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Heat waves in an electrothermal shock tube Armstrong, Bruce Allan 1978

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HEAT WAVES IN AN ELECTROTHERMAL SHOCK TUBE by BRUCE ALLAN ARMSTRONG B.E., U n i v e r s i t y o f Saskatchewan, 1972 M . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department o f P h y s i c s ) We a c c e p t t h i s t h e s i s as co n f o r m i n g t o the r e g u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J a n u a r y , 1978 © Bruce A l l a n A r m strong, 1978 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f i nanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 1 1 ABSTRACT The f l u i d f l o w i n an e l e c t r o t h e r m a l shock tube powered w i t h a c o n s t a n t c u r r e n t was compared t o t h a t p r e d i c t e d by a heat wave model and found t o be i n q u a l i t a t i v e agreement. An e m p i r i c a l h e a t i n g c h a r a c t e r i s t i c , o b t a i n e d e x p e r i m e n t a l l y from gas d y n a m i c a l measurements, was v e r i f i e d w i t h s p e c t r o s c o p i c and v e l o c i t y measurements i n d i c a t i n g the f l o w i s a l s o i n q u a n t i t a t i v e agreement w i t h the model. U s i n g a two s t e p c u r r e n t p u l s e we l a u n c h e d d o u b l e shock waves i n argon w i t h a s u b s o n i c heat wave. The i n t e r a c t i o n o f t h e s e shock waves agreed v e r y w e l l w i t h t h e o r y . I n c r e a s i n g the c u r r e n t i n the second s t e p c r e a t e d f a s t e r second shock waves u n t i l some l i m i t i n g v a l u e o f the c u r r e n t was r e a c h e d , a f t e r which a second shock c o u l d no l o n g e r be produced. R a t h e r , a s u p e r s o n i c heat wave was c r e a t e d which burned t h r o u g h the p r e v i o u s l y shock-compressed m a t e r i a l . The v e l o c i t i e s o f the second shocks were i n r e a s o n a b l e q u a n t i t a t i v e agreement w i t h p r e d i c t i o n s o f the heat wave t h e o r y , i n d i c a t i n g a g a i n the f l o w i n the shock tube i s w e l l d e s c r i b e d by the model. I t was , shown t h a t the r e s u l t i n g f l u i d f l o w , a f t e r a l l the waves had i n t e r a c t e d , d i d not depend on the h i s t o r y o f the power p u l s e , o n l y on i t s f i n a l magnitude. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES v LIST OF TABLES v i i i ACKNOWLEDGEMENTS i x CHAPTER I INTRODUCTION 1 CHAPTER I I THE PHYSICAL MODEL AND THE THEORY 8 A. ) The P h y s i c a l Model o f Heat Waves 9 B. ) Theory o f Heat Waves w i t h C o n s t a n t Power Input ... 15 C. ) Heat Waves w i t h a Two Step Power Input 23 D. ) The A p p l i c a b i l i t y o f the Heat Wave Model t o the E l e c t r o t h e r m a l Shock Tube 27 E. ) Assumptions o f the Theory 33 F. ) E l e c t r o t h e r m a l Shock Tubes as S i m u l a t o r s o f Other Inhomogeneous Energy A b s o r p t i o n Problems 36 CHAPTER I I I THE EXPERIMENTAL APPARATUS 39 A. ) The Shock Tube 41 B. ) The Power Supply 44 C. ) D i a g n o s t i c s 47 D. ) A u x i l i a r y Equipment and E x p e r i m e n t a l P r o c e d u r e ... 52 CHAPTER IV FLUID FLOW IN AN ELECTROTHERMAL SHOCK TUBE WITH CONSTANT POWER INPUT 54 A. ) Check o f Heat Wave Model Assumptions and Q u a l i t a t i v e V e r i f i c a t i o n o f the Theory 55 B. ) The D e t e r m i n a t i o n o f an E m p i r i c a l H e a t i n g C h a r a c t e r i s t i c 67 C. ) Q u a n t i t a t i v e V e r i f i c a t i o n o f the H e a t i n g C h a r a c t e r i s t i c 72 D. ) L i m i t i n g Heat F r o n t s 75 i v Page CHAPTER V STEPPED CURRENT EXPERIMENTAL RESULTS 79 A. ) Double Shocks 80 B. ) Burn Through Waves 89 CHAPTER VI CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK ... 94 A. ) The A p p l i c a b i l i t y o f the Heat Wave Model to the Flow i n an E l e c t r o t h e r m a l Shock Tube 94 B. ) The E l e c t r o t h e r m a l Shock Tube as a S i m u l a t o r o f Inhomogeneous Energy A b s o r p t i o n Phenomena 96 C. ) S u g g e s t i o n s f o r F u t u r e Work and O r i g i n a l C o n t r i b u t i o n s 97 BIBLIOGRAPHY 99 APPENDIX 1 RESULTING SHOCK VELOCITY 102 APPENDIX 2 METAL SHOCK TUBE 104 APPENDIX 3 TRANSMISSION LINES .106 APPENDIX 4 THE PRESSURE PROBE 113 APPENDIX 5 RESISTANCE OF THE SHOCK TUBE 116 APPENDIX 6 POWER DELIVERED TO THE HEAT FRONT 118 V LIST OF FIGURES F i g u r e Page 1 Heat wave w i t h p r e d e d i n g shock wave produced i n an e l e c t r o t h e r m a l shock tube 2 2 Schematic o f a heat wave w i t h o u t f l u i d motion 10 3 Schematic o f a s u b s o n i c heat wave w i t h i d e a l i z e d p r e s s u r e , d e n s i t y t e m p e r a t u r e , and p a r t i c l e v e l o c i t y p r o f i l e s . S=shock, HW=subsonic heat wave 12 4 Schematic o f a s u p e r s o n i c heat wave w i t h i d e a l i z e d p r e s s u r e , d e n s i t y , t e m p e r a t u r e , and p a r t i c l e v e l o c i t y p r o f i l e s . HW=supersonic heat wave, RW=rar e f a c t i o n wave 13 5 Response p l a n e w i t h l o c i : o f c o n s t a n t shock v e l o c i t y , c o n s t a n t heat wave v e l o c i t y , and c o n s t a n t p r e s s u r e r a t i o s 20 6 Schematic o f a double shock f l u i d f l o w w i t h p r e s s u r e , d e n s i t y , t e m p e r a t u r e , and p a r t i c l e v e l o c i t y p r o f i l e s . S 1 = f i r s t shock, S 2=second shock, H W R = r e s u l t i n g heat wave 25 7 W2 v e r s u s the second shock Mach number, w i t h the Mach number of the f i r s t shock as a parameter 26 8 The Mach numbers of the r e s u l t i n g shock v e r s u s the second shock w i t h the Mach number of the f i r s t shock as a parameter 28 9 An expected temperature p r o f i l e o f a one d i m e n s i o n a l plasma produced w i t h energy i n p u t a t the r i g h t end . 31 10 A comparison between a l a s e r c o m p r e s s i o n f l o w and the f l o w i n an e l e c t r o t h e r m a l shock tube 37 11 B l o c k diagram o f the e x p e r i m e n t a l f a c i l i t i e s 40 12 A t y p i c a l g l a s s shock tube and an e l e c t r o d e 42 13 O s c i l l o g r a m o f c u r r e n t p u l s e s from the t r a n s m i s s i o n . l i n e s . (a) bank I o n l y , (b) bank I I o n l y , and (c) both banks 45 14 Schematic o f the t r a n s m i s s i o n l i n e s 46 15 Temperature v e r s u s the l o g o f the f i l l i n g p r e s s u r e w i t h the e l e c t r o n d e n s i t y as a parameter 51 v i Figure Page 16 Framing photograph of a subsonic heat wave and i t s associated shock. At=10 //sec, p=5 t o r r , 1 = 20 KA . . . 56 17 Framing photograph of a supersonic heat wave. At=5 /usee, p=.6 t o r r , 1 = 22 KA 58 18 Framing photograph of a shock wave separating from a heat wave. <At=5 psec, p=2.5 t o r r , 1=28 KA 59 19 Streak photograph of a subsonic heat wave with shock r e f l e c t e d from a p i s t o n . S=shockwave, HW=subsonic heat wave, P=piston at x=27 cm 61 20 Streak photograph of a supersonic heat wave r e f l e c t e d from a p i s t o n . HW=supersonic heat wave, P=piston at x = 27 cm 62 21 The f i l l i n g pressure versus current for the t r a n s i t i o n p o i n t between the sonic and supersonic regimes 65 22 Oscillogram of a pressure p r o f i l e of a subsonic heat wave and shock wave 67 23 An e m p i r i c a l heating c h a r a c t e r i s t i c obtained from gas dynamical measurements 70 24 E l e c t r o n d e n s i t y as a f u n c t i o n of time, d i s t a n c e from the e l e c t r o d e s , and current 74 25 The log of the v e l o c i t y of the l i m i t i n g f r o n t versus the normalized i n t e n s i t y d e l i v e r e d to the f r o n t .... 77 26 Schematic of a streak photograph of a double shock. S-j = f i r s t shock, S2=second shock, S R = r e s u l t i n g shock, CS=contact s u r f a c e , RW=rarefaction wave, HW R=resulting subsonic heat wave, HW^=first heat wave 82 27 Streak photograph of a double shock. p=5 t o r r , If =15 KA, I 2=15 KA, At=50 £<sec, =5.7, M2=1.6, MR=9.7 ... 83 28 Oscillogram of a pressure p r o f i l e of double shocks followed by a subsonic heat wave 85 v i i F i g u r e Page 29 R a t i o o f the powers i n the stepped power p u l s e v e r s u s the Mach number o f the second shock. The c u r v e i s a computed v a l u e assuming the e n t h a l p y i s c o n s t a n t a t 3.5xl0 1 ergs/gram 86 30 Schematic o f a s t r e a k photograph o f a burn t h r o u g h . S - j = f i r s t shock, HW=subsonic heat wave 90 31 S t r e a k photograph o f a burn t h r o u g h . I-j =12 KA, I 2=15 KA, At=50 ysec 91 32 Schematic o f a s t r e a k photograph o f one shock wave o v e r t a k i n g another and the r e s u l t i n g f l u i d f l o w . S - | = f i r s t shock, S 2=second shock, S R = r e s u l t i n g shock, CS=contact s u r f a c e , R W = r a r e f a c t i o n wave 103 33 Photograph o f the m e t a l shock tube and an e l e c t r o d e s e c t i o n . PP=pressure p r o b e , AP=aluminum p l u g , E S = e l e c t r o d e s e c t i o n 105 34 Computed c u r r e n t waveforms. (a) c o n s t a n t c h a r g i n g v o l t a g e , (b) stepped c h a r g i n g v o l t a g e , (c) squared c h a r g i n g v o l t a g e , and (d) e x p o n e n t i a l c h a r g i n g v o l t a g e . N i s the number o f c a p a c i t o r - i n d u c t o r p a i r s i n the t r a n s m i s s i o n l i n e 107 35 Photograph o f an i n d u c t o r s i m i l a r i n d e s i g n t o a B i t t e r Magnet 110 36 Photograph o f the i n d u c t o r components. I = i n s u l a t o r , R=brass r i n g , CR<| and CR 2 =connecting r i n g s , EP=end p l a t e I l l 37 The p r e d i c t e d p r e s s u r e s behind shock waves v e r s u s the p r e s s u r e probe's s i g n a l 114 38 R e s i s t a n c e o f the shock tube v e r s u s c u r r e n t w i t h the f i l l i n g p r e s s u r e as a parameter 117 39 The l o g o f the shock v e l o c i t y v e r s u s the i n p u t power f o r f i l l i n g p r e s s u r e s o f 2, 5, and 10 t o r r . The s o l i d c u r v e s are computed v a l u e s . The s l o p e s o f the c u r v e s are g i v e n i n the b a l l o o n s 119 40 The l o g o f the shock v e l o c i t y v e r s u s the l o g o f the f i l l i n g p r e s s u r e f o r a c o n s t a n t c u r r e n t o f 20 KA ... 121 v i i i LIST OF TABLES T a b l e Page 1 C o m b i n a t i o n o f two shocks which produce a Mach 8 r e s u l t i n g shock and r e l a t i v e powers r e g u i r e d t o produce the c o m b i n a t i o n 88 2 The r e s u l t i n g shock speed produced by c u r r e n t s 1^  and I 2 whose sum i s 25 KA 88 i x ACKNOWLEDGEMENTS I w i s h t o e x p r e s s p r o f o u n d g r a t i t u d e t o my s u p e r v i s o r , Boye A h l b o r n , f o r s u g g e s t i n g t h i s work and f o r h i s c o n s t a n t h e l p and encouragement d u r i n g i t s c o m p l e t i o n . I would l i k e t o thank a l l members o f the plasma p h y s i c s group f o r t h e i r f r i e n d s h i p and the h e l p they have g i v e n me w i t h the e x p e r i m e n t . Frank Curzon has been e s p e c i a l l y h e l p f u l d u r i n g my d i s c u s s i o n s w i t h him. I would a l s o l i k e t o thank P h i l G r e g o r y , P e t e r R a s t a l l , and Greg Fahlman f o r t h e i r comments on t h i s document. Marj K i l / i a n and D a r y l Pawluk have h e l p e d me w i t h computing problems and Hannes Barnard has p r o v i d e d a program o f h i s own. A l a n Cheuck and Don Olson have been v e r y good i n d e s i g n i n g and m a i n t a i n i n g e l e c t r o n i c equipment. I a l s o w i s h t o thank the members o f the main machine shop and Cy Sedger of the s t u d e n t shop. Johnny Lees and E r n i e W i l l i a m s have been v e r y p a t i e n t when asked t o make s e e m i n g l y innumberable shock t u b e s . I g r a t e f u l l y acknowledge f i n a n c i a l s u p p o r t from the N a t i o n a l R esearch C o u n c i l , the K i l l a m s c h o l a r s h i p f u n d , and the plasma p h y s i c s group. T h i s work has been su p p o r t e d by g r a n t s from the N a t i o n a l Research C o u n c i l . 1 CHAPTER I INTRODUCTION Many n a t u r a l l y o c c u r r i n g and e x p e r i m e n t a l l y c r e a t e d phenomena are a r e s u l t o f inhomogeneous energy a b s o r p t i o n . These phenomena, o f t e n encountered when plasmas are produced, are c h a r a c t e r i z e d by a l o c a l i z e d d e p o s i t i o n o f energy which produces a p r e s s u r e g r a d i e n t , which i n t u r n a c c e l e r a t e s p a r t i c l e s away from the r e g i o n . When the power i s h i g h enough, a f a i r l y s h a r p l y d e f i n e d luminous r e g i o n i s formed which expands and h e a t s the s u r r o u n d i n g gas. T h i s expanding luminous r e g i o n w i l l be c a l l e d a heat wave. E v e n t u a l l y a shock wave forms, the time o f f o r m a t i o n depending upon the h i s t o r y and. magnitude of the energy d e p o s i t e d and the p r o p e r t i e s o f the m a t e r i a l . A photograph o f a heat wave and i t s a s s o c i a t e d shock, produced i n our e l e c t r o t h e r m a l shock t u b e , i s shown i n f i g . 1. Some b e t t e r known examples i n which inhomogeneous energy 1 9 "3 A a b s o r p t i o n are encountered are l a s e r s p a r k s i n gases ' ' ' , l a s e r plasmas a t s o l i d t r a g e t s 5 ' 6 ' 7 ; 8, HII r e g i o n s around hot e n e r g y u o to -i-> CD D > (D O f l ow F i g . 1. Heat wave with preceding shock wave produced i n an el e c t r o t h e r m a l shock tube. young s t a r s 9 ' 1 0 , and elec t r o t h e r m a l shock tubes 1^ . In order to understand these problems i t i s necessary to understand the hydrodynamics associated with a l o c a l i z e d d e p o s i t i o n of energy. The f i r s t three examples are d i f f i c u l t to study for various reasons. In the l a s e r experiments the time and space scales are nanoseconds and microns making time and space resolved measurements d i f f i c u l t to o b t a i n . In a d d i t i o n , i n the case of l a s e r plasmas near s o l i d t a r g e t s , v i s i b l e l i g h t cannot be used for d i a g n o s t i c s as the de n s i t y i s too high. A study of the e v o l u t i o n of an HII region would take sev e r a l tens of thousands of years. In a l l three examples noted, there i s l i t t l e or no c o n t r o l over the t o t a l amount of energy deposited or the shape of the power pulse which s u p p l i e s the energy. The a n a l y s i s of 3 the experimental r e s u l t s i s also complicated by the time varying power input, the poorly defined boundary c o n d i t i o n s , and the three dimensional geometries in v o l v e d . In c o n t r a s t , an e l e c t r o t h e r m a l shock tube i s a simple and convenient device for studying heat wave phenomena as the time and space s c a l e s are microseconds and centimetres. The shock tube produces a one dimensional flow with a w e l l defined boundary c o n d i t i o n . In a d d i t i o n , one has very good c o n t r o l over the shape and magnitude of the power pulse. By supplying constant current one can produce a quasi-steady f l u i d flow which aids i n understanding the p h y s i c a l processes involved and s i m p l i f i e s the theory and data a n a l y s i s . I f a s t e p - l i k e current pulse i s a p p l i e d , one can produce m u l t i p l e shock phenomena. This i s of i n t e r e s t since the promise of l a s e r f u s i o n to reach i g n i t i o n c o n d i t i o n s with l a s e r energies of the order of k i l o j o u l e s , hinges upon the assumption that i t i s p o s s i b l e to compress a deuterium t r i t i u m t a r g e t s e v e r a l orders of magnitude 12 1 3' 1 4 by means of m u l t i p l e weak shock waves ' 1 . One f i n a l advantage i n using an e l e c t r o t h e r m a l shock tube i s that i t i s r e l a t i v e l y inexpensive as compared to l a s e r experiments. Because of the r e l a t i v e s i m p l i c i t y and f l e x i b i l i t y that an e l e c t r o t h e r m a l shock tube o f f e r s , i t i s an a t t r a c t i v e t o o l to use i n the study of inhomogeneous energy absorption problems. The i n t e r p r e t a t i o n of the flow f i e l d i n an e l e c t r o t h e r m a l ' shock tube as a heat wave phenomenon c l a r i f i e s many c o n t r a d i c t o r y observations. The heat wave model described i n t h i s t h e s i s b u i l d s on previous work c a r r i e d out i n t h i s l a b o r a t o r y but d i f f e r s from other i n t e r p r e t a t i o n s of the flow 4 p r e v i o u s l y r e p o r t e d i n the l i t e r a t u r e . E l e c t r o t h e r m a l shock waves were not i n i t i a l l y u n d e r s t o o d because a l l f a s t luminous f r o n t s were i d e n t i f i e d as shock f r o n t s . T h i s was not u n r e a s o n a b l e s i n c e a Mach 50 shock f r o n t would e x c i t e the gas t o l u m i n e s c e n c e . I t was soon d i s c o v e r e d , however, t h a t the plasma v a r i a b l e s measured behind the v e r y f a s t f r o n t s d i d not agree w i t h the v a l u e s c a l c u l a t e d from shock t h e o r y 1 5 •>1 6 . Attempts t o e x p l a i n the d e v i a t i o n s by t a k i n g p r e c u r s o r e f f e c t s and r e l a x a t i o n i n t o account d i d not y i e l d c o n s i s t e n t r e s u l t s " ' '° 1 ' Cloupe au and o t h e r s found t h a t the s l o w e r moving luminous f r o n t s , M<20, were preceded by non-luminous f r o n t s which were l a t e r i d e n t i f i e d as shock f r o n t s 2 0 , > 2 1 . In a d d i t i o n , i t was demonstrated w i t h K e r r c e l l and image c o n v e r t e r p h o t o g r a p h s , t h a t the luminous f r o n t s i n i t i a l l y i n t e r p r e t e d i n s t r e a k photographs as shock f r o n t s , had an i r r e g u l a r s t r u c t u r e which i s i n c o m p a t i b l e w i t h the s t a b i l i t y o f a shock f r o n t 2 2 ' 2 3 . The l e a d i n g edges o f s l o w l y moving luminous f r o n t s w i t h p r e c e d i n g shock waves were b e l i e v e d t o be c o n t a c t s u r f a c e s , i . e . s u r f a c e s a c r o s s which the p r e s s u r e and p a r t i c l e v e l o c i t y a re c o n s t a n t but the temperature and d e n s i t y are d i s c o n t i n u o u s . The v e r y f a s t luminous f r o n t s were b e l i e v e d t o be shock f r o n t s w i t h an energy i n p u t . The i r r e g u l a r i t i e s o f t h e s e shock f r o n t s o 3 ?4 2 5 were e x p l a i n e d by R a y l e i g h - T a y l o r i n s t a b i l i t i e s 1 1 . A f t e r i t was d i s c o v e r e d t h a t shock tubes d i d not produce the d e s i r e d f l o w , l i t t l e work was done on them so the e x p l a n a t i o n o f the f l o w has changed l i t t l e i n the l a s t 10 y e a r s . The observed phenomena can be e x p l a i n e d n a t u r a l l y w i t h the heat wave model w i t h o u t h a v i n g t o i n v o k e p r e c u r s o r e f f e c t s , 5 R a y i e i g h - T a y i o r i n s t a b i l i t i e s , e t c . C e n t r a l t o t h i s model i s the concept o f a h e a t i n g c h a r a c t e r i s t i c , i . e . a r e l a t i o n s h i p between the i n p u t i n t e n s i t y , W, and the e n t h a l p y i n the luminous r e g i o n , h. T h i s h e a t i n g c h a r a c t e r i s t i c , c o u l d , i n p r i n c i p l e , be o b t a i n e d from a l a b o r i o u s e x a m i n a t i o n o f the energy s o u r c e and the a p p r o p r i a t e a b s o r p t i o n m e c h a n i s m s 2 6 . I t can be o b t a i n e d d i r e c t l y from a measurement o f the s p e c i f i c power f l u x W/P., , and 2 7 the f i n a l t e m perature . We w i l l show i t a l s o can be o b t a i n e d from gas d y n a m i c a l measurements. Once d e t e r m i n e d , the h e a t i n g c h a r a c t e r i s t i c can be used t o p r e d i c t p r o p e r t i e s o f the f l u i d f l o w , e.g. the maximum p o s s i b l e shock speed, o r , i f the net power i n p u t i s known, the p r e s s u r e s , v e l o c i t i e s and t e m p e r a t u r e s of the f l o w f i e l d . The work p r e s e n t e d i n t h i s t h e s i s i s a s t u d y o f the f l u i d f l o w produced i n an e l e c t r o t h e r m a l shock tube w i t h c o n s t a n t and stepped c u r r e n t i n p u t . The c o n s t a n t c u r r e n t e x p e r i m e n t s w i l l show t h a t the hydrodynamics are w e l l d e s c r i b e d by the heat wave model. The s t e p - l i k e c u r r e n t e x p e r i m e n t s w i l l demonstrate t h a t the shock tube can be used t o s i m u l a t e a s p e c t s o f o t h e r heat wave phenomena. A l t h o u g h the r e s u l t s o f the work are unique t o the shock tube s t u d i e d , the c o n c l u s i o n s drawn from the r e s u l t s are o f g e n e r a l a p p l i c a b i l i t y as they c l e a r l y i l l u s t r a t e many f e a t u r e s a s s o c i a t e d w i t h a l l heat waves. In c h a p t e r I I , a p h y s i c a l model o f a s t e p heat wave i s e x p l a i n e d b e f o r e the e q u a t i o n s which d e s c r i b e i t are p r e s e n t e d . From the model and the t h e o r y i t w i l l be seen t h a t t h e r e are two g e n e r a l c l a s s e s o f heat waves, the " s u b s o n i c " w i t h i t s a s s o c i a t e d shock, and the " s u p e r s o n i c " which has no shock wave 6 a s s o c i a t e d w i t h i t . U s i n g the t h e o r y , an e x p e r i m e n t a l method o f d e t e r m i n i n g an e m p i r i c a l h e a t i n g c h a r a c t e r i s t i c i s o u t l i n e d . I n the next two s e c t i o n s , we g i v e the reasons why we f e e l the f l o w i n an e l e c t r o t h e r m a l shock tube can be d e s c r i b e d by the heat wave model, and examine the r a m i f i c a t i o n s o f a d e p a r t u r e o f e x p e r i m e n t a l c o n d i t i o n s from those assumed i n the t h e o r y . F i n a l l y , a comparison o f an e l e c t r o t h e r m a l shock tube to a l a s e r c o m p r e s s i o n experiment i s made i n o r d e r t o i l l u s t r a t e t h e f e a t u r e s common t o both and t o heat waves i n g e n e r a l . T h i s comparison s u g g e s t s t h a t i t s h o u l d be p o s s i b l e t o l a u n c h m u l t i p l e shocks from a s u b s o n i c a b l a t i o n f r o n t i n an e l e c t r o t h e r m a l shock tube. Chapter I I I i s a d e s c r i p t i o n o f the e x p e r i m e n t a l a p p a r a t u s b u i l t t o t e s t whether the heat wave model, o u t l i n e d i n the p r e v i o u s c h a p t e r , can be p r o f i t a b l y a p p l i e d t o the f l o w i n our shock tube. The d i a g n o s t i c equipment and the e x p e r i m e n t a l p r o c e d u r e a re a l s o d e s c r i b e d i n t h i s c h a p t e r . Chapter IV i s an o u t l i n e o f the c o n s t a n t c u r r e n t e x p e r i m e n t a l r e s u l t s . The f i r s t s e c t i o n w i l l show t h a t the f l o w i n an e l e c t r o t h e r m a l shock tube i s i n q u a l i t a t i v e agreement w i t h the heat wave model. The r e m a i n i n g s e c t i o n s o f t h i s c h a p t e r demonstrate the f l o w i s a l s o i n q u a n t i t a t i v e agreement. To t h i s end, an e m p i r i c a l h e a t i n g c h a r a c t e r i s t i c i s determined e x p e r i m e n t a l l y from gas dy n a m i c a l measurements and i s v e r i f i e d w i t h s p e c t r o s c o p i c and v e l o c i t y measurements. For a f i x e d f i l l i n g p r e s s u r e t h e r e i s a l i m i t t o the i n p u t c u r r e n t , the "shock l i m i t " , w h i c h , i f exceeded, produces a s u p e r s o n i c heat wave and no shock. We s h a l l f i n d t h a t we cannot go f a r i n t o the 7 s u p e r s o n i c regime due to the energy t r a n s p o r t mechanism. The f l o w a t t h i s t r a n s i t i o n p o i n t i s examined and found t o be i n good q u a n t i t a t i v e agreement w i t h the f l o w p r e d i c t e d by the model. The p r e v i o u s c h a p t e r showed the heat wave model i s an adequate d e s c r i p t i o n o f the f l o w i n an e l e c t r o t h e r m a l shock tub e . Knowing t h i s , we can p r e d i c t the f l o w t h a t s h o u l d r e s u l t from a s t e p - l i k e power i n p u t . The r e s u l t s o f t h e s e e x p e r i m e n t s are p r e s e n t e d i n c h a p t e r V. We have shown t h a t i t i s p o s s i b l e t o produce e i t h e r two shock waves or a shock wave and a "burn-t h r o u g h " wave w i t h a two s t e p power i n p u t . The v e l o c i t i e s o f the second shock waves and the "shock l i m i t " c u r r e n t are i n q u a n t i t a t i v e agreement w i t h p r e d i c t i o n s based on the p r e v i o u s l y determined h e a t i n g c h a r a c t e r i s t i c . T h i s agreement i s another i n d i c a t i o n the heat wave model p r o v i d e s an adequate d e s c r i p t i o n o f the f l o w . The c r e a t i o n o f two shock waves w i t h a two s t e p power p u l s e shows the e l e c t r o t h e r m a l shock tube can be used t o s t u d y heat wave phenomena i n g e n e r a l . In c h a p t e r VI the c o n c l u s i o n s are summarized and s u g g e s t i o n s f o r f u t u r e work are g i v e n . My o r i g i n a l c o n t r i b u t i o n s are n o t e d . 8 CHAPTER I I THE PHYSICAL MODEL AND THE THEORY In t h i s c h a p t e r we p r e s e n t a p h y s i c a l and m a t h e m a t i c a l d e s c r i p t i o n o f a heat wave i n o r d e r t h a t we may a c h i e v e an u n d e r s t a n d i n g o f i t and det e r m i n e what f a c t o r s dominate i t s b e h a v i o u r . S e c t i o n A c o n t a i n s a d e s c r i p t i o n o f an i d e a l i z e d p h y s i c a l model t h a t c l e a r l y shows what parameters i n f l u e n c e the f l o w . U s i n g the model, i t i s shown t h a t the f l u i d f l o w can be d i v i d e d i n t o two regimes d i f f e r i n g from each o t h e r by the l a c k o f , or presence o f , a shock wave. A m a t h e m a t i c a l d e s c r i p t i o n o f the model i s p r e s e n t e d n e x t . S e c t i o n B d e a l s w i t h the c o n s t a n t power c a s e , w h i l e i n s e c t i o n C, the t h e o r y i s expanded to i n c l u d e a s t e p - l i k e power i n p u t . The two regimes mentioned i n s e c t i o n A are d e s c r i b e d g r a p h i c a l l y on the response p l a n e where the h e a t i n g c h a r a c t e r i s t i c can be c l e a r l y u n d e r s t o o d . S e c t i o n D e x p l a i n s why the t h e o r y developed i n the p r e v i o u s s e c t i o n s s h o u l d a p p l y t o the f l o w i n an e l e c t r o t h e r m a l shock tube. In s e c t i o n E, the i d e a l i z e d c o n d i t i o n s assumed i n the t h e o r y are 9 compared to the expected e x p e r i m e n t a l c o n d i t i o n s . The f i n a l s e c t i o n i n d i c a t e s why the e l e c t r o t h e r m a l shock tube can be used to s t u d y many phenomena c r e a t e d by inhomogeneous energy a b s o r p t i o n by making a d e t a i l e d comparison o f the shock tube and a l a s e r c ompression e x p e r i m e n t . CHAPTER I I SECTION A THE PHYSICAL MODEL OF HEAT WAVES In o r d e r t o understand heat waves, c o n s i d e r a v e r y s i m p l e experiment i n which c o l d gas, c o n t a i n e d i n a c l o s e d t u b e , i s sud d e n l y exposed t o an energy f l u x , W, e n t e r i n g the tube from the r i g h t t h r o u g h a window, f i g . 2. L e t us assume the c o l d gas i s opaque to the heat f l u x so the energy i s i n i t i a l l y absorbed i n a l a y e r o f gas next t o the window but. t h a t the gas becomes t r a n s p a r e n t t o the energy f l u x once i t has been h e a t e d . T h i s " b l e a c h i n g wave" b e h a v i o u r i s found i n many p h y s i c a l 2 8 29 s i t u a t i o n s ' , A f t e r the f i r s t l a y e r o f gas becomes t r a n s p a r e n t the second l a y e r o f gas can be h e a t e d . I n t h i s manner, the heat wave p r o p a g a t e s down the tube. To m a i n t a i n the s i m p l i c i t y o f the model, the f l o w i s assumed to be one d i m e n s i o n a l and no energy l o s s e s are c o n s i d e r e d . I t has a l s o been assumed t h a t the energy a b s o r p t i o n l e n g t h i s s h o r t i n comparison t o r e l e v a n t d i s t a n c e s . There are t h r e e f a c t o r s which a f f e c t the v e l o c i t y o f a heat wave, the d e n s i t y , P, i n t o which the heat wave t r a v e l s , the e n t h a l p y , h, the gas must a c h i e v e b e f o r e becoming t r a n s p a r e n t , and the i n t e n s i t y W. I f the i n i t i a l f i l l i n g d e n s i t y was h i g h e r . 10 F i g . 2. Schematic of a heat wave without f l u i d motion the heat wave would t r a v e l slower as there would be more gas to heat i n each l a y e r . S i m i l a r l y , i f the gas had to be heated to a higher temperature before i t became transparent, more energy per atom would be required and fewer p a r t i c l e s could be heated per u n i t time. Again, the heat wave would t r a v e l slower. The heat wave would t r a v e l f a s t e r i f the i n t e n s i t y was higher since more atoms could be heated per u n i t time. Eqn. 1 summarizes the three f a c t o r s . v (i) HW ph A c t u a l l y W/Pi and h are not independent parameters. Their r e l a t i o n s h i p w i l l be described l a t e r by the heating c h a r a c t e r i s t i c . Note that with high powers and/or low 11 d e n s i t i e s , i . e . h i g h W/0., , the v e l o c i t y o f the heat wave i s g r e a t e s t . The heat wave cannot be f u l l y u n d e r s t o o d i f the p a r t i c l e m o t i o n , induced by the p r e s s u r e g r a d i e n t , i s i g n o r e d . T h i s has u n f o r t u n a t e l y been done i n the d i s c u s s i o n o f "th e r m a l waves" by Z e l ' d o v i c h and R a i z e r 3 0 . I f the heat wave t r a v e l s s l o w l y , so t h a t i t s v e l o c i t y i s s u b s o n i c w i t h r e s p e c t t o the gas behind i t , the p r e s s u r e d i s t u r b a n c e , which t r a v e l s a t the speed o f s o u n d 1 , o v e r t a k e s the heat wave and precedes i t down the tube as a shock wave. I f the heat wave's v e l o c i t y i s s u p e r s o n i c w i t h r e s p e c t t o the gas behind i t , the p r e s s u r e d i s t u r b a n c e cannot o v e r t a k e the heat wave. I t f o l l o w s the heat wave i n the form o f a r a r e f a c t i o n wave. The two ty p e s o f heat waves are c a l l e d s u b s o n i c and s u p e r s o n i c i n accordance w i t h t h e i r exhaust and i n t a k e v e l o c i t i e s . F i g . 3 shows a s u b s o n i c heat wave, HW, and i t s a s s o c i a t e d shock, S. F i g . 4 shows a s u p e r s o n i c heat wave, HW, and i t s t r a i l i n g r a r e f a c t i o n wave, RW. P r e s s u r e , d e n s i t y , t e m p e r a t u r e , and p a r t i c l e v e l o c i t y p r o f i l e s a re drawn below the f i g u r e s . The n u m e r i c a l i d e n t i f i c a t i o n o f r e g i o n s w i l l be r e f e r r e d t o i n s e c t i o n B. The boundary c o n d i t i o n imposed by the window can a i d us i n u n d e r s t a n d i n g the i d e a l i z e d p r e s s u r e p r o f i l e s i n f i g s . 3 and 4. Next t o the window the p a r t i c l e v e l o c i t y i s z e r o . I f the heat wave i s s u b s o n i c , the p a r t i c l e v e l o c i t y must be zer o everywhere behind i t as no p r e s s u r e g r a d i e n t s can e x i s t which would c r e a t e A c t u a l l y , a s m a l l p r e s s u r e d i s t u r b a n c e t r a v e l s a t the speed o f sound. A l a r g e p r e s s u r e d i s t u r b a n c e p r o p a g a t e s a t a s u p e r s o n i c v e l o c i t y which depends on the magnitude o f the d i s t u r b a n c e . 12 HW RW 4 W PRESSURE DENSITY T E M P E R A T U R E PARTICLE V E L O C I T Y F i g . 3. Schematic of a subsonic heat wave with i d e a l i z e d pressure, d e n s i t y , temperature, and p a r t i c l e v e l o c i t y p r o f i l e s . S=shock, HW=subsonic heat wave. 13 S HW PRESSURE DENSITY T E M P E R A T U R E P A R T I C L E V E L O C I T Y F i g . 4. Schematic of a supersonic heat wave with i d e a l i z e d pressure, d e n s i t y , temperature, and p a r t i c l e v e l o c i t y p r o f i l e s . HW=supersonic heat wave, RW=rarefaction wave. 14 a d i f f e r e n c e i n p a r t i c l e v e l o c i t y . As a shock wave a c c e l e r a t e s p a r t i c l e s away from the window, t h e r e must be a p r e s s u r e drop a c r o s s the heat wave i n o r d e r t o b r i n g them to r e s t a g a i n . A s u p e r s o n i c heat wave w i l l a l s o a c c e l e r a t e p a r t i c l e s away from the window. The r a r e f a c t i o n wave, f o l l o w i n g the heat wave, i s r e s p o n s i b l e f o r b r i n g i n g the p a r t i c l e s t o r e s t i n t h i s c a s e . The d e n s i t y p r o f i l e s can be understood i f one r e a l i z e s t h a t b o t h the shock wave and the s u p e r s o n i c heat wave a c c e l e r a t e p a r t i c l e s away from the window. The d e n s i t y i m m e d i a t e l y behind them i s t h e r e f o r e h i g h e r than the f i l l i n g d e n s i t y . The m a t t e r swept up i s m i s s i n g from the r e g i o n next to the window and the d e n s i t y i n t h i s r e g i o n i s , t h e r e f o r e , lower than the f i l l i n g dens i t y . The temperature p r o f i l e s f o l l o w d i r e c t l y from the p r e s s u r e and d e n s i t y p r o f i l e s as the temperature i s p r o p o r t i o n a l t o the p r e s s u r e d i v i d e d by the d e n s i t y . The s u b s o n i c heat wave cannot be c o n s i d e r e d a c o n t a c t s u r f a c e s i n c e the p r e s s u r e and p a r t i c l e v e l o c i t y change a c r o s s i t and they do not a c r o s s a c o n t a c t s u r f a c e . E x p e r i m e n t a l e v i d e n c e p r e s e n t e d i n c h a p t e r IV shows t h e r e i s a p r e s s u r e drop a c r o s s the luminous f r o n t . T h i s c l e a r l y demonstrates t h a t the luminous f r o n t i s not a c o n t a c t s u r f a c e . Examining the s u b s o n i c heat wave from the l a b frame o f r e f e r e n c e , i t appears as a " l e a k y " p i s t o n . I t pushes matter ahead o f i t but l e t s some matter pass through i t due t o the p r e s s u r e d i f f e r e n c e . I n the s u b s o n i c heat wave's frame o f r e f e r e n c e , p a r t i c l e s s l o w l y e n t e r the h e a t i n g zone where they are heated and s i m u l t a n e o u s l y 15 a c c e l e r a t e d towards the energy s o u r c e , l e a v i n g w i t h i n c r e a s e d v e l o c i t y . The t h r u s t c r e a t e d from t h i s a c c e l e r a t i o n i s r e s p o n s i b l e f o r d r i v i n g the shock wave ahead o f the heat wave. One can a l s o t h i n k o f the l e a d i n g edge o f a s u b s o n i c heat wave as an a b l a t i o n f r o n t . The energy t r a n s p o r t e d f o r w a r d t h r o u g h the .heat wave t o the f r o n t a b l a t e s the shock-compressed m a t e r i a l q u i t e l i k e a h i g h power l a s e r a b l a t e s a s o l i d t a r g e t t h a t i t i s fo c u s s e d upon. There i s a c l a s s o f heat waves i n t e r m e d i a t e t o the s u b s o n i c and s u p e r s o n i c . I t o c c u r s over a narrow range o f parameters when the exhaust v e l o c i t y o f the heat wave i s s o n i c . The i n t a k e v e l o c i t y i s s t i l l s u b s o n i c . These heat waves have a shock ahead of them and p a r t o f a r a r e f a c t i o n wave a t t a c h e d t o the r e g i o n o f energy d e p o s i t i o n . The s o n i c heat wave w i l l be d i s c u s s e d i n more d e t a i l i n the f o l l o w i n g s e c t i o n . CHAPTER I I SECTION B THEORY OF HEAT WAVES WITH CONSTANT POWER INPUT A s p e c i a l case which f i t s the assumptions o f the model developed i n the p r e v i o u s s e c t i o n i s an i o n i z i n g r a d i a t i o n f r o n t i n hydrogen, where the p a r t i c l e s become t r a n s p a r e n t t o the i o n i z i n g r a d i a t i o n once the e l e c t r o n has been s t r i p p e d o f f . Steady r a d i a t i o n f r o n t s behind windows have been d i s c u s s e d by 31 A h l b o r n and Zuzak . The m a t h e m a t i c a l model developed i n t h a t r e f e r e n c e h o l d s f o r a l l heat waves i n g e n e r a l and i s t h e r e f o r e adopted h e r e . 16 The model i s based upon the c o n s e r v a t i o n e q u a t i o n s f o r mass, momentum and energy. P aV a " P b Vb (2) p a a F a b b b ( 3 ) l v 2 + h + -*L 2 a a p„V 1 Vb + hb a a (4) The s u b s c r i p t a r e f e r s t o q u a n t i t i e s ahead o f the d i s c o n t i n u i t y and b behind the d i s c o n t i n u i t y . R e f e r r i n g t o f i g . 3, a w i l l s t a n d f o r r e g i o n s [1] and [ 3 ] , and b f o r r e g i o n s [2] and [ 4 ] . I t i s n e c e s s a r y t o d i f f e r e n t i a t e between r e g i o n s [2] and [3] s i n c e the v e l o c i t i e s , V, o f the d i s c o n t i n u i t i e s are measured i n t h e i r own frame o f r e f e r e n c e . The o t h e r unknowns are i d e n t i c a l , e.g. p 3=P2• For a s u p e r s o n i c heat wave, f i g . 4, r e g i o n [1] i s i d e n t i c a l t o r e g i o n [3] so s u b s c r i p t a w i l l r e f e r o n l y t o [31 and s u b s c r i p t b t o [ 4 ] . P, 9 , and h are the e q u i l i b r i u m p r e s s u r e , d e n s i t y , and e n t h a l p y . No s o u r c e s o f mass or momentum, are i n c l u d e d i n the c o n s e r v a t i o n e q u a t i o n s . The e x t e r n a l source o f energy, r e p r e s e n t e d by W, i s zero f o r the shock f r o n t and p o s i t i v e f o r the heat f r o n t . A l l p h y s i c a l phenomena which can be d e s c r i b e d 17 by t h i s s e t o f c o n s e r v a t i o n e q u a t i o n s belong to the c l a s s o f heat waves. In a d d i t i o n t o the c o n s e r v a t i o n e q u a t i o n s , one needs an e q u a t i o n o f s t a t e r e l a t i n g h, p, and P . h = § £ (5) ( g - D P The c o e f f i c i e n t , g, e q u a l s the a d i a b a t i c exponent Y= c p / c v = 5/3 f o r argon when i t i s not a p p r e c i a b l y i o n i z e d and v a r i e s o n l y 32 s l i g h t l y as a f u n c t i o n o f p and P w i t h i n the l i m i t s l<g<Y . In a l l the c a l c u l a t i o n s t h a t f o l l o w , g i s assumed to be a c o n s t a n t . T h i s i s no r e s t r i c t i o n on g e n e r a l i t y s i n c e once the f i n a l s t a t e has been c a l c u l a t e d a new v a l u e o f g i s o b t a i n e d and the c a l c u l a t i o n s r e p e a t e d i f n e c e s s a r y . The i t e r a t i o n converges q u i c k l y 3 3. E l i m i n a t i n g the e n t h a l p y from eqn. 4 w i t h eqn. 5 and m a n i p u l a t i n g the c o n s e r v a t i o n e q u a t i o n s , we o b t a i n the r a t i o s o f v e l o c i t i e s , d e n s i t i e s , and p r e s s u r e s (6) \ P — = a a b 1 + 1 gb " I M 2 g, + 1 'a a 5b 2 g a \ 2 ( g b 2 - 1)W (Sa " " ^ W a 1 8 2 !b = 1 _ gb " g a M a  P a gb + 1 where the a b r e v i a t i o n s 2 g a 2 M a 2 ( g b 2 ' 1 3 W C§a " 1 ) C g b " g a M a 2 ) 2 p a V a h a (7) V 1 p V 2 M 2 = a = a a a 2 c g P and ^ a - ^ ^ b - ^ a 2 ^ have been used. M i s the Mach number o f the d i s c o n t i n u i t y measured w i t h r e s p e c t to the gas ahead o f i t . Eqns. 6 and 7 d e s c r i b e v a r i o u s t y p e s o f one d i m e n s i o n a l f l u i d f l o w . For example, the Rankine-Hugoniot r e l a t i o n s , which r e l a t e the f l u i d parameters behind the shock t o the Mach number of the shock, are o b t a i n e d i f we s e t W=0 and take the p o s i t i v e square r o o t s . There are f i v e unknowns i n the c o n s e r v a t i o n e q u a t i o n s , V a , v b ' P b ' Pb' a n d nb» w i s c o n s i d e r e d a g i v e n parameter o f the exp e r i m e n t . We have o n l y f o u r e q u a t i o n s , 2, 3, 4, and 5. The m i s s i n g e q u a t i o n , from which we c o u l d o b t a i n a h e a t i n g c h a r a c t e r i s t i c , would r e l a t e the i n t e n s i t y , W, t o the e n t h a l p y behind the heat wave, h4, by s p e c i f y i n g the energy t r a n s p o r t mechanisms and any atomic d a t a t h a t might be r e q u i r e d . The r e l a t i o n s h i p between h and W i s d i f f i c u l t t o o b t a i n from a 19 m i c r o s c o p i c view p o i n t . In our model i t w i l l be r e p l a c e d by an e m p i r i c a l r e l a t i o n . To compensate f o r the m i s s i n g e q u a t i o n , we must s p e c i f y one unknown and t h e n , a f t e r g i v i n g W, the o t h e r unknowns can be c a l c u l a t e d . The boundary c o n d i t i o n at the window, ex p r e s s e d i n eqn. 8, must a l s o be used i n the s u b s o n i c regime. \ = o = (v 3-v 1 +) + ( v r v 2 ) ( a ) When c o n s i d e r i n g - the heat f r o n t , the n e g a t i v e s i g n s i n eqns. 6 and 7 are used t o e l i m i n a t e e x p a n s i o n shocks. The r e s u l t s o f the c a l c u l a t i o n s are c o n v e n i e n t l y d i s p l a y e d on the response p l a n e , f i g . 5, which has the e n t h a l p y , h^, as the o r d i n a t e and W/Pf as the a b s c i s s a . In o b t a i n i n g f i g . 5, the i n i t i a l t e mperature was s e t e q u a l to 20 °C, w h i l e g 2 was s e t equal to g-j = 1.67 and g 4 e q u a l l e d 1.2. Drawn on the response p l a n e , which r e p r e s e n t s a l l p o s s i b l e f l u i d s t a t e s a c h i e v a b l e w i t h s t e p heat waves i n a tube s e a l e d at one end, are l o c i o f c o n s t a n t shock v e l o c i t y , c o n s t a n t heat wave v e l o c i t y , and c o n s t a n t p r e s s u r e (behind the heat wave). In e x p e r i m e n t a l s i t u a t i o n s i t i s v e r y d i f f i c u l t t o d e t e r m i n e W. T h e r e f o r e , i n p r a c t i c e two unknowns must be measured t o c o m p l e t e l y d e s c r i b e the f l o w . U s i n g the l o c i drawn on the response p l a n e , one can measure any two unknowns of the f l u i d f l o w , and, by n o t i n g where t h e i r l o c i i n t e r s e c t , determine the r e m a i n i n g unknowns. For example, " i f the shock and heat wave v e l o c i t i e s are 2.5 and .16 km/sec r e s p e c t i v e l y , the e n t h a l p y F i g . 5. Response p l a n e w i t h l o c i : ' o f c o n s t a n t shock v e l o c i t y , c o n s t a n t heat wave v e l o c i t y , and c o n s t a n t p r e s s u r e r a t i o s . 21 b e h i n d the heat wave i s about 6 . 5 x l 0 1 1 ergs/gram and the n o r m a l i z e d i n t e n s i t y needed to produce t h i s f l o w i s about 4 x l 0 1 6 erg-cm/gram-sec. Changing the i n i t i a l c o n d i t i o n s w i l l g i v e more p o i n t s on the response p l a n e . A l i n e j o i n i n g these p o i n t s i s c a l l e d the h e a t i n g c h a r a c t e r i s t i c . I f more than two unknowns o f the same f l o w are measured, the a p p l i c a b i l i t y o f the t h e o r y t o the f l o w b e i n g i n v e s t i g a t e d can be checked. T h i s was done f o r the f l o w i n an e l e c t r o t h e r m a l shock tube by measuring shock and heat, wave v e l o c i t i e s , p r e s s u r e s , and t e m p e r a t u r e s . The r e s u l t s are p r e s e n t e d i n c h a p t e r IV The response p l a n e i s d i v i d e d i n t o two reg i m e s , the s u p e r s o n i c and the ( s u b ) s o n i c . A p a r t i c u l a r l y s i m p l e h e a t i n g c h a r a c t e r i s t i c i n which the e n t h a l p y b e h i n d the heat wave does not change w i t h changing power i n p u t , i s r e p r e s e n t e d by ABCD. I f one s u p p l i e s the i n t e n s i t y r e p r e s e n t e d by p o i n t A, the heat wave w i l l t r a v e l s l o w l y and a shock wave w i l l precede i t . As the power i s i n c r e a s e d , the v e l o c i t y o f the s u b s o n i c heat wave i n c r e a s e s u n t i l a t some p o i n t the exhaust v e l o c i t y o f the heat wave i s j u s t s o n i c , p o i n t B. T h i s i s r e p r e s e n t e d m a t h e m a t i c a l l y by s e t t i n g the square r o o t s i n eqns. 6 and 7 e q u a l t o zer o w i t h the c o n d i t i o n t h a t M3<1. The l o c u s o f the s e p o i n t s s e p a r a t e s the s u b s o n i c and s o n i c regimes. As the power i s i n c r e a s e d s t i l l f u r t h e r , the exhaust v e l o c i t y remains s o n i c but a r a r e f a c t i o n wave, which i s a t t a c h e d t o the heat wave, d e v e l o p s . U4 i s no l o n g e r z e r o . The r a r e f a c t i o n wave b r i n g s the p a r t i c l e s t o r e s t . W ith i n c r e a s i n g power the heat wave's v e l o c i t y w i l l e v e n t u a l l y be the same as 22 the shock wave's and the r a r e f a c t i o n wave w i l l be c o m p l e t e l y d e v e l o p e d , p o i n t C. The heat wave behaves e x a c t l y l i k e a Chapman-Jouget (C.J.) d e t o n a t i o n a t t h i s p o i n t s i n c e the energy r e l e a s e zone i s t i e d t o the shock f r o n t . M a t h e m a t i c a l l y , t h i s c o n d i t i o n i s d e s c r i b e d by a g a i n s e t t i n g the square r o o t s i n eqns. 6 and 7 eq u a l to z e r o but w i t h M 3<1. The l o c u s o f the s e p o i n t s i s r e f e r r e d t o as the C.J. l i n e and s e p a r a t e s the s o n i c and s u p e r s o n i c regimes. A l l the f l u i d ' s parameters can be c a l c u l a t e d once the v e l o c i t y o f the d e t o n a t i o n i s known. The e q u a t i o n s t h a t r e l a t e the i n t e n s i t y , the e n t h a l p y , and the p r e s s u r e to the v e l o c i t y a r e 3 4 w l v 3 — = _ ±IAJL (9) P l 2 ( g 4 2 - l ) C.J. (10) (11) The heat wave becomes s u p e r s o n i c i f the power i s i n c r e a s e d s t i l l f u r t h e r , p o i n t D. I n t h i s regime, the boundary c o n d i t i o n a t the window has no e f f e c t on the heat wave s i n c e a p r e s s u r e s i g n a l from the window cannot r e a c h the heat f r o n t . T h e r e f o r e , a s u p e r s o n i c heat wave w i t h any o t h e r boundary c o n d i t i o n i s a l s o 23 d e s c r i b e d by the s u p e r s o n i c p o r t i o n o f the response p l a n e . No shocks are a s s o c i a t e d w i t h heat waves on s e c t i o n CD o f the h e a t i n g c h a r a c t e r i s t i c . CHAPTER I I SECTION C HEAT WAVES WITH A TWO STEP POWER INPUT The t h e o r y f o r c o n s t a n t i n p u t power was developed i n the p r e v i o u s s e c t i o n . I n t h i s s e c t i o n the t h e o r y i s expanded t o i n c l u d e a two s t e p power i n p u t . I t i s assumed t h a t the f i r s t s t e p o f the power p u l s e produces a s u b s o n i c heat wave w i t h i t s a s s o c i a t e d shock. The second s t e p i n the power p u l s e i s c o n s i d e r e d t o produce e i t h e r a second shock or a s u p e r s o n i c heat wave. The second shock case i s t r e a t e d q u a n t i t a t i v e l y but b e f o r e t h i s i s done, both cases are d e s c r i b e d q u a l i t a t i v e l y . When the power d r i v i n g a s u b s o n i c h e a t , f i g . 3, i s i n c r e a s e d , the v e l o c i t y o f , and the p r e s s u r e b e h i n d the heat wave i n c r e a s e s . I f the v e l o c i t y o f the heat wave remains s u b s o n i c w i t h r e s p e c t t o the gas behind i t , the p r e s s u r e d i s t u r b a n c e o v e r t a k e s the new s u b s o n i c heat wave and forms a second shock. On the o t h e r hand, i f the v e l o c i t y i n c r e a s e s so much t h a t the heat wave t u r n s s u p e r s o n i c , no shock w i l l be a b l e to form. T h i s s u p e r s o n i c heat, wave would t r a v e l f a s t e r than the p r e c e d i n g shock, b u r n i n g through the p r e v i o u s l y shock-compressed mater i a l . The case i n which a two s t e p power p u l s e produces two 24 shocks i s now t r e a t e d q u a n t i t a t i v e l y w i t h the aim o f d e t e r m i n i n g the r e l a t i o n s h i p between the magnitudes o f the two s t e p s i n the power p u l s e and the v e l o c i t i e s o f the shock waves. ( F i g . 6 i s a "snapshot" o f the f l u i d f l o w produced by a s t e p - l i k e power p u l s e sometime a f t e r the second shock has been c r e a t e d but b e f o r e i t has o v e r t a k e n the f i r s t shock. The numeric i d e n t i f i c a t i o n o f r e g i o n s , s i m i l a r t o t h a t o f the p r e v i o u s s e c t i o n i s , [1] the u n d i s t u r b e d g a s , [2] i m m e d i a t e l y behind , [3] i m m e d i a t e l y b e f o r e S 2 , [4] i m m e d i a t e l y a f t e r S 2 , [5] i m m e d i a t e l y b e f o r e HWR, and [6] a f t e r HWR. G i v e n the magnitude, W1, o f the f i r s t s t e p i n the power p u l s e , and the e n t h a l p y behind the f i r s t heat wave, the v e l o c i t y o f S-| can be c a l c u l a t e d u s i n g the t h e o r y p r e s e n t e d i n the p r e v i o u s s e c t i o n . In a s i m i l a r manner, i f we s p e c i f y the magnitude o f the second s t e p i n the power p u l s e , W2, and the e n t h a l p y b e h i n d the r e s u l t a n t heat wave, HWR, the v e l o c i t y o f S 2 can be c a l c u l a t e d i f we assume U 5 = 0. U 6 w i l l e v e n t u a l l y be z e r o as we have assumed t h a t the r e s u l t i n g heat wave i s s u b s o n i c , but i t may not be z e r o i m m e d i a t e l y a f t e r the a d d i t i o n o f power i s f e l t a t the heat f r o n t . T h i s p o i n t i s d i s c u s s e d f u r t h e r i n s e c t i o n E. F i g . 7 shows the r e s u l t s o f such c a l c u l a t i o n s when the 11 e n t h a l p y behind both heat waves i s f i x e d at 3.5x10 ergs/gram, a v a l u e suggested by e x p e r i m e n t a l r e s u l t s . The o r d i n a t e i s the n o r m a l i z e d i n t e n s i t y , W 2/P 1 , w h i l e the a b s c i s s a i s the Mach number o f S 2 w i t h r e s p e c t to the gas i m m e d i a t e l y i n f r o n t o f i t . The Mach number o f the f i r s t shock i s the parameter o f the c u r v e s on the gr a p h . The power r e q u i r e d t o produce the f i r s t shock i s o b t a i n e d from the c u r v e ' s y - i n t e r c e p t s i n c e o n l y an 25 HW R Wg P R E S S U R E DENSITY T E M P E R A T U R E P A R T I C L E V E L O C I T Y F i g . 6. Schematic of a double shock f l u i d flow with i d e a l i z e d pressure, d e n s i t y , temperature, and p a r t i c l e v e l o c i t y p r o f i l e s . S-|=first shock, S 2 = second shock, HWR = r e s u l t i n g heat wave. 26 F i g . 7. W 2 / P i v e r s u s the second shock Mach number, w i t h the Mach number o f the f i r s t shock as a parameter. 27 i n f i n i t e s i m a l a d d i t i o n o f energy i s r e g u i r e d t o produce a Mach 1 second shock. Changing the chosen e n t h a l p y from 3.5 to 6 x l 0 1 1 ergs/gram s h i f t s the c u r v e s a l o n g the o r d i n a t e by 10% and r a i s e s the r a t i o o f W2/W., 15% so the i n i t i a l c h o i c e o f e n t h a l p y i s not too i m p o r t a n t . I t i s , however, i m p o r t a n t t h a t the e n t h a l p y behind heat waves does not change w i t h changing power i n p u t as the c u r v e s i n f i g . 7 are much more s e n s i t i v e t o t h i s . Our c h o i c e o f e n t h a l p i e s i s su p p o r t e d by the e x p e r i m e n t a l l y d e termined h e a t i n g c h a r a c t e r i s t i c , c h a p t e r IV, s e c t i o n B. The i n t e r a c t i o n t h a t o c c u r s when one shock wave o v e r t a k e s another needs to be understood i f we are t o i d e n t i f y luminous f r o n t s on s t r e a k camera photographs as shock waves. G i v e n the f i r s t and second shock v e l o c i t i e s , the v e l o c i t y o f the shock a r i s i n g from the i n t e r a c t i o n can be c a l c u l a t e d . The c a l c u l a t i o n was done assuming an i n i t i a l t emperature o f 20 °C and a f i l l i n g p r e s s u r e o f 5 t o r r . The o r d i n a t e o f f i g . 8 i s the Mach number of the r e s u l t i n g shock wave; the a b s c i s s a i s the Mach number o f the second shock. The Mach number o f the f i r s t shock, the parameter o f the c u r v e s on the g r a p h , i s o b t a i n e d from the c u r v e ' s y - i n t e r c e p t . More d e t a i l s o f the i n t e r a c t i o n are g i v e n i n appendix 1. CHAPTER I I SECTION D THE APPLICABILITY OF THE WAVE MODEL TO THE ELECTROTHERMAL SHOCK TUBE. We now g i v e the reasons why we f e e l the heat wave model p r e s e n t e d i n the p r e v i o u s s e c t i o n s , i s s u i t a b l e t o d e s c r i b e the 28 F i g . 8. The Mach numbers o f the r e s u l t i n g shock v e r s u s the second shock w i t h the Mach number o f the f i r s t shock as a parameter. 29 flow ' i n an e l e c t r o t h e r m a l shock tube. This s e c t i o n compares general c h a r a c t e r i s t i c s of the flow produced i n the shock tube with the p r e d i c t e d flow. The p o s s i b l e energy tr a n s p o r t mechanisms from the arc to the heat f r o n t are examined to determine whether any of them are compatible with the heat wave model. Q u a l i t a t i v e l y one can see that both the subsonic and supersonic regimes can be produced i n an e l e c t r o t h e r m a l shock tube. At low currents and/or high f i l l i n g pressures, i . e . low W/P-j , a shock wave followed by a b r i g h t l y luminous zone i s p r o d u c e d 2 0 ' 2 1 . This b r i g h t l y luminous zone i s i n t e r p r e t e d as the region behind the heat f r o n t . As the current i s increased and/or the f i l l i n g pressure decreased, i . e . higher W/p^  , the shock speed increases and the distance between the shock and the leading edge of the luminous region decreases. At high currents and/or low f i l l i n g pressures, i . e . high W/P1, the h i g h l y luminous region propagates without a preceding shock. This f r o n t i s i n t e r p r e t e d as a supersonic heat wave. The f l u i d ' s parameters behind t h i s f r o n t cannot be p r e d i c t e d by the Rankine-Hugoniot r e l a t i o n s 1 5 , 1 6 . Often the power supplied to an e l e c t r o t h e r m a l shock tube i s of la r g e magnitude but short d u r a t i o n . Because of t h i s one should i n i t i a l l y see a supersonic heat wave which l a t e r decays i n t o a subsonic heat wave. As the heat wave passes through the C.J. p o i n t , a shock should break loose and propagate ahead of the heat wave. This type of flow 3 5 i s common and has been c l e a r l y demonstrated by B r i n k s c h u l t e Q u a n t i t a t i v e evidence that the b r i g h t l y luminous region behind a preceding shock wave i s a subsonic heat wave, was 30 o b t a i n e d by S t r a c h a n J ' . He measured the f l o w v e l o c i t y a t the luminous f r o n t t o be .6 km/sec. In 3 cm, however, i t decayed to l e s s than .1 km/sec. In compa r i s o n , the f l o w v e l o c i t y behind the shock wave was t y p i c a l l y 2 km/sec. I f t h i s luminous f r o n t was a c o n t a c t s u r f a c e the f l o w v e l o c i t y would have been c o n s t a n t a c r o s s i t . In o r d e r t o understand how a heat wave can be m a i n t a i n e d i n an e l e c t r o t h e r m a l shock t u b e , i t i s n e c e s s a r y t o examine the p o s s i b l e energy t r a n s p o r t mechanisms t h a t c o u l d be r e s p o n s i b l e f o r c a r r y i n g energy from the a r c t o the l e a d i n g edge o f the heat wave. Two p o s s i b l e mechanisms are e l e c t r o n t h e r m a l c o n d u c t i o n and r a d i a t i o n . The e l e c t r o n t h e r m a l c o n d u c t i v i t y i n a p a r t i a l l y i o n i z e d plasma r i s e s s h a r p l y w i t h temperature due t o the l a r g e r number of e l e c t r o n s , t h e i r i n c r e a s e d v e l o c i t y , and t h e i r i n c r e a s e d energy. F i g . 9 i s a p l o t o f the expected temperature p r o f i l e o f a plasma produced by heat s u p p l i e d to the r i g h t end. I t i s q u i t e s i m i l a r t o the expected temperature p r o f i l e o f a " b l e a c h i n g 28 2 9 wave" as d i s c u s s e d by S t e i n h a u e r e t . a l ' . When the temperature i s h i g h , the t h e r m a l c o n d u c t i v i t y i s h i g h and o n l y a s m a l l t emperature g r a d i e n t i s needed t o t r a n s p o r t the h e a t . As the t emperature i s lower near the l e f t end o f the plasma, the th e r m a l c o n d u c t i v i t y i s l e s s ; t o m a i n t a i n the energy f l u x , the temperature g r a d i e n t must i n c r e a s e c o r r e s p o n d i n g l y , thus the temperature f a l l s a b r u p t l y . I f the power i n t o the gas a t the r i g h t end i s i n c r e a s e d , 31 UJ cr < LLl CL LU J X F i g . 9. An expected temperature p r o f i l e o f a one d i m e n s i o n a l plasma produced w i t h energy i n p u t a t the r i g h t end. o n l y a s m a l l temperature i n c r e a s e i s needed to t r a n s p o r t the a d d i t i o n a l energy as the t h e r m a l c o n d u c t i v i t y r i s e s so s h a r p l y w i t h t e m p e r a t u r e . I n e f f e c t t h e n , the plasma i s " t r a n s p a r e n t " to the a d d i t i o n o f energy. The a d d i t i o n a l energy h e a t s the e x i s t i n g plasma o n l y s l i g h t l y , i n main, i t h e a t s the gas a t the l e a d i n g edge o f the heat wave. Because the plasma can be heated t o a h i g h e r t e m p e r a t u r e , the b l e a c h i n g cannot be c o n s i d e r e d complete. Any a d d i t i o n a l energy i n p u t would p r o b a b l y t r a v e l t o the f r o n t i n the form o f a s u p e r s o n i c heat wave. 32 The main features of an e l e c t r o n thermal conduction f r o n t are s i m i l a r to those of an i d e a l i z e d heat wave. The temperature, roughly constant behind the f r o n t , f a l l s r a p i d l y near the f r o n t . The plasma i s n e a r l y "transparent" to a d d i t i o n a l energy. U l t r a v i o l e t r a d i a t i o n from the arc i s another p o s s i b l e energy tr a n s p o r t mechanism. The i o n i z i n g r a d i a t i o n f r o n t would have a l l the main features of a heat wave, s i n c e , once the gas has been i o n i z e d , i t no longer s t r o n g l y absorbs the r a d i a t i o n . Experimental evidence from our work suggests that e l e c t r o n thermal conduction i s the dominant heat t r a n s f e r mechanism. The power deposited i n the shock tube by the arc was c a l c u l a t e d by measuring the voltage across, and the current through, the tube. T y p i c a l l y , only 2 to 4% of the deposited energy i s needed to d r i v e shocks whose speeds were measured 20 cm from the e l e c t r o d e s . A rough estimate of the f r a c t i o n of the r a d i a t i o n produced at the arc which reaches the heat f r o n t i s the r a t i o of the shock tube's c r o s s - s e c t i o n a l a r e a , T f r 2 , (r=1.25 cm), to the area of a sphere 20 cm i n r a d i u s , 4TT(20) . Fraction of energy ; delivered by r a d i a t i o n Since t h i s i s at l e a s t a.factor of 20 too low to support the measured shock speeds, we can conclude that r a d i a t i v e energy tra n s p o r t i s not dominant i n our shock tube. 3% of the energy deposited by the arc can be transported to TT(1.25) 2 = 4TT(20) 2 33 the heat f r o n t i f the energy t r a n s f e r mechanism i s therm a l c o n d u c t i o n . T h i s v a l u e i s e s t i m a t e d as f o l l o w s . The f r a c t i o n o f the energy d e l i v e r e d t o the heat f r o n t w i l l be the r a t i o o f the c r o s s - s e c t i o n a l a rea o f the t u b e , l t r 2 , t o the ar e a o f the w a l l s o f the column through which the energy must t r a v e l , 2 r r r ( 2 0 ) , as energy w i l l be l o s t t o the w a l l s through t h e r m a l c o n d u c t i o n . Fraction of energy delivered _ TT(1.25) 2 by thermal conduction " 2TT(1. 25) (20) = 3 ^ CHAPTER I I SECTION E ASSUMPTIONS OF THE THEORY P r e v i o u s l y we have c o n s i d e r e d o n l y i d e a l i z e d s i t u a t i o n s i n or d e r t o s e p a r a t e the main i d e a s from d e t a i l s . We now c o n s i d e r how w e l l the i d e a l i z e d c o n d i t i o n s are approximated i n an e l e c t r o t h e r m a l shock tube. More d e t a i l s are g i v e n i n Chapter IV, s e c t i o n A. The assumptions c o n t a i n e d i n the t h e o r y a r e : 1. ) The shock and heat wave can be c o n s i d e r e d d i s c o n t i n u i t i e s . 2. ) A s t e a d y s t a t e e x i s t s f o r a l l t i m e . 3. ) The f l o w i s one d i m e n s i o n a l . 4. ) Energy l o s s e s are n e g l i g i b l e . 5. ) The temperature i s u n i f o r m behind the heat wave. 6. ) There are no mass or momentum s o u r c e s . 7. ) The p a r t i c l e v e l o c i t y i s always z e r o behind a s u b s o n i c heat f r o n t . 34 1. ) To t r e a t the heat wave as a d i s c o n t i n u i t y , the energy a b s o r p t i o n l e n g t h s h o u l d be s h o r t i n comparison t o r e l e v a n t d i s t a n c e s (e.g. the d i s t a n c e from the e l e c t r o d e s t o the l e a d i n g edge o f the heat wave). From photographs p r e s e n t e d i n c h a p t e r IV, s e c t i o n A, i t appears t h a t the energy a b s o r p t i o n l e n g t h i s about 1 cm and t h a t the heat f r o n t i s p l a n a r t o h a l f a tube d i a m e t e r . The w i d t h o f the shock f r o n t i s unmeasureably s m a l l on the f r a m i n g camera photographs and i s p e r f e c t l y p l a n a r . 2. ) The t h e o r y d i d not c o n s i d e r the e v o l u t i o n o f the hydrodynamics from t=0, r a t h e r i t c o n s i d e r e d a s t e a d y s t a t e . The time a f t e r w hich t r a n s i t o r y e f f e c t s were no l o n g e r i m p o r t a n t was i n v e s t i g a t e d w i t h the f r a m i n g camera. D e t a i l s are g i v e n i n c h a p t e r IV, s e c t i o n A. 3. ) An a p p r e c i a b l e boundary l a y e r c o u l d p r e j u d i c e the assumption t h a t the f l o w i s o n e - d i m e n s i o n a l . S t u d i e s o f boundary l a y e r s i n d i c a t e they w i l l have n e g l i g i b l e e f f e c t i n 3 8 t h i s e x p e r i m e n t 4. ) The i n p u t power was h e l d c o n s t a n t and energy l o s s e s ' were s e t e q u a l t o zero i n the t h e o r y i n o r d e r t h a t the hydrodynamics would be s t e a d y s t a t e . No matter which energy t r a n s f e r mechanism i s e n v i s a g e d , a f r a c t i o n o f the energy d e l i v e r e d by the a r c w i l l not rea c h the heat f r o n t . As the power d e c r e a s e s w i t h d i s t a n c e from the e l e c t r o d e s , t h e r e g i o n behind the shock wave w i l l be i n f i l t r a t e d by r a r e f a c t i o n waves, w h i c h , when they c a t c h up to the shock, w i l l slow i t down. The r a r e f a c t i o n waves were o n l y e v i d e n t i n a m i n o r i t y o f the p r e s s u r e measurements, p a r t i c u l a r l y when the shock and heat 35 waves were f a r from the energy s o u r c e . To overcome the e f f e c t s o f the d e c r e a s i n g power, the shock and heat wave v e l o c i t i e s , when measured, were averaged over a d i s t a n c e o f 10 cm. P r e s s u r e measurements were averaged over a p e r i o d o f at l e a s t 10 mic r o s e c o n d s and o f t e n 30 or 40 microseconds i n the case o f the heat wave. 5. ) I f t h e r m a l c o n d u c t i o n i s the energy t r a n s p o r t mechanism, a t h e r m a l g r a d i e n t must e x i s t between the e l e c t r o d e s and the heat f r o n t . A t a f i x e d p o s i t i o n , the temperature i s exp e c t e d t o i n c r e a s e w i t h time i n o r d e r to m a i n t a i n a f l o w o f energy to the heat f r o n t which i s moving ever f u r t h e r away from the energy s o u r c e . Because o f the temperature i n c r e a s e w i t h time a t any p o s i t i o n , the p a r t i c l e s w i l l have a v e l o c i t y away 36 3 7 from the window. T h i s v e l o c i t y , measured by S t r a c h a n ' , was found t o be .1 km/sec, or 5% of the p a r t i c l e v e l o c i t y b ehind a t y p i c a l shock. The a d d i t i o n a l h e a t i n g and consequent e x p a n s i o n were assumed to be n e g l i g i b l e . 6. ) A mass or momentum source would g e n e r a t e a p r e s s u r e i n a d d i t i o n t o t h a t p r e d i c t e d by the t h e o r y . T h i s p r e s s u r e would m a n i f e s t i t s e l f i n f a s t e r shock waves. As i t i s d i f f i c u l t t o e s t i m a t e the magnitude o f such e f f e c t s , c a r e was taken to m i n i m i z e any mass or momentum source i n the e x p e r i m e n t . 7. ) When the power i n t o an e l e c t r o t h e r m a l shock tube i s i n c r e a s e d , the e f f e c t o f the i n c r e a s e d power w i l l not be im m e d i a t e l y p e r c e i v e d at the heat f r o n t . P r o b a b l y a s u p e r s o n i c heat wave w i l l p r opagate through the a l r e a d y heated gas; o n l y when t h i s s u p e r s o n i c heat wave reaches the f r o n t w i l l the 36 i n c r e a s e d power be f e l t . The p a r t i c l e s a c c e l e r a t e d by the s u p e r s o n i c heat wave w i l l be brought t o r e s t a g a i n by the c l o s e l y f o l l o w i n g r a r e f a c t i o n wave. I t i s f o r t h i s r e a son t h a t u6 was assumed t o be zer o i n s e c t i o n C. The momentum the s u p e r s o n i c heat wave c a r r i e s i s n e g l e c t e d . CHAPTER I I SECTION F ELECTROTHERMAL SHOCK TUBES AS SIMULATORS OF OTHER INHOMOGENEOUS ENERGY ABSORPTION PROBLEMS Some areas o f s c i e n c e i n which inhomogeneous energy a b s o r p t i o n i s i m p o r t a n t are mentioned i n the i n t r o d u c t i o n . These phenomena have much i n common. A l o c a l i z e d . energy d e p o s i t i o n o c c u r s , p r o d u c i n g a heat wave and the a s s o c i a t e d f l u i d f l o w . Because o f the s i m i l a r i t i e s among these phenomena, a s t u d y o f an e l e c t r o t h e r m a l shock tube f l o w s h o u l d y i e l d i n f o r m a t i o n about a l l f l o w s . To emphasize the s i m i l a r i t i e s among the phenomena mentioned, the f l o w i n an e l e c t r o t h e r m a l shock tube i s compared t o the f l o w i n a l a s e r compression e x p e r i m e n t . F i g 10a i s a c r o s s s e c t i o n o f one segment o f a l a s e r c o m p r e s s i o n e x p e r i m e n t . The l a s e r l i g h t i n i t i a l l y s t r i k e s the p e l l e t , but soon a corona i s formed. The l i g h t cannot p e n e t r a t e t h i s corona beyond the c r i t i c a l d e n s i t y and thus i s absorbed i n a r e g i o n (e) some d i s t a n c e from the p e l l e t . The energy t r a n s p o r t mechanisms between the c r i t i c a l d e n s i t y s u r f a c e and the p e l l e t ' s s u r f a c e are v e r y c o m p l i c a t e d but t h a t does not conc e r n us he r e . Some o f the energy absorbed i n r e g i o n (e) i s 37 F i g . 10. A comparison between a l a s e r compression f l o w and the f l o w i n an e l e c t r o t h e r m a l shock tube. t r a n s p o r t e d t o the p e l l e t and a b l a t e s i t s s u r f a c e c r e a t i n g a t h r u s t which d r i v e s a shock wave i n t o the p e l l e t . I f , a f t e r the f i r s t shock has been c r e a t e d , the l a s e r ' s energy i s i n c r e a s e d , the a d d i t i o n a l energy t r a n s p o r t e d t o the p e l l e t would, h o p e f u l l y , c r e a t e an i n c r e a s e d t h r u s t s u f f i c i e n t t o d r i v e a second shock ahead o f the a b l a t i o n s u r f a c e . I n summary, energy absorbed i n r e g i o n (e) and t r a n s m i t t e d through the c o r o n a , 38 r e g i o n ( f ) , produces an a b l a t i o n f r o n t a t the p e l l e t ' s s u r f a c e . The t h r u s t c r e a t e d by the a b l a t e d p a r t i c l e s d r i v e s a shock wave i n t o the p e l l e t . F i g . 10b i s a sch e m a t i c o f an e l e c t r o t h e r m a l shock tube. C u r r e n t from a c a p a c i t o r bank i s passed through two e l e c t r o d e s h e a t i n g the gas i n r e g i o n ( e ) . T h i s energy, t r a n s p o r t e d t h r o u g h r e g i o n ( f ) , a b l a t e s the rearward p o r t i o n o f the shock-compressed m a t e r i a l c r e a t i n g a t h r u s t which i s r e s p o n s i b l e f o r d r i v i n g a shock wave down the tube. We found t h a t an i n c r e a s e i n c u r r e n t some time a f t e r the f i r s t shock had been formed c o u l d l a u n c h a second shock by c r e a t i n g a d d i t i o n a l t h r u s t at the a b l a t i o n s u r f a c e . A g a i n we see t h a t inhomogeneously absorbed energy produces a heat wave, or a b l a t i o n f r o n t , and i s r e s p o n s i b l e f o r p r o d u c i n g p a r t i c l e m o t i o n . 39 CHAPTER I I I THE EXPERIMENTAL APPARATUS In o r d e r to t e s t the model p r e s e n t e d i n the p r e v i o u s c h a p t e r , i t was n e c e s s a r y t o b u i l d a shock tube which m i n i m i z e d any mass or momentum s o u r c e s and a l s o t o b u i l d a h i g h power c o n s t a n t c u r r e n t s o u r c e . The shock tube a l s o had t o a l l o w measurements o f p r e s s u r e s and ready a c c e s s t o the l i g h t e m i t t e d by the shock and heat waves. The a p p a r a t u s , d e s c r i b e d i n t h i s c h a p t e r , can be d i v i d e d i n t o f o u r main groups. A. ) The shock tube B. ) The power s u p p l y f o r the shock tube C. ) The d i a g n o s t i c equipment used t o measure: 1. Shock and heat wave v e l o c i t i e s 2. P r e s s u r e s 3. Temperatures D. ) A u x i l i a r y equipment 40 S H O C K TUBE POWER SUPPLY P R E S S U R E P R O B E S T R E A K C A M E R A F R A M I N G C A M E R A M O N O C H R O M A T O R F i g . 11. B l o c k diagram o f the e x p e r i m e n t a l f a c i l i t i e s They are shown i n r e l a t i o n s h i p to each o t h e r i n f i g u r e 11. The p r o c e d u r e f o l l o w e d when d o i n g the experiment i s g i v e n i n s e c t i o n D. 41 > CHAPTER I I I SECTION A THE SHOCK TUBE A t y p i c a l shock tube and e l e c t r o d e i s shown i n f i g u r e 12. The g l a s s tube has a 2.5 cm i n n e r d iameter and i s 50 cm l o n g , a l e n g t h s u f f i c i e n t t o i n s u r e t h a t r e f l e c t i o n s o f p r e s s u r e waves from the downstream end d i d not i n f l u e n c e the f l u i d f l o w d u r i n g t i m e s o f measurement. The shock tubes were i n i t i a l l y made o f pyrex but l a t e r q u a r t z was used as i t w i t h s t o o d the h i g h power l e v e l s b e t t e r . "Kovar" e l e c t r o d e s were a t t a c h e d 5 cm from one end. I f they were c l o s e r , t h e r m a l s t r e s s e s caused c r a c k s t o d e v e l o p . The b o d i e s o f the e l e c t r o d e s which c a r r i e d the c u r r e n t were made o f b r a s s w i t h 1/4" t u n g s t e n rods s o l d e r e d i n t o t h e ends. These e l e c t r o d e s were i n s e r t e d through the "kovar" e l e c t r o d e s , spaced 3/4", and then s o l d e r e d t o the "kovar" e l e c t r o d e s . The 5 cm space from the end o f the tube t o the t u n g s t e n t i p s o f the e l e c t r o d e s was f i l l e d w i t h a g l a s s p l u g , h e l d i n p l a c e by a p l a t e cemented, over the end o f tube. T h i s i n s u r e d t h a t p r e s s u r e d i s t u r b a n c e s c o u l d not propagate to the end o f the tu b e , be r e f l e c t e d , and l a t e r i n f l u e n c e the f l o w t r a v e l l i n g downstream. Other e l e c t r o d e d e s i g n s and m a t e r i a l s o t h e r than t u n g s t e n were used i n i t i a l l y , but a l l s u f f e r e d from e x c e s s i v e mass l o s s d u r i n g f i r i n g . A mass sour c e i s not i n c l u d e d i n the t h e o r y and any s o u r c e s s h o u l d be m i n i m i z e d . A l s o , m a t e r i a l e v a p o r a t e d from the e l e c t r o d e s i s d e p o s i t e d on the w a l l s o f the tube and r e n d e r s i t opaque, an u n d e s i r a b l e f e a t u r e . 42 F i g . 12. A t y p i c a l g l a s s shock tube and an e l e c t r o d e . 43 A c u r r e n t c a r r i e r i n a magnetic f i e l d i s s u b j e c t e d t o a L o r e n t z f o r c e . I n some shock t u b e s , c a l l e d T-tubes, t h i s f o r c e i s u t i l i z e d by f i x i n g the r e t u r n l e a d t o the c a p a c i t o r bank c l o s e t o the a r c 3 9 ' 4 0 . The momentum the c u r r e n t c a r r i e r s i n the a r c r e c e i v e from t h i s f o r c e i s used t o augment the p r e s s u r e i n o r d e r t o c r e a t e f a s t e r shock waves. I t was shown by A h l b o r n e t . a l , t h a t , a t a f i l l i n g p r e s s u r e o f 10 t o r r , the b a c k s t r a p had no e f f e c t on v e l o c i t i e s a f t e r the waves were 10 cm from the e l e c t r o d e s . In our case we w i s h to m i n i m i z e t h i s f o r c e as i t i s not i n c l u d e d i n the t h e o r y . T h i s was done by p l a c i n g the c u r r e n t c a r r y i n g l e a d s as f a r away as p o s s i b l e from the d i s c h a r g e . Because o f t h i s , i t i s expected t h a t the magnetic f o r c e s w i l l have a n e g l i g i b l e e f f e c t i n our experiment and i t i s t h e r e f o r e j u s t i f i a b l e t o i g n o r e them i n the t h e o r y . The shock tube d e s c r i b e d above was not used f o r p r e s s u r e measurements as the p r e s s u r e probes c o u l d not be i n s e r t e d . A m e t a l shock t u b e , which a l l o w e d the probes t o be mounted i n twenty d i f f e r e n t p o s i t i o n s a l o n g the tube and f l u s h w i t h the w a l l , was used. T h i s shock tube i s d e s c r i b e d f u r t h e r i n appendix 2. An i m p o r t a n t d i f f e r e n c e between the m e t a l and g l a s s shock t u b e s , which s h o u l d be noted h e r e , i s the e l e c t r o d e s e c t i o n o f the m e t a l tube had to be made of g l a s s i n o r d e r t o p r e v e n t the c u r r e n t i n the a r c from t a k i n g a " s h o r t c u t " t h r o u g h the tube and not through the gas. T h i s , u n f o r t u n a t e l y , meant t h a t where the shock tube and the the e l e c t r o d e s e c t i o n were j o i n e d t h e r e was an i r r e g u l a r i t y which caused a n o t i c e a b l e d i s t u r b a n c e i n the f l u i d f l o w under some c o n d i t i o n s . The i n s t a n c e s where t h i s happened are noted i n the a p p r o p r i a t e 44 s e c t i o n s . A l l shock tubes were jo i n e d to a 50 cm g l a s s s e c t i o n which i n turn was joi n e d to the metal p a r t s of the vacuum system. This e f f e c t i v e l y prevented discharges occuring between the anode and the vacuum system. CHAPTER I I I SECTION B THE POWER SUPPLY The shock tube was powered by c a p a c i t o r banks which can d e l i v e r current pulses as shown i n f i g . 13 where the parameters A t , I-j , and I 2 can be independently chosen. F i g . 13a i s a current pulse from bank I , f i g u r e 13b i s a current pulse from bank I I , and f i g u r e 13c i s a pulse when bank I I i s f i r e d 68 microseconds a f t e r bank I . The s h o r t i n g switch of bank I I was not working when t h i s photograph was taken. F i g . 14 i s a schematic of the two transmission l i n e s which supplied current to the shock tube. The current pulses from the transmission l i n e s are n e a r l y independent of each other as the impedance of the discharge i s much l e s s than the c h a r a c t e r i s t i c impedance of e i t h e r l i n e . Another way of s t a t i n g t h i s i s , the voltage drop across the shock tube i s much l e s s that the charging voltages so the anode of the discharge i s nearly at ground p o t e n t i a l from e i t h e r bank's p o i n t of view. When charged to 15 KV bank I provides 22.5 KA; bank I I gives 31.5 KA at the same charging voltage. I t i s d i f f i c u l t to estimate the current reguired to produce a shock wave of given strength before doing the experiment, as one does not know a p r i o r i what f r a c t i o n of the power supplied 45 F i g . 13. Oscillogram of current pulses from the transmission l i n e s . (a) bank I only , (b) bank I I only, and (c) both banks. reaches the heat f r o n t . I t i s for t h i s , and other reasons, that three other sets of transmission l i n e s preceded the set shown i n f i g . 14. Each set of transmission l i n e s took at l e a s t a month to b u i l d and adjust. The pulse lengths were easier to estimate. I f one wishes to produce a Mach 4 shock i n argon at 20 °C and observe t h i s shock 20 cm from the electodes the pulse duration must be greater than 150 microseconds or the shock w i l l not reach x=20 46 BANK I L = 4 . 1 J J H C = 1 0 U F T=160 usee I(15KV) = 22 KA SHORTING IGNITRONS 14 / SECTIONS 20 SECTIONS B A N K H L = . 8 J J H C = 5 J J F T=80 psec I(15KV) = 31 KA F i g . 14. Schematic o f the t r a n s m i s s i o n l i n e s . cm b e f o r e the power runs o u t . As bank I I i s used p r i m a r i l y t o p r o v i d e a stepped c u r r e n t p u l s e and i s t h e r e f o r e f i r e d sometime a f t e r bank I , i t s p u l s e need not be as l o n g as bank I ' s . The l e n g t h s chosen were 160 m i c r o s e c o n d s f o r bank I and 80 microseconds f o r bank I I . I g n i t r o n s , r a t h e r than spark gaps, were used i n a l l h i g h 47 v o l t a g e , h i g h c u r r e n t s w i t c h i n g a p p l i c a t i o n s as t h e y do not r e q u i r e any a d j u s t m e n t , as do spark gaps, when the c h a r g i n g v o l t a g e i s changed. Two i g n i t r o n s were used to connect the c a p a c i t o r banks t o the shock tube. As the t r a n s m i s s i o n l i n e s were not t e r m i n a t e d w i t h r e s i s t o r s , two more i g n i t r o n s were used t o " s h o r t " the l i n e s near the end o f the f i r s t h a l f c y c l e t o s t o p them from r i n g i n g through the shock tube. B e f o r e t r a n m i s s i o n l i n e s were chosen as the power so u r c e a l t e r n a t i v e power s o u r c e s were i n v e s t i g a t e d . The r e s u l t s o f the i n v e s t i g a t i o n are g i v e n i n appendix 3. Due t o the h i g h c u r r e n t s and the l o n g p u l s e d u r a t i o n , the i n d u c t o r s which connect the c a p a c i t o r s had t o be v e r y rugged. T h e i r d e s i g n , s i m i l a r t o t h a t o f a B i t t e r magnet, i s a l s o d e s c r i b e d i n appendix 3. CHAPTER I I I SECTION C DIAGNOSTICS As noted i n c h a p t e r I I , t o c o m p l e t e l y determine the f l o w i n an e l e c t r o t h e r m a l shock t u b e , we must measure two parameters o f the f l o w . We chose t o measure f o u r p a r a m e t e r s ; shock and heat wave v e l o c i t i e s , p r e s s u r e s , and temp e r a t u r e s so t h a t w i t h the e x t r a measurements we can check the t h e o r y . The equipment used t o measure these parameters i s d e s c r i b e d below. 1.) THE STREAK AND FRAMING CAMERAS. A s t r e a k camera, which makes space-time p l o t s o f luminous phenomena, was used t o measure shock and heat wave v e l o c i t i e s . The camera has been 48 d e s c r i b e d f u l l y e l sewhere . The sweep speeds are a c c u r a t e to 1%. To measure v e l o c i t i e s from a s t r e a k camera p h o t o , i t i s n e c e s s a r y t o know the m a g n i f i c a t i o n as w e l l as the sweep speed. I t was determined d i r e c t l y from the photographs by masking the shock tube a t 5 cm i n t e r v a l s , w i t h narrow b l a c k tape and measuring o f the s p a c i n g o f the l i n e s on the f i l m . The u n c e r t a i n t y i n the m a g n i f i c a t i o n i s e s t i m a t e d t o be l e s s than .4%. V e l o c i t i e s were o b t a i n e d from s t r e a k camera photos by measuring the a n g l e the luminous phenomena made w i t h the d i r e c t i o n o f sweep, which i s p a r a l l e l t o the image o f the b l a c k marker l i n e s on the f i l m . To maximize the a c c u r a c y , the sweep speed was a d j u s t e d so t h a t the a n g l e s measured were always c l o s e to 45 d e g r e e s . The e s t i m a t e d e r r o r o f .75 degrees i n the a n g l e measurement then g i v e s a 3% e r r o r i n the v e l o c i t y . Combining the e r r o r s i n the sweep speed, m a g n i f i c a t i o n , and a n g l e d e t e r m i n a t i o n g i v e s a r e s u l t i n g e r r o r o f 3%. A TRW f r a m i n g camera was used t o examine, q u a l i t a t i v e l y , the p l a n a r i t y o f the shock and heat waves. I t was a l s o used t o watch a shock s e p a r a t e from a heat wave and to f o l l o w the e v o l u t i o n o f the a r c d u r i n g the c u r r e n t p u l s e . 2.) THE PRESSURE PROBE. The p i e z o - e l e c t r i c probe used t o measure the heat wave p r e s s u r e was made by C e l e s c o , model LD-25. I t s s e n s i t i v i t y , d e t ermined e x p e r i m e n t a l l y , was .85 ± 20% v o l t s / a t m . 49 The r i s e t i m e o f the probe i s quoted as a microsecond i n the manufacture's s p e c i f i c a t i o n s when a p r e s s u r e p u l s e s t r i k e s the probe f a c e on. Our time r e s o l u t i o n , however, was about 10 mic r o s e c o n d s f o r two r e a s o n s : 1.) Our p r e s s u r e p u l s e (e.g. shock) t r a v e r s e d the f a c e o f the probe as i t was mounted f l u s h w i t h the w a l l o f the shock tube. The di a m e t e r o f the probe i s 1 cm so a t y p i c a l t r a v e r s a l time was 3 m i c r o s e c o n d s . 2.) The probe's s i g n a l i n i t i a l l y o v e r s h o t b e f o r e r e t u r n i n g t o , and o s c i l l a t i n g about, i t s e q u i l i b r i u m v a l u e . Because o f the o v e r s h o o t and the o s c i l l a t i o n s , a t l e a s t 10 mi c r o s e c o n d s a re needed t o determine the p r e s s u r e . F u r t h e r d e t a i l s on the p r o b e , and the pr o c e d u r e used t o c a l i b r a t e i t , are g i v e n i n appendix 4. 3.) TEMPERATURE DETERMINATION. One method o f e s t i m a t i n g the temperature o f a plasma i s t o a n a l y z e the l i g h t i t e m i t s . A f t e r c e r t a i n assumptions have been made, a temperature can be i n f e r r e d . To c o l l e c t the l i g h t , an o p t i c a l m u l t i c h a n n e l a n a l y z e r , OMA, was used i n c o n j u n c t i o n w i t h a 3/4 meter Spex monochromator. T h i s c o m b i n a t i o n r e c o r d s a wavelength spread o f about 140 Angstroms w i t h 500 c h a n n e l s o f i n f o r m a t i o n . The OMA can be used e i t h e r t o i n t e g r a t e the l i g h t s t r i k i n g i t or make time r e s o l v e d measurements. The temperature o f the plasma can sometimes be e s t i m a t e d i f one knows the p r e s s u r e and the e l e c t r o n d e n s i t y . Assuming t h e r m a l e q u i l i b r i u m , the e l e c t r o n d e n s i t y and p r e s s u r e can be c a l u l a t e d as a f u n c t i o n o f temperature u s i n g the Saha-Eggert e q u a t i o n 4 2 . 50 n n U 3 ^ = 2 n o u o o f ± ! ^ f e x p ( - E . / k T ) ,12, U + and U Q are p a r t i t i o n f u nctions for the ion and atom, m i s the mass of the atom, and Ej i s the i o n i z a t i o n energy. F i g . 15 shows the r e s u l t s of such a c a l c u l a t i o n for two cases, 100% argon ( s o l i d l i n e s ) and 95% argon and 5% tungsten (dotted l i n e s ) . The ordinate of f i g . 15 i s the temperature i n eV and the a b scissa i s the log of the t o t a l pressure i n t o r r . The curves on the graph are l o c i of constant e l e c t r o n d e n s i t y . The temperature can be found w i t h t h i s graph i f p and n e are known. This method of determining the temperature can be used only when the gas i s p a r t l y i o n i z e d , i n our case assuming pure argon, up to about 1.1 eV or a l i t t l e greater depending on the pressure. The method al s o s u f f e r s from the assumption of thermal e q u i l i b r i u m and the need to know the composition of the plasma. I f even 5% of an e a s i l y i o n i z e d i m p u r i t y , e.g. tungsten, i s present, the curves of constant e l e c t r o n d e n s i t y d r a s t i c a l l y change t h e i r shape i n some termperature ranges. F o r t u n a t e l y the plasma i n our e l e c t r o t h e r m a l shock tube appears to l i e w i t h i n the temperature range, about .8 to 1.1 eV, i n which the method i s u s e f u l . Above 1.1 eV one can obtain p a c c u r a t e l y from a measurement of n e even i f T i s known to l i t t l e accuracy. 51 LOG'(P) (torr) F i g . 15. Temperature v e r s u s the l o g o f the p r e s s u r e w i t h the e l e c t r o n d e n s i t y as a parameter. 52 CHAPTER I I I SECTION D AUXILIARY EQUIPMENT AND EXPERIMENTAL PROCEDURE The f l o w produced i n an e l e c t r o t h e r m a l shock tube depends s e n s i t i v e l y on the f i l l i n g p r e s s u r e and the power s u p p l i e d . The f i l l i n g p r e s s u r e , c o n t r o l l e d e l e c t r o n i c a l l y , was a c c u r a t e t o o n l y 2% of the r e a d i n g but was r e p r o d u c i b l e t o w i t h i n .001 t o r r . The c h a r g i n g v o l t a g e s o f the c a p a c i t o r banks were a l s o c o n t r o l l e d e l e c t r o n i c a l l y and were r e p r o d u c i b l e t o w i t h i n 50 v o l t s . The i g n i t r o n s , which s w i t c h e d the c a p a c i t o r banks, were t r i g g e r e d w i t h k r y t r o n c i r c u i t s which were themselves t r i g g e r e d by d e l a y u n i t s . The c o m b i n a t i o n had a j i t t e r o f l e s s than 1 m i c r o s e c o n d . When u s i n g e i t h e r the OMA or the s t r e a k camera i t . was the master o f the expe r i m e n t and c o n t r o l l e d the d e l a y u n i t s which i n t u r n t r i g g e r e d the r e m a i n i n g d i a g n o s t i c equipment. The e x p e r i m e n t a l p r o c e d u r e c o n s i s t e d o f c h a r g i n g the c a p a c i t o r bank(s) and f i l l i n g the tube s i m u l t a n e o u s l y . A f t e r each s h o t , t h e tube was evacuated w i t h a r o t a r y pump and t h e pro c e d u r e r e p e a t e d a f t e r a l l o w i n g a t l e a s t two minutes f o r the tube and c u r r e n t c a r r y i n g w i r e s t o c o o l . The experiment was performed t w i c e a t the l o w e s t c u r r e n t t o be used. The c u r r e n t was then i n c r e a s e d and the experiment performed t w i c e more. T h i s c o n t i n u e d u n t i l about 20 KA at which time the c u r r e n t was d e c r e a s e d , t a k i n g two s h o t s a t each s e t t i n g , u n t i l the o r i g i n a l v a l u e was re a c h e d . With c u r r e n t s above 20 KA the same p r o c e d u r e 53 was f o l l o w e d w i t h the e x c e p t i o n t h a t o n l y one shot was taken a t each s e t t i n g as the shock tube was i n danger o f b r e a k i n g a t the h i g h e r power l e v e l s . The run was r e p e a t e d t w i c e i n o r d e r t o o b t a i n the same amount o f d a t a . U s u a l l y the tube had t o be c l e a n e d d u r i n g t h i s p a r t o f the e x p e r i m e n t . I f the experiment c a l l e d f o r c u r r e n t s o f 40 KA or more, thes e s h o t s were t a k e n a f t e r a l l the d a t a a t lower c u r r e n t s had been o b t a i n e d . T h i s was n e c e s s a r y as i t was not uncommon f o r the t u b e , a c u r r e n t c a r r y i n g w i r e , or the e x p e r i m e n t e r ' s nerve t o break a t the h i g h powers. D u r i n g the c o u r s e o f t h i s work about two dozen shock tubes were consumed. I 54 CHAPTER IV FLUID FLOW IN AN ELECTROTHERMAL SHOCK TUBE WITH CONSTANT POWER INPUT The r e s u l t s p r e s e n t e d i n t h i s c h a p t e r i n d i c a t e the heat wave model a d e q u a t e l y d e s c r i b e s the f l o w i n an e l e c t r o t h e r m a l shock tube. The f i r s t s e c t i o n p r e s e n t s q u a l i t a t i v e e v i d e n c e o f the model's a p p l i c a b i l i t y . The next s e c t i o n shows how an e m p i r i c a l h e a t i n g c h a r a c t e r i s t i c can be e x t r a c t e d from gas d y n a m i c a l measurements. There i s no way o f c o m p l e t e l y c h e c k i n g the a c c u r a c y o f t h i s c h a r a c t e r i s t i c u n l e s s one has d e t a i l e d knowledge o f the energy source and a b s o r p t i o n mechanisms. However, s p e c t r o s c o p i c measurements gave p a r t i a l v e r i f i c a t i o n o f i t s a c c u r a c y . In a d d i t i o n , the measured v e l o c i t y o f the "shock l i m i t " heat f r o n t agreed w i t h the p r e d i c t e d v e l o c i t y based on the i n t e r s e c t i o n p o i n t o f the h e a t i n g c h a r a c t e r i s t i c w i t h the C.J. l i n e . In the l a s t s e c t i o n we show the "shock l i m i t " f r o n t s are i n q u a n t i t a t i v e agreement w i t h the C.J. p r e d i c t i o n s o f the t h e o r y . 55 CHAPTER IV SECTION A CHECK OF HEAT WAVE MODEL ASSUMPTIONS AND QUALITATIVE VERIFICATION OF THE THEORY In c h a p t e r I I the model assumptions were s t a t e d and d i s c u s s e d . The assumptions amenable to e x p e r i m e n t a l i n v e s t i g a t i o n are the w i d t h s o f the shock and heat f r o n t s , and the homogeneity o f the gas b e h i n d the heat f r o n t . I n t h i s s e c t i o n , f r a m i n g camera photographs are p r e s e n t e d which i n d i c a t e t h e shock and heat waves can be c o n s i d e r e d d i s c o n t i n u i t i e s . Framing camera photos a l s o show t h a t when a shock wave i s formed, i t u s u a l l y s e p a r a t e s from the heat wave when they are o n l y a few c e n t i m e t r e s from the e l e c t r o d e s . Other f e a t u r e s o f the f l u i d f l o w produced w i t h c o n s t a n t power i n p u t are i l l u s t r a t e d w i t h s t r e a k camera photographs o f s u b s o n i c and s u p e r s o n i c heat waves and w i t h an o s c i l l o r g r a m o f the p r e s s u r e produced by a s u b s o n i c heat wave. F i n a l l y , i t . i s shown t h a t the "shock l i m i t " c u r r e n t i s a f u n c t i o n o f p r e s s u r e . F i g . 16 i s a f r a m i n g camera photograph o f a shock f r o n t f o l l o w e d by i t s s u b s o n i c heat wave. F i f t y m i l l i t o r r o f a c e t y l e n e was added t o the argon i n o r d e r t o make the shock v i s i b l e . I t was assumed t h a t t h i s s m a l l amount of a c e t y l e n e , which was added any time i t was n e c e s s a r y t o photograph the shock, d i d not a p p r e c i a b l y i n f l u e n c e the f l o w . The b l a c k l i n e s i n t h i s , and a l l f o l l o w i n g p h o t o g r a p h s , are s e p a r a t e d 5 cm. The time i n t e r v a l between frames i s 10 microseconds and the exposure of each frame i s 200 nanoseconds. In the photograph i t appears t h a t t h e heat wave does not f i l l t he e n t i r e tube but t h i s i s an o p t i c a l d i s t o r t i o n caused by o v e r e x p o s i n g the camera i n o r d e r t o F i g . 16. Framing photograph of a subsonic heat wave and i t s associated shock. At=10 psec, p=5 t o r r , 1=20 KA. 57 see both the shock and heat waves. When the heat wave i s exposed p r o p e r l y i t f i l l s the e n t i r e tube. Note t h a t the shock f r o n t i s p e r f e c t l y p l a n a r . The t r a n s i t i o n between nonluminous c o l d gas behind the shock and luminous hot gas appears to be a b r u p t but the heat f r o n t i s o n l y p l a n a r t o h a l f a tube d i a m e t e r . F i g . 17 i s a f r a m i n g camera photo o f a s u p e r s o n i c heat wave. The i n t e r v a l between frames i s 5 m i c r o s e c o n d s and the exposure o f each frame i s 200 nanoseconds. The s t r u c t u r e o f the s u p e r s o n i c heat wave i s v e r y s i m i l a r t o the s u b s o n i c heat wave, f i g . 16. The change from nonluminous t o luminous gas o c c u r s i n l e s s than 1 cm and the f r o n t i s a g a i n o n l y p l a n a r t o h a l f a tube d i a m e t e r . In e i t h e r c a s e , the w i d t h o f the heat f r o n t i s on the o r d e r o f 1 cm, a l e n g t h s h o r t i n comparison t o the d i s t a n c e from the e l e c t r o d e s , 20 cm. We b e l i e v e t h a t the assumption o f t r e a t i n g the heat waves as d i s c o n t i n u i t i e s i s s u p p o r t e d by t h e s e two p h o t o g r a p h s . The t h e o r y p r e s e n t e d i n c h a p t e r I I , s e c t i o n B, assumes t h a t the f l o w i s i n a s t e a d y s t a t e . To check whether t r a n s i t o r y e f f e c t s were i m p o r t a n t , the e v o l u t i o n o f a s u b s o n i c heat wave and shock wave was s t u d i e d w i t h the f r a m i n g camera, f i g . 18. The i n t e r v a l between frames i s 5 m i c r o s e c o n d s and the exposure time i s 200 nanoseconds. The l i n e a t the r i g h t s i d e o f the p i c t u r e i s 5 cm from the e l e c t r o d e s . The f i l l i n g p r e s s u r e was 2.5 t o r r and the c u r r e n t was 28 kA. No shock i s v i s i b l e i n the bottom two frames. In s u c c e e d i n g frames the shock has formed and i s s e p a r a t i n g from the heat wave. I n t h i s photograph the shock wave forms 7 cm from the e l e c t r o d e s . By i n c r e a s i n g or 58 59 F i g . 18. Framing photograph of a shock wave separating from a heat wave. At=5 yfsec, p=2.5 t o r r , 1 = 28 KA. 60 decreasing W/p.j , the p o s i t i o n s at which the shock wave forms, could be moved f u r t h e r away or c l o s e r to the e l e c t r o d e s . One explanation of t h i s f a c t i s , t h a t , at p o s i t i o n s c l o s e to the e l e c t r o d e s , the power d e l i v e r e d to the heat f r o n t i s s u f f i c i e n t to form a supersonic heat wave. With i n c r e a s i n g distance from the e l e c t r o d e s the energy l o s s e s increase and e v e n t u a l l y the heat wave must turn sonic at which poi n t a shock wave forms. At i n i t i a l l y higher W/p., , the p o s i t i o n of shock formation would be f a r t h e r from the electrodes as more energy would have to be l o s t before the heat wave turned s o n i c . As most measurements of the flow were taken 20 cm from the e l e c t r o d e s , the few cm reguired for shock formation can be ignored and the flow can be assumed to be i n a steady s t a t e for a l l t i m e 1 . F i g s . 19 and 20 are streak camera photographs of a subsonic heat wave with shock and a supersonic heat wave, r e s p e c t i v e l y . In each case the flow was r e f l e c t e d from a p i s t o n located i n the shock tube near the l e f t side of the photos. Note that a l l f r o n t s , the shock f r o n t and both heat f r o n t s , t r a v e l with roughly constant v e l o c i t y due to the constant current input. When the shock wave i n f i g . 19 s t r i k e s the p i s t o n , a r e f l e c t e d shock i s formed which f i r s t t r a v e l s through the m a t e r i a l compressed by the o r i g i n a l shock and f i n a l l y through the gas behind the heat f r o n t . The r e f l e c t e d shock's v e l o c i t y i s constant as i t t r a v e l s through the shock compressed m a t e r i a l , As the energy l o s s e s i n the system increase with i n c r e a s i n g d i s t a n c e from the e l e c t r o d e s , both the shock and the heat waves' v e l o c i t i e s decrease with time and so the flow never achieves a steady s t a t e . This point was discussed i n Chapter I I , s e c t i o n E. 61 F i g . 19. S t r e a k photograph o f a s u b s o n i c heat wave w i t h shock r e f l e c t e d from a p i s t o n . p=6.5 t o r r , T=22 KA, S=shock wave, HW=subsonic heat wave, P = p i s t o n a t x=27 cm. 62 F i g . 20. Streak photograph of a supersonic heat wave r e f l e c t e d from a p i s t o n . p=.6 t o r r , 1=22 KA, HW=supersonic heat wave, P=piston at x=27 cm. 63 i n d i c a t i n g t h a t the r e g i o n behind the f i r s t shock i s homogeneous. When the r e f l e c t e d shock s t r i k e s the heat f r o n t i t s v e l o c i t y i m m e d i a t e l y changes and c o n t i n u e s t o change i n d i c a t i n g t h a t the r e g i o n behind the heat f r o n t i s not uni f o r m . 1 T h i s i s i n g r e a t p a r t caused by the c a p a c i t o r bank c e a s i n g t o p r o v i d e c u r r e n t as i n d i c a t e d by the u p t u r n i n g o f the s t r i a t i o n s i n the upper r i g h t c o r n e r o f the photograph. The r e f l e c t e d shock v e l o c i t y i s n e a r l y c o n s t a n t when i t t r a v e l s i n t o a heat wave s t i l l s u p p o r t e d by power i n p u t . The use o f a r e f l e c t o r i n the shock tube e n a b l e s one t o dete r m i n e whether t h e r e i s a non-luminous shock wave p r e c e d i n g the heat wave. A non-luminous shock i n c i d e n t on the p i s t o n can be d e t e c t e d as i t sometimes becomes luminous upon r e f l e c t i o n and i t always s t o p s the heat wave from r e a c h i n g the p i s t o n . The shock i n f i g . 19 was luminous so no p i s t o n was needed t o dete r m i n e i t s p r e s e n c e . The luminous f r o n t i n f i g . 20 reaches the p i s t o n i n d i c a t i n g t h e r e i s no shock wave p r e c e d i n g i t . The r e f l e c t e d heat wave's v e l o c i t y i s c o n s t a n t which s u g g e s t s t h a t the gas i n t o which i t i s t r a v e l l i n g i s homogeneous. I n o r d e r f o r t h i s t o be s o , t h e r a r e f a c t i o n wave must f o l l o w the s u p e r s o n i c heat wave v e r y c l o s e l y , i n d i c a t i n g t h a t the heat wave i s n e a r l y a C.J. d e t o n a t i o n . F i g s . 16 and 19, i n which t h e r e i s a shock f r o n t , were The f r o n t t r a v e l l i n g through the heat wave r e g i o n i s no l o n g e r an a d i a b a t i c shock due t o the energy f l u x f l o w i n g from the e l e c t r o d e end o f the shock tube. I t i s l i k e a r a d i a t i o n s u p p o r t e d shock t r a v e l l i n g i n t o a l a s e r beam. 64 o b t a i n e d a t r e l a t i v e l y low W/P1 w h i l e i n f i g s . 17 and 20, o b t a i n e d a t h i g h e r W/p1, t h e r e i s no shock wave. F i g . 21 i s a p l o t o f the f i l l i n g p r e s s u r e v e r s u s the "shock l i m i t " c u r r e n t . From i t we see t h a t a t h i g h c u r r e n t s and low f i l l i n g p r e s s u r e s no shock waves are produced. S i m i l a r l y a t lower c u r r e n t s , shock waves are produced a t a l l but the l o w e s t f i l l i n g p r e s s u r e s . T h i s graph shows t h e r e are two d i s t i n c t f l o w regimes. One can produce one or the o t h e r by changing e i t h e r the c u r r e n t or the f i l l i n g p r e s s u r e , i . e . W/p1 . The c u r v e i n f i g . 21 depends s e n s i t i v e l y on the smoothness of the i n t e r i o r o f . t h e shock tube used t o o b t a i n t h e d a t a . I t was n e c e s s a r y t o c l e a n the shock tube o f t e n d u r i n g t h i s e x periment s i n c e e l e c t r o d e m a t e r i a l on the w a l l s s t r o n g l y i n f l u e n c e d the r e s u l t s . I f the tube was d i r t y , up to 50% more c u r r e n t was r e q u i r e d to produce the s u p e r s o n i c heat wave than i f the tube was c l e a n . I t was noted l a t e r t h a t u s i n g a new shock tube a l s o had an e f f e c t on the r e q u i r e d c u r r e n t . The v e l o c i t y o f the l i m i t i n g heat wave was, however, independent o f the c o n d i t i o n o f the tube. T h i s i n d i c a t e s the e n t h a l p y behind the heat wave remained c o n s t a n t but the f r a c t i o n o f the energy r e a c h i n g the heat f r o n t depended on the shock tube's i n t e r i o r s u r f a c e . We have found, t h e n , t h a t w i t h i n c r e a s i n g power i n p u t the heat wave e v e n t u a l l y t u r n s , s u p e r s o n i c but t h a t the c u r r e n t r e q u i r e d i s s t r o n g l y dependent on the c o n d i t i o n o f the shock tube. L a t e r i n t h i s c h a p t e r the v e l o c i t i e s o f the l i m i t i n g f r o n t s w i l l be compared to the power d e l i v e r e d , to the f r o n t . A t y p i c a l example o f a p r e s s u r e s i g n a l o f a shock f r o n t and i t s s u b s o n i c heat wave i s shown i n f i g . 22. The shock a r r i v e d 65 C U R R E N T (KA) F i g . 21. The f i l l i n g p r e s s u r e v e r s u s c u r r e n t f o r the t r a n s i t i o n p o i n t between the s o n i c and s u p e r s o n i c regimes. 66 at t=48 m i c r o s e c o n d s , w h i l e the heat wave, c h a r a c t e r i z e d by a drop i n p r e s s u r e , a r r i v e d a t t=90 m i c r o s e c o n d s . T h i s i n t e r p r e t a t i o n was c o n f i r m e d by a s t r e a k camera photo made s i m u l t a n e o u s l y . Comparison o f f i g . 22 w i t h the p r e s s u r e p r o f i l e i n f i g . 3 shows t h a t the measured p r e s s u r e i s i n q u a l i t a t i v e agreement w i t h the model. F i g . 22 shows d e f i n i t l y t h a t the luminous f r o n t i n an e l e c t r o t h e r m a l shock tube i s not a c o n t a c t s u r f a c e . T h i s s u p p o r t s S t r a c h a n ' s measurement of the f l o w v e l o c i t y b ehind the luminous f r o n t ' . T h i s c o n c l u d e s the s e c t i o n on q u a l i t a t i v e v e r i f i c a t i o n o f the heat wave model. We have shown t h a t the shock and heat f r o n t s can be c o n s i d e r e d d i s c o n t i n u i t i e s . We have a l s o shown t h a t the "shock" and "no shock" regimes can be produced f o r v a r i o u s c o m b i n a t i o n s o f f i l l i n g p r e s s u r e and c u r r e n t . We have seen t h a t the p r e s s u r e p r o f i l e o f a s l o w l y moving luminous f r o n t i s c o n s i s t e n t w i t h the heat wave model and not w i t h a c o n t a c t s u r f a c e . I t s h o u l d be noted here t h a t the ragged s t r u c t u r e o f a s u p e r s o n i c heat wave can be e x p l a i n e d by a n i s o t r o p i c energy t r a n s p o r t a t the heat f r o n t . There i s no need t o propose mechanisms, i . e . R a y l e i g h - T a y l o r i n s t a b i l i t i e s , t o e x p l a i n i t s s t r u c t u r e s i n c e i t was not expected t o be p l a n a r . In the two f o l l o w i n g s e c t i o n s we show how an e m p i r i c a l h e a t i n g c h a r a c t e r i s t i c can be determined from gas d y n a m i c a l measurements. T h i s c h a r a c t e r i s t i c i s v e r i f i e d w i t h s p e c t r o s c o p i c and v e l o c i t y measurements. 67 F i g . 2 2 . O s c i l l o g r a m o f a p r e s s u r e p r o f i l e of a s u b s o n i c heat wave and shock wave. CHAPTER IV SECTION B THE DETERMINATION OF AN EMPIRICAL HEATING CHARACTERISTIC I t was p o i n t e d out b e f o r e t h a t f o r a f u l l d e s c r i p t i o n o f the f l o w f i e l d produced by a heat wave, the response f u n c t i o n o f the medium, or h e a t i n g c h a r a c t e r i s t i c , must be known. I n o r d e r t o d e r i v e t h i s f u n c t i o n t h e o r e t i c a l l y one would have to know i n v e r y g r e a t d e t a i l a l l the m i c r o s c o p i c heat t r a n s f e r and energy a b s o r p t i o n mechanisms as w e l l as a l l the l o s s p r o c e s s e s o f the 68 plasma i n the shock tube. In the absence o f t h i s knowledge the response f u n c t i o n can be o b t a i n e d from a measurement o f any two parameters o f the o n e - d i m e n s i o n a l n o n - a d i a b a t i c f l o w . T h i s s e c t i o n o u t l i n e s a g r a p h i c a l method of d e t e r m i n i n g the h e a t i n g c h a r a c t e r i s t i c on the response p l a n e , f i g . 5. For t h i s purpose the response p l a n e has been covered w i t h s e v e r a l n e t s o f parameters such as shock f r o n t v e l o c i t i e s , , heat f r o n t v e l o c i t i e s , and p r e s s u r e r a t i o s . I f any p a i r o f the above mentioned v a r i a b l e s are measured, the e n t h a l p y and net power i n p u t can be o b t a i n e d by n o t i n g where t h e i r l o c i i n t e r s e c t on the response p l a n e . By v a r y i n g the e x t e r n a l parameters o f the system, i . e . the f i l l i n g p r e s s u r e or the c h a r g i n g v o l t a g e , d i f f e r e n t p o i n t s on the response p l a n e are o b t a i n e d . A l i n e t h r o u g h t h e s e p o i n t s i s the h e a t i n g c h a r a c t e r i s t i c . I t w i l l be shown t h a t t h i s c u r v e i s unique t o the shock tube. To determine the response f u n c t i o n , l a b frame v e l o c i t i e s o f shock and heat waves were measured from s t r e a k camera ph o t o g r a p h s , e.g. f i g . 19. As mentioned p r e v i o u s l y , the f r o n t v e l o c i t i e s were averaged over 10 cm from x=15 t o 25 cm. The u n c e r t a i n t y i n the v e l o c i t i e s i s 3%. To determine the heat wave v e l o c i t y i n i t s own frame o f r e f e r e n c e , i t i s n e c e s s a r y t o measure i t s l a b frame v e l o c i t y and then s u b t r a c t from t h a t , the f l o w v e l o c i t y o f the f l u i d behind the shock wave. No d i r e c t measurement o f t h i s f l o w v e l o c i t y was made; r a t h e r , i t was c a l c u l a t e d from the shock f r o n t v e l o c i t y and the R a n k i n e -Hugoniot r e l a t i o n s . Because o f the s u b t r a c t i o n o f the two v e l o c i t i e s , the u n c e r t a i n t y i n the heat wave v e l o c i t y i s a p p r o x i m a t e l y ± .08 km/sec. 69 The i n t e r s e c t i o n s o f p a i r s o f v e l o c i t i e s are p l o t t e d on the response p l a n e i n round p o i n t s , f i g 23. The f i l l i n g p r e s s u r e was c o n s t a n t a t 2 t o r r and the c u r r e n t was v a r i e d from 11 t o 40 KA, t h e lower l i m i t d i c t a t e d by the n e c e s s i t y t o produce luminous shocks. Another p a i r o f parameters t h a t can be measured t o detemine the h e a t i n g c h a r a c t e r i s t i c are the shock v e l o c i t y and the p r e s s u r e o f the heat wave. The shock v e l o c i t y was o b t a i n e d u s i n g two p r e s s u r e probes; the f i r s t probe t r i g g e r e d an o s c i l l o s c o p e and the second c a l i b r a t e d probe read the p r e s s u r e and a l s o gave the t r a n s i t time between pr o b e s . The u n c e r t a i n t y i n the shock v e l o c i t y due t o the u n c e r t a i n t y i n the t r a n s i t t i me between probes can be as l a r g e as 5% f o r the f a s t e s t s h o c k s . The u n c e r t a i n t y i n the p r e s s u r e i s about 20%. As the shock wave no l o n g e r had t o be luminous i n o r d e r t o d e t e r m i n e i t s v e l o c i t y , t h e c u r r e n t was v a r i e d from 4 t o 30 KA. U n f o r t u n a t e l y , i t was found t h a t a t c u r r e n t s above 15 KA, p r e s s u r e d i s t u r b a n c e s propagated from the e l e c t r o d e end o f the tube and a l t e r e d the p r e s s u r e i n the heat wave. The d i s t u r b a n c e s c o u l d be e a s i l y d i s t i n g u i s h e d on an o s c i l l o s c o p e t r a c e o f a p r e s s u r e probe and c o u l d a l s o be seen on s t r e a k camera photographs t a k e n s i m u l t a n e o u s l y . Because of t h e s e d i s t u r b a n c e s , p r e s s u r e measurements taken a t c u r r e n t s above 15 KA were u n r e l i a b l e . The d a t a , a l s o t a k e n a t a f i l l i n g p r e s s u r e o f 2 t o r r , a re p l o t t e d (square p o i n t s ) on the response p l a n e , f i g 23. U s i n g t h i s d a t a and t h a t o b t a i n e d from shock and heat wave v e l o c i t y 70 8 W70, (erg-cm/gram-sec) F i g . 23. An e m p i r i c a l h e a t i n g c h a r a c t e r i s t i c o b t a i n e d from gas dyna m i c a l measurements. 71 11 measurements, one sees that the enthalpy r i s e s from about 10 11 to 3.5x10 ergs/gram and then remains constant with i n c r e a s i n g power input. The data obtained from pressure measurements appear to j o i n smoothly to the data from v e l o c i t y measurements. Heating c h a r a c t e r i s t i c s were a l s o obtained for f i l l i n g pressures of 5 and 10 t o r r but are not di s p l a y e d as there are no s i g n i f i c a n t d i f f e r e n c e s . This e m p i r i c a l heating c h a r a c t e r i s t i c i s s i m i l a r to an arc c h a r a c t e r i s t i c , which r e l a t e s the voltage and current i n an a r c . Both r e l a t e the response of a medium to an input of power. The macros c o p i c a l l y obtained c h a r a c t e r i s t i c , u n l i k e a m i c r o s c o p i c a l l y obtained one, does not contain any information about the s t r u c t u r e of the f r o n t . The heating c h a r a c t e r i s t i c i s very u s e f u l s i n c e , once i t i s known, one need measure only one parameter associated with the flow to determine the other unknowns. One can also use i t to p r e d i c t the maximum compression obtainable with m u l t i p l e shocks. This i s done i n chapter V. The accuracy of our e m p i r i c a l heating c h a r a c t e r i s t i c can be checked i n two ways. Knowing the enthalpy and the pressure a temperature can be determined. Thus, i f we can measure a temperature by spectroscopic technigues, we can compare the two values. A l s o , the v e l o c i t y of the l i m i t i n g f r o n t can be obtained from the i n t e r s e c t i o n p o i n t of the heating c h a r a c t e r i s t i c and the C.J. l i n e . I f the heating c h a r a c t e r i s t i c p r e d i c t s the " c o r r e c t " temperature and v e l o c i t y we have q u a n t i t a t i v e evidence the heat wave model a p p l i e s to our shock tube flow. Q u a n t i t a t i v e v e r i f i c a t i o n of the theory can a l s o be 72 o b t a i n e d by comparing the v e l o c i t y a t the "shock l i m i t " t o the C.J. v e l o c i t y . These checks on the t h e o r y are p r e s e n t e d i n the next two s e c t i o n s . CHAPTER IV SECTION C QUANTITATIVE VERIFICATION OF THE HEATING CHARACTERISTIC The h o r i z o n t a l p o r t i o n o f the h e a t i n g c h a r a c t e r i s t i c c o r r e s p o n d s t o an e n t h a l p y o f about 3.5x10 ergs/gram. Assuming pure argon a t .5 atm. p r e s s u r e t h i s e n t h a l p y c o r r e s p o n d s t o a temperature o f about 1.25 eV. ( I f the plasma i s composed o f argon and t u n g s t e n , the same e n t h a l p y would c o r r e s p o n d t o a lower temperature s i n c e a l l the t u n g s t e n atoms would be i o n i z e d . ) To check t h i s temperature and the assumption of s t e a d y f l o w , time and space r e s o l v e d s p e c t r o s c o p i c measurements were c a r r i e d o u t . R e l a t i v e l i n e i n t e n s i t y measurements were attempted i n i t i a l l y but proved unuseable due to the h i g h l e v e l o f i m p u r i t y ( m a i n l y tungsten) r a d i a t i o n . No A r i l l i n e s were p r e s e n t i n time i n t e g r a t e d s p e c t r a but a l l the s t r o n g A r l l i n e s , e.g. 7635, were found. I n a d d i t i o n , t h e t u n g s t e n s p e c t r a r e v e a l e d no WI but a l l the WII l i n e s were found. The l a c k o f A r i l and WI l i n e s e n a b l e s us t o p l a c e the temperature between .7 and 1.2 eV. Another temperature e s t i m a t e was o b t a i n e d from time r e s o l v e d , (2 m i c r o s e c o n d s ) , e l e c t r o n d e n s i t y measurements made by measuring the p r o f i l e o f the Hbeta l i n e and comparing i t t o computed p r o f i l e s 4 3 . For these measurements the f i l l i n g gas was 80% argon and 20% m o l e c u l a r hydrogen. The l a r g e a d d i t i o n o f 73 hydrogen changed the hydrodynamics but had no measureable e f f e c t on the e l e c t r o n d e n s i t y when compared to t e s t runs with only 1% hydrogen. The 20% l e v e l of hydrogen was chosen so that H$ would be s u f f i c i e n t l y strong under a la r g e range of c o n d i t i o n s . The measurements, taken at a t o t a l f i l l i n g pressure of 5 t o r r , are summarized i n f i g . 24. The ordinate i s the log of the e l e c t r o n d e n s i t y while the abscissa i s the time a f t e r the s t a r t of the discharge. The three s o l i d l i n e s were obtained with a current of 15 KA and were taken at 3 p o s i t i o n s , x=8, 13, and 18 cm. The dotted l i n e was obtained with a current of 20 KA at x=8 cm. For a l l measurements, the monochromator s l i t was imaged upon the a x i s of the shock tube. No attempt was made to determine whether the e l e c t r o n d e n s i t y was a n i s o t r o p i c . To determine the temperature from the e l e c t r o n d e n s i t y , one must know the pressure. For our purposes i t i s s u f f i c i e n t to f i x the pressure at .2 atm., log(p)=2.2, as the temperature's dependence on the pressure i s s l i g h t i n the region of i n t e r e s t . For an e l e c t r o n d e n s i t y of 1.6x10 cm , the corresponding temperature, obtained from f i g . 15, i s .95 eV. Although the temperature of .95 eV may not be accurate due to the assumptions made to get f i g . 15, i t should be safe to conclude the d i r e c t i o n of the temperature change from the e l e c t r o n d e n s i t y measurements, i . e . the temperature i s higher c l o s e r to the e l e c t r o d e s . We can al s o say that the temperature appears to be constant i n time. The three temperatures obtained from the response plane, the e l e c t r o n d e n s i t i e s , and the presence or absence of d i f f e r e n t P=5 t o r r t (Msec) F i g . 24. E l e c t r o n d e n s i t y as a f u n c t i o n o f t i m e , d i s t a n c e from the e l e c t r o d e s , and c u r r e n t . 75 i o n i z a t i o n s p e c i e s are i n r e a s o n a b l e agreement, and i n d i c a t e t h a t the temperature o f the plasma i s about 1.1 eV. The agreement among the tempe r a t u r e s i s a check on the gasdynamical d e t e r m i n a t i o n o f the h e a t i n g c h a r a c t e r i s t i c . CHAPTER IV SECTION D LIMITING HEAT FRONTS As W/P-i i s i n c r e a s e d , the shock speed i n c r e a s e s s t e a d i l y u n t i l the c u r r e n t reaches the "shock l i m i t " . The shock wave then d i s a p p e a r s and the heat wave p r o p a g a t e s i n t o u n d i s t u r b e d gas. From the model, one would expect the f r o n t at the t r a n s i t i o n p o i n t s h o u l d behave l i k e a C.J. d e t o n a t i o n . To t e s t t h i s p r e d i c t i o n , the v e l o c i t i e s o f l i m i t i n g f r o n t s , o b t a i n e d f o r f i l l i n g p r e s s u r e s r a n g i n g from .6 to 3 t o r r and measured from s t r e a k camera p h o t o g r a p h s , were compared to e s t i m a t e s o f the powers d e l i v e r e d t o the f r o n t s , f i g . 25. The e m p i r i c a l r e l a t i o n between the n o r m a l i z e d i n t e n s i t y W/Pi, and the c u r r e n t , I , r e s i s t a n c e o f the shock t u b e , R, and the f i l l i n g p r e s s u r e , p, i s (see appendices 5 and 6 ) . 76 w_ (I 2R) n - 6 9 (19) The s l o p e o f the b e s t f i t t o the p o i n t s i s .33 as expected from the model. The p r e s s u r e p r o f i l e o f the l i m i t i n g f r o n t a l s o s u p p o r t s the c o n c l u s i o n t h a t the f r o n t i s a C.J. d e t o n a t i o n . U n f o r u n a t e l y , the time r e s o l u t i o n o f the p r e s s u r e probe, due to i t s o v e r s h o o t and r i n g i n g problems, made measurements of the peak p r e s s u r e s i n the d e t o n a t i o n i m p o s s i b l e , so no comparison between the measured and p r e d i c t e d p r e s s u r e s i s p o s s i b l e . The i n t e r s e c t i o n p o i n t between the h e a t i n g c h a r a c t e r i s t i c o b t a i n e d a t 2 t o r r and the C.J. l i n e c o r r e s p o n d s to a l i m i t i n g f r o n t v e l o c i t y o f about 4.2 km/sec. In comparison, the measured l i m i t i n g f r o n t v e l o c i t y a t 2 t o r r was 4.15 km/sec which i n d i c a t e s the h e a t i n g c h a r a c t e r i s t i c i s c o r r e c t . The e x p e r i m e n t a l r e s u l t s o f t h i s s e c t i o n and the p r e v i o u s one i n d i c a t e the f l o w i n an e l e c t r o t h e r m a l shock tube i s i n q u a n t i t a t i v e agreement w i t h the heat wave model. The temperature determined from the h e a t i n g c h a r a c t e r i s t i c was c o n s i s t e n t w i t h s p e c t r o s c o p i c meaurements. The l i m i t i n g f r o n t v e l o c i t y was a c c u r a t e l y p r e d i c t e d by the t h e o r y and the l i m i t i n g f r o n t behaved l i k e a C.J. d e t o n a t i o n . Our e m p i r i c a l h e a t i n g c h a r a c t e r i s t i c can a l s o p r o v i d e i n f o r m a t i o n about the energy t r a n s p o r t mechanism. In f i g . 23 the h e a t i n g c h a r a c t e r i s t i c i n the s u p e r s o n i c regime l i e s v e r y 77 F i g . 25. The l o g o f the v e l o c i t y o f the l i m i t i n g f r o n t v e r s u s the n o r m a l i z e d i n t e n s i t y d e l i v e r e d to the f r o n t . 7 8 c l o s e t o the C.J. l i n e which r e p r e s e n t s heat waves whose exhaust v e l o c i t y i s the i o n ' s speed o f sound. The f a c t t h a t the f a s t e s t heat waves cannot be made to d e v i a t e f a r from the C.J. l i n e i n d i c a t e s the heat t r a n s f e r mechanism i s somehow c o u p l e d t o the i o n sound speed. I f the heat wave was s u p p o r t e d by r a d i a t i o n , no such c o u p l i n g would e x i s t . On the o t h e r hand, i f the energy was t r a n s p o r t e d by e l e c t r o n s , t h e r e would be a c o u p l i n g . I f the e l e c t r o n s t r i e d t o d i f f u s e f a s t e r than the i o n sound speed, a space charge would be c r e a t e d i n h i b i t i n g t h e i r freedom. In o t h e r words, energy c o u l d not be t r a n s p o r t e d f o r w a r d a t a v e l o c i t y s u b s t a n t i a l l y g r e a t e r than the i o n speed o f sound. 79 CHAPTER V STEPPED CURRENT EXPERIMENTAL RESULTS Having s t u d i e d the f l o w produced by a c o n s t a n t c u r r e n t , one i s equipped t o understand the e f f e c t s o f a changing power i n p u t . I n c h a p t e r I I , s e c t i o n D, we drew an ana l o g y between a s u b s o n i c heat wave and a b l e a c h i n g wave. T h i s a n a l o g y s h o u l d not be pursued f u r t h e r . I f the energy t r a n s p o r t mechanism i s t h e r m a l c o n d u c t i o n , the gas w i l l never be c o m p l e t e l y t r a n s p a r e n t t o an energy f l u x . To t r a n s p o r t a d d i t i o n a l energy t o the heat f r o n t , an i n c r e a s e i n t h e r m a l c o n d u c t i v i t y , i . e . t e m p e r a t u r e , i s r e q u i r e d . The a d d i t i o n a l energy s h o u l d c r e a t e a f a s t s u p e r s o n i c heat wave which would soon o v e r t a k e the p r e c e d i n g s u b s o n i c heat wave. Only when t h i s happens can a second shock be formed. The f a c t t h a t a s u p e r s o n i c heat wave i s c l e a r l y e v i d e n t on s t r e a k camera p h o t o g r a p h s , e.g. f i g s . 27 and 31, i s another i n d i c a t i o n t h a t the energy t r a n s p o r t mechanism i s e l e c t r o n t h e r m a l c o n d u c t i o n . In c h a p t e r I I , s e c t i o n F, we drew a p a r a l l e l between an 80 e l e c t r o t h e r m a l shock tube and a l a s e r c o m p r e s s i o n e x p e r i m e n t . In view o f the s i m i l a r i t i e s between the f l o w , we attempted to i l l u s t r a t e the dynamics o f a "s u p e r c o m p r e s s i o n " w i t h our shock tube. T h i s c h a p t e r , d e a l i n g w i t h stepped c u r r e n t p u l s e e x p e r i m e n t s , i s d i v i d e d i n t o two p a r t s . S e c t i o n A d e a l s w i t h double shock e x p e r i m e n t s i n which the second c u r r e n t s t e p does not produce a s u p e r s o n i c heat wave. S e c t i o n B d e a l s w i t h burn t h r o u g h e x p e r i m e n t s where the second c u r r e n t s t e p t u r n s the i n i t i a l l y s u b s o n i c heat wave s u p e r s o n i c . CHAPTER V SECTION A DOUBLE SHOCKS As e x p l a i n e d i n c h a p t e r I I , the second c u r r e n t p u l s e r a i s e s the p r e s s u r e ; i f t h i s new p r e s s u r e wave can t r a v e l f a s t e r than the r e s u l t i n g heat wave, a second shock wave i s formed. We have found t h a t we can c r e a t e two shocks w i t h a s u b s o n i c heat wave powered by a two s t e p c u r r e n t p u l s e , e.g. f i g . 13c. In t h i s s e c t i o n , a t y p i c a l s t r e a k photograph o f a double shock i s g i v e n and the i n t e r a c t i o n o f the shocks i s compared t o t h e o r y . P r e d i c t i o n s , based on the h e a t i n g c h a r a c t e r i s t i c , o f the second s h o c k s ' v e l o c i t i e s are found t o be i n r e a s o n a b l e agreement w i t h measured v e l o c i t i e s . An i m p o r t a n t r e s u l t i n t h i s s e c t i o n i s , the r e s u l t i n g shock v e l o c i t y i s a f u n c t i o n o n l y o f the t o t a l power i n p u t . I t does not depend upon the manner i n which the power i s a p p l i e d ; i t can be one l a r g e s t e p or two s m a l l e r s t e p s o f any r a t i o . 81 F i g . 2 6 i s a s c h e m a t i c o f a double shock f l u i d f l o w produced i n an e l e c t r o t h e r m a l shock tube. F i g . 27 i s a s t r e a k camera photograph o f d o u b l e shocks produced by f i r i n g t h e second c a p a c i t o r bank 5 0 m i c r o s e c o n d s a f t e r the f i r s t .bank. The c u r r e n t from each bank was 1 5 KA and the f i l l i n g p r e s s u r e was 5 t o r r . Assuming t h a t S-j and S 2 are s h o c k s , we c a l c u l a t e d , as o u t l i n e d i n appendix 1 , the l a b frame v e l o c i t i e s o f the r e s u l t i n g shock, S R, the c o n t a c t s u r f a c e , CS, and the head o f the r a r e f a c t i o n wave, RW, t o be V S R = 2 . 9 8 km/sec ( 2 . 9 4 + 3 % ) , V C S = 2 . 2 3 km/sec ( 2 . 2 5 ± 3%) and V R W = . 9 km/sec ( 1 . 0 ± 1 0 % ) . These v a l u e s agree v e r y w e l l w i t h the measured v e l o c i t i e s g i v e n i n the b r a c k e t s . The measured v e l o c i t y o f the r a r e f a c t i o n wave has a 1 0 % u n c e r t a i n t y as i t s t r a c e on the s t r e a k photograph i s s h o r t and c l o s e t o v e r t i c a l . Many s t r e a k photographs were a n a l y z e d and a l l showed the same s t r u c t u r e , a l t h o u g h on some, the r a r e f a c t i o n wave was not v i s i b l e . The c a l c u l a t e d v e l o c i t i e s are u n c e r t a i n by about 1 0 % , f o r , t o c a l c u l a t e them, i t was n e c e s s a r y t o know the v e l o c i t y o f the second shock i n i t s own frame o f r e f e r e n c e . T h i s v e l o c i t y was o b t a i n e d by s u b t r a c t i n g from the l a b frame v e l o c i t y , the f l u i d v e l o c i t y b ehind the f i r s t shock. The f l u i d v e l o c i t y was c a l c u l a t e d from the f i r s t shock's v e l o c i t y and the Rankine-Hugoniot r e l a t i o n s . The agreement between the measured and c a l c u l a t e d v e l o c i t i e s i s t a k e n as p r o o f t h a t we have produced two shocks by the a p p l i c a t i o n o f a stepped c u r r e n t p u l s e . Based on the measured shock v e l o c i t i e s , the d e n s i t y i n r e g i o n [ 4 ] i s 6.9 times g r e a t e r than i n r e g i o n [ 1 ] and the 82 F i g . 26. Schematic of a streak photograph of a double shock. S,=firstshock, S 2=second shock, S R = r e s u l t i n g shock, CS=contact surface, RW=rarefaction wave, HW R=resulting heat wave, HW«=first heat wave. 83 84 temperature i s o n l y 5300 °K. I t i s i m p o s s i b l e to r e a c h such a h i g h compression a t t h i s low temperature w i t h a s i n g l e s t r o n g shock. F i g . 28 i s an o s c i l l o g r a m o f the p r e s s u r e s i g n a l produced by double s h o c k s , s i m i l a r t o the one e x p e c t e d , f i g . 6. The f i r s t shock a r r i v e s a t t=26 m i c r o s e c o n d s , the second shock w i t h i t s i n c r e a s e i n p r e s s u r e a r r i v e s a t t=36 m i c r o s e c o n d s and the heat wave a r r i v e s around t=44 m i c r o s e c o n d s . The p r e s s u r e drop i n c r o s s i n g from the d o u b l y shocked r e g i o n to the heat wave r e g i o n i s e x p e c t e d , as the p a r t i c l e s are b e i n g d e c e l e r a t e d by the heat wave. The time i n t e r v a l s between the waves were c o n f i r m e d by a s t r e a k camera photograph t a k e n s i m u l t a n e o u s l y . The v e l o c i t y o f the second shock was c a l c u l a t e d as a f u n c t i o n o f power i n p u t i n c h a p t e r I I , s e c t i o n C, assuming the e n t h a l p y behind the heat f r o n t s was c o n s t a n t a t 3 . 5 x l 0 1 1 ergs/gram. The c o n s t a n c y o f the e n t h a l p y , independent o f the power i n p u t , i s c r i t i c a l t o the f o l l o w i n g comparison s i n c e the computed c u r v e i s s e n s i t i v e t o r e l a t i v e e n t h a l p y changes. The computed powers are compared to e x p e r i m e n t a l p o i n t s i n f i g . 29. The f i r s t shock was produced w i t h a c u r r e n t o f 15 KA and a f i l l i n g p r e s s u r e o f 5 t o r r . The second bank, f i r e d 55 mic r o s e c o n d s a f t e r the f i r s t bank, p r o v i d e d an a d d i t i o n a l c u r r e n t r a n g i n g from 6 to 25 KA g i v i n g a maximum c u r r e n t o f 40 KA. The v e l o c i t i e s o f the shock waves were measured from s t r e a k camera photographs s i m i l a r t o f i g . 27. The o r d i n a t e o f f i g . 29 i s the r a t i o o f powers a f t e r the second c u r r e n t s t e p to t h a t o f the f i r s t c u r r e n t s t e p . The e x p e r i m e n t a l powers were c a l c u l a t e d from eqn. 19. The a b s c i s s a i s the Mach number o f the second 85 F i g . 2 8 . O s c i l l o g r a m o f a p r e s s u r e p r o f i l e o f double shocks f o l l o w e d by a s u b s o n i c heat wave. shock wave. The s o l i d l i n e i s the computed c u r v e . I t would take o n l y a 10% i n c r e a s e i n c u r r e n t to make the e x p e r i m e n t a l p o i n t s f a l l on the computed c u r v e . The agreement between t h e o r y and experiment i n d i c a t e s the heat wave model a l s o a p p l i e s when t h e r e i s a s t e p - l i k e i n c r e a s e i n power and v e r i f i e s the c o n s t a n c y o f the h e a t i n g c h a r a c t e r i s t i c determined i n the p r e v i o u s c h a p t e r . The manner i n which a two s t e p power p u l s e i s a p p l i e d to a s t e p heat wave has l i t t l e e f f e c t on the r e s u l t i n g shock speed. 86 F i g . 29. R a t i o of the powers i n the stepped power pulse versus the Mach number of the second shock. The curve i s a computed value assuming the enthalpy i s constant at 3 . 5 x l 0 1 1 ergs/gram. 87 T h i s can be i l l u s t r a t e d i n v a r i o u s ways. T a b l e 1, e x t r a c t e d from f i g s . 7 and 8, g i v e s the Mach numbers of the f i r s t and second shocks and the r e l a t i v e powers r e g u i r e d to produce the c o m b i n a t i o n . The speeds o f the shocks were chosen so t h a t the r e s u l t i n g shock speed was Mach 8. Note t h a t the power r e g u i r e d t o produce the r e s u l t i n g Mach 8 shock wave v a r i e s o n l y 10%. As an a s i d e , note t h a t the r e s u l t i n g shock Mach number i s a p p r o x i m a t e l y g i v e n by the p r o d u c t o f the Mach numbers o f the f i r s t and second sh o c k s . The d a t a i n t a b l e 2 was o b t a i n e d e x p e r i m e n t a l l y by v a r y i n g the c u r r e n t s from banks I and I I w h i l e keeping the sum o f the c u r r e n t s c o n s t a n t . T h i s m a i n t a i n s a c o n s t a n t power i n p u t . The r e s u l t i n g shock v e l o c i t y i s n e a r l y independent of the manner i n which the power i s s u p p l i e d . The c u r r e n t s have a r e l a t i v e a c c u r a c y o f 3% and the r e s u l t i n g shock v e l o c i t y i s a c c u r a t e t o 3%. These t a b l e s i n d i c a t e t h a t the r e s u l t i n g shock, speed i s a f u n c t i o n o n l y o f the t o t a l i n t e n s i t y and not on the manner i n which i t i s a p p l i e d . I t i s i n t u i t i v e l y o b v i o u s t h a t t h i s s h o u l d be s o , f o r , on a time s c a l e l a r g e i n comparison t o the time i n t e r v a l between bank f i r i n g s , t h e i n i t i a l s t r u c t u r e o f t h e power p u l s e cannot have any major e f f e c t . The i n i t i a l s t e p i n c u r r e n t can be c o n s i d e r e d an i r r e g u l a r i t y i n the r i s e time of the c u r r e n t p u l s e . To summarize, we have found t h a t two shocks can be c r e a t e d by a two s t e p c u r r e n t p u l s e . The p r e s s u r e p r o f i l e o f the f l o w agrees g u a l i t a t i v e l y w i t h t h a t p r e d i c t e d by the t h e o r y . U s i n g 88 M B = 8 8 7 6 5 4 M 2 t: 1.13 1.31 1.58 2 \N/9A .95 .97 .98 1.01 1.05 Table 1. Combination to two shocks which produce a Mach 8 r e s u l t i n g shock and r e l a t i v e powers reguired to produce the combination. I,+I2=25 Ii 5 9 15 18 25 I 2 20 16 10 7 0 2.30 2.31 2.30 2.26 2.45 Table 2. The r e s u l t i n g shock speed produced by current I- and I 2 whose sum i s 25 KA. • 1 89 the heat wave model, i t i s p o s s i b l e t o p r e d i c t the c u r r e n t , t o w i t h i n 10%, needed t o produce a s p e c i f i e d second shock v e l o c i t y . I t has been shown t h a t the r e s u l t i n g shock v e l o c i t y i s independent o f the manner i n which the power p u l s e i s a p p l i e d . Knowing t h a t the r e s u l t i n g f l o w i s dependent o n l y upon the t o t a l power, we can s p e c u l a t e t h a t the s u p e r s o n i c regime i n a two s t e p c u r r e n t experiment s h o u l d be reached a t the "shock l i m i t " c u r r e n t t h a t would produce the s u p e r s o n i c regime when the c u r r e n t was a p p l i e d i n one l a r g e s t e p . T h i s w i l l be examined i n the next s e c t i o n . CHAPTER V SECTION B BURN THROUGH WAVES We found i n the c o n s t a n t c u r r e n t e x p e r i m e n t s t h a t i t was p o s s i b l e t o produce s u p e r s o n i c heat waves. As the f i n a l f l o w a c h i e v e d i s independent o f the manner i n which the power i s a p p l i e d , i t s h o u l d be p o s s i b l e t o produce a shock wave f o l l o w e d by a s u p e r s o n i c heat wave i f the c u r r e n t a f t e r the second s t e p exceeds the "shock l i m i t " . F i g . 30 i s a sc h e m a t i c o f a shock f o l l o w e d by a s u p e r s o n i c heat wave. F i g . 31 i s a s t r e a k photograph o f a "burn t h r o u g h " . The f i l l i n g p r e s s u r e was 2 t o r r , t he c u r r e n t i n the f i r s t s t e p was 12 KA, w h i l e the c u r r e n t from bank I I , f i r e d 50 mi c r o s e c o n d s a f t e r bank I , was 15 KA. I f c o n s t a n t c u r r e n t was a p p l i e d t o t h i s shock t u b e , the "shock l i m i t " was about 25 KA. No shock 90 X F i g . 30. Schematic of a streak photograph of a burn through S ^ f i r s t shock, HW1 = f i r s t heat wave. 91 92 precedes the supersonic heat wave as i t t r a v e l s through the p r e v i o u s l y shock-compressed m a t e r i a l . "Burn throughs", s i m i l a r to that i n f i g . 31, were produced f o r f i l l i n g p r e s s u r e s ranging from 1 to 4 t o r r . In each case, the "burn through" occurred at the "shock l i m i t " c u r r e n t f o r that f i l l i n g p r e s s u r e . I t appears that there i s no j u d i c i o u s way to apply a power p u l s e to avert a burn through. T h i s phenomena i s of i n t e r e s t i n l a s e r f u s i o n s i n c e they wish to achieve a very high compression by means of m u l t i p l e weak shock waves. Once a "burn through" occurs no f u r t h e r compression can be achieved. Knowing the i n t e n s i t y at which the "burn through" f i r s t occurs enables one to c a l c u l a t e the maximum c o m p r e s s i o n 4 4 . 9m/Q<\ i s the maximum a t t a i n a b l e compression that can be achieved i f one d r i v e s a heat wave i n such a manner so as to produce an a d i a b a t i c compression. W/P., i s the maximum normalized i n t e n s i t y and c 1 i s the speed of sound i n the f i l l i n g gas. Our heating c h a r a c t e r i s t i c allows us to c a l c u l a t e the maximum p o s s i b l e compression that can be achieved i n our shock tube. Our "shock l i m i t " i n t e n s i t y , determined from the i n t e r s e c t i o n p o i n t between 1 7 heating c h a r a c t e r i s t i c and the C.J. l i n e i s 1.2x10 erg-cm/gram-sec. The speed of sound i n argon at 20 °C i s 3.18xl0 4 cm/sec. S u b s t i t u t i n g these values i n eqn. 13 g i v e s 9m/9-\ = 2 1 . In 3 / 8 (13) input that can be used before the heat wave turns s u p e r s o n i c , 93 c o m p a r i s o n , the maximum compression a c h i e v e d w i t h a two s t e p power p u l s e i n our experiment was 11. 94 CHAPTER VI CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK The p r e s e n t a t i o n o f the e x p e r i m e n t a l r e s u l t s was d i v i d e d between c h a p t e r s IV and V to emphasize the p o i n t s we sought t o i n v e s t i g a t e . Namely, the f l o w i n an e l e c t r o t h e r m a l shock tube i s w e l l d e s c r i b e d by the heat wave model, and, the e l e c t r o t h e r m a l shock tube can be used to s t u d y phenomena c h a r a c t e r i z e d by a l o c a l i z e d d e p o s i t i o n o f energy. The f i r s t two s e c t i o n s o f t h i s c h a p t e r w i l l summarize the e x p e r i m e n t a l r e s u l t s . The f i n a l s e c t i o n c o n t a i n s s u g g e s t i o n s f o r f u t u r e work and summarizes my o r i g i n a l c o n t r i b u t i o n s . CHAPTER VI SECTION A THE APPLICABILITY OF THE HEAT WAVE MODEL TO THE FLOW IN AN ELECTROTHERMAL SHOCK TUBE I t was g e n e r a l l y b e l i e v e d t h a t v e r y f a s t luminous f r o n t s i n e l e c t r o t h e r m a l shock tubes were shock waves w i t h an energy i n p u t 95 and t h a t the slowe r moving luminous f r o n t s were c o n t a c t s u r f a c e s . The r e s u l t s p r e s e n t e d i n c h a p t e r IV suggest a more adeguate d e s c r i p t i o n o f the f l o w can be based on the heat wave t h e o r y . P r e v i o u s workers had shown t h e r e were two d i s t i n c t regimes o f f l o w . We have v e r i f i e d t h i s w i t h a c o n s t a n t i n p u t c u r r e n t and have shown how the two regimes can be r e a d i l y e x p l a i n e d w i t h heat wave t h e o r y w i t h o u t i n v o k i n g o t h e r e f f e c t s . In a d d i t i o n , we have measured a p r e s s u r e drop a c r o s s a s l o w l y moving luminous f r o n t which i s c o n s i s t e n t w i t h a s u b s o n i c heat wave but not w i t h a c o n t a c t s u r f a c e . From gas dy n a m i c a l . measurements we have determined an e m p i r i c a l h e a t i n g c h a r a c t e r i s t i c and v e r i f i e d i t w i t h s p e c t r o s c o p i c measurements. A p r e d i c t i o n o f a "shock l i m i t " f r o n t v e l o c i t y based upon the h e a t i n g c h a r a c t e r i s t i c agrees v e r y w e l l w i t h the measured v e l o c i t y . The l i m i t i n g f r o n t behaved l i k e a C.J. d e t o n a t i o n , a p r e d i c t i o n o f the model. These r e s u l t s , t a k e n t o g e t h e r , i n d i c a t e the heat wave model i s adeguate t o d e s c r i b e the f l o w i n an e l e c t r o t h e r m a l shock tube. T h i s i s not s u r p r i s i n g , f o r , i n d e v e l o p i n g the t h e o r y , i t was o n l y assumed t h e r e was an i n t e n s i t y , W, which was absorbed over a d i s t a n c e s h o r t i n comparison t o o t h e r r e l e v a n t d i s t a n c e s . No mention was made o f the energy t r a n s p o r t or a b s o r p t i o n mechanisms. Indeed, the t h e o r y i s independent o f t h e s e , and sh o u l d t h e r e f o r e a p p l y t o the many phenomena produced by inhomogeneous energy a b s o r p t i o n , the f l o w i n an e l e c t r o t h e r m a l shock tube b e i n g o n l y one o f them. 96 CHAPTER VI SECTION B THE ELECTROTHERMAL SHOCK TUBE AS A SIMULATOR OF INHOMOGENEOUS ENERGY ABSORPTION PHENOMENA In o r d e r t o demonstrate how the e l e c t r o t h e r m a l shock tube can be used t o examine a s p e c t s o f s p e c i f i c heat wave phenomena, we chose t o produce the hydrodynamics a s s o c i a t e d w i t h a m u l t i s h o c k c o m p r e s s i o n . With a two s t e p c u r r e n t p u l s e we produced a shock f o l l o w e d e i t h e r by a second shock or by a s u p e r s o n i c heat wave. In both c a s e s , a f t e r a l l waves had i n t e r a c t e d , the r e s u l t i n g f l o w was independent o f the h i s t o r y o f the power p u l s e , i . e . a s i n g l e s t e p c u r r e n t p u l s e produced the same f l o w as a two s t e p c u r r e n t p u l s e o f the same f i n a l magnitude. T h i s r e s u l t c o u l d be o f c o n c e p t u a l importance f o r the u n d e r s t a n d i n g o f l a s e r f u s i o n plasma hydrodynamics s i n c e i t i n d i c a t e s t h e r e is. no j u d i c i o u s way o f a p p l y i n g a power p u l s e t o a v o i d a "burn t h r o u g h " . The v e l o c i t i e s o f the second shocks were measured and compared to p r e d i c t i o n s o f the heat wave model expanded t o i n c l u d e a s t e p - l i k e change i n power. The good agreement between measured and p r e d i c t e d v e l o c i t i e s i n d i c a t e s the h e a t i n g c h a r a c t e r i s t i c i s c o r r e c t and un i q u e , i . e . i t does not depend on the power p u l s e ' s h i s t o r y . The above mentioned r e s u l t s show t h a t the e l e c t r o t h e r m a l shock tube can produce the hydrodynamics o f m u l t i s h o c k s and t h a t the expanded heat wave model a d e q u a t e l y d e s c r i b e s the f l o w . 97 CHAPTER VI SECTION C SUGGESTIONS FOR FUTURE WORK AND ORIGINAL CONTRIBUTIONS The hydrodynamics o f m u l t i s h o c k s i s o n l y one example o f phenomena which can be s t u d i e d i n an e l e c t r o t h e r m a l shock tube. Other problems a l s o l e n d themselves t o i n v e s t i g a t i o n . M i k o s h i b a and A h l b o r n 4 4 have c a l c u l a t e d the power p u l s e , which c o u l d be produced i n the manner o u t l i n e d i n appendix 3, t h a t would be r e g u i r e d t o produce a g u a s i - a d i a b a t i c c o m p r e s s i o n . The h i g h c ompression a c h i e v e d would m a i n l y be o f acedemic i n t e r e s t s i n c e the m u l t i s h o c k compressed r e g i o n would be s m a l l as the heat wave would c l o s e l y f o l l o w the l a s t shock. Gaseous e n d - p l u g g i n g o f a l i n e a r magnetic f u s i o n s o l e n o i d c o u l d a l s o be s t u d i e d w i t h an e l e c t r o t h e r m a l shock t u b e 4 5 . One approach would be to d i v i d e the shock tube i n t o two s e c t i o n s w i t h a t h i n membrane. The e l e c t r o d e s e c t i o n would be f i l l e d t o 1 t o r r or l e s s , w h i l e the r e m a i n i n g s e c t i o n would be f i l l e d t o p r e s s u r e s r a n g i n g from 10 to 100 t o r r . One c o u l d then d r i v e a s u p e r s o n i c heat wave from a r a r e f i e d gas i n t o a denser gas. The p o s s i b l e f l o w s , which depend on the i n t e n s i t y , W, and the r a t i o of f i l l i n g p r e s s u r e s , have been examined by A h l b o r n 4 5 . Another approach t o t h i s problem i s t o " p u f f " gas> i n t o the shock tube from the downstream end, and t o f i r e the c a p a c i t o r bank when the " p u f f e d " gas nears the e l e c t r o d e s . T h i s would e l i m i n a t e the membrane problem a s s o c i a t e d w i t h the p r e v i o u s approach but i t would be d i f f i c u l t t o a c h i e v e homogeneous downstream c o n d i t i o n s . N e v e r t h e l e s s , the s i m u l a t i o n , u s i n g e i t h e r o f the s e methods, would y i e l d i n f o r m a t i o n about gaseous 98 e n d p l u g g i n g . My own c o n t r i b u t i o n s can be summarized as f o l l o w s . I have measured a unique h e a t i n g c h a r a c t e r i s t i c f o r an e l e c t r o t h e r m a l shock tube. In v e r i f y i n g t h i s c h a r a c t e r i s t i c , I have shown the heat wave model a d e q u a t e l y d e s c r i b e s the f l o w i n the shock tube. In a d d i t i o n t o showing t h a t a stepped c u r r e n t p u l s e can produce m u l t i p l e shocks from a s u b s o n i c heat wave, I have produced "burn th r o u g h " waves and have e x p l a i n e d how they are a consequence of the unique h e a t i n g c h a r a c t e r i s t i c . D u r i n g the c o u r s e o f t h i s work I developed a r e l i a b l e h i g h c u r r e n t power sour c e which can s u p p l y e i t h e r a c o n s t a n t , or a stepped c u r r e n t p u l s e . 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Griem, ' S p e c t r a l L i n e Broadening by Plasmas', Acedemic P r e s s , Newyork (1974) 101 44 S. M i k o s h i b a and B. A h l b o r n , Phys. F l u i d s 17, 1198 (1974) 45 B. A h l b o r n , Can. J . P. 55, 1047 (1977) 46 R. Courant and K.O. F r i e d r i c k s , ' S u p e r s o n i c Flow and Shock Waves, I n t e r s c i e n c e , New York (1948) 102 APPENDIX 1 RESULTING SHOCK VELOCITY The method used t o p r e d i c t the r e s u l t i n g f l u i d f l o w when one shock o v e r t a k e s another i s o u t l i n e d i n t h i s a ppendix. Knowing what f l o w r e s u l t s from the i n t e r a c t i o n o f two shocks e n a b l e s us t o det e r m i n e whether luminous phenomena on s t r e a k camera photographs are shock f r o n t s . The reasons why a shock wave, S R, a c o n t a c t s u r f a c e , CS, and a r a r e f a c t i o n wave, RW, emerge from the i n t e r a c t i o n r e g i o n are d i s c u s s e d by Courants and F r i e d r i c k s 4 6 . We are concerned here w i t h p r e d i c t i n g t h e i r v e l o c i t i e s g i v e n the v e l o c i t i e s o f the f i r s t and second shocks and the i n i t i a l gas c o n d i t i o n s . The i n t e r a c t i o n i n the x - t p l a n e i s shown i n f i g . 32. U 4 and p 4 can be c a l c u l a t e d w i t h the Rankine-Hugoniot r e l a t i o n s once the mach numbers, M1 and M 2, and the i n i t i a l c o n d i t i o n s i n r e g i o n [1] are known. S i m i l a r l y p 7 and u 7 can be det e r m i n e d once M R i s known. The p r e s s u r e and p a r i c l e v e l o c i t y a re c o n s t a n t a c r o s s a c o n t a c t s u r f a c e , t h e r e f o r e p 7 = p g and u 7=u 6. Usi n g the f a c t s t h a t the J+ Riemann i n v a r i a n t i s c o n s t a n t a c r o s s a r a r e f a c t i o n wave and f l o w i s i s e n t r o p i c , U 6 and p^ can be w r i t t e n i n terms o f u 4 and p 4 . Thus we can w r i t e an e q u a t i o n i n v o l v i n g q u a n t i t i e s which are a l l s p e c i f i e d e x cept M R. U s i n g t h i s e q u a t i o n , and a computer, a v a l u e f o r M R can be o b t a i n e d . The v e l o c i t y o f the c o n t a c t s u r f a c e i s , by d e f i n i t i o n , u 7 . The head o f the r a r e f a c t i o n wave t r a v e l s a t a l a b frame v e l o c i t y o f u 4 - c 4 . Our c a l c u l a t i o n took i n t o account, the i o n i z a t i o n o f the argon. 103 f . X F i g . 32. Schematic o f a s t r e a k photograph o f one shock o v e r t a k i n g another and the r e s u l t i n g f l u i d f l o w . S ^ f i r s t shock, S2=second shock, S R = r e s u l t i n g shock, CS=contact s u r f a c e , R W = r a r e f a c t i o n wave. 104 APPENDIX 2 METAL SHOCK TUBE F i g . 33 i s a photograph o f the shock tube and e l e c t r o d e s e c t i o n used when p r e s s u r e measurements were made. An aluminum p l u g and a p r e s s u r e probe are shown b e s i d e the tube. P r e s s u r e probes can be mounted i n any o f twenty p o r t s a l o n g the top o f the tube. The probes were mounted i n l u c i t e p l u g s so t h a t no e l e c t r i c a l c o n n e c t i o n was made between the probe and the tube. The p o r t s w i t h o u t probes are f i l l e d w i t h aluminum p l u g s . V i s i b l e a c c e s s t o the i n t e r i o r o f the tube was p r o v i d e d by windows on both s i d e s o f the tube. B r a s s f l a n g e s were s o l d e r e d to both ends o f the 1" i . d . b r a s s tube. F l a t s 18" l o n g were m i l l e d on e i t h e r s i d e o f the tube a t a depth such t h a t 1/4" s l o t s were opened up. S i x 6" l e n g t h s o f pyrex p l a t e were f a s t e n e d over the s l o t s w i t h epoxy. The top o f the tube was m i l l e d t o such a depth t h a t a 3/8" s l o t was opened up. An aluminum b a r , machined to a c c e p t the p r e s s u r e p r o b e s , was f a s t e n e d over t h i s s l o t w i t h epoxy. Due t o the windows and the p r e s s u r e probe mount, a c r o s s -s e c t i o n o f the tube i s not c i r c u l a r but i t i s u n i f o r m over the l e n g t h o f the tube w i t h the e x c e p t i o n o f .5" a t e i t h e r end. The i r r e g u l a r i t y caused by b o l t i n g an e l e c t r o d e s e c t i o n t o one end o f the tube was r e s p o n s i b l e f o r some d i s t u r b a n c e s i n the f l u i d f l o w . F i g . 33. Photograph o f the metal shock tube and an e l e c t r o d e s e c t i o n . PP=pressure p r o b e , AP=aluminum p l u g , E S = e l e c t r o d e s e c t i o n . 106 APPENDIX 3 TRANSMISSION LINES T h i s appendix p r e s e n t s the r e s u l t s o f a s t u d y i n t o s u i t a b l e power s o u r c e s f o r the shock tu b e . A computer was used t o s o l v e the c o u p l e d d i f f e r e n t i a l e q u a t i o n s which d e s c r i b e the c u r r e n t from a c a p a c i t o r - i n d u c t o r network s i m i l a r t o one shown i n f i g . 14. The c u r r e n t waveforms, shown i n f i g . 34, were c a l c u l a t e d w i t h a l l i n d u c t a n c e s s e t e q u a l t o .8 m i c r o h e n r i e s and the c a p a c i t a n c e e q u a l to 5 m i c r o f a r a d s . The o r d i n a t e i s the c u r r e n t and the a b s c i s s a i s the time i n m i c r o s e c o n d s . The number of i n d u c t o r - c a p a c i t o r p a i r s used t o o b t a i n each p u l s e i s s t a t e d below the waveforms. The d i f f e r e n t waveforms were o b t a i n e d by c h a r g i n g d i f f e r e n t c a p a c i t o r s to d i f f e r e n t v o l t a g e s . The s w i t c h e s , needed t o i s o l a t e the c a p a c i t o r s , were a l l f i r e d s i m u l t a n e o u s l y . In p r a c t i c e , some compromise would have t o be a c h i e v e d between the r e g u l a r i t y o f the c u r r e n t p u l s e and the number of s w i t c h e s used between the c a p a c i t o r s . In f i g . 36a, a l l the c a p a c i t o r s were charged t o the same v o l t a g e as was done i n the a c t u a l e x p e r i m e n t . The s i m i l a r i t y between the computed c u r r e n t waveform, f i g . 36a, and the measured one, f i g . 13a, i n d i c a t e s t h a t the computer program works. The measured c u r r e n t does not o s c i l l a t e , as does the computed c u r r e n t , because the f i r s t few i n d u c t o r s o f the a c t u a l c a p a c i t o r bank were l a r g e r than the average v a l u e . T h i s was done p u r p o s e l y t o dampen the o s c i l l a t i o n s . In f i g . 36b, the c a p a c i t o r bank was d i v i d e d i n t o two s e c t i o n s w i t h a s w i t c h . The f r o n t s e c t i o n o f the bank was charged t o h a l f t h a t o f the r e a r s e c t i o n . T h i s a l t e r n a t i v e f o r p r o d u c i n g a stepped c u r r e n t p u l s e 107 1.08 * .72 ^ .36 N = 20 (a) 20 40 60 t (psec) 80 2.25 01 E x i i — • (b) F i g . 34. Computed c u r r e n t waveforms. (a) c o n s t a n t c h a r g i n g v o l t a g e , (b) stepped c h a r g i n g v o l t a g e , (c) squared c h a r g i n g v o l t a g e , and (d) e x p o n e n t i a l c h a r g i n g v o l t a g e . N i s the number of c a p a c i t o r - i n d u c t o r p a i r s i n the t r a n s m i s s i o n l i n e . 108 t (jjsec) (d) t (usee) 109 was not used s i n c e i t allows no f l e x i b i l i t y i n the timing of the second p u l s e . The waveforms of f i g s . 36c and 36d were obtained by charging the c a p a c i t o r s i n a square and e x p o n e n t i a l manner, i . e . V(J) i s p r o p o r t i o n a l to J and V(J) i s p r o p o r t i o n a l to e x p o n e n t i a l ( J ) where V(J) i s the charging v o l t a g e of the J 1 t h c a p a c i t o r . S i m i l a r c u r r e n t p r o f i l e s would be r e q u i r e d to achieve a q u a s i - a d i a b a t i c supercompression. From these waveforms i t appears that any shape of c u r r e n t p u l s e can be obtained simply by changing the charging v o l t a g e . In a d d i t i o n to v a r y i n g the charging v o l t a g e on d i f f e r e n t c a p a c i t o r s , one can a l s o manipulate the c u r r e n t pulse by changing the s i z e s of the i n d u c t o r s or c a p a c i t o r s . There i s no need for a l l s e c t i o n s to be i d e n t i c a l . Wire wound i n d u c t o r s proved s t r u c t u r a l l y inadequate when r e q u i r e d to c a r r y 20 KA peak c u r r e n t f o r about ten 160 microsecond o s c i l l a t i o n s of the t r a n s m i s s i o n e l i n e . Attempts to strengthen them with f i b e r g l a s s tape and epoxy proved to be a time consuming f a i l u r e . The inductor that was used to overcome the problem of of s t r u c t u r a l f a i l u r e i s shown i n f i g . 35. F i g . 36 i s a photograph of the inductor components. The body of the inductor i s made of i n s u l a t o r s ( I ) , and brass r i n g s (R). When s u f f i c i e n t r i n g s and i n s u l a t o r s have been a l t e r n a t i v e l y stacked to achieve approximately the wanted inductance, r i n g CR.| i s added to one end and r i n g CR 2 to the other end. I n s u l a t o r s are a l s o used with these r i n g s . Fine adjustment of the inductance i s accomplished by r o t a t i n g CR-^  . F i n a l l y , end p l a t e s are placed over each end and b o l t s used to clamp the assembly together. The e l e c t r i c a l connections between r i n g s are 110 F i g . 3 5 . Photograph of an i n d u c t o r s i m i l a r i n d e s i g n to a B i t t e r magnet. I l l F i g . 36. Photograph o f the i n d u c t o r components. I = i n s u l a t o r , R=brass r i n g , CRf and CR 2=connecting r i n g s , EP=end p l a t e . 112 maintained by the pressure. No problems were encountered when using these inductors as long as d i r t was not sandwiched between se c t i o n s for poor e l e c t r i c a l connections would r e s u l t i f i t was. 113 APPENDIX 4 THE PRESSURE PROBE The p r e s s u r e probes were c a l i b r a t e d by g e n e r a t i n g shock waves of Mach numbers r a n g i n g from 4 to 9 i n argon at f i l l i n g p r e s s u r e s r a n g i n g from 2 to 10 t o r r . The v e l o c i t i e s of the s h o c k s , g e n e r a t e d i n our e l e c t r o t h e r m a l shock t u b e , were c a l c u l a t e d from the t r a n s i t t imes between p r e s s u r e probes. The p r e s s u r e s behind the shocks were c a l c u l a t e d from the Rankine-Hugoniot r e l a t i o n s . The p r e d i c t e d p r e s s u r e s are p l o t t e d a g a i n s t the probe's v o l t a g e i n f i g . 37. The b e s t f i t to the p o i n t s i s P = 1.17 V + .008 where p i s the p r e s s u r e i n atm. and V i s the magnitude o f the probe's s i g n a l i n v o l t s . From the l i n e a r i t y o f the f i t t e d l i n e , i t appears t h a t the probe's response i s l i n e a r f o r p r e s s u r e s up to 1 atm. I t a l s o c o n f i r m s t h a t the luminous f r o n t s on the s t r e a k camera photographs are shock waves and t h a t the Rankine-Hugoniot r e l a t i o n s can be used to c a l c u l a t e the p r e s s u r e s behind them. The LD-25 p r e s s u r e probe has a c a p a c i t a n c e o f 240 pF. T h i s probe connected to an o s c i l l o s c o p e o f 1 Mohm i n p u t impedance would have an RC time c o n s t a n t o f 240 m i c r o s e c o n d s . To i n c r e a s e the p r o d u c t RC, one n o r m a l l y i n c r e a s e s the i n p u t impedance, R, by i n s e r t i n g a h i g h i n p u t impedance a m p l i f i e r between the probe and the scope. However, the s i g n a l from our probe i s r e f e r e n c e d t o i t s metal case which was i n c o n t a c t w i t h the plasma, and i t was f e a r e d t h a t t h i s c o n t a c t would cause the probe's s i g n a l to 114 1 2 1 .2 .4 .6 .8 PROBE S I G N A L (vo l t s ) F i g . 37. The p r e d i c t e d p r e s s u r e s behind shock waves v e r s u s the p r e s s u r e probe's s i g n a l . 115 be d i s t o r t e d . To t e s t t h i s p o s s i b i l i t y , the probe was a l s o used t o measure p r e s s u r e s i n a d e t o n a t i o n d r i v e n shock tube so t h a t the s i g n a l s c o u l d be compared. No d i f f e r e n c e s between the p r e s s u r e s i g n a l s c o u l d be n o t i c e d i f the b u f f e r a m p l i f i e r was not used between the probe and the scope. The plasma d i d a f f e c t the probe's s i g n a l when the a m p l i f i e r was used, thus i t was not p o s s i b l e t o use the a m p l i f i e r w i t h the e l e c t r o t h e r m a l shock tube. T h i s probe to plasma c o n t a c t , as w e l l as c a u s i n g the n o i s e problem, a l s o made i t n e c e s s a r y t o i s o l a t e the scope from ground t o a v o i d c u r r e n t from the d i s c h a r g e p a s s i n g t o ground through the scope. To i n c r e a s e the time c o n s t a n t RC i t was n e c e s s a r y t o i n c r e a s e the c a p a c i t a n c e o f the system by u s i n g 11' o f RG-58A/U c a b l e t o connect the probe to the scope. T h i s r a i s e d RC t o about 600 mic r o s e c o n d s but a t the expense o f the s e n s i t i v i t y which was o r i g i n a l l y near 2 v o l t s / a t m . 116 APPENDIX 5 RESISTANCE OF THE SHOCK TUBE I t was d e s i r a b l e t o know the power d e p o s i t e d i n the shock tube g i v e n the c u r r e n t and the f i l l i n g p r e s s u r e . To t h i s end, the v o l t a g e a c r o s s the e l e c t r o d e s was measured as a f u n c t i o n o f c u r r e n t and f i l l i n g p r e s s u r e . These d a t a were reduced to r e s i s t a n c e s and p l o t t e d a g a i n s t the c u r r e n t f o r f i l l i n g p r e s s u r e s o f .1, 1, 3, and 10 t o r r , f i g . 38. To measure the v o l t a g e a r e s i s t o r d i v i d e r network, composed o f a h i g h v o l t a g e 100 Kohm and a 1 Kohm, was connected a c r o s s the e l e c t r o d e s . The v o l t a g e a c r o s s the 1 K r e s i s t o r was connected t o an o s c i l l o s c o p e w i t h a s h i e l d e d c a b l e . The o s c i l l o s c o p e was i s o l a t e d from ground by an i s o l a t i o n t r a n s f o r m e r i n or d e r t o a v o i d c u r r e n t from the d i s c h a r g e p a s s i n g t h r ough the ground o f the scope and d i s t o r t i n g the s i g n a l . 1 1 7 . . . . . . . . . * 4 12 20 28 36 I (KA) F i g . 38. R e s i s t a n c e o f the shock tube v e r s u s c u r r e n t w i t h the f i l l i n g p r e s s u r e as a parameter. •118 APPENDIX 6 POWER DELIVERED TO THE HEAT FRONT Knowing the e n t h a l p y behind the heat wave, as determined from the h e a t i n g c h a r a c t e r i s t i c on the response p l a n e , f i g . 23, we can c a l c u l a t e , u s i n g the t h e o r y i n c h a p t e r I I , a r e l a t i o n s h i p between the shock v e l o c i t y and the i n t e n s i t y , W. I f we c o u l d measure W e x p e r i m e n t a l l y we c o u l d then g e t a check on the a p p l i c a b i l i t y o f the heat wave model t o the f l o w i n an e l e c t r o t h e r m a l shock tube. W, however, cannot be o b t a i n e d d i r e c t l y but some measure of i t can be o b t a i n e d by measuring the i n p u t power, WQ, t o the shock tube. Wc i s the v o l t a g e a c r o s s the e l e c t r o d e s m u l t i p l i e d by the c u r r e n t through the e l e c t r o d e s . In o r d e r t o determine the power i n p u t w i t h o u t always h a v i n g t o measure the v o l t a g e , the e f f e c t i v e r e s i s t a n c e o f the shock tube was measured f o r d i f f e r e n t c u r r e n t s and f i l l i n g p r e s s u r e s . The e x p e r i m e n t a l t e c h n i q u e and the r e s u l t s are p r e s e n t e d i n appendix 5. Knowing the r e s i s t a n c e , R, and the c u r r e n t , I , one can g e t the i n p u t power, WQ, from I 2 R . F i g . 39 i s the l o g o f the shock v e l o c i t y p l o t t e d a g a i n s t the l o g o f WO f o r f i l l i n g p r e s s u r e s o f 2, 5, and 10 t o r r . The b e s t f i t s t o the e x p e r i m e n t a l p o i n t s are i n d i c a t e d by the d o t t e d l i n e s . The s o l i d l i n e s , n o r m a l i z e d to f i t the s l o w e s t measured v e l o c i t i e s , are the t h e o r e t i c a l p r e d i c t i o n s assuming a f i n a l e n t h a l p y o f 3 . 5 x l 0 1 1 ergs/gram. The v e l o c i t i e s o f the s h o c k s , produced by c u r r e n t s r a n g i n g from 11 t o 40 KA, were o b t a i n e d from s t r e a k camera p h o t o s . The s l o p e s o f the l i n e s are g i v e n i n the b a l l o o n s next t o the l i n e s . 119 L O G (W) (arb. un i ts ) F i g . 39. The l o g o f the shock v e l o c i t y v e r s u s the i n p u t power f o r f i l l i n g p r e s s u r e s o f 2, 5, and 10 t o r r . The s o l i d c u r v e s are computed v a l u e s . The s l o p e s o f the c u r v e s are g i v e n i n the b a l l o o n s . 120 The e x p e r i m e n t a l p o i n t s c o n s i s t e n t l y f a l l below the t h e o r e t i c a l c u r v e . Assuming the t h e o r y to be c o r r e c t , the d i s c r e p a n c y can be accounted f o r i n two ways. F i r s t l y , the assumed e n t h a l p y behind the heat wave i s v e r y much i n e r r o r . S e c o n d l y , and more l i k e l y , the f r a c t i o n o f the i n p u t power r e a c h i n g the heat f r o n t d e c r e a s e s as the i n p u t power i n c r e a s e s . The l a t t e r i s not u n r e a s o n a b l e as energy l o s s e s from r a d i a t i o n w i l l i n c r e a s e a t h i g h e r power l e v e l s . A v e r a g i n g the e x p e r i m e n t a l exponents one f i n d s V - ( I 2 R ) - 4 2 ( 1 4 ) s A v e r a g i n g the computed exponents g i v e s 4 8 (15) Combining 14 and 15, we f i n d t h a t a t c o n s t a n t f i l l i n g p r e s s u r e — - ( I 2 R ) - 8 8 ( 1 6 ) P i The r e s u l t s o f h o l d i n g the i n p u t c u r r e n t c o n s t a n t a t 20 KA and v a r y i n g the f i l l i n g p r e s s u r e from 1 t o 20 t o r r are p l o t t e d i n f i g . 40. The shock v e l o c i t i e s were o b t a i n e d from the t r a n s i t t i mes between two p r e s s u r e probes as the slow e r shocks were not 121 0 .4 .8 1.2 L O G ( P ) ( t o r r ) F i g . 40. The l o g o f the shock v e l o c i t y v e r s u s the l o g of the f i l l i n g p r e s s u r e f o r a c o n s t a n t c u r r e n t o f 20 KA. 122 luminous. The o r d i n a t e i s the l o g o f the shock v e l o c i t y w h i l e the a b s c i s s a i s the l o g o f the f i l l i n g p r e s s u r e i n t o r r . The b e s t f i t to the p o i n t s has a s l o p e o f -.33. As the r e s i s t a n c e o f the a r c i s independent o f the f i l l i n g p r e s s u r e a t 20 KA, see appendix 5, the i n p u t power was c o n s t a n t d u r i n g t h i s e x p e r i m e n t . T h e r e f o r e , combining egns. 15 and 17 we f i n d w_ - . • 6 9 (18) Combining egns. 16 and 18 we have W_ a ( I 2 R ) - 8 8 (19) T h i s e m p i r i c a l r e s u l t i s used any time i t i s n e c e s s a r y to have an e s t i m a t e o f the power t r a n s p o r t e d t o the heat f r o n t . 

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