSURE XPLANE AT X = 1.00E+00 M

ALTITUDE M

ALTITUDE M

1440.

4.500E+00

1420. 1410. 1400.

1370. 1390. 1360. 1380. 1350. 1370. 1340. – 350. – 340. – 330. – 320. – 310. – 300. – 290. – 280. – 270. – 260. – 250. RANGE (Y) M SHAMRC: SMOKY CALCULATION WITH THERMAL LAYER – NO DUST RSM PROBLEM 30728.230 CYCLE 1718. TIME 200.000 MSEC

1360. 250.

260. 270. 280. 290. 300. 310. 320. 330. 340. 350. RANGE (Y) M SHAMRC: SMOKY CALCULATION WITH THERMAL LAYER – NO DUST RSM TIME 200.000 MSEC CYCLE 1718. PROBLEM 30728.230

Fig. 14.29 Close-ups of the effects of mild down and up slope terrain, 0.2 s with thermal layer

previous section on terrain effects. The same reference [9] will be used here to demonstrate the effects of the combined thermal layer, and rolling terrain. The pictures in Fig. 14.29 should be compared with those of Fig. 14.20. The only difference in the calculations was the inclusion of a thermal layer which was influenced by the slope of the local terrain. The thermal layer is hotter on the upslope side but the rising terrain decreases the volume into which the precursor can expand. Note that in both the down hill and up hill slopes, the triple point is higher than it was at the same time for the non-thermal interacting case. The thermal layer continues to influence the flow to late times and relatively low overpressures. Figure 14.30 shows the precursor shocks at a time of 0.7 s at a range of over 300 m. As the triple point rises above the thermal layer, it approaches the height that was seen in Fig. 14.21. The weak pressure wave seen about 200 m above the terrain surface is the signal generated by the sudden expansion of the thermal layer near the time of peak incident flux. This signal is traveling vertically at the ambient sound speed. The precursor in the direction of the up hill slope is significantly shorter than in the downhill direction even though the thermal layer is hotter on the up hill slope. The propagation of shocks, from tens of kilotons, over hill and dale, combined with the effects of thermal radiation is not readily predictable. While modern CFD codes have been very successful at predicting the entire flow fields for such complex flows, there are no simple rules of thumb. The dynamic pressure near the surface, especially on an uphill slope is very nearly zero because the flow is stagnated on the surface. At a height of just 1 m above the surface, the dynamic pressure may be 4–8 times what would be predicted by the Rankine–Hugoniot

240

14 Height of Burst Effects

a

b 4.000E–01

1.200E+00

2.000E+00

2.800E+00

8.000E– 01

BAR

1.400E+00

1700.

1660.

1660.

1620.

1620.

1580.

1580.

1540.

1540.

ALTITUDE M

ALTITUDE M

1700.

1500. 1460.

2.600E+00

3.200E+00

BAR

1500. 1460.

1420.

1420.

1380.

1380.

1340.

1340.

1300. –800. –760. –720. –680. –640. – 600. – 560. – 520. – 460. – 440. – 400.

1300. 450.

RANGE (Y) M SHAMRC: SMOKY CALCULATION WITH THERMAL LAYER – NO DUST RSM TIME 700.000 MSEC CYCLE 4688. PROBLEM 30728.230

2.000E+00

PRESSURE XPLANE AT X = 1.00E+00 M

PRESSURE XPLANE AT X = 1.00E+00 M

490.

530.

570.

610.

650.

690.

730.

770.

810.

850.

RANGE (Y) M SHAMRC: SMOKY CALCULATION WITH THERMAL LAYER – NO DUST RSM CYCLE 4688. TIME 700.000 MSEC PROBLEM 30728.230

Fig. 14.30 Close-ups of the effects of mild down and up slope terrain, 0.7 s with thermal layer

relations or from scaling an ideal surface result. (The R-H relations should not be expected to hold here because the overpressure and dynamic pressure peaks are at different locations in the flow field.) At the present time there are relatively simple predictive models for the temperature in the thermal layer as a function of the incident flux history. This can be combined with variable surface heights as a function of range and azimuthal angle to provide boundary conditions for modern three dimensional CFD code calculations.

14.4.3.2

Organic Surface Material

The previous section briefly reviewed some of the effects of a thermal layer over dusty surfaces. The thermal layer generated when organic material is on or above the surface changes the characteristics of the thermal layer dramatically. On the nuclear event MET, a 22 kt device detonated on a 400 ft tower, there were four different surfaces tested. These included: a “water” line, a desert line, a concrete line and an asphalt line. The water line was more like a mud line but had the response of a near ideal surface with little or no thermal effects. The data indicated that the precursor formation was stronger over the asphalt line than over the desert line. The precursor arrived sooner and the durations were longer over the asphalt surface. The dynamic pressure and dynamic impulse were higher on the desert line. How could this be? The major reason is that dust was suppressed over the asphalt line and, even though the temperatures and therefore the material velocities were higher over the asphalt, the dust contribution to the dynamic pressure on the desert line was greater than the contribution by the square of the velocity over asphalt.

14.4 Thermal Interactions

241

The temperature (sound speed) was greater over the asphalt because the volatile chemicals in the asphalt were released when the surface was heated and the temperatures due to thermal radiation were sufficient to ignite the volatile organics. The energy in the thermal layer therefore was increased by the energy released by the burning chemicals. Some of the solar furnace experiments mentioned earlier included organic material on the surface. If the organics were small sticks or dead brush, the thermal layers were enhanced. If the surface was covered with green grass, there was little heating of the near surface air layer. In addition to being hotter, the temperature in the thermal layer over organic materials is no longer decreasing with height above the surface. If we use the example of the asphalt layer, the neighborhood of any part of the surface is nearly uniformly heated. The gasses released from the surface expand upward and mix rather inefficiently with the oxygen in the air. When burning occurs, the flame is oxygen poor and a high percentage of soot (carbon) is generated. The distribution of temperature above the surface is more like that in a flame. The base is warm but the tip of the flame is hot. In the nuclear case, the dense smoke (soot) generated by the inefficient burning also absorbs the incident thermal radiation above the ground and increases the temperature above the surface. This layer may be 2–3 m thick with the peak temperature near the top of the layer. A dry hay field could generate a thermal layer that extends well beyond the radius that a desert thermal layer might grow. The temperatures would be higher and the layer much thicker than over a desert. The precursor blast in such a case could be extended to much lower overpressures and clean up might not occur until the half bar range. We might imagine a nuclear detonation over a dry pine forest. Directly below the detonation, much of the thermal radiation reaches the ground and a thermal layer is formed. As the distance from ground zero is increased the thermal radiation is absorbed in the canopy of the forest and the tops of the trees burn. Most of the radiation is absorbed in the canopy and little radiation gets to the ground. The thermal layer may be many meters thick with a relatively cool region in the lower 10 m near the surface. Such a temperature distribution would cause the blast wave to accelerate in the canopy and a shock wave would be generated both up and down from the hot layer.

14.4.4

Simulation of Thermal Layers

In Sect. 14.4.2 an example of a precursor was shown in which the high sound speed layer was generated by a helium filled region contained by a Mylar sheet. A similar method was used in the first large scale attempt at simulation of a precursor in a high explosive event. The General American Transportation Corporation (GATX) fielded an experiment using a helium filled balloon on the Distant Plain 1A shot conducted in Alberta Canada in the summer of 1966. At the time, the phenomena associated with precursor formation and propagation was not well understood and

242

14 Height of Burst Effects

the relative importance of dust and thermal effects were being debated. In the experiment a helium filled balloon in the form of a 50 ft long right circular cylindrical segment with a radius of 12.5 ft and a maximum height of 5 ft was placed 200 ft from ground zero. The width of the balloon at ground level was 20 ft. The explosive charge was a sphere of 20 tons (18,160 kg) of TNT at a height of burst of 85 ft. Positioning the balloon at a range of 200 ft insured that the triple point of the Mach stem would be higher than the top of the balloon. This was done so the interactions would be simplified to a single blast wave interacting with the high sound speed region. The predicted incident blast wave overpressure at the 200 ft range was 30 psi. The ideal blast wave would decay to 26 psi at the 250 ft range at the end of the helium bag. Overpressure and stagnation pressure gauges were placed inside the balloon at distances of 10, 25 and 40 ft from the leading end of the balloon with overpressure gauges at ground level and stagnation gauges at 2 ft above the surface. Remember that stagnation pressure is the sum of the overpressure and the dynamic pressure. An ideal 30 psi shock has a dynamic pressure of 16 psi and therefore a stagnation pressure of 46 psi. Good experimental data was collected on all gauges. Figure 14.31 shows the measured pressure waveforms from the Distant Plain precursor simulation experiment. The upper plot is the overpressure at 25 ft inside the balloon. The precursor pressure is about 10 psi and arrives 10 ms prior to the peak overpressure. The total or stagnation pressure shows the contribution of the dynamic pressure. The peak stagnation pressure reaches a peak of nearly 80 psi indicating a peak dynamic pressure in excess of 65 psi very near the time of minimum overpressure.

Fig. 14.31 Measured waveforms from distant plain precursor experiment

14.4 Thermal Interactions

243

Gauges 6 and 7 are the overpressure and total pressure at a point 40 ft inside the 50 ft balloon. The precursor overpressure has dropped to about 8 psi with the peak overpressure occurring some 16 ms later. The stagnation pressure reaches a peak of nearly 90 psi near the time of minimum overpressure thus indicating a dynamic pressure of about 70 psi. This 65–70 psi dynamic pressure is four times the value of the ideal wave at the same distance. The phasing of the overpressure and dynamic pressure correspond very closely to the explanation given in the shock tube example of Sect. 14.4.2. Many precursor experiments have been conducted since 1966. Experiments at scales from 0.5 g of explosive to 100 tons of TNT equivalent have used a variety of methods to generate the high sound speed layer. Helium filled balloons have been popular. The advantage of the helium balloon is that the sound speed can be well controlled from about three times ambient sound speed in pure helium to ambient sound speed for air in the balloon. Light Mylar curtains can divide the helium concentrations, thus providing a means of custom designing the effective temperature of the layer as a function of position. A major problem with Mylar balloons is the mass of the Mylar. A 2 mil Mylar sheet has the same mass per unit area as a foot of helium gas. A 2 mil Mylar sheet has twice the mass of the helium in a six in thick layer. While the helium gas provides an excellent simulation technique, the Mylar has a significant effect on the flow field in the precursor region. Examples of attempts to generate thermal layers without a container mass are numerous. These range from electrically heated surfaces, to thin sheet explosives and thermite layers to diffusion of helium through porous ceramic plates. The high explosives and thermites have significant mass that must be accounted for. Layers formed with high density energetic materials are also difficult to control and very difficult to customize. Both the electrically heated plate and helium diffusion techniques were developed at the Ernst Mach institute by Dr. Heinz Reichenbach and his associates in the mid 1980s. Figure 14.32 is a Schlieren photo of one of his experiments conducted in a shock tube at EMI. The helium gas was diffused through a ceramic plate in the floor of the shock tube and was siphoned off from the end of the experimental section of the tube, just above the floor. This technique provided a diffused helium layer with the highest sound speed near the floor of the tube and a decreasing sound speed with height. The thickness of the layer and the concentration of helium could be controlled by the rate of helium feed in and the siphoning rate. Although the ceramic diffuser was relatively smooth, multiple shocks were generated in the high dynamic pressure region at the bottom of the vortex and in the stagnation region near the surface behind the vortex flow. These shocks can be seen in the photograph of Fig. 14.32 as the curved signals tracking up and behind the slip line. Notice how clearly the slip line is defined. The weak concave upward shock is a reflection from the top of the shock tube. This experiment simulates the effects of a thermal precursor over a roughened surface. In the late 1990s, it was suggested that the LB/TS (the Large Blast and Thermal Simulator) could be used to generate a full scale precursor blast environment.

244

14 Height of Burst Effects

Photo Courtesy of Dr. Heinz Reichenbach, Ernst Mach Institute, Freiburg, Germany

Fig. 14.32 Precursor flow structure using an active diffused helium layer

Fig. 14.33 Precursor simulation in an 11 m radius shock tube

The LB/TS generates realistic blast waves for kiloton to megaton nuclear detonations in the 5–30 psi range. The idea was to modify these realistic blast waves using a helium layer which would then provide a precursor environment for full scale military equipment testing. A Mylar sheet was spread across the 22 m wide floor of the LB/TS and fastened to walkways on either side of the tube 6 in. (15 cm) above the floor of the tube. This Mylar bag extended from about 100 m prior to the test section to a few meters beyond the test section. The Mylar sheet was 22 m by 120 m. Figure 14.33 shows the calculated precursor blast configuration for a 30 psi incident shock simulating approximately a 10 kt detonation. The technique was never used for equipment testing because the precursor run up distance was about

References

245

one order of magnitude shorter than needed to get the duration of the precursor to realistic levels. A greater run up distance was not possible because the precursor shock was already approaching the roof of the facility and the reflected precursor shock would have generated improper loadings on the equipment being tested. At the 30 psi level, this method generated a precursor duration of 47 ms, whereas the Priscilla event measurements indicated 350 ms. Thus the simulation was more than a factor of seven shorter than duration needed for a realistic simulation.

14.4.4.1

Decursor Simulation

When the shock arrives prior to the peak overpressure, the blast phenomena have been called “precursors”. As we described in Sect. 14.3, a layer of snow can delay the arrival of the shock at the surface, thus the label “decursor”. It is not always possible to use snow in simulations so a search was launched to find a suitable substitute that would have at least some properties of snow under blast loading conditions. A few of the simulants tested were: very light weight concrete, Styrofoam balls, hollow glass beads, mixes of Styrofoam and concrete. Each of these had problems which prevented their use. The concrete and glass simulants compress to a density of about 2 gm/cc, rather than 1 for snow. Because of the high compressed density, too much energy would be transferred into the ground. The Styrofoam compressed to about the right density but burned and gave off noxious gasses when heated and was considered a potential fire hazard; not a good thing at an explosive test site. Some experiments were conducted with sulfur hexafluoride (SF6). This gas has a higher density than air and a much lower gamma than air. The resulting sound speed is about 1/3 of the sound speed of air. SF6 is very compressible, has about the right sound speed and could be contained in Mylar membranes. The primary reasons for simulating the decursor is that higher peak shock pressures are generated and much higher impulses for pressures above a damaging level are generated.

References 1. Brode, H.L.: Height of Burst Effects at High Overpressures, DASA 2506, Defense Atomic Support Agency, July, (1970) 2. Kuhl, A.L. and Scheklinski-Gluck, G.: On the Reflection of Spherical Blast Waves from Smooth and Rough Surfaces, Published in FESTSCHRIFT zum 65. Geburtstag von Dr. re. nat. Heinz Riechenbach by the Fraunhofer- Institute f€ ur Kurzzeitdynamik, >Ernst 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

248

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

1120

1725

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

0

20

70

95

170

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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

15 Structure Interactions

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

15 Structure Interactions

Maximum Overpressure (PSI)

50 45

7000

40

6000

35 5000

Z (ft.)

30 25

4000

20

3000

15

2000

10

1000

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

15 Structure Interactions

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

15 Structure Interactions

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

100. 1

40

70 100 130 160 190 220 250 280 310 340

399

90.

400 380 360 340

80.

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70.

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RANGE (Y) M

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50.

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10.

20.

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40.

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80.

90.

1 100.

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

15 Structure Interactions

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

0.48

360 340 310 270 230 190 150 110 70

0.36

range (y) km

0.24 0.12

1

0.00 –0.12 –0.24 –0.36 –0.48 –0.60 0.70

400 390 380

0.82

0.94

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

15.6 The Influence of Rigid and Responding Structures

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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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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

264

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

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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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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.

15.6 The Influence of Rigid and Responding Structures

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

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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

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15 Structure Interactions

102

101

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10 –1

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

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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.

16 External Detonations

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.

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16 External Detonations

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

16 External Detonations

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

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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

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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.

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16 External Detonations

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

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

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

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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

17 Internal Detonations

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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.

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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.

17 Internal Detonations

285

Fig. 17.6 Mach 1.31 shock through a baffle system, t ¼ 234.3 ms

Fig. 17.7 Mach 1.31 shock through a baffle system, t ¼ 354.3 ms

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

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17 Internal Detonations

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

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17 Internal Detonations

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

17 Internal Detonations

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

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18 Simulation Techniques

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

18 Simulation Techniques

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.

C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_19, # Springer-Verlag Berlin Heidelberg 2010

303

304

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

19 Some Notes on Non-ideal Explosives

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)

19.2 Combustion or Afterburning Dependency of Non-ideal Explosives

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

19.2 Combustion or Afterburning Dependency of Non-ideal Explosives

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.

19.2 Combustion or Afterburning Dependency of Non-ideal Explosives

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

19 Some Notes on Non-ideal Explosives

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

C.E. Needham, Blast Waves, Shock Wave and High Pressure Phenomena, DOI 10.1007/978-3-642-05288-0_20, # Springer-Verlag Berlin Heidelberg 2010

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

20 Modeling Blast Waves

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

20 Modeling Blast Waves

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

20 Modeling Blast Waves

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

20 Modeling Blast Waves

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

20 Modeling Blast Waves

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

20 Modeling Blast Waves

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

20.5 Blast Propagation Through an Opening in a Wall

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

20 Modeling Blast Waves

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

20.5 Blast Propagation Through an Opening in a Wall

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

20 Modeling Blast Waves

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