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Shock propagation into inhomogeneous media Strachan, James D. 1969

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SHOCK PROPAGATION INTO INHOMOGENEOUS MEDIA James D. Strachan B.Sc, University of British Columbia, 1 9 6 8 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE 'in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1 9 6 9 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e 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 , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The 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 V a n c o u v e r 8, Canada - i i -_ _ABSTRACT The propagation of a shock wave into a general inhomo-geneous flow f i e l d is studied. The equations for the shock velocity through such a region are developed in a general manner. A consequence of these equations is the development of the shock wave as a probe into unknown flow f i e l d s . The shock velocity is measured and the i n i t i a l parameters ahead of the shock are calculated. The unique advantage of the shock probe is that i t does not perturb the gas ahead of the front. An experimental application is described in which the shock probe is used to analyze the unknown flow f i e l d created by a constricted arc light source. The flow f i e l d is subse-quently identified as a radiation front at the Chapman-Jouguet point. i i i TABLE OF CONTENTS Page Chapter 1 INTRODUCTION . . . . . . . . . . . . . . . . 1 BASIC SHOCK THEORY 3 Chapter 2 SHOCK PROPAGATION INTO A DISCONTINUOUS INHOMOGENEOUS MEDIUM 8 Chapter 3 SHOCK PROPAGATION INTO A CONTINUOUS INHOMOGENEOUS MEDIUM 14 Chapter k PROBING A FLOW FIELD WITH A SHOCK WAVE . . . 2 5 Chapter 5 CONCLUSIONS • 4 4 BIBLIOGRAPHY " . . 4-7 Appendix A THE DOUBLY REFLECTED WAVE 4-9 Appendix B PARAMETERS AT THE BOUNDARY 5 2 Appendix C CONTINUOUSLY VARYING PARAMETERS 67 Appendix D THE SHOCK TUBE 74-Appendix E OTHER INTERESTING ASPECTS * 86 Appendix F THE EXPERIMENTAL SET-UP 90 Appendix G PIEZOELECTRIC PROBE CALIBRATION 93 - i v -LIST OP ILLUSTRATIONS Figure Page 1 Shock Wave i n Shock Frame . . . . . . . . . . . 3 2 E f f e c t i v e Adiabatic Constant f o r Argon . . . . 6 3 Shock - Shock C o l l i s i o n 10 4 Shock Polar 11 5 Thermal Discontinuity - Shock C o l l i s i o n . . . . 12 6 Shock Polars . 13 7 Shock Crossing an I n f i n i t e s s i m a l Discontinuity 17 8 Shock as Viewed by Observer Changing Frames . . 24 9 Shock - Flow F i e l d Intersection . . . . . . . . 27 10 Shock - Flow F i e l d Intersection Enlarged . . . 29 11 Shock - Flow F i e l d Intersection Enlarged . . . 30 12 Shock Wave Probing the Flow F i e l d 34 13 •4*"$ v s D i s t a n c e , 36 14 . M j vs Distance 36 15 Wj vs Distance 37 16 $ 3 vs Distance 38 17 Badiation Front V e l o c i t y vs Downstream Pressure 39 18 Behind the Radiation Front 40 19 ^ Behind the Badiation Front 40 20 Pressure Behind the Badiation Front . . . . . . 4 l 21 Pressure Behind the Radiation Front . . . . . . 4 l 22 Badiation Front at Low Downstream Pressure . , 45 - V -Figure Page A Doubly Reflected Wave . . . . . 50 B-l Graphical Solution for Intersection Problem , . 5^ B - 2 Error Bounds on State 4- 55 C Shock Velocity Behind Radiation Front 70 D-l Contact Surface Velocity 77 D-2 Contact Surface Velocity 77 D-3 Vertical Smear Camera 79 D-4- Shock Velocity vs V c 80 D-5 Shock Attenuation 81 D-6 Shock Attenuation . . . . . . . . . . . . . . . 82 D-7 Shock Attenuation 82 D-8 Crowbar Results . 84-D-9 Crowbar Results 85 E- l Radiation Effects in the Shock Heated Gas . . . 87 E - 2 Radiation Energy Addition at a Shock Front . . 88 F - l Apparatus 9 1 F - 2 Smear Camera . 9 2 G Piezoelectric Probe Calibration 9 3 - v i -ACKNOWLEDGEMENTS I am sincerely indebted to Dr. B. Ahlborn for his encouragement and his stimulating insight which was the motivation for this thesis. I am grateful for the many interesting discussions with a l l members of the Plasma Physics Group, and in partic-ular I thank J.P. Huni and R. Ardila. R. Ardila performed the pressure measurements which gave us confidence in the shock probe. I am also grateful to the National Research Council for their financial assistance. -1-Chapter 1 INTRODUCTION The propagation of shock waves into uniform gases is well understood, but no general study has attempted to understand the propagation of shocks into inhomogeneous and moving media. As a background, one needs to know the essentials of ( standard shock wave theory which are brie f l y outlined in the f i r s t chapter. Quite generally, the properties of the inhomogeneous media can change either abruptly or slowly. Shock propagation into abrupt changes has been thoroughly studied and is outlined in the second chapter. The propa-gation of a shock into slowly varying i n i t i a l conditions is presented in the third chapter and is one of the main contributions of this thesis. Prom the theoretical study of the penetration of shocks into inhomogeneous media, we draw one general conclusion which is discussed in chapter four. We conclude that i t is possible to probe an unknown flow f i e l d with a shock wave of known strength. By measuring the local variation of the shock front velocity one can obtain the local variations of the i n i t i a l parameters. Three shocks of different strengths are needed to encounter the reproducible but unknown flow f i e l d in order that the i n i t i a l pressure, density, and particle velocity can be derived. An experimental appli--2-cation which probes an unknown flow f i e l d i s described i n chapter four. A l l standard calculations and de t a i l e d descriptions are relegated to the appendix, of which, appendix D may be of s p e c i a l Interest to someone working with the same type of shock tube, and appendix E which discusses other i n t e r e s t i n g aspects w i l l be f a s c i n a t i n g f o r everyone. -3-BASIC SHOCK THEORY When the velocity of a f l u i d becomes comparable with or exceeds that of sound, effects due to the compressibility of the f l u i d become important. One of the most distinctive features of supersonic flow is that shock waves ( strong compres-sion waves) can occur in i t . Thermodynamlcally, a shock wave is a discontinuous jump in pressure, density, temperature, and entropy which, in the frame of the shock front, separates uniform equilibrium conditions of subsonic flow from uniform equilibrium conditions of supersonic flow. Consider one dimensional plane flow in a frame at which the shock front is at rest and label the quantities before the shock front as 1 and those behind the shock front as 2 ( f i g . 1). SHOCK DIRECTION. m ? is the density is the pressure Aj~ is the particle velocity in the shock frame DOWNSTREAM LGAS H is the enthalpy of a unit mass of gas SHOCK HEATED GAS Fig. 1 Shock Wave in the Shock Frame The t h r e e c o n s e r v a t i o n equa t i ons can be w r i t t e n as f o l l o w s t 1. C o n s e r v a t i o n o f Mass /IT, - ? Z A \ (1) 2. C o n s e r v a t i o n of Momentum 3. C o n s e r v a t i o n of Energy A/, + » Hx + i / ^ 7 (3) These c o n s e r v a t i o n equa t ions t o g e t h e r w i t h the equa t i on o f s t a t e a r e s u f f i c i e n t t o de te rmine the s o l u t i o n i n terms o f the shock parameters *Z1 , i » i and s i n c e the i n i t i a l c o n d i t i o n s ahead of the shock f r o n t a r e known; -yfi^  , » and 7^ a r e de te rmined once the shock f r o n t v e l o c i t y i s measured. F o r an i d e a l gas the e q u a t i o n of s t a t e i s " - £ 7 1  w where V i s the r a t i o of s p e c i f i c heats and i s a c o n s t a n t ( 5/3 f o r a r g o n ) . The c o r r e s p o n d i n g s o l u t i o n s a r e -5-?» O r - O n 1 , + <a (5) (6) (?) r, where i s the Mach Number and C i s the speed of sound. For monatomic gases i o n i z a t i o n and elec t r o n i c e x c i t a t i o n are the only processes which cause a departure from i d e a l i t y . In order to take i o n i z a t i o n into account, the energy equation i s usually changed so that Hx-Ht - T / ^ 2 - iAt%x*ce* (8) where i s the change i n energy due to i o n i z a t i o n z (Gaydon and Hurle, 1 9 6 3 ) . The other common way to take i o n i z a t i o n into account i s to introduce an e f f e c t i v e adiabatic exponent £ which i s defined by u- JL 't? ( 9 ) so that g s £ C-fi^f) "becomes a function of pressure and - 6 -2 0 0 0 1 0 , 0 0 0 2 0 , 0 0 0 1 0 0 , 0 0 0 TEMPERATURE (°K) Pig. 2 E f f e c t i v e Adiabatic Constant For Argon temperature which may be calculated f o r any gas ( f i g . 2 ) and i s , f o r instance, tabulated by Ahlborn and Salvat ( 1 9 6 7 ) . We use the e f f e c t i v e adiabatic exponent which has the advantage of separating the thermodynamic ca l c u l a t i o n s from the hydrodynamic c a l c u l a t i o n s . The hydrodynamics i s contained i n the Rankine - Hugoniot equations ( 1 ) , ( 2 ) , and ( 3 ) , while the thermodynamics i s confined to determining Q C-/3^7") with the i n c l u s i o n of a l l e s s e n t i a l i o n i z a t i o n and e x c i t a t i o n processes. The exact s o l u t i o n of the Rankine - Hugoniot equations (Ahlborn and Salvat, 1 9 6 7 ) now becomes - 7 -- « - . p r » j y + v / i ^ e - r do) 9t + i c J ( I D where The temperature i s found numerically from the equation of state, 7"- TCg^-p f o r the p a r t i c u l a r gas under study, assuming l o c a l thermal equilibrium. -8-Chapte r 2 SHOCK PROPAGATION INTO A DISCONTINUOUS INHOMOGENEOUS MEDIUM H i s t o r i c a l l y , a g r e a t d e a l o f work has been done on the i n t e r a c t i o n of shock waves w i th c o n t a c t s u r f a c e s i n v a r i o u s o b l i q u e c o n f i g u r a t i o n s (Taub, 1947). And t o o , the i n t e r a c t i o n o f shock waves w i t h shock waves has been w e l l u n d e r s t o o d . T h i s c h a p t e r i s a r e t a i l o r i n g of known theo r y t o s u i t our p a r t i c u l a r i n t e r e s t s . Suppose a shock runs th rough a r e g i o n i n wh ich some or a l l o f the i n i t i a l c o n d i t i o n s v a r y d i s c o n t i n u o u s l y . There a r e o n l y two p o s s i b i l i t i e s — 1, e i t h e r the p r e s s u r e i s i d e n t i c a l on b o t h s i d e s of the d i s c o n t i n u i t y , o r 2. the p r e s s u r e jumps a c r o s s the d i s c o n t i n u i t y . I f the p r e s s u r e i s c o n t i n u o u s , t hen the d i s c o n t i n u i t y i s a c o n t a c t s u r f a c e s e p a r a t i n g two d i f f e r e n t gases o r one gas a t two d i f f e r e n t t e m p e r a t u r e s . S i n c e d i f f u s i o n e f f e c t s must a lways occu r a t c o n t a c t s u r f a c e s they may no t be t r u l y d i s c o n t i n u o u s . Thus , c o n t a c t s u r f a c e s a r e t r e a t e d i n c h a p t e r 3-under con t i nuous inhomogeneous media and the t r ea tment i s such t h a t the s o l u t i o n h o l d s even i f a c o n t a c t s u r f a c e were i d e a l l y d i s c o n t i n u o u s . I f the p r e s s u r e changes d i s c o n t i n u o u s l y , the same g e n e r a l c o n s e r v a t i o n equa t i ons must h o l d as h e l d f o r a shock wave (chapter 1). Since the d i s c o n t i n u i t y surface i s more general than an adiabatic shock wave, source terms must be included i n the Rankine - Hugoniot equations. But to uncomplicate matters, mass source terms and force terms w i l l be ignored. Only an energy source term i s considered. In a c t u a l flow f i e l d s , mass sources at the edge of a pressure d i s c o n t i n u i t y seem very u n l i k e l y while external force terms appearing only at the edge of the d i s c o n t i n u i t y are possible but very uncommon. Energy source terms, on the other hand, occur i n r a d i a t i o n fronts, detonation waves, and i n r a d i a t i o n losses at shock f r o n t s . The conservation equations f o r the general pressure d i s c o n t i n u i t y become (15) (16) an f. J 2 where W i s a constant and i s the energy input per uni t mass. These equations have been solved exactly (Ahlborn, 1963) and the solutions are -10-Equations (18) and (19) quite generally describe the v a r i a t i o n of pressure and density across a thermal d i s c o n t i n -u i t y . We want to consider what happens when a shock front c o l l i d e s with a d i s c o n t i n u i t y across which equations (18) and (19) hold. As a s t a r t i n g point, we consider what happens when two one-dimensional plane shocks c o l l i d e (Shapiro, 1954-). See f i g u r e J . We l a b e l the various equilibrium regions by numbers and denote the inte r f a c e between two such regions, ju and i* , as <T-A V j * ) . DISTANCE F i g . 3 Shock - Shock C o l l i s i o n -11-In the above notation — ^ / N 3 * ^ i s the shock wave propagating towards the l e f t into region / . is the shock wave propagating towards the ri g h t into region / . is the shock t r a v e l l i n g towards the r i g h t into the already shock-heated gas i n region 3 . ^3^*/^ i s the shock t r a v e l l i n g towards the l e f t into the already shock-heated gas i n region *Sv . ^*FKS^ i s a contact surface which separates the two doubly shock-heated regions. The pressure and p a r t i c l e v e l o c i t y are continuous across contact surfaces, and i n this case, the contact surface v e l o c i t y equals the p a r t i c l e v e l o c i t y i n regions LF< and £1 PARTICLE VELOCITY* Pig. 4 Shock Polar -12-Th e problem i s completely s p e c i f i e d once the i n i t i a l v e l o c i t i e s of and are known and i s conveniently represented on a Shock Polar ( f i g . 4 ) . The l i n e s connecting the various states are unique and are obtained by using the Rankine - Hugoniot equations with the shock v e l o c i t y as a parameter. The states and S" are determined as the i n t e r s e c t i o n of the polars from states and 3 . Now we return to the more general case where the shock , runs into a general pressure d i s c o n t i n u i t y , 0 \ 3 ^ and the s i t u a t i o n appears as i n figur e 5t DISTANCE F i g . 5 Thermal Discontinuity - Shock C o l l i s i o n -13-are as previously-defined. But is a pressure discontinuity with, energy input V/^ and C T ^ ^ O is a pressure discontinuity with energy input V/^ • If these energy inputs are known, then there are fourteen unknowns* ? x , , , ^ , •^1 * "Pi ' ' "**H ' ~f*H 1 ' ' • -f°i » \y N ; with fourteen equationsj four sets of the three Rankine - Hugoniot equations plus PH » and ^ « ^ , Therefore, i t is possible to f ind ^ and probably the best method is to again draw the shock polars ( f i g . 6 ) . Instead of applying the shock polar from state / to state 3 use the shock polar with energy input V7j , and similarly, in going from state *Z to state V use the shock polar with energy input W. . SHOCK POLARS WITH ENERGY INPUT PARTICLE VELOCITY* Fig. 6 Shock Polars - 1 4 - -Chapter 3 SHOCK PROPAGATION INTO A CONTINUOUS INHOMOGENEOUS MEDIUM The solution for continuously varying i n i t i a l pressure and density distributions has been attacked with the view of understanding certain astrophysical phenomena such as colliding stars (De Young and Axford, 19^7) and the emission of mass from the surface of stars (Nadezhin and Frank-Kamenetskii, 1965) , The usual methods have been the Chisnell method (Chisnell, 1955; 6 n o , Sakashita, and Yamazaki, i 9 6 0 ) and the Whitham Rule (Whitham, 1958) which requires the writing of the characteristic equation. The method developed here is a generalization of the Chisnell method. We consider the propagation of a shock wave through a general one-dimensional, continuous, and inhomogeneous medium, where the i n i t i a l density, pressure, and particle velocity each vary as a function of position in a certain region. The propagation of a certain i n i t i a l shock through the region is physically well defined and a unique physical process occursx-that i s , the velocity remains single valued throughout the inhomogeneous region. The shock wave velocity i s , in general, a function of the i n i t i a l density, i n i t i a l pressure, i n i t i a l particle velocity, and the driving mechanism. The shock velocity is completely specified i f these variables are precisely given. Usually, -15-of course, the exact nature of the d r i v i n g mechanism i s not c l e a r l y known,and thus the shock v e l o c i t y i s usually taken as a parameter. S t i l l , the shock v e l o c i t y can be written i n fu n c t i o n a l form; V s V ( ? n ^ i N ^ i J ^ (20) where £ corresponds, i n some sense, to the d r i v i n g mechanism. If we assume that the d r i v i n g mechanism i s constant, the rate of change bf the shock v e l o c i t y throughout the inhomogeneous region i s given by d i f f e r e n t i a t i n g equation (20)« cTx * 2?, + *x * ^ ( 2 1 ) where , , and are treated as completely independent parameters, i n l i n e with the i n i t i a l assumption of a completely general flow f i e l d . Integrating equation (21) y i e l d s X - o The problem of f i n d i n g the shock v e l o c i t y as a function of p o s i t i o n i n the inhomogeneous region has been reduced to the problem of f i n d i n g three functions i -16-1. a t c o n s t a n t and - o , 2. a t c o n s t a n t 9, and - o , 3. — a t c o n s t a n t ~fil and ? C o n s i d e r the p h y s i c a l s i t u a t i o n s which c o r r e s p o n d to the above p a r t i a l d e r i v a t i v e s . The ma themat i ca l p rob lem of f i n d i n g i s e q u i v a l e n t t o the p h y s i c a l prob lem of f i n d i n g the mot ion o f a normal shock wave th rough a non-un i f o rm , one-d imens iona l medium of c o n t i n u o u s l y chang ing d e n s i t y , p r e s s u r e , o r p a r t i c l e v e l o c i t y . I f we f o l l o w the method of C h i s n e l l , the non-un i form r e g i o n i s r e g a r d e d as a s u c c e s s i o n of s m a l l d e n s i t y , p r e s s u r e , o r p a r t i c l e v e l o c i t y d i s c o n t i n u i t i e s s e p a r a t e d by u n i f o r m r e g i o n s , Assume t h a t the i n i t i a l parameter , i n c r e a s e s m o n o t o n i c a l l y w i t h d i s t a n c e i n a c e r t a i n r e g i o n and i s u n i f o r m o u t s i d e t h i s r e g i o n , A p l ane shock moves i n the ^ - d i r e c t i o n th rough the r e g i o n x < 0 w i th c o n s t a n t s t r e n g t h and u n i f o r m f l o w beh ind i t . When the shock passes th rough the r e g i o n o f the chang ing pa ramete r , i t s s t r e n g t h changes and a wave i s r e f l e c t e d b a c k -wards from i t . In a d d i t i o n , the mot ion o f the r e f l e c t e d wave th rough the non-un i fo rm r e g i o n gene ra tes a n o t h e r " doub l y r e f l e c t e d " wave moving i n the same d i r e c t i o n as the i n c i d e n t s h o c k . The ma themat i ca l c o m p l i c a t i o n s encoun te red by c o n s i d e r --17-ing the doubly r e f l e c t e d wave are enormous, so we w i l l l i m i t the usefulness of our r e s u l t s by making the approximation that t h i s doubly r e f l e c t e d wave can be ignored (appendix A ) . F i r s t of a l l , the physical s i t u a t i o n to consider i n deri v i n g — , i s that of an i n f i n i t e s s i m a l density d i s c o n t i n u i t y ( f i g , 7 ) . The contact d i s c o n t i n u i t y , <'» N5 >> , has no di s c o n t i n u i t y i n pressure or f l u i d v e l o c i t y , although there i s an i n f i n i t e s s i m a l jump i n the density, such that f s " • * d ? (23) while - / a 3 * - / a Y \ - / a , * ~F*S (24) 3 •= ^ Y \ y w , = (25) DISTANCE F i g . 7 Shock Crossing an I n f i n i t e s s i m a l Discontinuity -18-Th e strength of the disturbance i s taken as where ^ i s the r a t i o of the pressure i n region to that i n region , Therefore, the pressure r a t i o of the incident shock i s ^. •= — . By expressing the Rankine - Hugoniot * equations i n terms of the parameter and then solving, we obtain where and ^U^-z ^ , + ^ (?IX n ^ (27) V • + V TT^T" ( 2 8 ) • f t * / Similar equations hold f o r the elements of the r e f l e c t e d wave, and d i f f e r with those f o r a shock only i n t h i r d and higher powers of - { , where i s the strength of the disturbance. Provided t h i s r e s t r i c t i o n i s remembered, the shock equations may be used f o r the small disturbances to save formulating a second set of equations. -19-Returning to the conditions on the v e l o c i t y and pressure (equations 24- and 25), obtain . (29) * W - * W ( 3 0 ) Note that the increase i n strength of the penetrated shock must be i n f i n i t e s s i m a l so that • ~*» * <** ( 3 1 ) By i n s e r t i n g equations ( 3 0 ) , ( 3 D . ( 2 6 ) , and (23) into equation (29) and taking i t to the f i r s t order i n smallness i n terms of f and d ^ , obtain s dj J L - _ — ' — • + JL. 11 + ^ ' ( 3 2 ) but, from equation (28) and d i f f e r e n t i a t i n g with respect to at constant -ja, and ^ M , » obtain -20-(33) Therefore, a y (34) 3 where (35) so that f i n a l l y (36) which i s Chlsnell's r e s u l t . The physical s i t u a t i o n to consider i n de r i v i n g ~* , i s that of an i n f i n i t e s s i m a l pressure d i s c o n t i n u i t y across which the p a r t i c l e v e l o c i t y and density are continuous. Although a rather unnatural type of d i s c o n t i n u i t y , imagine some force f i e l d maintaining i t u n t i l the shock wave crosses i t , a f t e r which time the force f i e l d i s removed. -21-Again, as i n figur e 7, an incident shock wave generates at the jump p o s i t i o n OyS^ f a r e f l e c t e d wave £"5^ a*> , a penetrated shock £ £ N , and a contact surface £ 1 ^ ^ The p h y s i c a l quantities before and behind the shock are connect ed by the Hankine - Hugoniot equations (18 to 28). The following boundary equations also applyt (37) <j ~ » r U ^ (38) (39) -Pi = (4-0) (4-1) And, as previously I9L > (4-2) and (*3) where (44) -22-Therefore, (45) or (46) and by taking to the f i r s t order of smallness, By i n s e r t i n g equations (37 to 41, and 47) into equation ( 4 2 ) , D i f f e r e n t i a t i n g equation (28), « v , v r J L „ _ ! _ « t± "1 and i n v e r t i n g (28), (50) -23-then by entering equations (48) and ( 5 0 ) into equation (4-9), obtain ( % , ) . - A C * , v > Equation ( 5 1) has been derived i n a manner very smilar to the previous derivation of — by the C h i s n e l l method. F i n a l l y , the physical s i t u a t i o n corresponding to the "3V p a r t i a l d e r i v a t i v e , i s a d i s c o n t i n u i t y surface across which only the i n i t i a l p a r t i c l e v e l o c i t y changes. An analagous experimental s i t u a t i o n would be f o r an observer to watch the propagation of a shock wave from some i n e r t i a l frame at which JUx -Ju/ • The observer sees the shock propagation with some v e l o c i t y N / and with strength parameters r and £ , Then, at time 7= O l e t the observer change ' i to a d i f f e r e n t frame— one i n which the p a r t i c l e v e l o c i t y i s ' ^ -J „ ( 5 2 ) where di~**t i s n o n - r e l a t i v i s t i c . To that observer just described, the flow of the shock wave appears as i n figur e 8. Since the only process to occur i n f i g u r e 8 has been that of the observer changing frames, the shock w i l l have the same strength parameters ^ and ~ , i even though the shock v e l o c i t y has changed. The change i n shock v e l o c i t y i s O C J I^ » and therefore ( 5 3 ) -24-s M OBSERVER IN A FRAME IN WHICH , . OBSERVER IN A FRAME IN WHICH DISTANCE F i g . 8 SHOCK AS VIEWED BY OBSERVER CHANGING FRAMES A l l the p a r t i a l d e r i v a t i v e s have f i n a l l y been c a l c u l a t e d and the i n t e g r a l e x p r e s s i o n f o r the v e l o c i t y as a f u n c t i o n o f p o s i t i o n i s x=o (54) T h i s e q u a t i o n g i v e s the a t t e n u a t i o n o f the o r i g i n a l shock f r o n t v e l o c i t y \/Co\ a f t e r t r a v e l l i n g a d i s t a n c e X i n t o a g e n e r a l inhomogeneous r e g i o n . : -25-Chapter 4 PROBING A FLOW FIELD WITH A SHOCK WAVE. The study of shock propagation into inhomogeneous media has an i n t e r e s t i n g a p p l i c a t i o n . Since one knows the r e l a t i o n between the i n i t i a l conditions and the f i n a l v e l o c i t y , a measurement of the shock front v e l o c i t y can be used to diagnose an unknown plasma flow f i e l d . Most methods of measuring the parameters i n an unknown flow f i e l d depend upon, either measurements of the ra d i a t i o n given o f f , or probes which are assumed to perturb the flow f i e l d by a n e g l i g i b l e amount. We chose the other extreme and l e t a shock wave in t e r a c t with the unknown flow f i e l d . The shock i s a large amplitude perturbation, but i t s i n t e r a c t i o n with any variatio n s i n a flow f i e l d i s understood from chapters two and three. The unique advantage of t h i s probing technique l i e s i n the f a c t that the unknown flow f i e l d does not experience the probing t o o l (the shock wave) u n t i l the measurement i s completed. This phenomenon i s a r e s u l t of the shock wave t r a v e l l i n g ' f a s t e r than the s i g n a l v e l o c i t y (speed of sound) i n the unknown medium. Thus, a technique which r e l i e s upon measurement of the shock fr o n t v e l o c i t y w i l l always be measuring the properties of the unper-turbed gas, since the shock always flows into unperturbed gas. Suppose the parameters i n a supersonic flow f i e l d are unknown. A supersonic flow f i e l d w i l l be bounded by a d i s c o n t i n -u i t y of the type discussed i n chapter two. The parameters -- 2 6 -immediately behind the boundary can be obtained by i n v e r t i n g the discussion i n chapter two and measuring the v e l o c i t i e s of the four fronts, <K"a"> , <\V> , <aN«V>, and <Xs> . By observing the v a r i a t i o n of the shock v e l o c i t y throughout the unknown flow f i e l d and in v e r t i n g the r e s u l t s of chapter three, the dependence of the plasma parameters on p o s i t i o n can be obtained throughout the flow f i e l d . In our experiment, we use the shock wave to investigate the properties of the flow f i e l d produced by a Bogen l i g h t source. Ahlborn and Zuzak ( 1 9 6 8 , 1 9 6 9 ) u t i l i z e d the Bogen l i g h t source to create a r a d i a t i o n front behind a window. In our case we removed the window to observe the flow f i e l d creat-ed by the l i g h t source. One important question i s rai s e d , " i s the flow f i e l d a r a d i a t i o n front created by the intense u l t r a v i o l e t r a d i a t i o n from the l i g h t source, or i s i t a b l a s t wave, perhaps r a d i a t i o n supported, driven by the escaping arc heated gas from the l i g h t source?" Details of the experimental set-up are presented i n appendix F. The shock waves are formed by a hybrid electro-thermal-magnetic shock tube designed by P.R. Smy (appendix D), and the measuring device i s a smear camera. The shock wave and flow f i e l d from the l i g h t source c o l l i d e head-on and the smear camera i s at r i g h t angles to the d i r e c t i o n of flow ( f i g . 9 ) . The experiment requires intersections of the flow f i e l d created by the l i g h t source, with shock waves. This i s quite a simple procedure. D i f f e r e n t shock strengths are acquired by -27-SHOCK WAVE UNKNOWN DISTANCE * * SHOCK WAVE UNKNOWN WAVE = 1.5 TORR A r P i g . 9 S h o c k — Flow F i e l d I n t e r s e c t i o n -28-changing the spark gap separation, S t , i n the d r i v e r section and thus changing the capacitor bank f i r i n g voltage. The in t e r s e c t i o n p o s i t i o n of the flow f i e l d and the shock wave are. al t e r e d by adusting the r e l a t i v e delay s e t t i n g on the delay units f o r the shock d r i v e r and Bogen l i g h t souroe, which are . triggered by the ro t a t i n g mirror of the smear camera. With such an easy experiment, there was nothing to do but to proceed. F i r s t , by using colour f i l m , some i n i t i a l r e s u l t s were obtained ( f i g . 9 and 11) . In fi g u r e 9 . the shock wave (i n blue from the l e f t ) intersects an i n v i s i b l e preceding shock from the l i g h t source (in green from the r i g h t ) . On many of the p i c t u r e s , the separation of the shock wave from the main luminous mass i s v i s i b l e (due to impurity radiation) and the separation occurs where the flow f i e l d changes colour from white to blue i n figur e 9 . In f i g u r e 11 the i n t e r s e c t i o n i s seen i n greater d e t a i l since the s l i t on the smear camera has been gre a t l y reduced. The photograph i s very i n t e r e s t i n g because a l l the regions that were discussed q u a l i t a t i v e l y i n f i g u r e 3 (chapter 2) can be i d e n t i f i e d . The regions / , SL , 3 » ' */ • S" i n fi g u r e 10 are p r e c i s e l y those indicated i n fi g u r e 3« i O s a " ^ and Ok3*^ are shock waves. and ^"iyS"^ are the ref r a c t e d shock waves, i s the contact surface separating the two doubly heated shock regions. This surface i s ^ v i s i b l e because of the temperature difference across S"^  . . . F i g . 11 Shock — Flow F i e l d I n t e r s e c t i o n E n l a r g e d - 3 1 -a n _ 3^^ r7*>> are contact surfaces separating the driver gas from the shock heated gas, <fcsB>and O n 7 > are the refracted shocks which are refracted on passing through the contact surfaces ^ a ^ " ) and ^VO • ^4^9*} and Cs^y are the refracted contact surfaces An experimental check of our results is possible in this case, since we can predict theoretically and measure independent ly the particle velocity in the doubly shock heated region. FRONT VELOCITY 4 - . 2 + 8 # -0.604-+10# 3.13+10^ 2.2+10# F i r s t l y , consider the parameters immediately at the edge of the flow f i e l d by observing the change in velocity of the shock front at the boundary. The situation is the same as that i n figure 5. The velocities of the four fronts, 0 N"Sf> , O s3"> . ^*3> X «0 i and £ls$y are known; the i n i t i a l parameters, i ?! i 77 » a r e also known. The energy inputs N/s/j and are unknown, and the desired information is -fQ^ , ? j , -*irlj , and T . As the problem is stated, there are enough -32-equations with the four sets of Rankine - Hugoniot equations plus and JU^ - JU^ . Unfortunately, there i s a s l i g h t complication. In general, there w i l l "be an extra energy input term f o r state S ( )» and eith e r an extra equation or an extra measurement must be used. The a n a l y t i c s o l u t i o n to the above set of equations seems very d i f f i c u l t ; therefore, numerical methods were employed (appendix B). In applying the theory, we need one more equation since there are three unknown energy terms, VVj , , and V/^ . In our s p e c i f i c experiment, the energy absorbed i n each state should depend upon the equilibrium properties of that state, and states V and S" should be very s i m i l a r ; thus, we make the assumption that Although t h i s assumption i s a r b i t r a r y , i t does not seem, too unreasonable. With t h i s assumption, the equilibrium parameters f o r the smear picture i n fig u r e 11 are calcu l a t e d by the program i n appendix B. STATE <? lo £ T 3 -J1- T 9 W / 3.224 0.0 1.5+4* 298 1.666 0.0 SI 19.8+10^ 3.518+10# 399+10* 10908+10* 1.381 0.0 3 7.4?+30# 1.364+30# 81+30* 6927+30* 1.648 0.009 V 37.7+30^ 2.1+10^ 950+40* 13300+10^ 1.234 0.350 5" 32.7+30^ 2.1+10^ 950+4o# 14000+10$ 1.197 0.350 -33-The measured contact surface v e l o c i t y should equal the ca l c u l a t e d p a r t i c l e velocities-X**/ and . In t h i s case, JUH=JUC*' t to°/o kA~//i*c The agreement, i s well within the error bounds and thus provides a v a l i d i t y check on the methods followed i n appendix B. According to the c a l c u l a t i o n s , V/^is very much smaller than and NA/^ . . This f a c t i s quite p l a u s i b l e since the i o n i z a -t i o n i n state i s very low while that i n states */ and 5" i s quite high. The bound electrons i n state 3 absorb only i n s p e c i f i c wavelengths while the free electrons i n states */ and X absorb i n any wavelength because of inverse bremstraul-ing. Also, notice that the v e l o c i t y ^ < / N 8 ^ i n figur e 11 i s not constant but rather i t increases. This observation corres-ponds to an expected r a r e f r a c t i o n wave caused by the high absorption i n state *4 . A l l i n a l l , f i g u r e 11 has provided a good check on the f i r s t part of the .probing technique, and more generally, the theory i n chapter two. With t h i s knowledge established, we then increased the energy into the l i g h t source and probed the r e s u l t i n g flow f i e l d at several positions down the tube ( f i g . 12). The four v e l o c i t i e s were measured f o r each c o l l i s i o n and the parameters were obtained as outlined above. The r e s u l t s f o r many i n t e r -sections were calcul a t e d and are pl o t t e d as a function of -34-18.8 microsec DISTANCE L SHOCK WAVE 10 cm LIGHT SOURCE P i g . 12 Shock Wave P r o b i n g the Flow F i e l d 1.5 TORR Ar - 3 5 -distance i n figures 13 to 1 6 . The most i n t e r e s t i n g feature i s that the absorbed energy, V / j i i n each measurement i s very close to the Chapman-Jouguet energy, w - . and thus, implies that the flow f i e l d i s at the Chapman-Jouguet point. The only e f f i c i e n t way of transmitting energy of t h i s amount r i g h t into the fr o n t i s by r a d i a t i v e absorption. We therefore i d e n t i f y t h i s unknown wave as a Chapman-Jouguet-type r a d i a t i o n f r o n t . As a further check on the v a l i d i t y of t h i s f a c t , the i n i t i a l density ?| was varie d and the front v e l o c i t y was measured at f i x e d distances ( f i g . 1 7 ) . The points f a l l reasonably well on the expected p r o p o r t i o n a l i t y (Ahlborn and Zuzak, 1 9 6 9 ) 1 - '3 V <* ? i Since the p a r t i c l e s leave a Chapman-Jouguet front with sonic v e l o c i t y , a r a r e f r a c t i o n wave i s expected to follow t h i s f r o n t . This means that the pressure, density, and p a r t i c l e v e l o c i t y are expected to decay behind the f r o n t . I t i s checked by applying the second part of our shock probe a n a l y s i s . We w i l l show that the r e s u l t s are s e l f - c o n s i s t e n t and agree with Independent piezo probe measurements. - 3 6 -800 7 0 0 J 6 0 0 A 5 0 0 4oo i 3 0 0 J o EH CO 7J 5 4 4 J 3H EH M o _ S o B3 i l l a " 14 30 DISTANCE (cm) A 5D ' 2 0 * 3 0 DISTANCE (cm) IJo 5 0 F i g . 13 and 14 - f l , and - U j vs D i s t a n c e -37-3.0 2.0 1.0 r-1 ^ CHAPMAN-JOUGUET ENERGY 6 w 3 5 • » •I '2o r3C) '4-0 "" r'5'0" DISTANCE (cm) P i g . 15 W, vs D i s t a n c e -38-12-11-10-9-8. 7-6 d 3-1 2 U 1 d o EH M CO A X 20 DISTANCE (cm) 30 F i g . 16 40 50 ^ v s D i s t a n c e -39-100-r 10 4 14 J EH M O O > .01 .10 PRESSURE (TORR) 1 1.0 10.0 Pig. 17 Radiation Front Velocity vs Downstream Pressure -4-0-O c/ > 5 v I T f o Hr—T7 T2 ri I T 1 r 1 1 1 r 0 1 2 3 4- 5 6 '7 8 9 10 11 12 13 iJ TIME (microsec) Fig. 18 >Uy Behind the Radiation Front V A £ ~ — r — r — r - — r 1 1 1 1 1 1 1 1 1 r 0 1 2 3 4- 5 6 7 8 9 10 11 12 13 i i TIME (microsec) Fig. 19 ? 2 Behind the Radiation Front 700J 6 0 0 - & 5 0 0 J • 4ooi 3 0 ( H 2 0 0 H These are two traces of the pressure p r o f i l e behind the r a d i a t i o n front at i d e n t i c a l i n i t i a l conditions PIEZOELECTRIC PRESSURE TRACE (R. A r d i l a ) TIME (microsec) 10 ' 1 1 ' 1 2 ' 1 3 'l4 6 0 0 400-3 0 0 2 0 0 C a l i b r a t i o n of the absolute pressure probes i s shown i n appendix G, PI-RilOELECTRIC PRESSURE TRACE (R. A r d i l a ) _ 0 l l ' 2 ^ '4 *5 T6 TIME (microsec) Fig. 2 0 and 2 1 Pressure Behind the Radiation Front "8 ~~1—1—1— 1 1 1 2 13 14 -42-To apply the second part of the probing technique, we measured the shock v e l o c i t y as a function of time through the continuously varying region behind the r a d i a t i o n f r o n t . We chose intersections which are approximately 35 cm from the l i g h t source. Prom chapter three, we have equation ( 5 4 ) , i n which the i n t e g r a l s are extremely d i f f i c u l t to perform. Therefore, the s o l u t i o n must again come numerically, using a simple predictor numerical i n t e g r a t i o n formula, such as and obtaining * V * 6 ^ + ^ +4 ( 5 5 ) There are three unknowns— &jAt » & » a n d - ^-^/ '' therefore three equations are required. Consequently, we must probe the flow f i e l d with three d i f f e r e n t shock v e l o c i t i e s and obtain the set of equations, (56) -43-These equations are then inverted, to complete the so l u t i o n (appendix C). The exact s o l u t i o n of the equations developed i n chapter three i s highly unstable, but i t i s r e l a t i v e l y easy to obtain l i m i t s on the v a r i a t i o n of the plasma parameters throughout the flow. The density, p a r t i c l e v e l o c i t y , and pressure p r o f i l e are p l o t t e d against time i n figures 18 to 21. The pressure p r o f i l e i s compared with some absolute p i e z o e l e c t r i c pressure probe measurements. These quantitative pressure measurements* provide a completely independent check on the r e s u l t s . The p r o f i l e s agree both i n shape and i n magnitude, and are of the type expected behind r a d i a t i o n fronts at the Chapman-Jouguet point (Zuzak, 1968). The pressure r e s u l t s are conclusive evidence that the flow f i e l d i s a r a d i a t i o n front and not a bl a s t wave. If the flow f i e l d were a b l a s t wave, the pressure at the edge would be governed by the Rankine-Hugoniot equations, i n which case the pressure would be the same as the equilibrium pressure behind a shock wave t r a v e l l i n g at the same v e l o c i t y . However, the pressure found by both the shock and pressure probes i s one-half that of the shock pressure ( i e . p r e c i s e l y the pressure expected behind a r a d i a t i o n front t r a v e l l i n g at the Chapman-Jouguet p o i n t ) . * Measurements were c a r r i e d out by R. A r d i l a . -44-Chapter 5 CONCLUSIONS The i n e v i t a b l e r e s u l t of t h i s t h e s i s i s t h a t we have r a i s e d more q u e s t i o n s than we have answered. One of the answer -ed q u e s t i o n s i s t h a t c o n c e r n i n g the p r o p a g a t i o n o f a shock wave i n t o an inhomogeneous and moving med ia . An e x t e n s i o n o f the equa t i ons i n v o l v e d has been the development o f the shock wave as a t o o l f o r the p r o b i n g o f unknown f l ow f i e l d s . In our e x p e r i m e n t a l a p p l i c a t i o n we have p robed a f l o w f i e l d p roduced by a Bogen l i g h t s o u r c e . We have found our shock probes to be u s e f u l and a c c u r a t e . We have d i s c o v e r e d the f l o w f i e l d to be a r a d i a t i o n f r o n t a t the Chapman-Jouguet p o i n t . The q u e s t i o n s r a i s e d i n c l u d e those i n u n d e r s t a n d i n g the behav i ou r o f the r a d i a t i o n f r o n t . More e x p l i c i t l y , a s t r o -p h y s i c i s t s f e e l t h a t weak R-type r a d i a t i o n f r o n t s do e x i s t . Weak R-type f r o n t s have energy i n p u t s g r e a t e r than the Chapman-Jouguet ene rgy . But , so f a r we have not obse rved any energy -i n p u t s g r e a t e r than the Chapman-Jouguet ene rgy . We a l s o b e l i e v e t h a t the r a d i a t i o n f r o n t a t lower i n i t i a l p r e s s u r e s may be q u i t e i n t e r e s t i n g ( f i g , 2 2 ) , but b e f o r e any i n v e s t i g a t i o n , the o p e r a t i o n o f the shock tube must be improved so t h a t we can be c e r t a i n of i t s p r o p e r t i e s a t these low p r e s s u r e s . We would a l s o p r e f e r a more r e p r o d u c i b l e l i g h t s o u r c e . We a r e c u r i o u s as t o why the energy i n p u t i s ma in t a i ned even a f t e r the l i g h t sou r ce has s topped r a d i a t i n g . We. suppose -45-F i g . 22 R a d i a t i o n Front a t Low Downstream Pr e s s u r e -46-that the energy d i f f u s e s from the l i g h t source through the gas. Indeed, a spectroscopic study (perhaps time resolved) should be enlightening. Another feature i s the s i m i l a r i t y between the r a d i a t i o n front which we observe and the flow f i e l d produced by a stand-ard electromagnetic T-tube. Perhaps the T-tube flow f i e l d i s not a b l a s t wave but rather a r a d i a t i o n front. Perhaps, too, r a d i a t i o n fronts are quite common but .unrecognized laboratory occurrences. F i n a l l y , we have become quite excited about applying our probe technique to other unknown flow f i e l d s , such as la s e r sparks. - 4 7 -BIBLIOGRAPHY 1. Ahlborn, B., IPP 3 / 1 2 ( 1 9 6 3 ) 2. Ahlborn, B., Salvat, M., IPP 3/2 ( 1 9 6 2 ) 3. Ahlborn, B., Salvat, M., Z. Naturforschung 22a, 2 6 0 ( 1 9 ^ 7 ) 4 . Ahlborn, B., Zuzak, W., Rev. S c i . Instr. 3 8 , 1 9 4 ( I 9 6 7 ) 5. Ahlborn, B., Zuzak, W., Can. J. Phys. 4j7_, 4 7 0 9 ( 1 9 6 9 ) 6. A r d i l a , R., M.Sc. Thesis U.B.C. (to be published) 7. Axford, W.I., DeYoung, D.S., II Nuovo. 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(to be published) -49-Appendix A THE DOUBLY REFLECTED WAVE In d e r i v i n g the p a r t i a l d e r i v a t i v e s (ft.) 1 ^ 1 N - ^ l and the a n a l y s i s was c o n s i d e r e d o n l y to the f i r s t o r d e r . C h i s n e l l ( i 9 6 0 ) a l s o c o n s i d e r e d the e f f e c t of the wave r e f l e c t e d by the v a r i a b l e medium when the s i n g l y r e f l e c t e d wave passes back th rough i t . The doub ly r e f l e c t e d an expans ion wave. The p h y s i c a l s i t u a t i o n c o r r e s p o n d i n g to the p a r t i a l d e r i v a t i v e s now becomes t h a t i n f i g u r e A . The f o l l o w i n g phenomena need t o be c o n s i d e r e d 1 1. r e f l e c t i o n by the medium of an element o f the s i n g l y r e f l e c t e d wave fo rm ing an element of the doub l y r e f l e c t e d wave 2. the i n t e r a c t i o n o f the e lements of the s i n g l y and doub l y r e f l e c t e d waves, 3. the v a r i a t i o n o f the s t r e n g t h o f an e lement o f the doub l y r e f l e c t e d wave as i t t r a v e l s th rough the v a r i a b l e r e g i o n 4. the c a t c h i n g up of the i n c i d e n t shock by the expans ion wave. wave moves i n the same d i r e c t i o n as the i n c i d e n t shock and i s -50-comparing the theory f o r shock wave-contact surface interactions with h i s r e s u l t s (applying both s i n g l e r e f l e c t i o n and correct-ions due to double r e f l e c t i o n ) . He found good agreement f o r the s i n g l e r e f l e c t i o n theory whenever the r e f l e c t e d disturbance was small. With a p p l i c a t i o n of the double r e f l e c t i o n theory, the agreement was always excellent even when the strength of the r e f l e c t e d wave was large. In other words, the s i n g l e r e f l e c t i o n analysis used i n chapter three w i l l l i k e l y break down i f the gradients of i n i t i a l pressure or density are very large. - 5 1 -Ohyama ( 1 9 6 1 ) extended the method o f Ono, S a k a s h i t a , and Yamazaki ( i 9 6 0 ) (which i n t u r n i s j u s t the C h i s n e l l method w i t h s i n g l e r e f l e c t i o n ) t o the case o f doub l y r e f l e c t e d waves, by t a k i n g the e f f e c t s of the the second-orde r waves i n t o c o n s i d e r a t i o n . He d i d not i n v e s t i g a t e the behav iou r o f weak shocks , but d i d f i n d t h a t the r a t e of growth of shock s t r e n g t h o b t a i n e d by the f i r s t a p p r o x i m a t i o n i s l i t t l e i n f l u e n c e d , as f a r as the s t r o n g shock l i m i t i s c o n c e r n e d . We can c o n c l u d e t h a t the a n a l y s i s i n c h a p t e r t h r e e w i l l h o l d i f e i t h e r o f the f o l l o w i n g c o n d i t i o n s i s mett 1 . the i n c i d e n t shock i s s t r o n g , or 2. the d e n s i t y and p r e s s u r e g r a d i e n t s a r e not too s t r o n g . -52-Appendix B PARAMETERS AT THE BOUNDARY The numerical methods used are quite straightforward. " The equilibrium properties of the four states, *»V , 3 , *Y , and 5* (chapter 2) are calculated iteratively. Each state is calculated from three subroutines. Subroutine PEA2 calcu-lates § , -jP' , and /U" using the solutions of the Rankine-Hugoniot equations (18) and (19). Subroutine TEA2 calculates T using the equilibrium equation of state, where cl, is the degree of f i r s t ionizations and - 3 -f3-where I , is the ionization potential and is the atomic mass. Subroutine GEE2 calculates ^ .essentially by reproduc-ing figure 2. The energy input terms, Wj , \Jq , and are used as parameters and the resulting sets of solutions are shown in figure B-l. When becomes very large, the iterative procedure often breaks down because the square root in equations (18) and (19) w i l l sometimes become negative. Therefore, instead of using W as the parameter, i t is more convenient to use the -53-e n t i r e r o o t as the pa ramete r . The d e s i r e d v a l u e of i s o b t a i n e d g r a p h i c a l l y by i n s e r t i n g the c o n d i t i o n s , ^ "= -^ <r (B-3) F i g u r e B-l i n d i c a t e s the s o l u t i o n f o r the smear p i c t u r e i n f i g u r e 10. One impor tan t f e a t u r e of the s o l u t i o n i s t h a t the c o n d i t i o n @H - i s a c t u a l l y found by v a r y i n g o n l y W3 . Sma l l v a l u e s o f W a have a l a r g e e f f e c t on -fig , w h i l e l a r g e v a l u e s of and V/j- have v e r y l i t t l e e f f e c t on and -^Q-j-r e s p e c t i v e l y . T h e r e f o r e , s i n c e we a r e i n t e r e s t e d i n f i n d i n g o n l y , we can remove the a r b i t r a r y a ssumpt ion t h a t V/q »W S i and v a r y on l y V/3 i n o r d e r t o f u l f i l l the c o n d i t i o n , We used t h i s c o n d i t i o n t o o b t a i n the r e m a i n i n g r e s u l t s i n c h a p t e r f i v e . F i g u r e B-2 i n d i c a t e s the t ype of e r r o r s r e s u l t i n g i n and <AJ^ due t o e r r o r s i n measur ing ^ci^^ and V>/<iv2> • A g a i n , an e r r o r i n ^OsX) w i l l cause q u i t e a n o t i c e a b l e change i n --jQ.^  . - 5 4 -8 0 0 0 9 0 0 0 1 0 , 0 0 0 1 1 , 0 0 0 1 3 , 0 0 0 1 5 , 0 0 0 PRESSURE (TORR) Fig. B-l Graphical Solution for Intersection Problem -55-• « 1 1 1 1 • \ 1 8000 9000 10,000 11,000 12,000 13,000 14,000 PRESSURE (TORR) F i g , B-2 E r r o r Bounds on S t a t e */ -56-1 F 0 R M A K 3 F 1 5 . 8 ) 2 F O R M A T ( 5 X , 2 H U 1 , 9 X , 2 H P 1 , 1 0 X , 2 H T 1 , 1 0 X , 2 H G 1 , 9 X , 2 H D 1 , 1 1 X , 2 H W 1 ) 3 F O R M A T ( 5 X , 2 H U 2 , 9 X , 2 H P 2 , 1 0 X , 2 H T 2 , 1 0 X , 2 H G 2 , 9 X , 2 H D 2 , 1 1 X , 2 H W 2 ) 4 F O R M A T ( 5 X , 2 H U 3 , 9 X , 2 H P 3 , 1 0 X , 2 H T 3 , 1 0 X , 2 H G 3 , 9 X , 2 H D 3 , 1 1 X , 2 H W 3 ) 5 F O R M A T ( 5 X , 2 H U 4 , 9 X , 2 H P 4 , 1 0 X , 2 H T 4 , 1 0 X , 2 H G 4 , 9 X , 2 H D 4 , 1 1 X , 2 H W 4 ) 6 F O R M A T ( 5 X , 2 H U 5 , 9 X , 2 H P 5 , 1 0 X , 2 H T 5 , 1 0 X , 2 H G 5 , 9 X , 2 H D 5 , 1 1 X , 2 H W 5 ) 7 F O R M A T ( 5 X , 4 F 1 2 . 3 ) 8 F O R M A T ( 1 1 X , 4 H V S 1 2 , 8 X , 4 H V S 1 3 , 8 X , 4 H V S 2 4 , 8 X , 4 H V S 3 5 ) 9 F O R M A T ( 4 F 6 . 3 ) 10 F O R M A T { 5 X y 5 H A L F = , 5 X , F 8 . 1 ) 11 F O R M A T ( 5 X , F 6 . 3 , 5 X , F 9 . 1 , 5 X , F 9 . 1 , 5 X , F 6 . 3 , 5 X , F 8 . 6 , 5 X , F 1 0 . 6 ) 12 F O R M A T ( 5 X , 6 H A L F 2 = , 5 X , F 8 . 2 ) 1 3 F O R M A T ( 5 X , 6 H A L F 3 = , 5 X , F 8 . 2 ) R E A D ( 5 , 1 )W3U,W4U,W5U R E A D ( 5 , 1 ) W 3 L , W 4 L , W 5 L R E A O ( 5 , 1 ) W 3 M , W 4 M , W 5 M 3 8 R E A D ( 5 , 9 ) V S 1 2 , V S 1 3 , V S 2 4 , V S 3 5 W R I T E ( 6 , 8 ) W R I T E ( 6 , 7 ) V S 1 2 , V S 1 3 , V S 2 4 , V S 3 5 W3 = W3U D l = 0 . 0 0 3 2 2 4 2 8 G l = 1 . 6 6 6 Wl = 0 . 0 U I = 0 . 0 T l = 2 9 8 . 0 P I = 1 . 5 0 0 W R I T E ( 6 , 2 ) W R I T E ( 6 , 1 1 ) U 1 , P 1 , T 1 , G 1 , D 1 , W 1 P I = 1 3 3 . 3 * P 1 G2 = G l V I = V S 1 2 - U I SHM = V l / 0 . 3 2 1 W = 0 . 0 1 0 0 C A L L P E A 2 ( P 1 , D 1 , G 1 , G 2 , W , V 1 , S H M , D R , D 2 , P 2 , V 2 ) C A L L T E A 2 ( P 2 , D 2 , G 2 , T 2 , A L F , A L F 2 , A L F 3 , A L F 4 , A L F 5 ) C A L L G E E 2 ( T 2 , P 2 , G Z ) G2 = ( G Z + 3 . 0 * G 2 ) / 4 . 0 I F ( A B S ( G 2 - G Z ) . G T . 0 . 0 0 0 1 ) GO T O 1 0 0 P2 = P 2 / 1 3 3 . 3 W R I T E ( 6 , 3 ) U2 = V S 1 2 - V 2 W2 = 0 . 0 W R I T E ( 6 , 1 1 ) U 2 , P 2 , T 2 , G 2 , D 2 , W 2 W R I T E ( 6 , 1 0 ) A L F W R I T E ( 6 , 1 2 ) A L F 2 DX = D2 GX = G2 UX = U 2 -57-PX = P 2 * 1 3 3 . 3 T X = T 2 DB = 0 1 G B 1 = G l 9 9 9 9 W3 = W3-W3M P I = 1 . 5 0 0 P I = 1 3 3 . 3 * P 1 W4 = W4U W5 = W5U W = W3 V I = V S 1 3 - U 1 D l = DB G l = G B 1 G2 = G B 1 SHM = V l / 0 . 3 2 1 2 0 0 C A L L P E A 3 ( P 1 , D 1 , G 1 , G 2 » W , V 1 , S H M , D R , D 2 , P 2 , V 2 ) C A L L T E A 2 ( P 2 , D 2 , G 2 , T 2 , A L F , A L F 2 T A L F 3 , A L F 4 , A L F 5 ) C A L L G E E 2 ( T 2 , P 2 , G Z ) G2 = ( G Z + 3 . 0 * G 2 ) / 4 . 0 I F ( A B S ( G 2 - G Z ) . G T . 0 . 0 0 0 1 ) GO T O 2 0 0 W R I T E ( 6 , 4 ) U 3 = V S 1 3 - V 2 G 3 = G2 D3 = 0 2 T 3 = T 2 P 3 = P 2 / 1 3 3 . 3 W R I T E ( 6 , 1 1 ) U 3 , P 3 , T 3 , G 3 , D 3 , W 3 W R I T E ( 6 , 1 0 ) A L F W R I T E ( 6 , 1 2 ) A L F 2 2 8 D l = 0 3 W R I T E ( 6 , 6 ) 9 9 W5 = W5-W5M W = W5 G l = G 3 P I = P 3 * 1 3 3 . 3 T l = T 3 V I = V S 3 5 + U 3 A l = G 1 * P 1 / D 1 SHM = 1 0 0 0 . 0 * V 1 / ( S Q R T ( A l ) ) J L = 0 4 0 0 C A L L P E A 2 ( P I , D 1 , G 1 , G 2 » W , V 1 , S H M , D R , D 2 , P 2 , V 2 ) J L = J L + 1 C A L L T E A 2 ( P 2 , D 2 , G 2 , T 2 , A L F , A L F 2 » A L F 3 , A L F 4 , A L F 5 ) C A L L G E E 2 ( T 2 , P 2 , G Z ) G2 = ( G Z + 3 . 0 * G 2 ) / 4 . 0 I F ( J L . G T . 5 0 ) G 0 T O 5 0 0 I F ( A B S ( G 2 - G Z ) . G T . 0 . 0 0 0 1 ) GO T O 4 0 0 7 0 0 0 5 = D2 -58-U5 = VS35 - V2 G5 = G2 P5 = P2/133.3 T5 = T2 WRITE (6,11)U5,P5,T5,G5,D5,W5 WRITE ( 6 , 1 0 ) A L F WRITE ( 6 , 1 2 ) A L F 2 WRITE ( 6 , 1 3 ) A L F 3 WRITE(6,13)ALF4 WRITE (6,13>ALF5 IF (W5.LT.W5L) GO TO 99 WRITE (6,5) 999 W4 = W4-W4M W = W4 J L = 0 D l = DX G l = GX PI = PX T l = TX VI = VS24+UX A l = G1*P1/D1 SHM = 1000.0*V1/(SQRT(AI ) ) 300 CALL PEA3(P1,01,G1,G2,W,V1,SHM,DR,D2,P2,V2) J L = J L + 1 CALL TEA2(P2 ,D2 ,G2,T2,ALF,ALF2,ALF3,ALF4,ALF5) CALL GEE2(T2,P2 ,GZ ) G2 = (GZ + 3.0*G2)/4.0 IF ( J L . G T . 5 0 ) GO TO 500 IF (ABS(G2-GZ).GT.0.0001) GO TO 300 900 04 = 02 G4 = G2 U4 = V2-VS24 T4 = T2 P4 = P2/133.3 WRITE (6,11)U4,P4,T4,G4,D4,W4 WRITE ( 6 , 1 0 ) A L F WRITE ( 6 , 1 2 ) A L F 2 WRITE ( 6 , 1 3 ) A L F 3 WRITE(6,13)ALF4 WRITE ( 6 , 1 3 ) A L F 5 IF (W4.LT.W4L) GO TO 999 IF (W3.LT.W3L) GO TO 9999 IF (VS12.GT.0.0000) GO TO 38 500 STOP END - 5 9 -S U B R O U T I N E T E A 2 ( P 2 , D 2 , G E , 1 2 ) G2 = G E H = G 2 * P 2 / ( ( G 2 - 1 . 0 ) * 0 2 ) R E A L MA MA = 6 . 6 3 E - 2 6 A L F = ( 2 . 3 3 2 7 E - 6 ) * ( H - 2 . 5 * P 2 / D 2 ) I F ( A L F . G T . 1 0 0 . 0 ) A L F = 1 0 0 . 0 A L F 2 = ( 1 . 3 1 7 6 E - 6 ) * ( H - 2 . 5 * P 2 / D 2 ) - 0 . 5 6 4 8 * A L F I F ( A L F 2 . G T . 1 0 0 . 0 ) A L F 2 = 1 0 0 . 0 A L F 3 = ( 8 . 9 7 5 8 E - 7 ) * ( H - 2 . 5 * P 2 / D 2 ) - . 3 7 2 5 * A L F - . 6 8 1 2 * A L F 2 I F ( A L F 3 . G T . 1 0 0 . 0 ) A L F 3 = 1 0 0 . 0 A L F 4 = ( 5 . 9 9 6 1 E - 7 ) * ( H - 2 . 5 * P 2 / D 2 ) - . 2 4 8 8 * A L F - . 4 5 5 1 * A L F 2 - . 6 6 8 0 * A L F 3 I F ( A L F 4 . G T . 1 0 0 . 0 ) A L F 4 = 1 0 0 . 0 B = ( 4 . 6 8 9 1 E - 7 ) * ( H - 2 . 5 * P 2 / D 2 ) A L F 5 = B - . 1 9 4 6 * A L F - . 3 5 5 9 * A L F 2 - . 5 2 2 4 * A L F 3 - . 7 8 2 1 * A L F 4 A L F A = ( A L F + A L F 2 + A L F 3 + A L F 4 + A L F 5 ) / 1 0 0 , 0 T 2 = P 2 / ( D 2 * 2 0 8 . 1 * ( 1 . 0 + A L F A ) ) R E T U R N END -60-SUBROUTINE PEA3 ( PI ,D1 ,G1 ,G2 , W ,V 1, SHM , DR , D2, P2, V2) 207 A = G2/(SHM*SHM*G1) - 1.0 C = W ' P2 = P1*(1.0-G1*SHM*SHM*A*(1.0+C)/(G2+1.0)) 307 V2 = Vl*(1.0+A*(l.0+C)/(G2+1.0)) DR = V1/V2 D2 = DR*D1 407 RETURN END - 6 1 -SUBROUTINE PEA2(P1 tDl,G1,G2,W,V1,SHM,DR,02,P2,V2) 207 A = G2/(SHM*SHM*G1) - 1.0 B = 2.0*(G2*G2-1.0)*W/(Dl*(V1**3 )) B l = (Gl-1.0)*((G2-G1*SHM*SHM)**2) A l = 2.0*(G2+1.0)*(G1-G2)*SHM*SHM*G1 EP = A l / B l D = 1.0+EP-B/(A*A) C = SQRT(D) P2 = Pl*(l.0-Gl*SHM*SHM*A*(1.0+C)/(G2+1.0)) 307 V2 = V1*(1.0+A*(1.0+C)/(G2+1.0)) DR = V1/V2 D2 = DR*01 407 RETURN END - 6 2 -SUBROUTINE GEE2(T2,P2,GZ) P2 = P2/133.3 IF (T2.LT.15000.0) GO TO 15 IF (T2.LT.20000.0 ) GO TO 25 IF (T2.LT.21000.0) GO TO 35 IF (T2.LT.22000.0) GO TO 45 IF (T2.LT.23000.0) GO TO 55 IF (T2.LT. 24000.0) GO TO 65 IF (T2.LT.25000.0) GO TO 75 IF (T2.LT.26000.0) GO TO 85 IF (T2.LT. 27000.0) GO TO 95 IF (T2.LT.28000.0) GO TO 105 IF (T2.LT.29000 0) GO TO 115 IF (T2.LT.30000.0) GO TO 125 GA = 1.125 GB = 1.130 GC = 1.140 GD = 1.160 GE = 1.193 GO TO 1000 IF (T2.LT.16000.0) GO TO 135 IF (T2.LT.17000.0) GO TO 145 IF (T2.LT.18000.0) GO TO 155 IF (T2.LT.19000.0) GO TO 165 GA = 1.118 - 0.175*AL0G(T2/20000.0) GB =1.148 - 0.039*AL0G(T2/20000.0) GC = 1.162 + 0.078*AL0G(T2/20000.0) GD = 1.170 - 0.039*AL0G(T2/20000.0) GE = 1.207 - 0.234*AL0G(T2/20000.0) GO TO 1000 IF (T2.LT.9000.0) GO TO 175 IF (T2.LT.10000.0) GO TO 185 IF (T2.LT. 11000.0) GO TO 195 IF (T2.LT.12000.0) GO TO 205 IF (T2.LT.13000.0) GO TO 215 IF (T2.LT.14000.0) GO TO 225 GA = 1.131 + 0.072*ALOG(T2/15000.0) GB =1.135 GC = 1.157 - 0.420*AL0G(T2/15000.0) GD = 1.257 - 0.826 *ALOG(T2/15000.0) GE = 1.367 - 1.000*AL0G(T2/15000.0) GO TO 1000 IF (T2.LT.3000.0) GO TO 235 IF (T2.LT.4000.0) GO TO 245 IF (T2.LT. 5000.0) GO TO 255 IF (T2.LT.6000.0) GO TO 265 IF (T2.LT.7000.0) GO TO 275 IF (T2.LT.8000.0) GO TO 285 -63-GA = 1.363- 1.452*AL0G(T2/9000.0) GB = 1.546 - 0.637*AL0G(T2/9000.0) GC = 1.609 - 0.375*ALOG(T2/9000.0> GD = 1.660 - 0.051*ALOG(T2/9000.0) GE = 1.665 - 0.008*ALOG(T2/9000.0> GO TO 1000 125 GA = 1.125 GB = 1.130 GC = 1.140 GD = 1.160 - 0.118*AL0G(T2/30000.0) GE = 1.193 - 0.059*AL0G(T2/30000.0) GO TO 1000 115 GA = 1.125 GB = 1.130 GC = 1.140 GD = 1.164 - 0.142*AL0G(T2/29000.0) GE = 1.195 - 0.057*AIOG(T2/29000.0) GO TO 1000 105 GA = 1.125 GB = 1.130 + 0.055*AL0G(T2/28000.0) GC = 1.140 GD = 1.169 - 0.165*AL0G(T2/28000.0) GE = 1.197 - 0.055*AL0G(T2/28000.0) GO TO 1000 95 GA = 1.125 GB =1.128 + 0.053*AL0G(T2/27000.0) GC = 1.140 - 0.106*ALOG(T2/27000.0) GO = 1.175 - 0.159#A10G(T2/27000.0) GE = 1.199 - 0.026*ALOG(T2/27000.0) GO TO 1000 85 GA = 1.125 GB =1.126 + 0.051*AL0G(T2/26000.0) GC = 1.144 - 0.127*AL0G(T2/26000.0) GD = 1.181 - 0.051*AL0G(T2/26000.0) GE = 1.200 GO TO 1000 75 GA = 1.125 + 0.049*AL0G(T2/25000.0) GB =1.124 + 0.024*AL0G(T2/25000.0) GC = 1.149 - 0.171*AL0G(T2/25000.0) GD = 1.183 GE = 1.2000 + 0.122*AL0G(T2/25000.0) GO TO 1000 65 GA = 1.123 + 0.094*ALOG(T2/24000.0) GB =1.123 - 0.094*ALOG(T2/24000.0) GC = 1.156 - 0.164*AL0G(T2/24000.0) GD = 1.183 GE = 1.195+0.047*AL0G(T2/24000.0) GO TO 1000 -64-55 GA = 1.119 + 0.045*AL0G(T2/23000.0) GB =1.127 - 0.157*AL0G(T2/23000.0) GC = 1.163 - 0.067*ALOG(T2/23000.0) GD = 1.183 + 0.067*AL0G(T2/23000.0) GE = 1.193 - 0.022*ALOG(T2/23000.0) GO TO 1000 45 GA = 1.117 + 0.021*ALOG(T2/22000.0) GB =1.134 - 0.193*AL0G(T2/22000.0) GC = 1.166 + 0.043*AL0G(T2/22000.0) GD = 1.180 + 0.086*AL0G(T2/22000.0) GE = 1.194 - 0.107*ALOG(T2/22000.0) GO TO 1000 35 GA = 1.116 - 0.041*AL0G(T2/21000.0) GB =1.143 - 0.1O2*ALOG(T2/21OOO.O) GC = 1.164 + 0.041*AL0G(T2/21000.0) GD = 1.176 - 0.123*AL0G(T2/21000.0) GE = 1.199 - 0.164*AL0G(T2/21000.0) GO TO 1000 165 GA = 1.127 - 0.148*ALOG(T2/19000.0) GB =1.150 GC = 1.158 + 0.037*AL0G(T2/19000.0) GD = 1.172 - 0.111*AL0G(T2/19000.0) GE = 1.219 - 0.444*AL0G(T2/19000.0) GO TO 1000 155 GA = 1.135 - 0.070*AL0G(T2/18000.0) GB =1.150 + 0.052*ALOG(T2/18000.0) GC = 1.156 + 0.122*ALOG(T2/18000.0) GD = 1.178 - 0.280*AL0G(T2/18000.0) GE = 1.243 - 0.630*AL0G(T2/18000.0) GO TO 1000 145 GA = 1.139 + 0.049*AI_OG(T2/17000.0) GB =1.147 + 0.115*AL0G(T2/17000.0) GC = 1.149 + 0.033*ALOG(T2/17000.0) GO = 1.194 - 0.363*AL0G(T2/17000.0) GE = 1.279 - 0.660*AL0G(T2/17000.0) GO TO 1000 135 GA = 1.136 + 0.077*ALOG(T2/16000.0) GB =1.140 + 0.077*AL0G(T2/16000.0) GC = 1.147 - 0.155*AL0G(T2/16000.0) GD = 1.216 - 0.635*AL0G(T2/16000.0) GE = 1.319 - 0.743*ALOG(T2/16000.0) GO TO 1000 225 GA = 1.126 + 0.054*AL0G<T2/14000.0)' GB =1.135 - 0.189*AL0G(T2/14000.0) GC = 1.186 - 0.661*AL0G(T2/14000.0) GD = 1.314 - 1.093#AL0G(T2/14000.0) GE = 1.436 - 0.877*AL0G(T2/14000.0) GO TO 1000 -65-215 GA = 1.122 - 0.012*ALOG(T2/1300G.O) GB =1.149 - 0.412*ALOG(T2/13000.0> GC = 1.235 - 1.012*AL0G(T2/13000.0) GD = 1.395 - 1.124*AL0G(T2/13000.0) GE = 1.501 - 0.837*AL0G(T2/13000.0) GO TO 1000 205 GA = 1.123 - 0.310*AL0G(T2/12000.0) GB =1.183 - 0.839*AL0G(T2/12000.0) GC = 1.316 - 1.253*ALOG(T2/12000.0) GD = 1.485 - 0.942*AL0G(T2/12000.0) GE = 1.568 - 0.517*ALOG(T2/12000.0) GO TO 1000 195 GA = 1.150 - 0.703*ALOG(T2/11000.0) GB =1.255 - 1.416*AL0G(T2/11000.0) GC = 1.425 - 1.228*AL0G(T2/11000.0) GD = 1.567 - 0.283*AL0G(T2/11000.0J GE = 1.613 - 0.346*AL0G(T2/11000.0) GO TO 1000 185 GA = 1.217 - 1.386*AL0G(T2/10000.0) GB =1.390 - 1.481*AL0G(T2/10000.0) GC = 1.542 - 0.636*AL0G(T2/10000.0) GD = 1.640 - 0.190*ALOG(T2/10000.0) GE = 1.646 - 0.085*ALOG(T2/10000.0) GO TO 1000 285 GA = 1.537 - 0.524*ALOG(T2/8000.0) GB =1.621-0.195*AL0G(T2/8000.0) GC = 1.651 - 0.075*ALOG(T2/8000.0) GD = 1.666 GE = 1.666 GO TO 1000 275 GA = 1.607 - 0.175*AL0G(T2/7000.0) GB =1.647 - 0.078*AL0G(T2/7000.0) GC = 1.661 - 0.032*AL0G(T2/7000.0) GD = 1.666 GE = 1.666 GO TO 1000 265 GA = 1.634 - 0.093*AL0G(T2/6000.0) GB =1.659 - 0.038*ALOG(T2/6000.0) GC = 1.666 GD = 1.666 GE = 1.666 ' GO TO 1000 255 GA .= 1.651 - 0 .036*AL0G ( T2/5000. 0) GB = 1.666 GC = 1.666 GD = 1.666 GE = 1.666 GO TO 1000 -66-2 4 5 G A = 1 . 6 5 9 - 0 . 0 2 8 * A L 0 G ( T 2 / 4 0 0 0 . 0 ) G B = 1 . 6 6 6 G C = 1 . 6 6 6 G D = 1 . 6 6 6 G O T O 1 0 0 0 2 3 5 G A = 1 . 6 6 6 G B = 1 . 6 6 6 G C = 1 . 6 6 6 G D = 1 . 6 6 6 G E = 1 . 6 6 6 1 0 0 0 I F ( P 2 . L T . 7 6 . 0 ) G O T O 1 6 I F ( P 2 . L T . 7 6 0 . 0 ) G O T O 2 6 I F ( P 2 . L T . 7 6 0 0 . 0 ) G O T O 3 6 G 2 = G D + 0 . 6 2 1 * ( G E - G D ) * A L 0 G ( P 2 / 7 6 0 0 . 0 ) G O T O 2 0 0 0 3 6 G 2 = G C + 0 . 4 3 4 * ( G D - G C ) * A L 0 G ( P 2 / 7 6 0 . 0 ) G O T O 2 0 0 0 2 6 G 2 = G B + 0 . 4 3 4 * ( G C - G B ) * A L 0 G ( P 2 / 7 6 . 0 ) G O T O 2 0 0 0 1 6 G 2 = G A + 0 . 4 3 4 * ( G B - G A ) * A L 0 G ( P 2 / 7 . 6 ) 2 0 0 0 G Z = G 2 P 2 = 1 3 3 . 3 * P 2 R E T U R N E N D -6?-Append ix C CONTINUOUSLY VARYING PARAMETERS The n u m e r i c a l method t o d e s c r i b e c o n t i n u o u s l y v a r y i n g parameters i s ex t r eme l y s i m p l e . E s s e n t i a l l y , the s e t o f e q a t i o n s , • <w Y * A-W, + fa  6 f >  + (C- l ) a r e i n v e r t e d t o o b t a i n 4*m4< ( c _ 2 ) Us ing a s i m p l e p r e d i c t o r i n t e g r a t i o n f o r m u l a f o r ^ and proved t o be t oo u n s t a b l e a method. I t i s u n s u c c e s s f u l because equa t i ons (C-2) i n v o l v e the d i f f e r e n c e s of d i f f e r e n c e s , and because the f l o w f i e l d produced by the l i g h t sou r ce i s no t as r e p r o d u c i b l e as we would l i k e . I n s t ead o f f i n d i n g an exac t s o l u t i o n , s i m p l e l i m i t s were found on the v a r i a t i o n o f the -68-parameters. The lower l i m i t was found by assuming that one parameter alone was responsible f o r the v a r i a t i o n i n shook v e l o c i t y . For example, while the upper l i m i t f o r J^>y would then be (c - 3 ) \ (c-4) since the other two parameters would account f o r the complete shock v a r i a t i o n on the other two measurements. In our p a r t i c u l a r case, the di s c o n t i n u i t y "^3^ *^  i s not a shock wave because there i s energy input into state 5" . In f a c t , VJj. whould be very close to the Chapman - Jouguet energy. Therefore, we re-derive the p a r t i a l d e r ivatives ~ and r"^ , assuming that the front / i s at the Chapman - Jouguet point. V = -• * J^[[ci-M^(c-5) -69-*S>V At the Chapman - Jouguet point, Therefore, Ol ^ c . x » ° * ^ S H O C * (C-7) * - i * (C-8) C « J . SH©C< (C-9) (C-10) and Ot^t.y j_ ? " S H O C K where °L*SHOC* a n d ^ S t f o c * as derived in chapter three. Figure C-l indicates the variation of four shock velocities throughout the flow f i e l d . These velocities have been normal-ized for the distance which they travel into the flow f i e l d . -70--71-1 F O R M A T ( 5 X , I 2 , 5 X , F 6 . 3 , 5 X , F 6 . 1 , 5 X , F 8 . 6 , 5 X , F 8 . 1 , 5 X , F 6 . 3 ) 2 F O R M A T ( 5 X , I H I , 6 X , 2 H U 3 , 9 X , 2 H P 3 , 9 X , 2 H D 3 , 1 1 X , 2 H T 3 , 9 X , 2 H G 3 ) 3 F O R M A T ( I 2 » 4 F 6 . 3 ) 5 F 0 R M A T ( 5 X , I 2 , 5 X , F 6 . 3 , 5 X , F 6 . 3 , 5 X , F 6 . 3 ) 6 F O R M A T ( 1 2 ) 7 F O R M A T ( F 6 . 3 ) R E A D ( 5 , 7 ) G E IM = 1 7 D I M E N S I O N V I ( 1 7 ) , V 2 ( 1 7 ) , V 3 ( 1 7 ) , V X ( 1 7 ) , V 4 ( 1 7 ) D I M E N S I O N U I ( 1 7 ) , P I ( 1 7 ) , D 1 ( 1 7 ) , T 1 ( 1 7 ) , G 2 ( 1 7 ) R E A D ( 5 , 1 ) I , U 1 ( 1 ) , P 1 ( 1 ) , D 1 ( 1 ) , T 1 ( 1 ) , G 1 G 2 ( I ) = G l W R I T E ( 6 , 2 ) W R I T E ( 6 , 1 ) I , U 1 ( I ) , P 1 ( I ) , D 1 ( I ) , T 1 ( I ) , G 1 P K I ) = 1 3 3 . 3 * P 1 ( I ) R E A D ( 5 , 3 ) I , V 1 ( I ) , V 2 ( I ) , V 3 ( I ) t V 4 ( I ) 1 0 0 R E A D ( 5 , 3 ) I , V 1 ( I ) , V 2 ( I ) , V 3 ( I ) , V 4 ( I ) G 2 ( I ) = G l V X ( I ) = V K I ) C A L L F E E ( G 2 ( 1 - 1 ) , V X ( I ) ,U1 ( 1 - 1 ) , D 1 ( I-•1 ) , D 1 ( I -•1) , D 1 ( I -•1) , D 1 ( I -•1) , D 1 ( I -• 1 ) , D 1 ( I -•1) , D 1 ( I -) , P 1 ( ) , P 1 ( ) t P K ) , P 1 ( ) , P 1 ( ) , P 1 ( ) , P 1 ( ) , P 1 ( - 1 ) - 1 ) C A L L G E E ( G 2 ( 1 - 1 ) , V X ( I ) , U 1 ( I - 1 ) , D 1 ( I -F G 1 = G X F l = F X V X ( I ) = V 2 ( I ) C A L L F E E ( G 2 ( I - 1 ) , V X ( I ) , U 1 ( I C A L L G E E ( G 2 ( 1 - 1 ) , V X ( I ) , U 1 ( I F 2 = F X F G 2 = GX V X ( I ) = V 3 ( I ) C A L L F E E ( G 2 ( 1 - 1 ) , V X ( I ) , U 1 ( I C A L L G E E ( G 2 ( I - l ) , V X ( i ) , U l ( I F 3 = F X F G 3 = GX V X ( I ) = V 4 ( I ) C A L L F E E ( G 2 ( 1 - 1 ) , V X ( I ) , U 1 ( I C A L L G E E ( G 2 ( I - 1 ) , V X ( I ) , U 1 ( I F G 4 = GX F 4 = F X U K I ) = U 1 ( I - 1 ) + V 1 ( I ) - V 1 ( I - 1 ) D l ( I ) = D l ( 1 - 1 ) + ( V 4 ( I ) - V 4 ( 1 - 1 ) ) / F 4 - ( U l ( I ) - U l ( I XX = ( U I ( I ) - U l ( 1 - 1 ) ) / F G 2 + ( D l ( I ) - D l ( 1 - 1 ) ) / F G 2 P K I ) = P I ( 1-1 ) - ( V 2 ( I ) - V 2 ( 1 - 1 ) ) / F G 2 + X X P2 = P I ( I ) D2 = D l ( I ) C A L L T E A 2 ( P 2 , D 2 , G E , T 2 ) T l ( I ) = T 2 P K I ) = P K I ) / 1 3 3 . 3 W R I T E ( 6 , 1 ) I , U 1 ( I ) , P 1 ( I ) , D K I ) , T 1 ( I ) , G 1 P K I ) = 1 3 3 . 3 * P 1 ( I ) I F ( I . L T . I M ) GO T O 1 0 0 S T O P END - 1 ) , F X ) - 1 ) , G X ) - 1 ) , F X ) - 1 ) , G X ) - 1 ) , F X ) - 1 ) , G X ) , F X ) , G X ) •1) ) / F 4 -72-S U B R O U T I N E G E E ( G 2 , V X , U 1 , D 1 , P 1 , G X ) L A M = ( G 2 - 1 . 0 ) / ( G 2 + 1 . 0 ) Z = ( 1 . 0 - L A M ) * ( ( V X - U 1 ) * * 2 ) * D 1 / P 1 - L A M C A = S Q R T ( Z * ( 1 . 0 + L A M * Z ) / ( 1 . 0 + L A M ) ) C O = ( Z - 1 . 0 ) / ( 2 . 0 * ( L A M + Z ) ) C C = - C A / Z - 1 . 0 C B = ( Z - 1 . 0 ) / 2 . 0 OZ = ( C A + C B ) / ( P 1 * ( C C + C 0 ) ) DZ = 0 . 5 * D Z G X = ( V X - U 1 ) * ( 0 . 5 / P 1 + 0 . 5 * D Z / ( Z - 1 . 0 ) ) R E T U R N E N D -73-SUBROUTINE FEE(G2,VX,U1,Dl,P1,FX) LAM = (G2-1.0)/(G2+1.0) Z = (1.0-LAM)*((VX-U1)**2)*D1/P1-LAM CA = -SQRT((1.0+LAM*Z)/(Z*(l.O+LAM))) CB = 1,0-(Z-1.0)/(2.0*(LAM+Z)) DZ = (Z-1.0)/(2.0*D1*(CA+CB)> DZ = 0.5*DZ FX = (VX-U1)*(-.5/Dl+0.5*DZ/(Z-1.0)) RETURN END - 7 4 -Appendix D THE SHOCK TUBE In this work, the original "low attenuation" shock tube of P. R. Smy (1962, 1965) was used. The driver gas is el e c t r i c a l l y heated and accelerated by a standard back-strap configuration, and a mylar membrane is employed to separate test and driver gases. This shock tube has been used in several experiments (ref. 4 , 11, 12, 13, 25, 3 0 ) . Details of design and performance are given in Smy's a r t i c l e (19^5) and also in M. P h i l l i p s ' thesis (1969). In order to use this shock tube for the testing of some other gas flow, one needs to ascertain that a shock is actually formed. Smy's explana-tion was not entirely satisfactory and when coupled with the criticisms of the conventional electromagnetic shock tubes by Muntenbruch (1969), and the discussion of flow i n s t a b i l i t i e s in electric arc-driven tubes by Barach and Vermillion ( 1 9 6 5 )1 serious doubt arose about the description of the shock tube and the predictability of any shock waves. The recent work of Phillips has helped to understand the performance of such an arc-driven membrane shock tube. However, since Phillips introduced several modifications in the design of the tube, which were not incorporated in our tube, we have establised, independently, that in our tube, a reproducible and well behaved shock is formed. In order to intensify our understanding of the performance of the arc-driven shock tube, we f i r s t derive the shock velocity -75-as a function of the energy stored i n the d r i v i n g capacitor bank. The t o t a l k i n e t i c energy, ^ M^. J U e , introduced through the membrane into the t e s t section of the shock tube must be a f r a c t i o n f> of the capacitor bank energy, C V c , a M . v U 0 2 * / i 4 C V C * (D-l) M*. i s the mass o r i g i n a l l y expelled by the discharge. w>u0 i s the average p a r t i c l e v e l o c i t y at the membrane, but since more and more mass i s accelerated and the t o t a l momen-tum i s constant, the p a r t i c l e v e l o c i t y w i l l gradually decrease; where A i s the cro s s - s e c t i o n a l area, and X i s the distance from the membrane. Thus, one obtains the contact-surface v e l o c i t y as a function of p o s i t i o n Smy derived an i d e n t i c a l expression, t r e a t i n g the d r i v e r gas as a b u l l e t and considering only integrated magnetic d r i v i n g e f f e c t s . With t h i s background and the motivation of understanding the properties of the shock tube, we have two objectives» f i r s t l y , to e s t a b l i s h that a shock f r o n t i s formed and that i t s properties correspond to those given by s o l u t i o n of the - 7 6 -Rankine - Hugon io t e q u a t i o n s , and s e c o n d l y , t o t e s t the f u n c t i o n a l dependence sugges ted i n e q u a t i o n (D-3) . In o r d e r t o t e s t the q u a l i t y o f the shock waves fo rmed , we measure the c o n t a c t s u r f a c e and shock v e l o c i t i e s ( P h i l l i p s , 1 9 6 9 ) . I d e a l l y , the c o n t a c t s u r f a c e v e l o c i t y s h o u l d equa l the p a r t i c l e v e l o c i t y i n the shock hea ted g a s . In p r a c t i c e , bound -a r y l a y e r s a t the w a l l of the shock tube r e s u l t i n a mass f l o w out o f the r e g i o n between the c o n t a c t s u r f a c e and shock f r o n t . T h u s , the p a r t i c l e s have a ne t v e l o c i t y towards the c o n t a c t s u r f a c e so t h a t the p a r t i c l e v e l o c i t y i s somewhat s m a l l e r than the c o n t a c t s u r f a c e v e l o c i t y . In t y p i c a l smear p i c t u r e s , the c o n t a c t s u r f a c e and shock f r o n t can be e a s i l y i d e n t i f i e d ( f i g . 9 ) , There fo re , bo th v e l o c i t i e s a re measu reab l e . In f i g u r e s D-l and D - 2 t he c o n t a c t s u r f a c e v e l o c i t y i s p l o t t e d a g a i n s t the shock v e l o c i t y . The c o n t a c t s u r f a c e v e l o c i t y has been o b t a i n e d i n t h r e e waysi 1 . by measur ing the c o n t a c t s u r f a c e v e l o c i t y , 2. by measur ing the shock v e l o c i t y ; c a l c u l a t i n g the p a r t i c l e v e l o c i t y f rom the Rankine - Hugon io t equa t i ons and i d e n t i f y i n g t h i s w i t h the c o n t a c t s u r f a c e v e l o c i t y , as f o r the i d e a l c a s e , and 3 . by measur ing the shock v e l o c i t y and a p p l y i n g the wel l-known boundary l a y e r t h e o r y o f Roshko ( i 9 6 0 ) t o c a l c u l a t e the c o n t a c t s u r f a c e v e l o c i t y . The agreement i s q u i t e good , so t h a t on the b a s i s o f these measurements, the f l o w p a t t e r n l o o k s l i k e a shock wave w i th -77-3 y <y rH EH M O o > o CO EH o < " 5 O o MEASURED • A AAA A* CALCULATED WITH BOUNDARY LAYERS ' F i g . D-l Con tac t Su r f a ce V e l o c i t y 3 ' I 7 SHOCK VELOCITY <K»*?a«c^ MEASURED. # • A \ A • A • A CALCULATED IDEALLY >H EH M o a w t > w I2 D W EH O o o • A A A* • A • A •4? • A A • F i g . D-2 Con ta c t S u r f a c e V e l o c i t y "»2 «3 SHOCK VELOCITY ( jt>~ /^CL) -78-boundary layers i n the investigated i n i t i a l pressure range of 1 to 10 Torr. A further i n d i c a t i o n of the formation of a shock front i s provided by a v e r t i c a l smear picture ( f i g . D-3) which c l e a r l y shows a planar shock front followed by a d i f f u s e contact surface. We now proceed to test the attenuation equation (D-3). There are two convenient ways to test t h i s r e l a t i o n , since the contact surface v e l o c i t y depends upon the two external parameters, V c and ^, . Figure D-4 demonstrates the v a r i a t i o n of the contact surface v e l o c i t y with the bank voltage \£. and indicates p r i m a r i l y that there i s a large v a r i a t i o n i n the shock v e l o c i t y formed under i d e n t i c a l conditions. The expected proportion i s also roughly indicated. Figures D-5, D-6, and D-7 show the v a r i a t i o n of the contact surface v e l o c i t y with the downstream pressure. The s t r i k i n g r e s u l t i s that Mr*, cannot be the t o t a l mass of d r i v e r gas as Smy has predicted. In t r y i n g to acquire a better f i t , i t was noticed that no s i n g l e value of would cause equation D-3 to correspond to the experimental curves. However, at lower downstream pressures, a good f i t could be obtained by taking fir*. - Mf the mass of d r i v e r gas o r i g i n a l l y between the two electrodes. At higher pressures, a better f i t was obtained by taking tl^ to be larger; t\_*Mx one-half of the o r i g i n a l t o t a l d r i v e r gas. A seemingly reasonable explanation f o r t h i s behaviour would be that capacitor r i n g i n g causes a perturbation i n the d r i v i n g mechanism. These r i n g i n g e f f e c t s are a common SHOCK DIRECTION II IMAGE OP ENTRANCE SLIT 1 1 JJ SHOCK TUBE LENS DOVE PRISM ENTRANCE SLIT OP SMEAR CAMERA F i g . D 3 V e r t i c a l Smear P i c t u r e -80-6 J F i g . D-4 Shock V e l o c i t y vs V c F i g . D-5 Shock A t t e n u a t i o n -82-0 1 2^ r3 ^ 5 1 7 DOWNSTREAM PRESSURE (TORR) -83-prob lem i n e l e c t r o m a g n e t i c shock tubes (Muntenbruch, 1 9 6 9 ) . In o r d e r to t e s t t h i s h y p o t h e s i s , we a p p l i e d a crowbar t o the d i s c h a r g e ( f i g . D - 8 ) . A l t hough the crowbar was not c o m p l e t e l y s u c c e s s f u l , one of the smear p i c t u r e s y i e l d e d j u s t the d e s i r e d i n f o r m a t i o n . As can be c l e a r l y s e e n , the p a r t i a l crowbar reduces the secondary r i n g i n g e f f e c t s enough so t h a t the t a k e -over o f the d r i v i n g mechanism i s d i s t i n c t . The p a r t i a l crowbar r e s u l t s compare q u i t e w e l l w i th the m o d i f i e d t h e o r y ( f i g . D - 9 ) , and e s t a b l i s h no t on l y tha-t the f l o w f i e l d i s p e r t u r b e d by c a p a c i t o r r i n g i n g , but a l s o t h a t e q u a t i o n D-3 h o l d s . In c o n c l u s i o n , the shock tube p r o v i d e s a low a t t e n u a t i o n , h i g h v e l o c i t y shock wave w i t h p r o p e r t i e s p r e d i c t a b l e by the Rankine - Hugon io t e q u a t i o n s . There a r e c e r t a i n l i m i t a t i o n s due t o the f o l l o w i n g cond i t ionst 1 . c a p a c i t o r r i n g i n g i n the d r i v e r s e c t i o n 2. i m p u r i t y con t en t f rom bu rn t my la r which coa t s the w a l l s o f the shock tube 3 . j i t t e r i n g i n the shock speeds f rom about 1 0 t o 1 5 p e r c e n t , even when the bank v o l t a g e and downstream p r e s s u r e a r e c o n s t a n t . As a s i d e n o t e , we have o f t e n s p e c u l a t e d whether the c a p a c i t o r r i n g i n g e f f e c t s may s i g n i f y two d i f f e r e n t d r i v i n g mechanisms; e l e c t r o m a g n e t i c a c c e l e r a t i o n a t low downstream p r e s s u r e s and e l e c t r o t h e r m a l a c c e l e r a t i o n a t h i g h t e s t gas p r e s s u r e s . - 8 4 -ROGOWSKI COIL CAPACITOR CROWBAR BANK G A P DRIVER GAP (with backstrap) The crowbar and d r i v e r gap are t r i g g e r e d independently,with the crowbar gap t r i g g e r e d 11 microsec a f t e r the d r i v e r gap. ROGOWSKI COIL MEASUREMENTS WITHOUT CROWBAR WITH CROWBAR SMEAR PICTURE (wi t h crowbar) CLEARLY INDICATING THE CHANGE IN THE DRIVING MECHANISM F i g . D-8 Crowbar R e s u l t s CONTACT SURFACE VELOCITY - 8 6 -Appendix E OTHER INTERESTING ASPECTS Our experimental s i t u a t i o n i n i t s e l f i s very i n t e r e s t i n g . We have noticed several phenomena which suggest another experiment. One of our f i r s t observations was that under c e r t a i n conditions (-f>, >$T TORR, ^ ^ 1 5 ) , things would happen i n the shock heated region of the known shock when the l i g h t source was turned on. Figure E - l i s a smear picture which i l l u s t r a t e s t h i s phenomenon The luminous wedge s t a r t s when the l i g h t source i s turned on. The e f f e c t seen i n figur e E - l i s not reproducible i n the sense that i t i s seen at every shot but i t i s predictable i n the sense that whenever observed, i t behaves i n much the same way. There was always the same c o r r e l -a t i o n between the s t a r t of the l i g h t source and the s t a r t of the luminous wedge. Other s i m i l a r luminous phenomena were also observed, although with even less r e p r o d u c i b i l i t y . When the shock front separated from the r a d i a t i o n f r o n t , a luminous wedge would appear i n the known shock heated region ( r e c a l l that the colour of the r a d i a t i o n front changed when the shock separated from i t , f i g . 9 ) . Often times, instead of seeing a luminous wedge, a t h i n luminous l i n e would appear some distance behind the known shock f r o n t . The luminous l i n e appears very close to, and t r a v e l s s l i g h t l y slower than the contact surface. -87-- 8 8 -I t appears obv ious t h a t t hese phenomena have someth ing t o do w i t h the a b s o r p t i o n of r a d i a t i o n i n the shock hea ted g a s . As a f i r s t gues s , they c o u l d be r a d i a t i o n f r o n t s , o r e x c i t a t i o n f r o n t s , o r l u m l n o s c i t y caused by some dynamica l p e r t u r b a t i o n which i s a r e s u l t o f the abso rbed r a d i a t i o n . T h i s t ype of p rob lem has been t r e a t e d t h e o r e t i c a l l y by R. G. Rehm ( 1 9 6 8 ) and B. T . Chu ( 1 9 5 5 ) . Rehm f o l l o w e d the method o f s t r a i n e d c o o r d i n a t e s to c a l c u l a t e the f l o w f i e l d when l a s e r r a d i a t i o n o f s m a l l amp l i t ude i s abso rbed ove r a f i n i t e r e g i o n beh ind an e s t a b l i s h e d shock wave. He p r e d i c t s the f l o w f i e l d as drawn i n f i g u r e E - 2 . The main shock i s expec ted to speed up , and a second shock i s formed some t ime T a f t e r the l i g h t i s t u rned on and some d i s t a n c e b e h i n d the shock f r o n t . SECONDARY SHOCK L a s e r tu rned on a t T=0 LASER LIGHT F i g . E-2 Rad ia t i ve Energy A d d i t i o n a t a Shock F r o n t -89-Unfortunately, the accuracy of the v e l o c i t y measurements from the smear pictures i s not s u f f i c i e n t to measure the expected v e l o c i t y increase. The expected secondary shock wave can be seen only on very few photographs, and then only by an observer who has f a i t h i n the theory. But something i s happening and, to understand just what, could develop into an i n t e r e s t i n g experiment. -90-Appendix F THE EXPERIMENTAL SET-UP The Bogan l i g h t source consists of an arc c o n s t r i c t e d through a narrow channel i n a polyethylene rod. I t radiates as a black body with an e f f e c t i v e temperature of the order of 6o Nooo°tsfor a period of about ten microseconds (Zuzak, 1968). The measuring device i s a smear camera (J.P. Huni). The smear camera makes use of a r o t a t i n g parabolic mirror driven by a high speed e l e c t r i c motor. When the motor i s turned on, a pulse i s generated every revolution and when the time between these pulses becomes as small as a prescribed time i n t e r v a l (sweep speed s e t t i n g ) , a t r i g g e r i n g pulse i s sent out. The remaining t r i g g e r i n g apparatus can have a j i t t e r as high as tens of microseconds because of the large amount of time a v a i l -able on the f i l m (of the order of 300 microseconds). The t r i g g e r i n g pulse i s fed into a 162 Tektronix Waveform Generator which y i e l d s a negative sawtooth of several m i l l i -seconds duration. The negative sawtooth i s fed into two I63 Tektronix Pulse Generators, each y i e l d i n g pulses a f t e r v a r i a b l e and independent f r a c t i o n s of the sawtooth duration. These two pulse generators y i e l d pulses which are delayed with respect to each other. Each pulse i s fed into a separate thyratron and theophant-s doubling agent which i n i t i a t e s the breakdown i n the d r i v e r section of the shock tube and i n an external series spark gap f o r the l i g h t source. The apparatus i s i l l u s t r a t e d i n figures F - l and F-2. -91-PULSE OUT TL SMEAR CAMERA HI CAPACITOR BANK EkTER NAL LENS DRIVER SECTION WITH BACKSTRAP MYLAR DIAPHRAGM BOGEN LIC SHOCK TUBE ISOLATION TRANSFORMERS _ THEOPHANlT jr™L\ h DOUBLING AGENT Li 1 - 1 0 kv ii CAPACITOR BANK IP HT SOURCE EXTERNAL SPARK GAP WAVEFORM GENERATOR F i g . F-l Appara tus -92-F i g . F-2 Smear Camera - 9 3 -Appendix G 400 -3 0 0 -2 0 0 -1 0 0 -o EH « CO DETAILS OF PROBES width = 1 ris e time mm •• = 1 yfcsee reading time = 25yfcsec sensitive crystal = The pressure probe was calibrated by measur-ing the pressure jump across known shock waves (R. Ardila) SLOPE = 10 TORR " T 20 l30~ VOLTAGE (mV) 40 50 60 70 Fig. G Piezoelectric Probe Calibration 

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