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Investigation of the dynamics of radiation fronts Zuzak, William W. 1968

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•  INVESTIGATION OF THE DYNAMICS OF RADIATION FRONTS  -  by WILLIAM W, ZUZAK  ,E. (Eng.Sc.Phys.) University of Saskatchewan, M.Sc., University of Saskatchewan, 1965  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the department of PHYSICS V/e accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1968  In p r e s e n t i n g this thesis  in partial f u l f i l m e n t of the  requirements  f o r a n . a d v a n c e d d e g r e e at the U n i v e r s i t y of British C o l u m b i a , I a g r e e  that the Library shall m a k e it freely a v a i l a b l e for r e f e r e n c e and  Study.  I further agree that p e r m i s s i o n for e x t e n s i v e copying of this  thesis for s c h o l a r l y p u r p o s e s m a y be granted b y the Head of m y  Department or by h ils r e p r e s e n t a t i v e s .  It is u n d e r s t o o d that  copying  or p u b l i c a t i o n of this thesis for financial gain shal1 not b e a l l o w e d  without my written  D e p a r t m e n t of  permission.  J?//  5  The U n i v e r s j t y o f British V a n c o u v e r 8 , Canada Da te  Z-f,  Columbia  S&6&  ABSTRACT  A theoretical investiation of steady radiation fronts was carried out for the experimentally realistic situation in which ionizing or dissociating radiation passes through a transparent window into an absorbing gas.  It was shown that five different types of radia-  tion fronts may occur -depending on the ratio of photon flux to absorber density. It was possible to calculate the flow in each case provided the final temperature behind the radiation front was assumed. This final temperature may be calculated if the structure and all reactions within the radiation front are taken into accountc An analytic expression can be obtained if particle motion and recombination are neglected , and the radiation is assumed to be monochromatic. This ideal case corresponds closely to a weak R-type radiation front. A first order relativistic correction indicates that the width of the front decreases as the velocity of the front approaches the speed of light. In an associated experimentt radiation fronts in oxygen and iodine were produced by an intense light pulse from a constricted arc. The experiment in iodine demonstrated the beginning of the formation of a radiation  front during the 1 0 ^ sec light pulse. Radiation induced shock waves were observed in oxygen after the decay of the light pulse. These Mach 1.1 shocks were considered theoretically as unsteady one-dimensional flow and were treated by the method of characteristics, which was modified to include the energy input. The agreement between the theoretical and experimental results was satisfactory.  iv  TABLE  OF  CONTENTS page  Chapter 1 1.1 1.2 Chapter 2 2.1 2.2 2.3 2.4 2.5  Chapter 3 3.1  INTRODUCTION The problem An outline of the thesis  9  The possible types of flow Conservation equations for a discontinuity in one-dimensional flow Rarefaction waves in a one-dimensional flow The equations of state Estimation of the temperature behind the radiation front  9  PROPERTIES OF STEADY RADIATION FRONTS Idealized propagation of a radiation front  3.1.2  3.3 3.4 3.5  1 7  BASIC EQUATIONS AND ASSUMPTIONS  3.1.1  3.2  1  Case of one frequency and one absorption cross section, Case of black body radiation F ( P ) and continuous absorption cross section j, oc(  13 17 20 23 29  29 30 36  Relaxation of restrictions on particle motion and recombination  42  3.2.1 3.2.2  42 43  The coefficient, 5The energy input, W/ S, v7  Weak R-type front R-critical front Weak D-type front preceded by a shock wave  44 48  3.5.1 3.5„2  53 55  General relations Iterative procedure for calculations  52  mmmmcmiv  Table of Contents —Continued page 3.6 3.7 Chapter 4  4.1  4.2 4.3  D-critical front preceded by a shock M-critical front preceded by a shock THE STRUCTURE OF STEADY RADIATION FRONTS C o n s e r v a t i o n e q u a t i o n s of mass,  4.3.1 4.3.2 4c3c3 4.4 Chapter 5 5.1 5.2  THE BOGEN LIGHT SOURCE Description of light source Measurement of intensity  5.2.2 5.2.3  6.1 6.2 6.3  Conservation equations for absorbing particles The rate of energy input per unit volume, / ( x , t) q (x,t) Calculation of the front structure  Concluding remarks on Chapters 2, 3 and 4  5.2.1  Chapter 6  monentum,  and energy within the radiation front Reactions within a radiation front Special case of a dissociation front in oxygen  o Absolute intensity at 5000 A with discharge voltage at 3.0 kV Intensity as a function of wavelength at 3.0 kV Intensity as a function of discharge voltage .  EXPERIMENTS AND RESULTS Beginning of formation of dissociation front in iodine . Shock fronts in oxygen Attempts to measure ionization in the test chamber  59 64  69  70 71 74 75 78 80 82 85 86 88 89 92 92 94  95 101 107  vi  T a b l e of C o n t e n t s —  Concluded  page Chapter 7  7.1  UNSTEADY ONE-DIMENSIONAL FLOW WITH ENERGY INPUT .. Method of characteristics 7.1.1 7.1.2  7.2 7.3  Physical characteristics in Eulerian and Lagrangian coordinates State characteristics  Method of finite differences Application of the two methods to dissociate fronts in oxygen 7„3 „ 1 7.3.2  Chapter 8  Shock formation for time dependant radiation from Bogen source Structure of a steady dissociation front  SUMMARY AND CONCLUSIONS  108 109  111 112 114 116 117 119 120  Appendices A  NUMERICAL CALCULATION OF A STEADY RADIATION FRONT IN OXYGEN  124  B  SCALED DRAWINGS AND DATA FROM LITERATURE  125  C  EQUATIONS FOR SPECIAL REACTION SCHEME  127  D  METHOD OF CHARACTERISTICS AT FIXED TIME INTERVALS  132  E  METHOD OF FINITE DIFFERENCES  139  REFERENCES  144  vii  LIST  OF  FIGURES  Figure 1.1 1.2  page Classification of conditions encountered by radiation fronts Hypothetical experimental situation  2.1  Schematic representation of flow velocities for various values of F Q /N 2.2(a) Steady discontinuities in an R-critical front (b) Steady discontinuities in an M-critical front 2.3 Propagation of a rarefaction wave 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8  Radiation front travelling in + x direction with velocity vr Plot of the radiation equations for various values of F Q /CN Q Idealized radiation front in oxygen for black body radiation F( Weak R-type radiation front R-critical radiation front Weak D-type radiation front preceded by a shock D-critical radiation front preceded by a shock M-critical radiation front preceded by a shock N  4.1  Plot of velocities versus  5.1 5.2 5.3  Schematic representation of light source Light pulse from Bogen source Experimental setup for absolute intensity measurements Intensity of Bogen source as function of discharge voltage  5.4 6.1 6.2  0/FQ  Schematic representation of experiment with iodine Typical oscilloscope traces for measurements in iodine  4 5 11 14 14 19 31 35 40 47 51 58 63 68 83 87 89 90 93 96 100  viii  List of Figures —  Concluded page  Figure 6.3 6.4 6.5 6.6 6.7 7.1 7.2 7.3 7.4 B.l B.2  Increase in light intensity during time of light pulse Schematic of experiment in oxygen Oscilloscope traces of piezoelectric probe Shock strength as function of d at 400 Torr oxygen Velocity of shock at 400 Torr oxygen Mach lines and path lines of characteristic net Computer profiles, 1.0 atm Computer profiles, 0.1 atm Computer profiles, 0.01 atm (method of finite differences)  100 102 105 105 106 109 118 118 119  Scale drawing of Bogen light source Iodine absorption cross sections (Rabinowitch and Wood (1936)) Oxygen absorption cross sections (Metzger and Cook (1964))  126  D.1  Calculation of an ordinary point D  132  E.1  Lagrangian mesh for finite difference calculations  139  B.3  125 126  XX  LIST  OF  SYMBOLS  A list of the symbols which appear several times throughout the thesis is given below.  Symbols  used only in isolated instances and those appearing in the appendices are not listed.  a  velocity.of sound defined by a = (gp// ) 2  c  velocity of light  c„ fa  velocity of sound defined by• c„ o = ( •Pp/oV)co.  D  dissociation energy  E  ionization energy  £,  internal energy per gram  F  • - F(x s t) ~ / F ( ^ , x f t ) d ^  8  photon flux (eq'n (3.16))  FQ  - F(o,t)  g  ==. h/£ , the effective adiabatic exponent (eq'n (2.12))  G(t)  S F(x s t)/F'(x), the time dependance of the photon flux  h  enthalpy per gram\ Planck's constant  i  subscript index  j  subscript index  k  recombination coefficient*} Boltzmann1 s constant  k(j  collisional dissociation coefficient  mmmmcmix  List of S y m b o l s —  Continued.  k^  three body recombination coefficient  m  mass  M  mass (usually 0 2 molecule); third particle in 3 body recombination  N  ss N(x 51), particle density  Nq  particle density if no dissociation were present j = N(x,0), particle density at time zero  p  pressure  q  » q(x,t), rate of energy input per unit mass  Q,  artificial viscosity (see eq'n 7.12)  t  time  T u  temperature particle velocity in lab frame of reference  v  particle velocity the closest  in frame of reference  of  discontinuity  V  velocity of radiation front  W  energy  x  Eulerian spatial  y  degree of  (eq'ns  (3.2),  flux co-ordinate  dissociation  Lagrangian spatial co-ordinatej C (3o20)  (see eq'n  (3.25'))  xi  List of Symbols —  oC y'  Concluded.  photoabsorption coefficient isentropic exponent (see eq'n (2.7)) right flowing Mach line  fC  coefficient of thermal conductivity  2  wavelength coefficient of viscosity  f  frequency  $  left flowing Mach line J coefficient (see eq'n (1 0 1)  f  density  X.  ionization or dissociation energy  *  denotes a molecule in a vibrationa11y excited state  ACKNOWLEDGMENTS  The author wishes to acknowledge the stimulating supervision of Dr. B. Ahlborn for the past three years. Special thanks are due to Dr. J. H. Williamson for his assistance in preparing the thesis and especially for suggesting the calculation procedure used in section 3,1,2,  Finally, the author is indebted to  Ricardo Ardila who carried out most of the measurements in section 6.2. It has been a pleasure being associated with friendly and stimulating people comprising the Plasma Physics group at the University of British Columbia.  C H A P T E R  :  1.1  I  INTRODUCTION  The Problem In most plasmas produced in the laboratory,  radiation is considered an undesirable energy loss mechanism, of interest only to spectroscopists for analysis of the conditions within the plasmas.  However, absorp-  tion of radiation may be used to produce plasmas.  This  n  was first illustrated by Stromgren (1939) in his investigation of expanding H II regions in interstellar space. These H II regions a.re produced by a hot star emitting ionizing radiation into a rarified cloud of hydrogen atoms. Kahn (1954) and Axford (1961) have made extensive theoretical studies of the ra.dia.tion fronts which presumably occur at the edges of these H II regions. With the advent of the giant pulsed lasers, it has become possible to study radiation produced laser spark plasmas in the laboratory.  Following the early  work of Ramsden and Savic (1964) there has been a flood, of investigations of the breakdown mechanisms and dynamics of these laser sparks.  The absorption of the radiation  in this case is of a special nature and does not correspond  2  to the single photon absorption mechanism.  It is, there-  fore, perhaps, surprising that these laser sparks exhibit properties of detonations or Chapman-Jouguet waves which is a singular point on the manifold of radiation fronts which Kahn predicted to exist. Let us consider the single photon absorption mechanisms which occur at the edge of a radiation produced plasma. When ionizing or dissociating radiation is incident upon an absorbing gas, a radia/tion front tends to form and propagate into the gas such that ahead of the front the gas is in its original state while behind it the gas is ionized or dissociated (i.e. a plasma).  Behind the radiation front the gas  is a,t a considerably higher temperature and there are more particles per unit mass than ahead of the front.  The result-  ing pressure gradient across the radiation front may result in considerable motion of the plasma. Most of the theoretical work in the literature on radiation fronts deals with interstellar H II regions and consequently, the equations used are expressly adopted for conditions found in interstellar space.  One of these  equations which is used by many workers is a relation between the particle density No,  the- p h o t o n flux  the velocity of the radiation front,"V^ (e.g. (1961)).  Vf  '_ -  and  Goldsworthy  F  Fo/N  o  This relation assumes that each photon ionizes (or dissociates) exactly one particle and the speed of light (see section 3.1)«  F  C  , wherecis-  Since we.wish, to  3  consider recombination and collisional ionization, we introduce a coefficient,5, which is the average number of photons required to ionize one particle.  (We shall consider  this coefficient in more detail in section 3.2). We thus write *u<F ^  1  •  *1  The terminology in this thesis has been adopted from the definitive work of Kahn (1954):  Supersonic radia-  tion fronts which compress the gas weakly are called weak R-type fronts since they occur if the radiation front propagates into a Rarified gas. Subsonic radiation fronts which heat and expand the gas are called weak D-type fronts since they occur if the radiation front propagates into a Dense gas.  Radiation fronts across which the flow switches from  supersonic to subsonic are called strong R-type, whereas radiation fronts across which the flow switches from subsonic to supersonic are called strong D-type fronts.  These only  occur under very specialized conditions, and are not encountered for the conditions described in this thesis.  In  general, weak R-type radiation fronts occur when the ratio . of radiation flux to particle density,  is large com-  pared to the speed of sound behind the radiation front and weak D-type fronts occur when this ratio T?0/N0  is  small.  Conditions in the Middle between these two extremes where the ratio F 0 / N 0 is of the order of the speed of sound of the gas behind the radiation front are referred to as M-type. The singular point which separates the M-type and weak R-type conditions is called R-critical and the point which separates the M-type and weak D-type conditions is called  D-critical.  In both of these singular cases, the radiation  fronts propagate at exactly sonic speed with respect to the gas behind them.  This scheme is illustrated in Fig. 1.1  below. R-critical weaic R-type Pig. 1.1  D-critical weak D-type  M-type  Classification of conditions encountered by radiation fronts The reader may be familiar with a classification  of isolated discontinuities in the literature in which the relative velocity? v, is compared with the local speed of sounds a.  Using the subscripts 1 and 2 to refer to condi-  tions ahead of and behind the radiation front respectively, this classification may be conveniently tabulated as follow  D- type type  I weak j v, < O- ,  | j  iT, > a, ^g. >- & a  critical -zr, < ex, =  •Vk = fti  strong 1/; < a, ir^ >ftj •p~, > (X,  In this thesis, we wish to make further theoretical and experimental investigations of the development and propogation of radiation fronts and phenomena/associated with such fronts.  For this purpose, we consider an  experimentally realistic situation in which ionizing or dissociating radiation passes through a transparent window into a semi-infinite tube containing the absorbing gas. The boundary conditions for this situation permit unique solutions' to be obtained.  These experimental conditions  differ from laser spark experiments in two ways, First, Measured in the frame of reference of the closest discontinuity  the radiation front is considered in plane geometry. Secondly, the incident radiation may have any frequency distribution and is of long time duration.  Corresponding to this idealized experimental situation, let us consider a tube containing N 0 absorbers per unit volume with absorption cross section «<(v), which are dissociated (we use the term dissociation generally to include ionization) by photons in the frequency interval Vi to?/2 . At time iz = o, a steady parallel beam of F 0 photons/ cm2sec .in the interval ~V\ toV^ and with average energy  is directed into the absorbing gas, see Fig. 1.2.  -transparent window  Y  /  absorbing gas N 0 (cm )  /Ekaions) oicm sec }  Fig. 1.2  /- radiation front  Hypothetica,l experimental'situation.  A radiation front will form and propagate away from the window into the undisturbed gas. the velocity  According to eq'n (1.1)  of the front w i l l be proportional to the  ratio of the photon flux F 0 to the particle density N"0 of the absorbing gas.  It is one aim of this thesis to show  that the properties of the radiation front which develops F depend critically upon the magnitude of - ^ c o m p a r e d with the speed of sound behind the front.  We feel that with the completion of this thesis, we have achieved three major points.  First, it is now poss-  ible to predict the flow pattern for any value of F 0 /N 0 in this experimental situation and (by assuming a reasonable temperature behind the radiation front) to calculate the velocities and thermodynamic quantities associated with the steady radiation front.  This was not possible from the  existing literature, where due to the lack of definite boundary conditions only general statements about the possible fronts had been obtained.  Secondly, we have pointed  out that the final temperature behind steady radiation fronts can (at least, in principle} be obtained from a detailed analysis of the structure of and mechanisms occurring within the radiation front„  A knowledge of this  temperature makes unique solutions possible.  Thirdly, the  experiments performed here, indicate the existance of radiation fronts in agreement with our theoretical investigation For this, we have modified the theory of unsteady one-dimensional flow to include energy input.  The method of character  istics at fixed time intervals or the method of finite differences may now be applied to predict the flow for any developing or unsteady radiation front and for any time varying photon flux, F „  The main requirement for an experiment to observe radiation fronts in the laboratory is an extremely intense light source which radiates a large number of photons in the frequency interval in v/hich the test gas has a high photoionization or photodissociation cross section and which radiates for as long a period of time as possible. Our light source was an arc v/hich was forced to pass through a narrow channel in a polyethylene rod similar to that described by Bogen et al (1965).  This source radiated with  an effective black body temperature of the order of 105 for a period of 1 0 ^ s e c «  Iodine and oxygen which have large o  photodissociation cross sections in the region 5000 A and o  1420 A. respectively were used as the absorbing gases. 1  •2  An outline of the thesis. The thesis consists of two main sections?  a theor-  etical investigation of steady radiation fronts and an experimental part.  •  In Chapter 2, we list the various steady radiation fronts which we expect to occur and then we develop the equations necessary to describe the flow for each case. There is always one more unknown than equations. .Thus, in order to obtain unique solutions, it is either necessary to assume the final temperature behind the radiation front or to calculate the detailed structure of the front. In Chapter 3, we carry out the calculations for a simplified model.  Also, by assuming the final temperature,  we calculate the flow for each of the cases v/hich are expected to occur, in Chapter 4, we outline how to obtain the  detailed structure of a radiation front and the temperature behind it.  .  The radiation source is described in Chapter 5. •Experiments and results are discussed in Chapter 6. In order to understand details of the experimental results, in Chapter 7, we develop the theory of unsteady one-dimensional flow with energy input and apply it to the temporal development of the shock fronts observed experimentally. The main results of the thesis are summarized in Chapter 8.  C H A P T E R BASIC  EQUATIONS  AND  2 ASSUMPTIONS  Let us now consider in more detail^ the experimental situation illustrated in Pig 1.2 for various values of F 0 / N 0 . 2• 1  The possible types of flow We assume that after a certain length of time, a  radiation front forms and that the flow associated with it approaches a steady state.* We then may treat the radiation front as a discontinuity across which the standard conservation equations of mass and momentum may he applied.  The  energy equation must he .modified to include the radiantenergy absorbed within the front.  Thus, the problem may  be treated as steady one-dimensional flow with energy input For values of F 0 / N 0 either large or small compared with the speed of sound behind the radiation front, a^ (we always use the subscript 4 to refer to quantities behind the radiation front) there is no ting the type of flow which will occur.  difficulty in predic For F 0 / N 0 > ^ a 4 ,  the radiation front propagates so rapidly that the particles do not have an opportunity to react to the pressure  * This assumption is never strictly true; its validity will be discussed in section 2.5 and the following two chapter  10 gradient across the front and consquently, there is only weak compression and little particle motion behind the front. At the other extreme, F 0 / H 0 « £4, the particles can, and do, react to the pressure gradient.  The compression wave over-  takes the radiation front and becomes a shock which propagates ahead of the radiation front.  The radiation front  in this case is an expansion wave since the gas entering it is in a compressed state and is heated and expanded as it passes through the front. However, conceptual difficulties arise in the transition region where Po/ffo85* a 4 •  I»et us envisage what  occurs as we decrease the radiation flux F 0 from a value at which  v  r = ^0/5  in Fig. 2.1.  a  4* -?he various cases are illustrated  Initially, the radiation front will propagate  supersonically with weak compression and little particle motion, as discussed above.  Since there is a small driftV  velocity v p imparted to the particles passing through the front a rarefaction wave will be set up (see Fig. 2.1) which eventually brings the particles to rest.  The head  of the rarefaction v/ave travels at the speed of sound relative to the particles entering (v H = V p + a ^ ) w h i l e  the  tail travels at the speed of sound of the stationary particles behind it ( v f = a^, where the subscript 5 refers to quantities behind the rarefaction v/ave).  Following Kahn  (1954) we describe such a front as weak R-type (weak compression wave) followed by a, rarefaction v/ave.  velocity Fig. 2.1  Schematic representation of flow velocities for various values of P 0 /N 0  As the incident radiation flux is decreased, we eventually reach a point where the front travels at the speed of sound relative to the gas behind it (Vp = where v p i s the particle drift velocity).  v  p +a 4  The head of the  rarefaction wave travels with the same speed and is in conjunction with the radiation front.  This is called an  R-critical front. If the intensity of the incident radiation is decreased still further, a shock wave travels ahead of the front causing the gas between the shock and radiation front to be compressed and heated.  The slower moving radiation  front now enters a gas of higher density; the gas passing through the front is heated and expanded.  We expect the rare  faction wave to follow the radiation front in the same manner as in the R-critical case.  In this case, we adopt  the terminology "M-critical front preceded by a shock wave". As we further decrease the radiation intensity, the velocity of the radiation front decreases until it equals the velocity of the tail of the rarefaction wave.  The rare-  faction wave is thus merged with the radiation front.  The  front travels with the speed of sound, a^., relative to the gas behind it which itself is stationary in the lab frame of reference (vp= a/f).  This case is called a D-critical  front preceded by a shock wave. Finally, for still lower values of F 0 , w,e have the low velocity extreme Fo/3%<<a4.»  which the radiation front  travels at subsonic speed relative to the stationary particles behind it (Vp< a ^ and a shock front propagates ahead of it.  The discontinuities appear in the same order  as for the D-critical case.  We call this a weak D-type  front preceded by a shock wave. We note that in the above scheme, there are three regions of solution (weak R-type, M-critical, weak D-type) separated by two point solutions (R-critical and D-critical). Two other types of fronts, the strong R-type and strong D-type, which we mentioned previously and which are mentioned in the literature (liahn (1954), Axford (1961)) do not occur in our case. 2.2.  Conservation equations for, a discontinuity in one dimensional flow We have assumed that the flow associated with the  radiation front reaches a steady state such that the radiation front may be considered as a discontinuity across which the conservation equations of mass and momentum.and the modified energy equation are valid.  We label all  quantities immediately behind the radiation front with the subscript 4, see Pig. 2.2, and the initial undisturbed quantities carry the subscript 1.  Similarily, the sub-  script 2refers to quantities behind the shock front and subscript 5 refers to quantities entering an M-critical, D-critical or weak D-type front.  The thermodynamic  14  4  f,  K  K h, If,  (a)  iTr^CLs  % &  fA  «  pA  F  .  ^  1 1  r* = =  f, p, • Pzh, h3 <  7  (b)  Pig. 2.2  (a) Steady discontinuities in an R-critical front (b) Steady discontinuities in an M-critical . front-  quantities with the subscripts 2 a n d 3  are  assumed to be  identical, the velocities v 2 a nd v 3 are, of course, different in their respective frames of reference.* If a rarefaction wave exists, the quantities behind it are labelled with the subscript 5. In the frame of reference of the discontinuity, the conservation equations of mass, momentum and energy may be 'written as (e.g. for an R-critical or weak R»type front)  2.1 2.2 jC  Compare footnote on page 4  _  L  JL tr  Z.  z  2.3  v (em/sec) is the velocity of the ga,s particles relative to the discontinuity,y7 (gm/cm^) i s the mass density, p (dynes/cm 2) is the pressure, h (ergs/gm) is the enthalpy and W (ergs/cm 2sec) is the energy flux which is absorbed by the gas, W  —  W is defined by < hr>  =  </r  2  *4  We have neglected the radiation pressure in eq'n (2.2) since it is negligible for any cases that we consider. With proper choice of the indices all discontinuities associated with the radiation front can be described by these conservation equations.  For an M-critical, D-  critical or weak D-type front, the indices on the right hand side of eq'ns (2.1) to (2/3) have to be changed to 3f (compare Figs 2.2(a) a.nd 2.2(b)).  For shocks which precede  the radiation front, the quantities on the left hand side of these equations are labelled with the subscript 2 and the energy flux W Is zero.. These conservation equations, however, cannot be applied to rarefaction waves which are treated in the next section. From these conservation equations and the equation of state, see section 2.4, it is possible to express the compression ratio in terms of an effective adiabatic exponent, g (Lun'kin(1959)).  For an ideal gas g is analo-  gous to the ratio of specific heats £" (see Zel'dovich and Raizer (1966), p 207). and 1.7  The value of g varies between 1.06  and often ma»y be estimated quite accurately a priori.  16  Ahlborn and Salvat (1967) show the compression ratio (for an R-critical or weak R-type front) is ^ ML= * - -  /  p, JLJLJEI  !,  —| ji'I/I-(-&!•)  ^p1'^  2.5 where we have used the equation of state in the form h = (g/g-1) (P/VO to eliminate the enthalpy from the energy equation (Lun'kin (1959)). Note that aq'n (2.5) has two roots signifying the mathematical possibility of two different compression ratios.  The negative root corresponds  to the weak R-type solution; the positive root corresponds to the strong R-type solution .which does not occur in.our case.  •• We have pointed, out previously that for the crit-  ical cases (R, M and D-critical) the radiation front travels with the speed of sound, a^, relative to the gas behind it. We refer to the quantity a4 as a thermodynamic speed of sound since it is defined as .i  a  4  ~  (S/^/fA)2  2  <b  Usually, the speed of sound, cs„ is defined by a differential along an isentrope cs ~ Vr' 's 3?or a polytropic  gas (p  where  this equation  17  reduces to e^'n (2.6).  In a plasma (e.g. behind a rad-  iation front) the isentropic exponentr" is not, in general fi  equivalent to the effective adiabatic exponent g (see Zel'dovich and Raizer (1966), p 207); however, it has been shown by Ahlborn (1966) that the approximate speed of sound given by eg,'n (2.6) differs by less than 10$ from more accurate calculations based on eq'n (2.7). In this thesis we will use the speed of sound as defined by eq'n (2,6). With this definition of the speed of sound, the term inside the square root sign of eg'n (2.6) becomes identically zero for the three critical cases.  This yields  an extra relation and simplifies the solutions considerably. 2.3  Rarefaction, wave s in a one-dimensional flow' For the boundary conditions considered in this thesis  fast rad.ia.tion fronts are always followed by rarefaction waves.  Though the properties of such waves are well known  from the literature, for the convenience of the reader, we summarize the important facts in this section. Consider a semi-infinite tube of gas closed at the left end, travelling in the +X direction (i.e. to the right) with the speed Vp^. complete stop.  At time t = o, the tube comes to a  What is the motion of the gas at the left  end? This is the age old problem of Riemann and the  18  solution is well kn-own (see von Mises (1958) or C our ant and Friedrichs (1948)).  A rarefaction wave is formed which  causes the particles to decelerate through an expansion fan as illustrated in Pig. 2.3.  The head of the rarefaction  wave travels at the speed, v^ = vp/f + a^ (where  is the  speed of sound in the gas in region 4); the tail of the wave travels at the speed, v-j- =  The expansion through  the rarefaction fan occurs isentropically and it can be shown that in an isentropic expansion, the quantity 2a/(g~1)is conserved at every point along the expansion (we use the effective adiabatic exponent g in place of the isentropic exponent g~").  Thus we may write _ ^  =  -  2.8  For a polytropic gas the quantity p f ^xs conserved; hence P5/5  P4 A  •  2.9  We now assume g^ = g^ = g (an approximation which is not  ,  generally true for a plasma) and combine eq'ns (2.8) and (2.9) to obtain ' V  ^  r  z  -  /  fJL •  2.10  To obtain the temperature T5 we use the relation 7V _ ^  -T% -  Jl 4  AS*  '  9  n  where M4 and M5 are the initial and final molecular weights of the gas in regions 4 and 5.  rarefaction  rarefaction  4  £  a n  Pig. 2.3  I * I P+ j* •  s~  | •—:—s- _  Propagation of a rarefaction wave We have tailored our treatment above to apply-  directly to the rarefaction fans which occur behind M-critical, R-critical and weak R-type radiation fronts One may expect problems to arise when the assumption g  = g^ is not valid and when g differs markedly from /  If this occursf  one must use more accurate values and  iterative techniques.  20  2.4  The ..'equations-.;of state An equation of state relates the thermodynamic  quantities such as the pressure, density, enthalpy, temperature, I, and internal energy,£ . Throughout this thesis, we find it convenient to use several different forms of the equation of state.  We have already had occasion to use the  equation which relates the enthalpy to the pressure and density "by means of the effective adiabatic exponent, g, J -/  /j —  .  2.12  Similarily, for the internal energy,£ , we may write LI  k. = since  -f  y  2.13  . We note that the temperature does not appear  explicitly in these equations, however, the adiabatic exponent g is a weak function of temperature and pressure (or density).. For a monatomic gas such as argon g varies from 1.67 at 300 °K to 1.13 at 2Q000 °K; while for a diatomic gas such as oxygen it varies from 1.4 at 300 °K to 1.06 at 10,000  Curves of g (p.(T)plotted versus T may be found  for various gases in Ahlborn and Salvat (1966) and Kuthe and Neumann (1964).  (See Zel'dovich and Raizer (1966),  p 207 for a different approach.) The pressure in a multicomponent plasma may be written as the sum of the partial pressures of the individual  components. =  2.14  where for a diatomic gas the index j = m, a, e, i = 1,2,3?.. refers to molecules, atoms, electrons and degree of ionization respectively, n- is the particle density of the d  ( component, T. the tranj is the temperature associated with  slational degrees of freedom of the j1* component and k ji c  (=1.38 X 10"  ergs/°K molecule) is Boltzmann's constant.  She internal energy, £ , is defined to be the sum of the energies in the various degrees of freedom of all the components of the gas.  For a diatomic gas £(ergs/gm)  may be written as 'yk— Z- J J ' ,/jfg  2.15 ^ tti  where the quantities k n - j and T j are defined as in eq'n (2.14).  m-i (gm) is the mass of the  component particle,  D (ergs) is the dissociation energy- of the diatomic molecule (ergs) is the ionization potential of the 1th stage of ionization and ^  is the partition function of the f a com-  ponent excluding the electrons.  (The terra containing -j  gives the energy in the vibrational and rotational states of the molecule and the excited electronic states of the atoms and ions.).  The denominator of eq'n (2.15) is the  0  mass density,7-= S tim^.tJ By combining eq/ns (2.14) and '(2.15 ) it is possible to obtain an equation of the form of eq'n (2.13) and thus  obtain an explicit expression for the effective adiabatic exponent, g.  To illustrate, let us consider the gas in  a dissociation front in oxygen initially at room temperature.  We assume that there is no ionization and we neglect  the energy in the excited electronic states of the atoms and molecules.  At room temperature, the rotational degrees  of freedom of the molecules are in full excitation and contribute nfflkT to the internal energy of the gas.  The  excitation of the vibrational degrees of freedom becomes appreciable at temperature larger than 1000  Assuming  . the molecule vibrates as a harmonic oscillator the contribution to the internal energy is given by (see Zel 1 dovich and Raizer (1966),"p 181) - V — J6 M ^ L^ h-vy^ £/ -tr-Lli where for oxygen hY'/k = 2228  J •  )  2 .16  This expression assumes  that the vibrational degrees of freedom are in complete equilibrium with the translational degrees of freedom. We may thus write eq'n (2.15) as  M where M is the mass of the oxygen molecule (M=nim=2ma) and' where the degree of dissociation y is defined as  y^-  '  2.17  Similarily, eq'n (2.14) may be written as r =  f  4f  ;  •  2.14'  .;  23  where /=  (n m + &n a > M. Combining eq'ns (2.14') and (2.15') we obtain an  expression for £, in the form  •L~-  J  2.131  where the term in the square brackets corresponds to (g-1) of eq'n (2.13). We note that in our simplified example  ^  is a function of temperature a,nd the degree of dissociation y only.  In general, one should also include the degree  of ionization and invoke equilibrium relations or rate equations to relate the various particle concentrations.  2.5  Estimation of the temperature behind the.radiation front  -  Let us consider a non-relativistic weak R-type •front across which the conservation equations (2.1) to (2.3) can be applied.  If we assume that the rate of energy in-  put W is known, then these three equations plus an equation of state and an equilibrium relation*g.i.ve us five equa.tions with six unknowns v^, v/f ~f ^,  T^  (one may  argue that v^ = v F is known from eq'n (1 .1); however, this is no help since then the coefficient j is unknown.) Similarily, if we consider any other type of front we always have one more unknown than equations.  We must, there-  fore, obtain a further specifying equation or fix a parameter  * for the particle concentrationsy  to solve  the In  must  problem.  order  be a b l e  ticles  as  to  calculate  collision and bilities,  transfer  forth.  In a d d i t i o n  further  problem rates,  the  the  structure  in that  T/m  m a y be  of t h e  If we  consider  g a s at  room  or g i v e  the  by  considering  a dissociation  t e m p e r a t u r e , the  behind  the  where  D is t h e  plete  dissociation  will  radiation  show that  J H = 7? if,  values  the  front  but no  —  ?  since front,  proba-  and  or  so  for  the  to  of  instead, temper-  order  the r a d i a t i o n equation  entering  a  diatomic  (5kT4 +  energy and where we  m a y be  front  (2.3).  by  of the m o l e c u l e .  ionization.  the  obtain  final  A first  energy  a  unknown.  e n t h a l p y h-j is g i v e n  term ¥ / / 1 v 1  5<^> "  task  i n the  but w i l l ,  is g i v e n b y h 4 =  dissociation  the  of m a n y  attempt  behind  the  is  complexity, we have  front.  M is t h e m a s s  front  It  of  of r a d i a t i o n  front  limits  par-  transition  inaccurate  temperature  h-| = 3 . 5 kT-j/M, w h e r e enthalpy  obvious  radiation  to the  obtained  sections,  either  front.  occurring  present, we w i l l not  behind  approximation  rates  one  of the  structure  quantitative  etc. are  state  is a f o r m i d a b l e  by means  to the  s h o w h o w to e s t i m a t e ature,  this  cross  equation,  radiation  detailed  the r e a c t i o n  excitation  energy  Por  the  the  In general,  consider  specifying  in d e t a i l the  through  to k n o w  discontinuity.  reaction  obtain the  they pass  also necessary  one m u s t  to  The  D)/M  assume  In s e c t i o n  com-  3.2 we  written  3 '29'  T h u s e q ' n '(2.3)  becomes  3 . 5 kl-i/M + h ^ w h i c h m a y he  solved  dissociation  front  * The  "  the  atoms.  evident  the f i n a l  minimum  assume  zation front discontinuity are still  a  2.18  ionization front  e n e r g y a n d m„  ci  z  is  )  i  2.18-  is the m a s s  ( 2 . 1 8 ) or  of f a n d v^ a r e  the  limits.  of  ( 2 . 1 9 ' ) it  required to  temperature  is  calculate  behind  are  the  equilibrium  must  ionized  be at after  all  , at w h i c h  passing  the  must  " is 9 9 . 8 $ ) .  such  that  through  relaequi-  be  larger  virtually  (for e x a m p l e , w e  temperature*  least  front  discontinuity  dissociated  that"virtually the  t o be b e t w e e n  For a dissociation  temperature.,' T m i  all the p a r t i c l e s rarily  behind  for an  £v42  + D)/M +  end temperature  (1.1) and  can expect  temperature  some  (5k T 4  -•£• + -j -m*. : :  5'<"  ionization  values  tively narrow  that  the  =  temperature.  One  librium  5<h^/M  5  From-eq'ns  that  +  temperature  2-s^A >7 • ~ ~ is the  for  ~  corresponding  where E  2  arbit-  F o r an  behind a. 1 1 the  ioni-  the particles  the r a r e f a c t i o n  wave.  In f a c t , e v e n i f r a d i a t i o n l o s s e s f r o m the p l a s m a b e h i n d the d i s c o n t i n u i t y a r e i g n o r e d , a n e q u i l i b r i u m t e m p e r a t u r e c a n n o t be r e a c h e d s i n c e t h e r e is a f i n i t e p r o b a b i l i t y o f p a r t i c l e s r e c o m b l n i n g e v e n at e x t r e m e l y h i g h t e m p e r a tures. On t h e o t h e r h a n d , i f r a d i a t i o n l o s s e s a r e c o n s i d e r e d , t h e n it is n o t p o s s i b l e f o r the r a d i a t i o n f r o n t a n d t h e f l o w a s s o c i a t e d w i t h it to r e a c h a s t e a d y s t a t e u n l e s s ( p e r h a p s ) t h e r a d i a t i o n f l u x i n c r e a s e s in t i m e in some s p e c i a l w a y . N e v e r t h e l e s s , the c o n c e p t of an' e q u i l i b r i u m t e m p e r a t u r e is n e c e s s a r y in the " s t e a d y s t a t e " a p p r o x i m a t i o n of r a d i a t i o n f r o n t s .  For a dissociation we  choose* T m i n  the  gas  front  = 6000  in  o x y g e n at a t m o s p h e r i c  °K at w h i c h  is i n m o l e c u l a r  form and  temperature  pressure,  only 0 . 2 #  0 . 0 0 5 ^ is i o n i z e d  of  (Landolt-  BOrnstein II.4, p 717). On the ture, one v1  other hand, we  This  obtains = v4  from  a n d f. =  sociation  energy  If  that  eV.  there  In t h i s  is l i t t l e  time  for  the  °K,  photon  one  dissociates  case, we  front  to be  T  =  N max  If the tively  low  D-type  front  there (v3  o  Also,  case,  f l u x JP  - v^  expect only  ( such  consider  front  a  enter-  °K.  (v^ = v ^ ) a n d  Also,  dis-  dissociation  ( such  travels  atoms  with  to  so  rapidly  there  is  dissociate  at a t e m p e r a t u r e  recombination.  Thus  of one  one molecule,, s u c h t h a t ^ = 1 . In  the  temperature  behind  the  radiation  8900  that  preceded  of the  is p l e n t y  8 . 8 eV" r a d i a t i o n  3 ? 0 / N 0 « a^) we by a s h o c k .  expansion  ) is s m a l l  there  front  is e x t r e m e l y h i g h  little  v/hich  (8.8 e V ) photons  dissociated  intensity  is a l a r g e  p  hot  tempera-  of T 4  T _ ^8900 max  compression  expect  let us  1420 A  In. c o l l i s i o n s .  could  value  molecules which have a  the p h o t o n  other molecules  this  by  the w e a k R ~ t y p e  little  8900  As an example,  produced  that  is t h a t  another  (2.18) for a w e a k R-type  oxygen  of  of 5.1  eq'n 1.  front  ing a cloud  temperature  can define  shall  In t h i s  (such that  is  obtain case,  for  v ^ > > v ^ ) the  the h o t  a  weak  although  c o m p a r e d v/ith ? < h ' Z / / - D i n e q ' n of t i m e  rela-  term (2.18),  dissociated  * For convenience of calculation v/e assume the temperature behind the radiation front, T 4 , rather than the temperature behind the rarefaction wave, Tg, This assumption is of little consequence since the temperature drop across the rarefaction wave is small,,  27 particles to cool by colliding with and dissociating other molecules.  Thus, one photon will dissociate more than  one molecule such that 5< 1 .  In this case we expect T 4 __sl m i n  = 6000 For intermediate values of  (such that P o /.N o ^-a 4 )  for which we obtain D, M or R-critical fronts, the situation is more uncertain. 2  In this case, the term (v.,2- v , 2 ) or  2  (v1 -v^ ) is appreciable.  Collisions In the radiation  front result in both dissociation and recombination.  If  three body recombination is predominant over collisional dissociation, t h e n 1 > 1 and we expect T. 4  „ . max  If collis-  ional dissociation is predominant over recombination, then j<1 '  and we expect T . < 1 A • T , . uiJ-Xi _ [flci.A  Unfortunately, there  seems to be no criteria by which one could predict the relative importance of collisional dissociation and three body recombination within the front. another a idealized example, consider radiation at 912 AoAs entering gas of hydrogen atoms with an ionization potential of 13.6 eY.  There is no excess energy of the  ionization <fhf V - Es=o such that T = •§• T. « T . and, \ , • max i idj.D therefore, on the average, one would require substantially more than one photon (i.e. 5- > 1) to ionize one atom and heat the gas to a. minimum temperature of the order of 10,000 The results of this section may be summarized as follows s T. = T  4  T  4  =  T  max  min  if if  or  ^JiT^k  , f=/  ,• (ii)- <(h^>~~?f > f o / N < , « ° L < t  S  ^  T%<i  T T  4 < \>ax ' i f  4 > T max  if  <*>*>  I ,  - ^ , S < , f'/rt*  7  5-  where"X-is the dissociation energy for a dissociation front and the ionization energy for an ionization front.  C H A P T E R  ?  PROPERTIES OP STEADY RADIATION FRONTS  In the discussion of radiation fronts in a gas filled tube closed on one side by a transparent window, we postulated the existance of five different types of radiation fronts assuming that the flow in each case would reach a steady state.  We also developed the equations  which will enable us to calculate the thermodynamic properties and the flow velocities of the gas for each of the five types o ptions o  Por this calculation,, we must make two assum-  First, we assumed that the flow is steady in  every case (we consider a rarefaction wave with its head and tail both travelling at constant but different speeds as being a steady state situation). the temperature»  Secondly, we assume  is -known (either by assumption or  by a calculation of the detailed structure of the front).  3.1  Idealized propagation of a radiation front In our hypothetical situation, the dissociating  radiation of F Q photons/cm2sec passes through a transparent window into a semi-infinite tube containing N Q absorbing particles per unit volume,  let us now make the further  assumptions that all particles are stationary and that  there is no recombination of the dissociated particles9 such that there is a 1:1 correspondence between photons absorbed and. absorbers depleted.  (We assume that the dis-  sociated particles are transparent to the incoming radiation.) It turns out that this situation is closely approximated by a supersonic (weak R-type) radiation front. After a sufficient length of time , we expect the radiation front which forms and propagates down the tube with velocity v f to approach a steady state.  Let us con-  sider such a steady front for the case where the radiation F q consists of photons of one frequency and also for the case of black body radiation F( ^ ) with absorption cross sections o<r( y ). 3.1.1 cross section,,  Casej3i one fr e que nc .—  an d one absorpt ion  The diagrams in Fig. 3.1 illustrate  the radiation front as a discontinuity on one side of which there are only absorbers and no photons and on the other side of which there are photons but no absorbers. 3 lab frame of reference F c /c ™ c/V0 photons/cm  In the  travelling  with the speed of light, c, enter a stationary gas of N Q absorbers/cm3.  The velocity ox the front is v,, „  Vle may  make a Lorentz transformation into the frame of reference in v/hich the front is stationary.  Following Schwarz (1964)t  p 392,and considering the photons as a flux of relativistic particles, we find the flux of atoms entering from the right is - tfvF N 0 (absorbers/cm2sec) , while the flux of photons  31  MB  SYSTEM Ir C  o  0  6  w  o cm^sec  /absorber^ ~x  V2=c  v^O  FRONT SYSTEM —to—  0 (  /  —  F  <2/0  absorbers oVcm^sec  —Jso—  V-C  Fig, 3<>1  -X'  vj=0  Radiation front travelling in tx direction with velocity v F . \  entering the front from the left is F^ (photons/cm 2 sec ) , where  y(l - v^/c)W Q  X s (X » v ^ 2 / c 2 ) * .  Equating the photon flux to the particle flux, we obtain  ?  which may be solved, for v F to give  We note that for very high intensities such that  I^/N^c,  the f r o n t velocity approaches the speed of light as we expect ( v p ~ > c ) .  For low intensities such that F 0 /N q  we obtain the expected non relativistic relation, vF = ^  c,  Since the photoabsorption cross section  is not  infinite, the radiation front will have a finite thickness. The intensity F(x) of the radiation at any point within the front will vary from 'F at the extreme right. X ) varies from N extreme left.  at the extreme•left to zero  Similarily, the absorber density at the extreme right to zero at the  We define the position of the front to be  the point where  = £ = U(x)/N .  By" equating the.  number of absorbers depleted in an interval of length A ^ to the number of photons absorbed, in the interval, we obtain*  Substituting for  from eq'n (3.2), eq'n (3.3) becomes —-TJ- =  -n 51 a. -n't /  o -4  .  The usual exponential decay equation in the lab frame of reference, F(x ; t) =  exp (~of|jx) , should be  generalised in our case to be PCX,*). = 'Ac exF  ,  I  3.5  /-irp/<r  since the number of absorbers I is a function of position, X, and time, t.  Also, the source is moving away from the  front with velocity the first order  so, in general, we must include  correctionf F' - (1 - w / c )F.  Thus instead  of the usual differential equation for exponential decay * Since we are not particularly interested in the relativistic regime v/e now and in the following equations assume r= i.  £F(X)/£X«= -C<N F(7f), we must write 3 /Yx,t£) 3>T- =  At(y>£) Ffx,*)  ,  3.6  which by virtue of eq'ns (3.2) and (3.4) becomes Sf - A -  - - ^  .  3.7  This differential equation is readily solved by separation of variables using the transformation u = (F/F 0 ) y = u - !•  f ^  = ^M,  {/-f-  i-?/-?*) -  and then  fc/K />/«£-//„ ) x4  ( /  or, retransforming /  3.8 where the proportionality constant y Q (t) can still be a function of time. So far, we have ignored the time variable.  Since  this steady wave must necessarily satisfy the wave equation Q x' the function y Q (t) may be evaluated. A solution which satisfies the wave equation and the boundary conditions  d FK.-t) F0  /  L  t  for  °  >  for-  '  is f(Xj-t)  fof X •  K  -  ,  = o  >  zfr ~ >  X—^ /  _  3.10  From eq'ns (3.4) and (3.10) we obtain the particle density / —  These two equations will hereafter be referred to as. the " radiation front equations!'  Eq'ns (3.10) and(3.11) are  derived for an ideal case but they correspond closely to a weak R-type front which has little particle motion and no recombination. (i)  Thickness of an ideal radiation front. —  We  define the thickness of radiation front, <Tx as the distance between the points where F/F 0 is O.S and 0.1.  Using these  values in eq'n (3.10). we obtain  We note that for high intensities and low particle densities, such that F 0 / N 0 » c , the width of the front is inversely proportional to the intensity F Q .  In this case, the front  travels with the speed of light.  Conversely for low inten-  sities and high particle densities such that F 0 / N 0 « c , the width of the front is inversely proportional to the particle density N 0 .  The front speed is given by the ratio F 0 /N 0 .  These dependencies are illustrated in Fig. 3.2 in whibh F/F 0 and N/No of eq'ns (3.10) and (3.11) are plotted versus «<N0* for F 0 /cN 0 =o, 1 and 10, and for t = o. (ii)  Typical values. —  Radiation fronts occur  under extremely varied conditions.  The high  intensity,  low density extreme is illustrated by ionization  fronts  associated with H II regions in interstellar space.  In this  O  case, a star radiates photons at wavelengths below 912 A into a cloud of hydrogen atoms.  Typical values are  35  , F q = 10 20 cm  N q = 10 cm  2  sec"^ and photoionization cross  section for hydrogen at 912 A, <*„=. 6.3 X 10~ 1 8 cm 2 .  This  ionization front, if it satisfies the assumptions made in this section, would travel with the speed of light and have a width of 2 X 10 8 cm. Our own experiment described later in this thesis, in whach photons in the 1400 A wavelength region dissociate oxygen molecules is an example of the low intensity, high density extreme. F 0 = 10 22  cm  2  Here, typical values are N  = 10  1q  cm  ,  sec~^ and the photodissociation cross sec—1 °  tion o(Q2 - 15 X 10  2  cm .  Thus (neglecting recombination,  etc.} the dissociation front should travel at 10^ m/sec and have a width of 0.03 cm.  3.1.2  Case of black body radiation F(^) and con-  tinuous absorj>tion_ cross section <*(?'), —  Most radiation  sources which are available in the laboratory or which occur in nature have a continuous spectrum over a wide range of frequencies.  Furthermore, the absorption cross section cxr(^)  of the absorbing gas varies widely over the frequency spectrum.  Eq'n (3.12) indicates that the width of the radiation  front is inversely proportional to the absorption cross section.  For example, if at the maximum absorption cross section  o ^ at some frequency ^  the the thickness of the radiation  front is 1 cm, then at some frequency where thickness will be 100 cm.  is 0.01 o ^ tho  Thus, it is often necessary to  ignore the absorption cross section outside a chosen frequency region.  A good rule of thumb is to consider only photons  •with absorption cross sections in the range  ^ «:(-//)<o. 2><ro.  The differential equations may be set up in a manner similar to the preceding subsection.  We pick two frequencies  and /*2 between which the absorption cross section is finite and outside of which it is negligible.  (We assume  the gas is transparent to photons with frequencies outside this region.  In general, this is not the case and it is  necessary to consider several such frequency intervals). The total number of photons contributing to the front is K pr^) J f 3.13 The velocity of the front is again given by eq'n (3.2). The equation corresponding to eq'n (3.3) is  [> -  ^ (C^  ^. 14  x ) J r  such that again  — —  ^  ;  3.15  where v;e emphasize that this equation is valid across a steady radiation front propagating at a velocity vF, Corresponding to eq'n (3.5) and (3.6) we may write in the frame of reference of a semi-infinite tube with a transparent window at x = o, r  and  Integrating over the frequency and. using the identity  .16  38 (1- vp/c)  1  = (l+F 0 /cN 0 ) from eq'n (3.2) v/e obtain  -Vl  [h fW*,*) jr) = -{<+ tH/) N  M  .  3.18  The following procedure proves convenient in obtaining a solution of these equations.  The procedure consists of  transforming the x co-ordinate in which N(x) varies to a  ^  co-ordinate in which N (z) is a constant (we set N(») <= N 0 ) . The problem then corresponds to black body radiation into a non-depleting cloud of absorbers in which the intensity at each frequency^decays exponentially with a decay length which depends on In this terminology eq'n (3.16) may be rewritten as  where ZfX,*) •== and where range ^  No  7£ f ^/rrJ*]  +  3.20  is the maximum absorption cross section in the to^-  Also, we have written  o, t)  —  since the incident radiation is constant in time.  o  )  We now  integrate eq'n (3.20) over the frequency range and normalize to obtain  ^  ^  ^  **  3.21  This equation must be solved (numerically, if necessary)to determine F(Z') as a function of ^ for any functions F(/%o) an We now invert eq'n (3.20) to solve for  X  39 •Now from eq'n (3.15) we may write Hil)  _  ._  _  ft*)  " t i *  o o. zo  £  where we have used the fact that the number of photons passing a point in the x co-ordinate system is the same as that passing the same point in the ^co-ordinate system.  Eq'n (3.22)  has been derived in a frame of reference where 2 = o when x = o.  For numercial solutions, it is more convenient to  shift to the frame of reference of the radiation front with boundary condition x = o when x = -<~and x = o where F(55-)/F0 = o. 5. Substituting eq'n (3.23) in eq'n (3.22) we obtain for positive x Xf- -  (I + ib<„  A  / -  *  /j  )  3.24  and fox' negative x d z' 1  J  - 3.24  where we have broken the integration up into two parts for' convenience in carrying out numerical caculations. ^0.5 indicates the value of  where F(2-)/F  = 0.5.  Care must be  taken that the limit of integration.in eq'n (3.24') approaches but never reaches £—>0. We now have F(2-)/FQ as a function of (3.21) and x as a function of  from eq'n  from eq'ns (3.24).  We may  thus plot F(x)/F 0 as a function of x to obtain the overall structure of the radiation front.  The individual frequencies  F C ^ x ) / F 0 may also be plotted since we know that they decay exponentially in the ^-co-ordinate system.  Fig.3.3 Idealized radiation front in oxygen for black body radiation.F(^)  An example is plotted in Fig. 3.3 to demonstrate the technique.  It shows a dissociation front in oxygen  produced by black body radiation from a source at a temperature of 6 X 1Q4 °K.  The photo dissociation cross sec-  tions in the Schumann - Runge region from 1280 A to 1800 A were taken from Metzger and Cook (1964), see Fig.B.3, Appendix B.  The program to carry out the numerical cal-1  culation of F(xj/F 0 and the integration of (1-F(2-)/F0) to obtain z-(x) appears in Appendix A. We note in Fig. 3.3 that photons with low absorption cross sections penetrate substantially further than photons with high values of absorption cross section. In concluding this section on idealized radiation fronts, we should like to point out that although we have assumed that there is no particle motion, if Langrangian co-ordinates are adopted, the results obtained here are valid regardless of the flow of the gas.  Of course, it is  still necessary to transform back to Eu/erian co-ordinates. Secondly, the radiation front equations (3.10 and (3.11) which are derived for planar symmetry can easily be adapted to consider spherical symmetry such as expanding H II regions in interstellar space.  42 3  •2  of restrictions on particle motion and recombination The assumptions in the last section that all particle  are stationary and that recombination and collisional ionization or dissociation are negligible permitted us to obtain the radiation equations (3.10) and (3.11).  We now  wish to relax these assumptions and, in particular, redefine the front velocity vF and the coefficient $ . 3.2.1  The coefficient.5 . —  In the last section,  we used the relation (1 - ^L )FQ = vA N 0 to equate the number of photons entering the front from the left to the number of absorbers entering the front from the right.  There, we  insisted, that one photon dissociate or ionize only one absorber.  However, if we allow collisions to occur such  that the energetic, dissociated or ionized particles either dissociate or ionize other absorbers or recombine to become absorbers again, then on the average, we could have one photon dissociate or ionize either more than one absorber or less than one absorber.  We now define—the coefficient,  S , to be the number of photons required to dissociate or ionize one absorber ( o < j '  Thus instead of eq'n (3.1)  we write (j -<t*/*)F 0 =  jX  3.25  or acF  =  — — - — — ~ / + ^ / c $Wo  3 .25'  We emphasize that vr is the rate at which the front is receding from the source.  The photon flux density is de-  fined by eq'n (3.13). 3.2.2  The energy imput,  v-| . —  The last  term W//^ v-^ in eq'n (2.3) is the net energy imput per unit mass inside the front.  If we neglect radiation losses, we  may write 0 - ^ / c )  whei'e F (1  -fp/c)  O  -  f j  1  ^  3.26  /, h//F(^)d^/ is the energy flux and  <Th=  /'I  is the usual relativistic correction.  If the par-  ticles ahead of the front are stationary with resjject to the source then v = -/f  M  .  The density  <w0 -t .  where the subscript  may be written as  --••/;  3.27  refers to the absorbers (of density  0  N ) and the index j refers to the dissociated or ionized particles and to any impurity particles which may be present in the absorbing gas but which are not affected by the incoming radiation. xij is the particle density of j t h particles and m refers to the mass. "^f^T ~  Thus we may write )Ar/=  " - w . +  >  3.28  which with the help of eq'n(3.25') becomes  -wlc,  I  M,  v/here we have written V]_ - v F .  /  f  3.29  44 We note that if there are no impurities in the absorbing gas and if there are no  dissociated or ionized  particles ahead of the front then the term  m- in ''J.J  eq'n (3.29) is zero and we may write  3.29' We have used this equation previously in section 2.5 where we shov/ed how the temperature behind the radiation front can be estimated.  The coefficient f and eq'n (3.29) are  especially useful when considering R-critical, or weak R-type radiation fronts. 3.3  Weak R-type front  A weak R-type radiation front moves supersonically relative to the gas ahead and behind it. pressed (1  with a compression ratio  /4//1 < 2).  The hot gas is com-  between 1 and 2  The asymptotic solution ( v ^ — > c ) of this  type of front corresponds to the idealized front described earlier in section 3.1.  When the front velocity is compar-  able with the speed of light one finds  -f  = 1, an  approximate value for T 4 may be obtained from eq'n (2.18) or (2.18' ) with vj_ = v 4 (assuming otherwise for  <{h i /)>-'X ^  kT  min>  5 - 1, T  4  T  for  min^  (b^-J- > kT rnin . and  the  Pressure  P 4 may be obtained from eq'n (2.14). In the non relativistic region (where v? = F 0 / S N Q ) we may make various approximations to simplify eq'n (2.5). Et For all weak R-type fronts the terms ^ P3.//1 and ~ ~  45 may be neglected with respect to the term  '-^-Vj2. This  corresponds to the standard notation of gas dynamics M ^ » 1, where M^ is the Mach number.  For front velocities at least  10 times larger than the speed of sound in the gas behind the front (i.e. F D / J N q > 10a 4 ) we can make the further approximation  s W/K - « 1.  We may now expand the square root  of eq'n (2.5) (taking the negative root to correspond to the weak R-type solution) to obtain / * -tr^i) ^  •A- ^  «  * £  •- •  i M  frg)  •• • 3.30  where K =  * ~f \ V 1 •  Substituting for W/ "?f vj from eq'n (3.29') A  vf  ~  , ^ fklz'll^f!  [_  where v/e have let  Mors J  h  = N Q M and  z  '[  we obtain  fe^lfe*-')  =»  •  3.30'  as defined in eq'n (3.25  For the non relativistic case (vA = F 0 /5N 0 ) we obtain to first order  which illustrates the relationship between the compression ratio and F 0 , N 0 and % . The particle velocity behind the front (v is related to the density through the relation v p  =  v  i~v 4 )  y(/^//^-l  46 Thus from eq'n (3.30*) we obtain 2 /  3.31 and for the non relativistic case to first order we obtain _  . iV. 3.31'  Note that the particle velocity behind the front is inversely proportional to the velocity of the radiation front, Vfz .  The coefficient J and the function g 4 depend upon  the assumed temperature of the gas behind the front. Finally, the pressure ratio for the non relativistic case may be written from eq'ns (2.2) and (3.31') as —  / ^  fezOJ^*  ^  '  3.32  where we have used the eq'n of state p^^ = N 0 kTi and n Q M.  =  This ratio should check with the value obtained from  the equations of state. Obtaining numerical solutions from the above equations is straight forward.  With the assumed temperature T 4  (e.g. we choose T 4 = Tfflax) we approximate the enthalpy by h 4 ^ (5kT4 + X ) /M and the internal energy by f  4  where X is the ionization or dissociation energy. ctive adiabatic exponent is then g 4 = h 4 / £  ^ (3kT4  +ZJ/M  The effe-  With this  value of g 4 and f = 1 (if T 4 = T m a x ) approximate values of />4 and  are obtained from eq'ns (3.30") and (3.32).  It  is then possible to obtain g 4 accurately either by calculation or from curves of g (p ; T) vs T.  Accurate values of all-  quantities may then be obtained either from the asymptotic  47  r v.=1 .1 TxlOf-cm/sec v =.456x10 cm/sec  X(10 cm)  F_=2.69x10  25  ph/cm 2 sec  F -2.69x10  .Fig. 5-,4 Weak R-type radiation front  ph/cm sec  formulae above or directly from eq'n (2.5} and the conservation eq'ns (2.1) to (2.3J. An example of a weak R-type radiation front is shown in Fig. 3.4. to  For this and other examples (Figs 3.4  3.8 inclusive) we use oxygen at a pressure of 0.01 1 HI  atm (N 0 = 2.69 X 10 Tj = 300 °K.  Q  particles/cm )and a temperature of  We work out the examples for two final tem-  peratures T m i n ~ 6000 °K and T m a x » 8900 °K.  For T r a i n we  use h 4 = 2.31 X 1 0 1 1 ergs/gui, g 4 « 1.146 and a 4 = 1.835 X 10  5  cm/sec.  For T m a x >  h 4 -=5: 2.78 X 1 0 1 1  g 4 = 1.228 and a 4 = 2.517 X 10 5 cm/sec.  ergs/gm, The dissociating  photons have an average energy of 8.8 e K 5.08 e V  and a value of  is used as the dissociation energy of oxygen.  The  upper diagram in these examples is a plot of time,t versus position X showing the velocity of propagation of the various steady discontinuities in the flow.  The lower three  diagrams are plots of the pressure, density and temperature as a function of x at a constant time, t = 1 sec. In Fig. 3.4 we assume a value of F 0 / N 0 = 10 s cm/sec. At T m a x we find v x = 1.17 X 10 6 cm/sec ( 5  F Q /v A N 0 - 0.855)  and at T m i n , v x = 1.33X10 6 cm/sec ( $ = 0.752).  Notice that  although the pressure and temperature rise sharply behind the front, the density is almost constant. 3.4  R-critical front  If the velocity of a supersonic radiation front is reduced, either by reducing F Q or by increasing N q , one will  49  reach the point where the velocity relative to the hot gas behind the front is exactly sonic, but the velocity relative to the undisturbed gas ahead of the front is still supersonic.  This is an R-critical front.  It corresponds to the  Chapman - Jouguet point and can be considered the high density limit of weak R-type radiation fronts.  The compression  ratio is always slightly below 2-. The structure of the front is quite complicated - a shock starts to develop  in  the radiation front and the head of the rarefaction wave which follows is merged with the front. Approximate analytical relations are readily obtained 1/2 for this case.  The condition v 4 = (g4 f  -f ^)  s= a 4  (a4 is the speed of sound in the gas behind the front)  re-  sults in the quantity under the square root in eq'n (2.5) being identically zero.  Thus after some algebra we obtain  the compression ratio =  ^  -  1  ^  1  3.33  2 v  where we note that p x / -fx l ^ (M12 »  bv  1 for  an  R-critical front  1).  The particle velocity behind the front is given • Vr  ^ ^  _  f  ^  ft  1  which to first order in p ^  2  may be written as  The pressure behind the front as given by eq'n (2.2) is ft. = Fft^Kust  7  3*35 where we have substituted from eq'ns (3.33) and (3.34) and retained only first order terms in Pj/V-jV-^2. The condition that the square root in eq'n (2.5) is zero gives us an extra relation between v-^, -Z7^ and W. Neglecting terms of p ^ / / ^ we obtain 3 • ^ (z^-')^,) or, solving for  r  "  *  3.36  and substituting for v^ from eq'n (3.33),  ~  •  3.37  Finally, we obtain a relation for the coefficient $ from eq'ns (2.3) and (3.29') -  /1+-  I tr,  ) - A,  }  from which with the help of eq'ns (2.12) and (3.33) we obtain  S  »  ^  '^  -  3.38  The R-critical case is a point solution separating the M-critical fronts and weak R-type fronts.  For a given  set of conditions eq'n (3.37) is useful in predicting the type of front which one can expect to occur.  For this pur-  pose, one must approximate the value of a 4 by the relation  51  52 where we emphasize that m is the average value of the mass of the particles behind the front, m == s-n-i A value f J m./^nj. j J J of g4 is obtained as outlined previously for the weak R-type case. A complete numerical solution for the R-critical case is straightforward once an accurate value of g 4 is obtained. We treat the rarefaction wave following the front as an isentropic expansion in a manner outlined in section 2.3.  first  For a  approximation we assume g5• « g 4 . (Quantities behind the  tail of the rarefaction carry the subscript 5). temperature T & , pressure p 5 , density  Once the  -f5 and enthalpy h 5 are  approximately known, a more accurate value of g 5 may be obtained and the final properties of the gas calculated more accurately. An example of an R-critical radiation front calculated for the same conditions as used in the weak R-type case (see Fig (3.4)) is shown in Fig. (3.5).  We find that for T m a x ,  we require F 0 = 9.50 X 1 0 2 2 photons/cm  2  sec and the front-  travels at a velocity v x = 4.55 X 10 5 cm/sec ( S -0.777). For T • F = 6.25 X 1 0 2 2 photons/cm2sec and Vj = 3.41 X 10 5 • A ^mm' o cm/sec ( 5 =  0.680).  Notice that the pressure ratio is large  whereas the compression ratio is still quite small.  The rare-  faction wave is dominant. 3  •5  Weak D"typ_e__fi'_Qnt preceded b y _ a _ s h o c ^ v a ^  W e a k D - t y p e fronts lie on the opposite extreme on the density scale from weak R-type fronts, they occur  for  53 high densities'and relatively low radiation intensities. This subsonic radiation front has many similarities with combustion zones.  It moves subsoncially with respect to  the gas ahead and behind it.  As explained previously, a  shock discontinuity propagates .ahead of the front compressing the gas, the slower moving radiation front heats and expands this compressed gas.  A rarefaction wave would travel at  sonic speed and overtake the radiation front and, therefore, does not exist.  We now have two discontinuities to consider-  the shock wave with no energy imput, and the weak D-type front with energy input. 3.5.1  General equations. —  The conservation equa-  tions across the shock corresponding to eq'n (2.5) with W = o give  ^  r - r - ' l i W y c  ___  —  *  -Tn  J 3.40  where we have chosen the positive root. We note in passing 2 2 that for strong shocks such that v-j_ » p \ / f \ (or Mj_ ^ 1) we obtain the well known approximate relation -A  ~  3.40'  If we assume a velocity for the shock front v x we may solve  for all the parameters behind the front as out-  lined in Gaydon and Hurle (1963), Chapter 3 or Ahlborn and Salvat (1967).  (Preferably we use plots of ^  ,^  and-in-  versus Mach number as given in Gaydon and Hurle , page 52). At any rate the solution is straightforward and we shall  not comment on this point any further. The conservation equations across the weak D-type front corresponding to eq'n (2.5) give is.  1  ^  1// //  (  //  ^3.41  where we have again chosen the positive square root.  (The  negative root corresponds to a strong D-type front —  an  expansion shock with energy input, which does not occur in our case.) Since the partic3.es are stationary ahead of the shock front and behind the radiation front we have the following relationship for the particle velocity between the shock and radiation fronts:  o .42 where the velocities are defined as in Fig. scripts  s  2.2 ; the sub-  and. p refer to shock and front respectively.  The final pressure, obtained from a momentum equation corresponding to eq'n (2.2), is 3.43  where the particle velocity v is defined in eq'n (3.42). For future reference we note that the final pressure p 4 must fall in the limits  p  £ P 2 <^^4 ^  The equations in this weak  2'  D-type case do not  lend themselves to approximate solutions as easily as in the weak R-type and R-critical cases. low a numerical method of attack.  We, therefore, fol-  As usual, we assume the  55 final temperature T 4 .  We then assume a reasonable shock  velocity v^, calculate the thermodynamic quantities behind the shock front, calculate the velocity of and thermodynamic properties behind the radiation front and finally, calculate the radiation intensity required to produce the velocity.  An iterative procedure is required to obtain exact  solutions. For potential users of this technique, we will outline the iteration procedure in more detail in the next section.  Readers who are not particularly interested in details  may omit this subsection and proceed to Fig. 3.6 at the end of the section. 3.5.2  Iterative procedure for calculations.— The  prbcedure is as follows: and calculate p 2 , f2 ,  We choose a shock velocity, v-j,,  T 2 , h 2 , g 2 , and the velocities v 2 and  Vp. (For oxygen we use curves of  T / r  p2//?1,  2 " l  Mach number given by Gaydon and Hur.le/page 52).  versus  We will see  below that the final pressure p 4 must fall between  £ P 2 < P4_ < P 2  and can thus calculate a value of h 4 within 5% (using assumed T4) and. g 4 quite accurately. The term beneath the square root sign in eq'n (3.41) must have a numerical value between o and 1.  By assuming  that it is zero we obtain a maximum value of 2 ^ _I±.  /^V ^  )Z  •  3.44  56 where we have used the energy equation —r  -f- h 4. —  JL- J- In s- ——  and eq'n (2.12) for the enthalpy.  Since v 4 = v p + v 3  we may solve eq'n (3.44) by iterating to obtain an upper limit for V3. With the square root equal to zero the rest of eq'n (3.41) gives a minimum value for ...  *  / s v  i ^  W  '  3.45  which we solve using the value of v 3 obtained from eq'n (3.44). We then obtain a minimum value for V3 ^  =  7 T T ^ Z ^ )  The numerical values of v 3 m a factor two.  . and v 3 m i n usually agree within  We substitute the mean of these two values  into eq'n (3.41) to obtain a first approximation for and then utilize eq'n (3.46^to obtain a better value of v . We repeat this iterative procedure until we obtain a self~ consistant value of v 3 and -f^. (The solutions of v^ tend to oscillate about the final value and it is thus best to average the initial value of v 3 v/ith the result of the iteration as a starting point for the next iteration.  Two or three  iterations are usually sufficient to obtain an accurate value of v 3 .) (3.43).  We then calculate v 4 and P 4 from eq'ns (3.42)and  57 With a relatively accurate value of P4 we can obtain accurate values of h^ and g^. accurate values of V3, v^,  Similarily, one can calculate  f A and P^.  Finally v/e use eq'n (2.4) and the energy equation corresponding to eq'n (2.3) to obtain the radiation intensity required to produce the observed front:  where we assume V3/C  1.  If one wants to calculate v^ and other quantities for an experimentally given - f a n d F 0 , one must vary the assumed  (eventually by interpolation) until the value of  F 0 calculated from eq'n (3.47) agrees with the experimentally given flux density.  An example of a weak D-type radiation front (for our standard conditions as in Fig. 3.4) preceded by a Mach 3 shock front is shown in Fig. 3.6.  For T m a x , the  photon flux, F Q = 4.72 X 10 2 1 photons/cm2sec, enters the shocked gas with a velocity v 3 = 0.4.17 X 10 4 cm/sec <S =  F  o A / v 3 ^2 N o - 0.719).  For T m i n , F 0 = 4.62 X 10 2 1  photons/cm2sec and v 3 = 0.491 X 10 4 cm/sec (f =• 0.549). We notice that in this case the pressure ratio is not as large as in the other cases.  However, if we study the den-  sity distribution for this weak D-type (subsonic) front, it is seen that these radiation fronts act like "leaky" pistons, pushing the shocked gas away from the radiation  58  x(l0 5 cm) ]? o =4.62x10 21 ph/cm 2 sec  5 1 x( 1 Cr c m ) — 1^=4..72x10 21 ph/cm sec  Fig.3.6 V/eak D-type. radiation front preceded by a shock  59 source into the undisturbed gas.  Behind the radiation  front the density of the gas (which is completely at rest) is substantially lower than the initial density N 0 .  This  behavior is markedly different from the weak R-type (supersonic) radiation fronts.  3.6  D-critical front preceded by a shock  We will now discuss the low density limit of subsonic (weak D-type) radiation fronts. is called the D-critical front.  This limiting solution  The appearance of a D-cri-  tical radiation front is exactly the same as for a weak Dtype frontJ  the only difference is that the front travels  at sonic speed with respect to the gas behind it.  At  slightly higher velocities a rarefaction wave begins to form.  The D-critical case represents a singular solution  which separates weak D-type fronts from M-critical fronts. As for the weak D-type fronts we must consider two discontinuities.  Approximate analytical relations are again  more difficult to obtain than in the R-critical case.  As  before, the condition v 4 = a 4 results in the quantity under the square root in eq'n (3.41) being identically zero. Corresponding to eq'n (3.33) the compression ratio is _ -4 ^  _  %i±l  ~  ">  L {' +  >•  3.48  where we note that contrary to the R-critical front the term P 2 / / 2  v  32 ^  1  (°r M 3 2 -£<1) for a D-critical front.  Eq'n (3.48) may be rearranged and solved for vo  60  --I1—,  ^r  T,u  w h e r e we have used the definition a ^  2 « ggP^/'r,.  The term  in the square brackets is approximately equal to unity, The condition that the square root in eq'n  (3.41)  is zero gives u s an extra relation involving the energy flux W.  We solve this equation for v g  •v* -  sv  to obtain  -irJr  ' -syp/  3 5 0  where again a^ is the speed of sound ahead of the front and the terms in the square brackets may be considered  to  be correction factors which are set equal to unity in a first approximation.  We now equate.eq'ns  (3.49) and (3.50) and  solve for the pressure p 2 to obtain  t  e*  - 'ggv'.l't&fr  -  W  . 3.51  We note that except for the small correction factors the pressure ahead of the front depends only on the energy  flux  W. The relationships for the particle velocity  be-  tween the shock and the radiation front and for the pressure d r o p across the radiation front are the same as for the weak D - t y p e case  (see eq'ns  these equations and eq'n  (3.42) and  (3.43)). From  (3.48) we may obtain the  pressure  ratio  3.52  If typical numerical values are inserted we find P 4 From this equation and equation behind the radiation  ^  g- P 2 -  (3.51) we obtain the pressure  front  +  [_  - ^ / J  ~  3.53  With a lengthy calculation we can obtain the shock front velocity associated with a D - c r i t i c a l front.  For  purpose we u s e the relation v p = a 4 - v 3  eq'n  to rewrite  this  (3.49) as  ^  =  ^  _  f/  H+ 3.54  We must now write Vp and a 2 in terms of the velocity v^. For this purpose v/e assume that the effective  adiabatic  exponents ahead and behind the shock are identical, g a - g^, and use the ordinary shock equations for an ideal gas.  The  particle velocity behind the shock is  vp - ( * / * * • ! ) v/here M-^ 3  -/^J/  vlf  v ^ / a ^ is the Mach number of the shock.  shocks the term in brackets approaches unity. a22 where  = ax2  For  strong  We may write  (TaAj.) , • • o  Substituting these equations into eq'n  (3.54) we obtain a  quadratic equation,the solution of which is  \  O t 00  62 w i t h the correction Jfcj^  terms ri^/y^J  3.56  /  h q  =  In calculating numerical values of v-^ oC),  one  first  obtains an approximate value by neglecting the correction terms and then an accurate value by repeating the w i t h the correction terms  calculation  included.  T h e c r i t i c a l density may be calculated by using pressure  ratio  w h e r e again we assume, g 2 = g±. a12  the  w e  - gx  Jfps)  From the definition  obtain  f<t*+i\ %rJL=Y-T). ft'tixr)  >  3 / 5 8  where the initial pressure P 2 ( ^ ) and. velocity v x  ( P C ) are  defined in eq'ns  We  (3.51) and  (3.55) respectively.  that in the first approximation upon the photon flux F c .  depends  note  linearly  Since the D - c r i t i c a l case is a  singular solution separating the weak D - t y p e and M-critical cases eq'n  (3.58) is u s e f u l in predicting which front  occur for a given set of  shall  conditions.  A numerical solution for a specific set of condi-  I  tions may be obtained from the above equations.  However,  for exact solutions it is preferable to apply the basic  10  0 10  0 10  10  K to 0 O T-=6000 °K  1  x(10 cm)• P 0 =2.32x10 2 2 ph/cm 2 sec  T,=8900 °K 4 0  0  I cx(10 cm) •  P 0 = 3 • 96x10 2 2 ph/cm 2 sec  Pig.3.7' D-critical radiation front preceded by a shock  64 equations across the shock and radiation front such as eq'ns (3.40) and (3.41).  The procedure is similar to that  used for calculating weak D-type fronts. Fig. 3.7 shows an example of a D-critical radiation front (for our standard conditions as in Fig. 3.4).  In  this case for T m a x , a Mach 8.10 shock (vx = 2.67 X 10 5 cm/sec) precedes the radiation front. 22 F Q = 3.96 X 10  The radiation flux,  2 photons/cm sec enters the shocked gas with  a velocity, v 3 = 0.183 X 10 5 cm/sec ( 1.02).  F Q / ? 1 /v 3 / 2 N Q =  For T m i n ; a Mach 6.02 shock (vi = 1 . 9 8 X 10 5 cm/sec)  precedes the front. F Q ^ 2.32 X 10 2 2 photons/cm2sec and v «J ^ 0.162 X 10 5 cm/sec (  0.893).  We notice that the  appearance of the flow is very similar to the weak D-tyBe case| however, the shocked region is narrower and the pressure and compression i-atio are higher.  A D-critics,! front  "sweeps up" less gas than a weak D-type radiation front. 3.7  M-critical front preceded by a shock  We have studied radiation fronts at low and high densities and have given limiting densities for these super and subsonic radiation fronts„  If one calculates numerical  examples with given F Q , N Q and Ta it is found that the high density limit  /j  of the supersonic radiation fronts  is still considerably below the low density limit of the subsonic radiation fronts (see Fig 1.1).  /]_ (0C)  The region  between  (*<r) and  conditions of Kahn.  ^  (^r) corresponds to the M~type  Thus radiation fronts which occur in  this region are called M-critical reminding us that these fronts exist over an extended range of densities which lie in the Middle between the Rarified and Dense conditions. A radiation front which travels slightly faster than a D-critical front, wave following it.  >aA!  must have a rarefaction  On the other extreme when the front  velocity is slightly less than the R-critical velocity, a shock must propagate ahead of the radiation front and a rarefaction wave follow it.  We thus see that an R-critical  front can also be described as a D-critical front slightly preceded by (or merged with) a shock but with both travelling at the same velocity. erical calculations.  This can be confirmed by num-  We assume that in the M-critical  region a radiation front propagates at sonic velicity rel™ 7 /? ative to the gas behind it such that v^. - a^  (g/.p^/ f ^ Y  and such that the term under the square root in eq'n (3.41) is identically zero. The velocity relationships are now slightly more complicated.  Corresponding to eq'n (3.42) we now have 3.59  where v^, is the velocity of the particles leaving the radiation front measured in the lab system.  From the equa  tion of conservation of mass,  and  59) we obtain  v g **  from eq'n  66  = . irp-  flk _ / J  3o60  Corresponding to eq'n (3.43) the pressure behind the front is .  3.01  We note that for all M-critical fronts P 4 - (l/2)p2 to about 5% accuracy.  To obtain a numerical solution we follow roughly the same procedure as outlined in section 3.5.2 for weak D-type fronts.  Assuming a reasonable shock velocity v^ we calculate  all the thermodynamic quantities behind the shock.  Since  P4 ~ fl/2j P 2 we can calculate h^ and g 4 quite accurately (using an assumed value of T^)„  A value of Vg is obtained  by iteration from eq'n (3 „ 44) using a value of a 4 obtained from eq'n (3.39).  The compression ratio,  - f  t  is  obtained from eq'n (3.48), and the particle velocity behind the front, v p A f is obtained from eq'n (3.60).  Finally an  accurate value of p 4 is obtained from eq'n (3.61) and an accurate value of a^ from the definition, a^ J  g^p^//^.-.  If there is insufficient accuracy the whole procedure is repeated. The radiation intensity associated with the initially assumed shock velocity 15 found from eq'n (3.47).  The rare-  faction wave is treated in exactly the same manner as described in section 3.4 for the R-critical case. .  67 Fig. 3 . 8  illustrates an M-critical shock preceded  by a Mach 9 shock front.  (Again the calculations are for our  standard conditions as for the weak R-type case in Fig„ For T m a x , the photon  flux, F q » 4 . 6 3 x 1 0 2 2  ph/cm2secs  enters the shocked gas with a velocity, v,, = 0.200 x cm/sec and v 3  ( $  0.968).  For T m i n ,  0 . 2 7 7 x 1 0 5 cm/sec  F0 = 4087 x 1022  (S =  0.735).  3.4.)  105  ph/cm 2 sec  The pressure and  compression ratios are still higher than for the D-critical case;  the appearance, however 9 is similar except for the  rather weak rarefaction wave which follows the radiation front.  68'  Jig.3.8 M-critical radiation front preceded "by a shock  69  C H A P T E R  4  THE STRUCTURE OF STEADY RADIATION  FRONTS  In the last chapter- we t r e a t e d r a d i a t i o n as ideal discontinuities  behind which  dissociated,,  It w a s n e c e s s a r y  ture  to c a l c u l a t e  in o r d e r  front.  could  a l l d e t a i l s of recombination and a s t e p w i s e  calculation.  (at l e a s t ,  processes within  it is  that  final one  this  the f r o n t  dossociation are  i n t e g r a t i o n a c r o s s the front chapter  to o u t l i n e  Again, w e consider steady  if and  considereds is c a r r i e d  such a  A s a n e x a m p l e we w i l l d i s c u s s a  as a one-dimensional  the  in p r i n c i p l e ) be o b t a i n e d  the r a t e s of i o n i z a t i o n ,  in o x y g e n .  tempera-  if the  W e f e e l that  of t h i s t h e s i s to r e a l i z e  It is the aim o f . t h i s  all  the f l o w a s s o c i a t e d w i t h  be c a l c u l a t e d .  contribution  temperature can  front  to a s s u m e a f i n a l  T h e p r e v i o u s r e s u l t s w o u l d be u n i q u e ,  temperature major  the g a s w a s  fronts  out.  detailed  dissociation  the r a d i a t i o n  state discontinuity  front  with  energy  input. . Unfortunately, required  rate  the n u m e r i c a l actually  it t u r n s out  coefficients integration  finally  not  are not  that m o s t  yet k n o w n ,  is q u i t e d i f f i c u l t  successful).  Therefore,  of  the  and a l s o (and the  that  was merits  70  of this chapter lie more in the outline of a procedure to obtain T 4 than in the production ox numerical results. The cohesion of the thesis is not lost, if the reader turns over to section 4.4,  He may later return to some  parts of this chapter in order to study two definitions which are used in Chapter 7, namely, the local power input and the local degree of dissociation.  4„1  Conservation equations of massp momentum and energy.  Similar to the treatment of shock front structures (Zel'dovich and Raizer (1966)f Chapter VII), we will include viscous forces and heat transfer in our discussion of the radiation front structure. be written  f (X) Vfx)  PCX) -h ffx)  The conservation equations may  •  4.1  =  ~  e/jrfxl J  z  X  JTCx) •f c/ X  ^  \ i dirod\ ' "ITx/ - 7T>.  4.2  71  The terms on the right hand side of these equations are constants of integration, expressed in terms of the initial values of the flow variables, distinguished by the subscript "o".  v Q is the front velocity relative to the particles  ahead of the front. ^c and 7< are coefficients of viscosity and thermal conductivity respectively (one usually assumes that these coefficients are constant. )  The term  is the  rate of energy input per unit mass such that far behind the front  /_  '  /A/ 8  ~  ;  4.4  where W / ^ v ^ is the total energy input per unit mass as defined in eq'ns (2.3) and (3.29'}.  All the other variables,  are defined as in eq'ns (2.1) to (2.3). We note that these equations are valid at any point inside the radiation frontj  in fact, far behind the radia-  tion front these equations are identical with eq'ns (2.1) to (2.3) since the terms containing viscosity and heat conduction vanish. 4.2  Reactions within a radiation front In general, many kinetic reactions occur within a  radiation front.  The photodissociated particles tend to  recombine either directly by two body or three body recom-  72  bination or indirectly through a chain process in which "intermediate" stable or metastable compounds are formed. Negative a.s v/e 11 as positive ions may occur, atoms and molecules are found in various stages of electronic, vibrational and rotational exitation.  Collisions between hot  particles within the front tend to cause further dissociations .  Finally, at sufficiently high temperatures and  radiation intensities the gas in the radiation front absorbs and radiates as a grey body (see Zel'dovich and Raizer, Chapter IX), presumably through an inverse bremsstrahlung mechanism with the free electrons. If the incident radiation has a black body frequency distribution one could expect each type of particle to absorb in some region of the frequency spectrum.  Further-  more, the front may produce its own radiation through freefree or free-bound two body collisions or radiational deexcitations.  "Trapping"of resonant radiation may occur.  The various typos of particles in the radiation front are generally not in equilibrium with each other such that equilibrium relations (e.g. Safia relations) must be used with caution,if at all.  Thus the concentration of each  type of particle must, in general, be described by a separate conservation equation. Let us consider the various mechanisms which occur in a diatomic gas.  As pointed out by Zel'dovich and Raizer,  73  Chapter VI, studies of relaxation times of the various processes behind a shock front indicate the following: Complete equilibrium between the translational, rotational and electronic degrees of freedom is reached after less than ~9 20 collisions per particle (3 X10 sure ) .  sec at atmospheric pres-  It takes a much longer time to reach equilibrium  between the vibrational and translational degrees of freedom„ Blackmail (1956) estimates that 2 . 5 X 1 0 7  collisions 19 at  300 ° K (9.6/ftiec at a standard density of 2.69 X 10  o u  cm  )  and 1.6 X 10 3 collisions per particle at 3000 °K (0.083^sec at a density of 2.69 X 10 1 9 cm~ 3 ) are required to reach equilibrium in oxygen.  On the other hand, equilxbriation between  the individual vibrational states is extremely rapid (of the order of 20 collisions).  Mathews (1959) has determined that  behind, shock fronts the dissociation time is an order of magnitude larger than the vibrational relaxation time.  The  collisional dissociation mechanism seems to be due to collisions between a particle in a highly excited vibrational state and a particle with high translational energy.  Col-  lisions between molecules in the ground state rarely produce dissociation.  Conversely the three body recombination mech-  anism presumably leaves the molecule in a highly excited vibrational state. Various types of reactions may occur to produce complex molecules in the radiation front.  For example, in  oxygen at low temperatures and low degrees of dissociation  atoms tend to combine with molecules to form ozone.  (If  ionization were present we would also have to consider 0"'" A and C>2 particles. ) /  4.3  Special case of a dissociation front in oxygen  To illustrate the concepts let us consider a dissociation front propagating in pure oxygen caused by black o body radiation above 1280 A (we assume that there is no ionization).  Y/e choose a sufficiently high particle density  such that excited oxygen atoms are collisionally de-excited and the dominant recombination mechanism is by means of three body collisions.  According to the mechanisms outlined  in the previous section, the following reactions aredominant : /  /& OO  ^ ^  * & O &  £>x +  &  —^  At ^  ^  sM •  —  +  M ^ ^  +Af  . 4.5  where M, the third body in the collision stands for any of 0„ (%t  O'o , the superscript "*" denotes a molecule in a ° * 1800 vibrationals excited state, the notation h ^ 1 2 8 o indicates that the 0o molecule has a high photodissociation cross ^ o o section in the wavelength regions 1280 A to 1800 A.  75 The set of reactions above is perhaps not completej but these reactions clearly illustrate the principle and concepts which we wish to emphasize later.  Unfortunately, the  reaction rate constants for most of the reactions in eq'n (4.5) are not known.  For calculation purposes we further simplify  the reaction scheme of eq'ns (4.5) as follows  that is, we neglect ozone formation and vibrationally excited molecules.  (A general treatment for the reaction scheme of  eq'ns (4.5) is given in Appendix C.) 4.3,1 icles.  Conservation equations for absoxMng ;part-  Since we only have two types of particles In the  reactions eq'ns (4.6), the conservation equations for the atoms and molecules differ only by a factor 2, «?//  '  guAh  _  +  f JHi ^ ^t  /  /  4.7  where u is the flow velocity, N is the particle density and the subscripts j and  2  denote 0 and 0 2 particles respectively.  The conservation equation for the molecules may be written  ^  M h r  =  _ m f  +  __  ( f l , )  m)*  -  3  ' ,  4.8  where k is the reaction rate constant and the subscripts d and /t donote dissociation and recombination respectively.  The term  SfJ in eq'n (4.8) is equal to the number  of molecules which have been destroyed by absorbing and is defined by eq'ns (3.18 and (3.16).  photons  We assume  that the photon flux is sufficiently small so that the term (1 + F 0 /cN 0 ) may be neglected.* may be written as L  tj.x&Jr - i  The solution of eq'n (3.18) •  ' ^  e  „ fix *  * '  4j9  where for a black body radiating at a constant temperature the photon flux entering the absorbing gas is f [If^^jjf/ ^s  cnst 6  j  4.10  where (for future reference) we have separated the time variation, G (t)f from the frequency dependure F(^ / )d/ / . The mass density of the gas for the case under consideration is  ;  where M is the mass of the 0 2 molecule.  4.11,, The degree of dis-  sociation y is defined as 4  M  '12  such that we may write //,_ -  ,  4.13  Using eq'n (4.13) in the left hand side of eq'n(4.8) we have  .  jua/z __ ^ JO-;/) )  *  -1 Actually the term (1 » vF/c) » (I + F 0 A N Q ) „:LS„ In ropriate only if we have a steady radiation iroac general, one should omit this term and write tne photon flux as F ( x f t/ ) j where t' is the regarded tine, c « (t - x/cj„  since 3 / V a t + oVu)/^x « o.  Thus eq'n (4.8) can he  written in terms of the more usual thermodynamic variables  ,7  4.14 where we have differentiated eq'n (4.9) to obtain the first term on the right hand side. Collisional dissociation and three body recombination coefficients in oxygen have been measured by various workers (Rink et al (1961), Cemac and Vaughan (1961), Mathews (1958}}. The reaction rate constants depend only on the temperature and are related to each other by the principle of detailed balancing  1  ^  ' 4.15  where K(T) is the equilibrium constant, which determines the equilibrium degree of dissociation y at a given temperature and density; A is a constant.  Although eq'n (4.15)  is strictly valid for an equilibrium situation, presumably it is also, at least approximately, valid for non equilibrium situations (Hurle (1967)}. The dissociation coefficient is assumed to be of the form £  [ exr {-Mf)  j •/.*<*< 3.0  •  '  where E is stfme constant, D is the dissociation energy and the exponent n is believed to have a value between one and  78  three.  The value ox n is difficult to determine since  the temperature dependence is swamped by the exponential tex'm. The oxygen atom is roughly three times as effective as a molecule in recombination reactions, thus using the •  •  •.  "3  values of Rink et at (1S61 and a value of A - 115 X 10 we obtain  4.17  We must emphasize that these values were obtained from shook wave .studies in oxygen near 4000 °K and the exponent (~I) in the equations for k^ was used by Rink et al to obtain an approximate temperature dependence.  It may, in fact, be as  large as (-3) and small as (- -t). Consquently, at room temperature the values of k^. obtained from eq'ns (4.17) may be significantly in error.  Nevertheless we shall use these  values for calculations In the thesis. 4,3.2 / ( X; t) % ( X,  The rajte of energy input per unit volurae — T h e energy input for radiation fronts  or radiation produced shocks is through absorption of photon For high radiation intensities and low number densities (corresponding to weak R-type conditions) there is little part icle motion so a knowledge of the energy input at any  point in space and time is not necessary.  Nevertheless,  if recombination of the particles behind the radiation front is negligible this energy input may be easily calculated in a manner analogous to the methods outlined in section 3.1. If recombination is not negligible then the calculation is much more complicated and furthermore depends upon whether the recombination is due to two body collisions with resulting photon emmission or due to three body collisions with no emission. Recombination of the particles would tend to broaden the radiation front and distort the energy input across the front since the photons of high absorption cross section would tend to be absorbed by the recorobined particles which presumably are formed relatively far behind the leading edge of the front.  Also, if the particle density is relatively  low (1016crif3) the dominant recombination mechanism is by two body collisions with photon emission.  This results in  substantial "diffusion" of radiant energy in the vicinity of the front and it is necessary to employ the theory of radiative transfer (Chandrasekhar (I960)) to obtain the net energy input at any point in space and time. in weak D»type fronts preceded by shocks there is substantial motion of the gas and therefore a knowledge of the energy input at every point in space and time is of dominant importance if one wishes to analyze the development of a radiation or shock front or the structure of a  80  steady state radiation front. The rate ox energy input for the reaction scheme in eq'n (4.6) is obtained directly from the j-^J term as defined in eq'ns (4,8) and (4.9) simply by replacing the photon flux F with the energy flux E.  (See /jppendix C for the general  case.) . The relation between the energy flux and photon flux /  ,  • ~ U-  ' sy -tr  st  e .  , 4.18  where we have used eq'n (4.9) and where  t) df is de-  fined by eq'n (4.10)., Thus differentiating eqfn (4.18) we obtain /  4. Ii/  which in the notation used in eq'n (4.14) is ^  In general, if there is more than one type of absor-  bing particles in the radiation front, an equation similar to eq'n (4.19} must be written for each type.  This is ill-  ustrated in Appendix C for the reaction scheme shown in eq'ns (4.5). 4.3,3  ra illation of the front .structure. -- For  given boundary conditions one can, in principle, calculate the structure of a steady radiation front from eq'ns (4.1), (4.2) and (4.3) where to evaluate the term  //(*) fl.OOdx  in eq'n (4.3) it is necessary to use eq'ns (4.19*) and (4.14).  81 For a steady radiation front one replaces the time derivative with the spatial derivative d/dt—-  v>  ,where  is the velocity of the front,such that the left hand side of eq'n (4.14) becomes A m  f ij^) I  u  j^ji 3X /  where -tr- u - y^  A/  2- (t-^)- sjr^ cPX /yf  . 9n ^ , 4,<1U  is the particle velocity relative to the  front. An attempt was made to calculate the structure of a weak D-type front in oxygen preceded by a Mach 3 shock front.  For this we used eq'ns (4.1), (4.2), (4.3), (4.14)  and (4.19') as well as the equation of state.  In this man-  ner we hoped to obtain a value of the temperature behind tho radiation front which we had assumed for the calculations In Chapter 3.  The procedure was to divide up the radiation  front into equal sections (in L&grangian co-ordinates) with the first section at the point where the photon flux was 1% of the initial value.  Calculations were then carried out  for each succeeding section.  First,, the degree of dissocia-  tion was calculated from eq'n (4,14), -/q was calculated from eq'n (4.19') and v 2 /2 from eq'n (4.3). the iterative procedure did not converge  Unfortunately, negative values  of the density and imaginary values of the velocity always occurred.  Perhaps this is hardly surprising since the v 2 /2  term is about 10" times smaller than the V q and the h terms in eq'n (4.3) —  our iterative procedure could hardly be  expected to produce such accuracy.  Perhaps some other cal™  82 culation procedure would prove to be more satisfactory. However, further work in this direction was abandoned. We shall return briefly to this proplem in Chapter 7 where we will use the equations developed in this section.  4  •4  Concluding rem^^jon_Chapters 2, 3 and 4  In the previous chapters we have treated steady radiation fronts propagating in a semi-infinite tube and showed that five different types of fronts were possible. In Chapter 3 we carried out detailed calculations (for an assumed temperature, T 4 ) for each of the five types of radiation fronts which occur,  We would like to stress one  of the most interesting phenonema:  Radiation fronts may  act like driving pistons to accelerate the gas ahead of them.  The results are best presented by plotting the vel-  ocities vp5J v P and vs as a function of N 0 /F 0 , in Fig. 4.1 show such a plot.  The diagrams  The values of these curves  were obtained from Figs. 3.4 to 3,8 which were calculated for standard conditions as outlined in section 3.3. In Chapter 4 we introduced concepts and equations to calculate the structure of any steady radiation front wit! given boundary conditions, Tn this way it is possibly in prxncipl to calculate the final temperature behind  the front so as  to make the solutions of Chapter 3 and the relations presented in Fig 4.1 unique in terms of the final temperature.  Pig.4.1 Plot of velocities versus N q / F 0 for  and T  This, in effectj yields an additional equation so that now there are as many equations as unknowns (see section 2.5) andP therefore, no assumptions are necessary. Although we failed to obtain a numerical solution for a simplified case we believe that the ideas developed in this chapter and Appendix C will point the way to succesful calculation in the future. We have intimated several times that in the strictest sense of the word steady radiation fronts do not occur in real gases.  All radiation fronts will possess non  steady state characteristics to some degree.  The application  of steady state equations to radiation fronts will yield approximate results - in some cases quite accurate and in others, less reliable.  Howevers even in obviously non steady  state situations, the results of these chapters are useful in estimating the properties of and thermody.ua.mic quantities associated with the radiation front0 In the next tv;o chapters we describe an experiment which matches the geometry, which we have considered throughout this thesis.  In trying to understand the details of our  experimental results we found it necessary to consider aspect of non steady radiation fronts.  Consequently, in Chapter 7  we develop a method to consider such fronts0  C H A P T E R  THE  BOGEN  LIGHT  5  SOURCE  Having treated steady radiation fronts in the first part of this thesis we will now focus our attention on an experiment to produce radiation fronts in the geometry of Fig. 2.1.  An extremely intense light source radiating in a wavelength region where the photoabsorption cross section of the test gas is large is a necessary requirement for experimental work on radiation fronts.  An ideal source would  be a powerful pulsed laser radiating at the desired frequenc and for a period of several tens of microseconds.  Compari-  son of the experimental results with the theory for such a monochromatic source would be much simpler than for a black body source. Unfortunately such ideal lasers are not available at present.  For our experiments we choose a light source  similar to that described by Bogen et al (1965).  This sourc  consists of an arc constricted through a narrow channel in a polyethylene rod and radiates as a black body with an effective temperature of the order of 105 °K for a period of about 10 yu. sec.  86 5.1  Description of light source  The light source is illustrated schematically in Fig. 5.1  A 25 ji. F capacitor bank capable of being charged  to 20 k K i s discharged through a 2 - 4 nun diameter hole drilled through a 4.2 cm long polyethylene rod.  The dis-  charge, squeezed through the hole, vaporizes the polyethylene at the walls and produces an extremely hot, high density plasma which radiates along the axis of the hole as a black body.  The radiation passes into the test  chamber either directly or through a glass, quartz or LiF window. Unfortunately much of the polyethylene plasma consists of vaporized carbon which tends to settle on the walls of the chamber and on the window.  Consequently it is nec-  essary first to remove the test chamber as far from the source as practical,  secondly, to insert baffles between  the source and the test chamber and thirdly to use large dump chambers to disperse the spent plasma.  Otherwise, the  window must be cleaned after every one or two shots. The sequence of events in firing the light source is as follows:  The system is pumped down to below 0.05  Torr which is sufficiently low to ensure that breakdown does not occur«  The condenser bank is charged to the desired  value (usually 3 k V ).  The light source is fired by directing  a jet of helium onto the hole in the polyethylene.  This  raises the pressure until for the applied voltage a point  on the Paschen curve is reached where breakdown occurs. The spent plasma and excess helium are pumped out and the whole pi-ocess may be repeated every 30 to 60 seconds. After about 1000 shots the discharge channel becomes enlarged and the polyethylene must be replaced. An alternate method of triggering the discharge would be supplying a pulse of approximately ~ 12 k / at the negative electrode by means of a brush cathode.  This method,  was not used since the electrical noise associated with the triggering pulse tended to trigger the oscilloscope prematurely. Various designs of the light source were tried before the design illustrated in Fig, B.I, Appendix B was successful.  It consists of two electrodes embedded in and  polyethylene  dump chambers  window test  inlet  gas  for  helium  \ test chamber baffles 25AB\20kV capacitor bank over clamped  Fig. 5.1  Schematic representation of light  source  88 separated  by e p o x y s t r e n g t h e n e d  diameter  threaded  such that  polyethylene  with fibreglass , rod  is s c r e w e d  Although polyethylene  symmetric  into the  this design was quite satisfactory  high discharge  voltages;  channel  in the p o l y e t h y l e n e  ible from  the  light  shot  The  after many  shots  were  dump chambers consisted  tubing  with 0  in d i a m e t e r .  the p e a k i n t e n s i t y  of v a r i o u s  lengths  Con-  reproduc-  tended  to  fired„ of 6 inch  diameter  (2 i n c h e s to 12  inches)  - rings.  T h e LiT? a n d q u a r t z w i n d o w s w e r e 1/4" t h i c k by diameterj  the a c t u a l a p e r t u r e  for the r a d i a t i o n  the test c h a m b e r w a s 1 . 7 cm d i a m e t e r . consisting from  5.2  the test  chamber  .  to s t o p the  It w a s  1"  entering  A mechanical  of s h e e t m e t a l , w a s i n s t a l l e d  entering  outside  the  discharge  i n t e n s i t y w a s not s t r i c t l y  to shot and  delayed  increases  the  Also,  f r e q u e n c y of the b a n k d e c r e a s e s as the  sequently,  of  especially  ringing  sealed  epoxy  apparatus.  t e n d s to c r a c k a f t e r m a n y s h o t s  at r e l a t i v e l y  aluminum  3/4"  the 2 - 4 m m d i a m e t e r h o l e s e r v e s a s the a x i s  the c y l i n d r i c a l l y  become  A  operated  shutter light from  the c h a m b e r by m e a n s of a m a g n e t .  Measurement of intensity A typical oscilloscope  shown in Fig. 5.2,  The  t r a c e of the light  peak i n t e n s i t y  pulse  of the light  is  pulse  89 was measured as a function of wavelength and as a function of discharge voltage.  2 v/divj g jisec/div X =s 5000 A', discharge voltage = 2. 5kV  Fig. 5.2  Light pulse from Bogen source  5.2.1  Absolute intensity at 5000 A with discharge  voltage at 3.0 k V. --  The absolute intensity was measured  by comparison with a standard carbon arc (made by Leybold, with Ringsdorf RW 202 anode and RW 401 cathode).  The arc  was operated as prescribed by Null and Lozier (1962) experimental setup is indicated in Fig. 5.3.  The  Care was taken  to ensure that the optical systems were identical for the two light sources.  This was accomplished by means of a mirror —  first measuring the Intensity of one system, rotating the mirror by 90° and measuring the intensity of the other system.  monochromator  source aperture  mirror chopping wheel  Pig.5.3 Experimental setup for absolute intensity rneasuremen  91  By adjusting the size of the source aperture (see Fig. 5.3} it v/as possible to adjust the effective size of the light sources.  By adjusting the solid angle aperture it  was possible to measure the intensity of the Bogen light source as a function of solid angle.  Measurements show that  the intensity per unit cross-section tends to decrease slightly with the size of the hole in the polyethylene insert. optimum size (at a discharge voltage of 3.0kV  The  and at a wave-  o  length of 5000 A ) was found to be approximately 4 mm. Measurements also show that the light from the Bogen source is concentrated in quite a narrow beam in the axial direction since,the intensity per unit solid angle decreases markedly for large solid angles (perhaps by a factor 3 for •_/!=»0.1 sterad).  o  Measurements indicated that at 3.0 kV and 5000 A the average intensity of the Bogen light source for a solid angle of 0.1 sterad was (1.9 + 0.2)' X 103 times as bright as the carbon arc. Along the axis this value is roughly o three times larger. Since the carbon arc intensity at 5000 A is 200 watts/(cm2ster \i.) we calculate that at the source p.'l  aperture we have a photon flux of about 3.6 X 10*" photons/ (300 A cm2sec) (for _/L = 0.064 and area magnification of 5.3). From Stefan's law the effective black body temperature of the Bogen source is in the region 60,000 °K to 150,000 °K depending on the solid angle,  92 5.2«2 To  :Ll£J£L.  Intensity as a function of wavelength at measure' the  intensity  in the  wavelength  o  region from  2500 A a procedure  above was used.  However,  similar  no l e n s e s w e r e u s e d  neutral d e n s i t y  filters were replaced  quartz windows  (the t r a n s m i s s i o n  was measured  the  Bogen  value  of  4.2 X  10 times  I06 watts/  larger  measurements  increases  (cm2ster  is v e r y  difficult.  are  the o r d e r  At  to i n c r e a s e  wavelength (i.e.  Unfortunately  the  o  Neverthe  of  to a  |i) at 2 5 0 0 A  about  i.ntenaccurate  less bs? c o m p a r i s o n  the v a l u e s of the  the i n t e n s i t y of the B o g e n  more  slowly  around  at  larger  the polyethylene discharge  the i n t e n s i t y  intensities of  °K.  linearly.  being reached„ s e e m s to be  intensity  Intensity as function pf_discharge voltage. ---.  low v o l t a g e s  reases quite  with  to  an e f f e c t i v e b l a c k b o d y t e m p e r a t u r e  of 4 0 , 0 0 0  5,2.3  frosted  s m a l l at 2 5 0 0 A and  w i t h P l a no c k b l a c k b o d y r a d i a t i o n at 2 5 0 0 A i n d i c a t e  prior  s h o w that  than at 5 0 0 0 A).  s i t y of the c a r b o n arc  the  of the various, c o m b i n a t i o n s  The measurements  source gradually  and  by a set of  as a f u n c t i o n of w a v e l e n g t h  measurements),  described  to that  voltage  However,  f o r larger  insert. is s h o w n  source  at h i g h e r v o l t a g e s  indicating 6 kV.  light  a saturation  it  level  A l s o this s a t u r a t i o n  level  d i a m e t e r s of the c h a n n e l  Fig,  5,4.  From  this  tends is  A t y p i c a l c u r v e of i n t e n s i t y  in  inc-  in versus  curve  it  93  discharge  Fige  5.4  appears 6 kV. the  Intensity discharge  that  of B o g e n l i g h t voltage.  t h e optinura d i s c h a r g e  Unfortunately, u n l e s s  light  tages and  it w a s p r e f e r a b l e  place  the w i n d o w  source  chipped to u s e  closer  (kV)  as a function  voltage  the w i n d o w  s o u r c e , it g e t s b a d l y  consequently,  voltage  is a r o u n d  is very at  these  far  5 to  from  voltagesj  lower - discharge  to t h e  light  of  source.  vol  94  C H A P T E R EXPERIMENTS  AND  6 RESULTS•  Throughout our theoretical investigations we have considered steady radiation fronts which are generated in a semi infinite tube sealed by a transparent window.  For experiments with ionization fronts one must use  a window which transmits photons of energy below the ionization potential of the test gas (esg, the ionization o  limit of hydrogen atoms is 912 A).  We know of no material  which transmits radiation at such. low wavelengths.  Lith-  ium fXouride, which transmits radiation down to a waveo length, of about 1200 A has the lowest cut off limit. Consequently we could only study dissociation fronts in test gases which have photodissoeiation cross sections in o • a wavelength region above 1200 A.  Iodine and oxygen ful-  fill this requirement and were used as test gases. When we examine the temporal variation of the light pulse from the Bogen source, v/e find, that it is of much too short a duration for a steady dissociation front to develop . Therefore it was decided to study .two phenomena (i) the beginning of the formation of the radiation front at low absorber densities during the time of the light pulse and (ii) the formation of shocks at high absorber densities after the light pulse was over.  Such experimental  95 investigations  6.1  are d e s c r i b e d  below.  Beginning of formation of dissociation front. in iodine  An experiment  to o b s e r v e  m a t i o n of a r a d i a t i o n finite  length  monochrornators,  it is p o s s i b l e through amount ative  the test  to  the  front.  passing  radiation indicates  Furthermorep  weak R-type  case  a steady radiation  i o n s is  wide  (such  (i.e.  on it  if the c o n d i t i o n s  front  that  3 ) or  10/M sec  light  of the r e s u l t s  is  To carry a s the t e s t  pulse  the relin  correspond  to a  if the predicted, w i d t h to such  gradient  is  and, therefore,  the  condit-  small), during  interpretation  simplified. out such an e x p e r i m e n t  gas since  o  iodine was  it is p h o t o d i s s o c i a t e d  by  chosen  radiation  °  in the r e g i o n 4 6 0 0 A to 5 0 0 0 A. c r o s s s e c t i o n of  use  increase  an  t h e n t h e r e w i l l b e little m o t i o n of the p a r t i c l e s the  which  a-dissociation  of  corresponding  the p r e s s u r e  in  the test c h a m b e r  the d e v e l o p m e n t  (see C h a p t e r  gas  passing  A n i n c r e a s e w i t h time  incident  of  photomultipliers),  f i l t e r s and  through  for-  (where v/e m a y  the a m o u n t of light  chamber.  of r a d i a t i o n  transmission)  of  to m e a s u r e  region  chamber  a test  If v/e c h o o s e  absorl3S in the v i s i b l e w a v e l e n g t h conventional  test  front requires a  Fig.6.1).  (see  the b e g i n n i n g of the  It h a s a  2 . 4 X 1 0 ~ 1 8 c m 2 at 4 9 9 5 A  photoabaorption (see r e s u l t s  of  Fig.6.1 Schematic'representation of experiment with Iodine  97  Rabinowitch and Wood (1936) in Pig.B.2, Appendix B) and a recombination coefficient of 7,6 X 1Q~ 3 0 C m 6 /moIecuIe 2 sec for three body recombination with l 2 particles as the third body (Porter and Smith (1961)).  There were three conditions  to satisfy in the choice of the length of the test chamber and the initial pressure so as to obtain a maximum in the variation of the transmission.  First we wanted about 90%  of the radiation at 4995 A to be absorbed within the chamber since for this case the signal to mission:  transmission) was large.  noise ratio ( ^ transSecondly, we wanted to  use a short focal length lens to focus the light into the test chamber and thus obtain a large photon flux, F . • dictated the use of a short, test chamber.  This  On the other hand,  the chamber could not be too short since this would require the use of high particle densities at which three body recombination would not be negligible.  To satisfy these con- .  di.tions we chose a test chamber 10 cm in length and used a particle density of 1.12 X 1 0 1 7 particles/ cm 3 . The test chamber was a 3.5 cm diameter evacuated glass cell (containing iodine crystals) enclosed in a brass container which was equipped with heating elements to control the temperature.  The particle density of the iodine  vapor was regulated by adjusting the temperature of the cell. We used 70 ± 0,5 °C which corresponds to jfl^j =s (1,12 t 0.03) X 10  cm  S8  The experimental has an optical system ing the a b s o l u t e A 2.5" focal b e r of the  the B o g e n  to the a r r a n g e m e n t  of the B o g e n s o u r c e  lens was used  light F passed  inserted  to d i v e r t  be p l a c e d  A  5.3)  cham-  the r a d i a t i o n of 5.3).  into  It  a differential  is a s f o l l o w s :  of the i n c i d e n t  of or b e h i n d  b e t w e e n the  radiation which  the  signals F  reproducible,  technique.  it  was  the amount  cell.  and F Q  is  procedure  is p l a c e d  of light, F  fil-  necessary  The experimental  The neutral density filter  i o d i n e c e l l and  was  and p h o t o m u l t i p l i e r  in front  The  was  4.0 neutral density gelatin  the d i f f e r e n c e not  and  A plane glass plate  D~  either  small and, furthermore,  of t h e  to f o c u s  a small fraction  as a m o n i t o r ,  Since  to u s e  measur-  (see F i g .  into a m o n o c h r o m a t o r  into a s e c o n d m o n o c h r o m a t o r  ter c o u l d  for  to c l e a n t h i s lens after e v e r y f o u r s h o t s .  by photomultiplxer  served  It  to seal the d u m p i n g  (with a n a r e a m a g n i f i c a t i o n  measured  F0,  is s h o w n in F i g . 6 . 1 .  s o u r c e and a l s o  test c h a m b e r  transmitted  similar  intensity  length  was necessary  setup,  in  front  , entering  °  p h o t o t u b e ^"2 (at 4 S 9 5 A ) is a d j u s t e d and  equalized  to F e n t e r i n g  ional neutral density  difference.of scope,  (ideally  practise  filter  the p r o c e d u r e  phototube  filters  difference  is d i s p l a y e d  is then placed  If  behind  there  addit-  6.1).  on an  The  oscillo-  should be z e r o , but  o c c u r s ) and r e c o r d e d  repeated.  levels  by m e a n s of  (not s h o w n in Fig,  two s i g n a l s  this  this never  The 4,0 N.D, and  those  to r e a s o n a b l e  in  on polaroid, film. the iodine  is s u b s t a n t i a l  cell  depietio:  CQ  of the  1-2 m o l e c u l e s  pulse5the  ia the cell during the time of the  s i g n a l from phototube  s i g n a l of the m o n i t o r . is a m e a s u r e  Typical oscilloscope Unfortunately  should be larger t h a n the  The d i f f e r e n c e  of the development  light  in these two  of the r a d i a t i o n  signals  front.  traces are shown in F i g .  and in order  the s i g n a l s are not r e p r o d u c i b l e  to o b t a i n a m e a n i n g f u l m e a s u r e m e n t  6.2.  it w a s n e c e s s a r y  to  aver-  age m e a s u r e m e n t s over Fig.  6.3.The solid  1 2 shots. The r e s u l t s are shown in indicate the standard d e v i a t i o n s . error bars/ The dashed error bars indicate  the r e s u l t s o b t a i n e d with no iodine vapor in the cell p l i s h e d by k e e p i n g the cell at liquid  nitrogen  for w h i c h w e s h o u l d o b t a i n a straight  line along the  o n t a l axis.  temperatures)  T h e d e v i a t i o n from the expected result  large error b a r s are testimony of the d i f f i c u l t y  the  radiation  front  in this  (accom-  horizand  in  the  detecting  experiment.  The two solid c u r v e s in F i g . 6,3 give upper and  limits a  for  the  p h o t o n flux  expected  F Q = 1.44  flux half  this v a l u e  5,2 u s i n g  solid  dians). Chapter  t h e o r e t i c a l results; 1 X  1G 22 ph/300 A cm2sec)  (corresponding  a n g l e s ox  H e r e we  and 0 . 1 2 8  T h e c a l c u l a t i o n s are carried, out as outlined 7.  (Drift m o t i o n and d i f f u s i o n of the  as well as wavelength  to be p a r a l l e l , )  Also,  the radiation  section  sterain  particles  dependence of the absorption  section were neglected.  used  and a photon  to the r e s u l t s of  0.256 steradians  lower  v/as  cross  assumed  D e s p i t e the obvious s h o r t c o m i n g s of  tho  R a t h e r than p l o t t i n g the original and the increased flux w h i c h d i f f e r only by about 5%, we gave the e x p e c t e d d e f e r e n c e s o f ' b o t h s i g n a l s in Fig. b.4. .  100  monitor, 2.0 v/div ) CP - F c ), 0.5 v/div)  4>Q ±n fr  filter  °nt °f  Cel1  monitor, 2.0 v/div ) , ' j N.D. 4.0 filter (F - F 0 ), 0.5 v/div) behind cell  Fig. 6.2  Typical oscilloscope traces for measurements in iodine.  250-  200 -  150 -  100 -  8  10  t(/<sec)-  solid error bars -- with iodine vapor in_cell dashed error bars' bars-- no iodine vapor in cell / S ""I* =1 .44x10^ph/cm^sec & ^ theoretical curves o ' " ' ' ~ '  0  Fig.6.3 Increase in light intensity during of light pulse  101 measurements and theoretical curves there is general agreement between theory and results.  We would like to emphasise  the difficulties encountered in these measurements —  the  intensity and time duration of the light pulse from the Bogen source were simply insufficient to measure the development of the radiation front precisely. In concluding this section the author would like to suggest that an experiment similar to the one described above but using a strong d.c. light source be attempted.  (Possibly  a large carbon arc such as are used as projectors in drive-in theatres would be satisfactory).  Also, other gases (or mix-  tures of gases} such as chlorine, bromine and sulphur dioxide may be preferable as test gases.  6.2  Shock fronts in oxygen  In the experiment in iodine a low density was used such, that little particle motion could be expected.  In this  section we wish to accentuate the dynamics of the test gas so as to produce shocks  As shown in Chapters 3 and 4 one has  to use a high absorber density and a test gas with a high absorption coefficient in order to produce significant particle motion over short periods of time. as test gas for this purpose.  Oxygen was chosen  It has a high photodissocia-  tion CSPOSS section, in the Schumann Runge region from about o o 1280 A to 1800 A (see results of Hetzger and Cook (.1934 Fig. 8.3, Appendix B) j, its maximum value of 14,9 X l c T 1 8 c m 2 Similar shocks were reported by Elton, (1964).  o at 1 4 2 0 A is six  O  at  4-995  A.  r e g i o n of higher  1018cm  t h a n for  tion front iodine  Also  .  times  larger  the p a r t i c l e  cm"3 s  the case of iodine.  t e n d s to be m u c h n a r r o w e r  Schematic  of e x p e r i m e n t  in  Consequentlys  a radia-  than in the case  oxygen.  of a 2" diameter  in F i g .  pyrex  iation passes through  in  g a s and p r o d u c e s •  the 1 . 7 cm d i a m e t e r window,  is absorbed  a shock w h i c h t r a v e l s  opening in the  in the  6.4.  T-juncThe  flouride (LiF)  of  larger.  t i o n filled, w i t h o x y g e n at the desired p r e s s u r e .  lithium  the  substantially  s e t u p is i l l u s t r a t e d  test c h a m b e r c o n s i s t e d  iodine  d e n s i t i e s used w e r e in  ° to 2,69 X 1 0 1 9  The experimental The  than the v a l u e for  and the p r e s s u r e g r a d i e n t s m u c h  Fig. 6.4  102  -  radthe  oxygen direction  S ^ n c e t h e r e are no c o n t a i n i n g w a l l s it also tends to d:l p e r s e o u t w a r d s in the r a d i a l direction, h o w e v e r , this Tseems to h a v e no effect on the a x i a l propagation of -cue s h o c k s i n c e a 1 . 8 cm I.D. tube inserteo co prevent xa?.s d i f f u s i o n r e s u l t e d in n o d e t e c t a b l e d x f i e r e n c c xn w e s t r e n g t h of the s h o c k .  103  A piezo electric pressure probe, (pressure transducer LD-15/B9 of the Atlantic Research Corp., Alexandria, Va.) was placed directly facing the incoming radiation, the distance between it and the LiF window could be adjusted to any desired value by means of a threaded screw.  The face of the piezo probe  was coated with aluminum paint to prevent the radiation from falling directly onto the crystal.  This probe measured the  time of arrival and strength of any shocks or compression , waves which were formed. The procedure was simply to set the piezo probe at any desired distance d, fire the Bogen source and record the signal from the piezo probe as displayed on an oscilloscope. (It was necessary to clean the LiF window after every 6 shots.) Typical traces are shov/n in Fig. 6.5.  V/e notice that the  sharp shock signals are superposed on a long duration slowly .decaying signal.  This signal is presumably due to thermal o  heating of the crystal when radiation (above 2000 A) strikes and is absorbed by the face of the probe.  In fact, the  amplitude of this signal proved to be a convenient way of monitoring the intensity of the radiation passing through the LiF window.  The secondary peak which appears after the  primary signal is due to the reflected shock (from the piezo probe, back to the LiF window and back to the piezo probe). From these signals we may calculateinitial the speed and point of formation of the shock at various/pressures. V/e find that at high pressures (600 Torr) the shock forms very near the LiF window while at low pressures, (20 Torr) the  distance d must be at least one centimeter before a signal can be detected.  In general, as the distance d is increased  the. amplitude of the signal first increases to a maximum and. then decreases gradually.  Presumably this indicates that the  compression wave initially builds up in strength to a maximum and then slowly decays.  This is illustrated in Fig. 6.6  for an initial pressure of 400 Torr at which the maximum is at about 0.5 cm.  We will examine these results in more detail  in Chapter 7. The velocity of the shocks at all pressures is 364 t 8 m/sec.  In fact, the velocities at low pressures seemed to be  slightly larger than at high pressures but certainly no more than 8 m/sec0  The time of arrival of the shock as a function  of distance d is plotted in Fig. 6.7 for an initial pressure of 400 Torr.  From the slops we obtain a velocity of 368 rn/sec  while from the reflected shock the velocity is 363 iii/'sec, Notice that there is a slight bend in the curve at 0 o 5 cm, indicating that near the window the velocity may be different than the measured value.  Unfortunately, it is difficult to  obtain, reproducible results in this region. results with theory in Chapter 7.  We compare these  105  d a 3.0 cm  Pressure «= 400 Torr Oxygen 0.05 v/div 50 |isec/div  d = 5.0 cm  Fig. 6.5  Oscilloscope traces of piezoelectric probe.  24 -  20  16  >  3 12 <D tJ S 8 A B cfl H  Qj so 4 •H CQ  0. 0  d( cm)-  Fig.6.6 Shock strength as function of d at 400 Torr oxygen  Fig,6.7 Velocity of shock at 400 Torr oxygen  107 6  •3  ionization In the test chamber If one removes the window which was used in the  two previous experiments then it is possible for ionizing radiation to enter the test chamber.  Indeed large signals  of the order of 100 volts were measured by means of electrodes inserted into the test chamber.  However, these  signals did not seem to be correlated with the light pulse in any way, seeming to start at or just after breakdown of the Bogen source whereas the light pulse is delayed 2 or 3 y^sec.  Also, the signals depended upon the grounding of the  dump chamber and polarity of the test chamber.  Furthers  work along these lines was abandoned. We also observed the photoeffect from metal sur™ o  faces due to radiation in the range of 1200 to 2000 A^ With the intense Bogen light source it seems to be easy to produce a cold electron plasma, ideally suited for the measurements of electron-neutral collision cross sections. However5 no systematic investigations were carried out.  C H A P T E R  7  UNSTEADY ONE-DIMENSIGNAL FLOW WITH ENERGY INPUT  In the theoretical section of this thesis we considered only the steady state cases in which the radiation front was fully developed and the incident radiation was constant as a function of time.  In this chapter we will  consider the development of radiation fronts with the incident radiation varying with time in an arbitrary manner. In particular, we will set up the theory to calculate the development of the shock fronts in oxygen which were observed experimentally in Chapter 6. The boundary conditions.again are a tube bounded at one end by a window.  The motion of the gas may be des-  cribed as unsteady one-dimensional flow with energy input. If the energy input as a function of time and position along the tube is known, the evolution of the flow along the tube may be calculated by the .method of characteristics or by the method of finite differences.  The rate of energy-  input q(x,t) may be calculated quite generally according to the treatment outlined in Appendix C for the case of oxygen.  However, we will base our calculations of this  quantity on the simplified treatment outlined in section  109  7.1  Method of characteristics  A detailed explanation of the method of characteristics is given by Shapiro (1954), Chapters 23-25 and Oswatitsch (1957), Chapter 3.  Hoskin (1964) describes a method  of calculation at fixed time intervals which is particularly applicable to our case.  In order to show the limitations  of this method we will first give a brief explanation. Consider the x-t plane shown in Fig. 7.1(a).  Let  us assume that  Fig. 7.1  Mach lines and path lines of characteristic net.  the complete state of the gas at points 1 and 2 is known. Any disturbance travelling to the right from point I will propagate with the speed Uj + c g i j any disturbance travelling to the left from point 2 will propagate with the speed u 2 - c s 2 (where u is the particle drift velocity and c s is the speed of sound at the point in question).  We refer to  the loci of right travelling waves as  characteristics or  Mach lines and to the loci of left travelling waves as J characteristics or Mach lines.  The loci of the individual  particles are called path lines.  The  and j characteris-  tics intersect at some point 3. The basis of the method of characteristics rests on the fact that along the Mach lines and path lines the thermodynamic quantities vary according to certain specified equations (see below) such that the state (and velocity) of the gas at point 3 may be calculated,* The characteristics net is constructed as illustrated in Fig 5.1 (b).  Using our case as an example we choose  equally spaced points along the x~axis where the particle velocity is zero and the speed of sound is constant. then find the intersection points  of  We  the 71 and J"  characteristicsp and determine the thermodynamic quantities at these points.  We then simply repeat the procedure to  obtain the next set of points.  As we feed in energy the  characteristics net becomes distorted indicating the formation of compression and rarefaction waves.  A shock forms  at a point where two or more characteristics of the same family intersect. The method of characteristics at fixed time intervals  *  Thj s is strictlv true only if the n. and J characteristics can be drawn as straight linesj however, we can satisfy this condition to as high an accuracy as v/e wish simply by decreasing the distance between points 1 and 2.  Ill  is similar in concept to the above explanation except that the points at which the state of the fluid is calculated are selected beforehand.  For this purpose one usually  selects a rectangular mesh in time -Lagrangian space coordinates and uses the required differential equations in Lagrangian form.  The Mach lines are drawn backwards in  time from the pre-selected point into the region where the state of the gas has already been calculated, 7.1.1  Physical characteristics in Eulerian and  Lagrangian co-ordinates, —  In Eulerian co-ordinates the  equations of the Mach lines is m k i  where the upper sign of i refers to the ^ characteristic and the lower sign refers to the Jj characteristic.  The equa-  tion of the path lines is simply  ,=  -„"' • ^  ...7  UA m / d-l<;) P^-th  In Lagrangian co-ordinates z , eq'ns (7.1) and (7.2) may be written as (dS- j  -  ±  f£5-  /I  J  7.3  and ( S ) ? ^  7.4  where / i s the mass density, c s is the speed of sound and is a constant reference density (e.g. the density at t « 0 when  (x) is constant).  fQ  112 7.1.2  State characteristics. —  In Lagrangian co-  ordinates the properties of the fluid along the path lines are described by the second law of thermodynamics )  _  fa?) r.Utl* = $  7 .5  where h is the enthalpy per unit mass, p is the pressure.  is the density and  is the rate of energy input per unit  mass as given in eq'n (4.19'). In Lagrangian co-ordinates the equations of momentum and mass may be written as (Hoskin (1964))  .  , i + 7f  ju  ^  _ ^  ;  7.6  and J^  Df  . f  Jji  .  1.7 where  is a reference density defined in eq'n (7.3). (We  neglect thermal conductivity and viscosity as was done in If we multiply eq ? n (7.7) by the speed of sound  Chapter 4.3  c„ and use the equation of state in the form of eq'n (2.12) s  h ~  , - ^ >'F  2i 12  eq'n (7.7) may be written in the form  Using the energy equation and the equation of state (eq'ns (7.5) and (2,12)) we may write the right hand side of eq'n (7.9) in terms of the rate of energy input (7.9) becomes  Thus eq'n  11c  7.10 We  now add and subtract eq'n (7.6) from eq'n (7.10)  to obtain two equations in characteristic form +  ' -jr  Sc53F.\ + f 3^u -yr j^J -(  -  ^  -^.J  Thus along the characteristics (d. ^ / d t =  ± /Cg/^0  we have  £Mks  iMk,  4  W .  . 7.«  We emphasize that the differentials on the left hand side are evaluated along the Mach lines whereas the differentials on the right hand side of eq'n (7.11) are evaluated along the path lines. There are two points in eq'n (7.11) which we would like to discuss.  First, the term containing ( dg/ cK)^  is  usually small (though not necessarily negligible) compared to the term (g ~ 1)/^-.  For calculations in this thesis we  will neglect this term.* Secondly, the speed of sound c g was introduced ad hoc and has not yet been defined.  The  problem here is that for a system of particles not in thermodynamic equilibrium the speed of sound depends on the frequency of the sound wave (see Zel'dovich and Raizer (1966) chapter VIII).  (We should point out that we are really  interested in the velocity of propagation of a disturbance *  see addendum Appendix D, page 138  at some point in the radiation front which we assume to be equal to the speed of sound at that point rather than in the speed of sound itself.)  Classically the speed of sound is  defined as the rate of change of pressure with respect to density at constant entropy (see eq'n (2.7))  where  is the isentropic exponent.  Chapter 2 that replacing  We pointed out in  with tbeeffective adiabatic  exponent g may not be a very good approximation.  For non-equilibrium  situations the validity of  such an approximation is still more questionable.  Neverthe~  2  less j for calculations in this thesis we will assume c g « gp/y 7  such that the term containing {a-f/Jt)z  is zero.  in eq'n (7.11)  With these two approximations we obtain a simp-  lified form of eq'n (7.11)  7.2  Method of finite_differences in Lagrangion co-ordinate If we use a constant energy input then the method  of characteristics will determine the evolution of a radiation front and eventually the steady state structure as given in Chapter 3.  However, once a strong shock has formed;  a special procedure is required to calculate the thermodynamic auantities across it.  If the structure of the shock  115 is of no importance it is more convenient to use the method of finite differences to calculate the flow. A treatment of the method of finite differences in one space variable and no energy input is given by Richtmyer and Morton (1967), Chapter 12.  Once their equations are  modified to include the energy input, their treatment is directly applicable to our case for numerical solution on a computer. One drawback of the method of finite differences is that it is incapable of handling shock discontinuities and sharp gradients in the thermodynamic quantities.  The pre-  sence of such a discontinuity results in an oscillatory solution.  To overcome this difficulty Richtmyer and Morton  introduce an artificial viscosity which "smears out" the discontinuity over a finite distance and thus eliminates or reduces the oscillations in the solution. This artificial viscosity Q s is of the form  •  ~  y  fa  7.12  where "a" is a numerical constant (a>"l), the value of which one chooses at ones convenience and change over the space interval  fa is the velocity  between lattice points.  This viscosity appears in the momentum and energy equations (see eq'ns ution p  (7.6)  and  5>p + Qc, .  (7.5)).  One simply makes the substit-  Since wo do not develop any new concepts in using this method we relegate the differential equations for this treatment to Appendix e.  7  '3 .  the two methods to dissociation fronts  in oxygen The equations given in the two sections above (and in Appendices D and E), together with eq'n (4.19') for the energy input and eq'n (4.19'J for the energy input and eq'n (4.14) for the degree of dissociation, permit us to calculate the development and flow of any radiation front.  One drawback is that in many cases the amount of  computer time necessary for such calculations is prohibitively long (and expensive).  Perhaps the procedures  outlined in Appendices D and E could he modified to make more efficient use of computer time.  (One possibility  is to obtain a better first approximation in the iterative procedures by extrapolating the values of the variables from the previously calculated values.) Nevertheless, these methods were used to help explain the results obtained previously.  For the calcul-  ations we assumed that the incident radiation had a black body spectrum corresponding to a temperature of 6 X 10^ °K.  117 71,3 1,1  for time_dependgnt radia-  tion from Bogen source.— The results of section 6.2 indicate that the shock front in oxygen at a high pressure forms more rapidly and nearer the LiF window than at low pressure. Also , near the LiF window the speed of the shocks may he lower at high pressures than at low pressures. , It was not known if the signal measured by the piezo probe was due to a shock or a compression wave.  In order to compare these  results with the theory, the development of the shock was treated by the method of characteristics at fixed time intervals.  (The method of finite differences cannot be used in  this case since the artificial viscosity "smears out" any shocks which may form.) The calculations were carried out for pressures of 1.0 atm and 0.1 at®,  The various constants, the difference  intervals and the computer programme which were used are given in Appendix D, used.)  (A computing time of 10 minutes was  Figs..7.2 and 7.3 show computer plots of the various  thermodynamic quantities as a function of dimensionless distance X at various times. The pressure profiles are of special interest since this is the quantity which produces the signal measured in section 6.2.  At 1.0 atm the pressure rises to a maximum of  p » 1,9 p 0 within 3.0 |jisec, then decreases as the pressure wave propagates away from the window.  At 8 |j.sec the compress  wave is at 0,295cm and is travelling at a velocity of 460 ta/sec.  At 0.1 atm the maximum pressure is p = 3.8 p G within  3.2 (isec, is 3.3 p Q at 8 jisec and 3.2 p Q at 16-jisec. At 16 [isec the comjjression wave is at 0.85 cm and. is travelling at a velocity of about 562 m/sec.  In section  6.2 we obtained velocities of 364 m/sec at distances far from the LiF window.  The fact that the calculated  values are consistantly higher than the measured value indicates that the photon flux (F  = 1.16 X 10 2 2 ph/cm2sec)  which'we used in our calculations was too high.  On the  other hand the velocity measurements were taken relatively far from the LiF window whereas the calculations were carried out to distances relatively near the LiF window.  Finally, it is possible that our programme gives  a systematically high value for. the velocity. These pressure profiles indicate that the compression waves do not become shocks within the computing time.  However, we should consider this statement with  caution since it was.not practicable to show that if sufficient time were allowed the compression waves do become shocks.  .7.2 Computer profiles, 1.0 atm. Method of characteristics at fixed time Energy input from Bogen light source. Peak photon flux, F 0 =1.16 x 10 22 .ph/52oS Time, t=(N-l  , with ^t=0.10 (isec.  Distance from window, x=0.00249 X [cm] . Curves plotted for (N/10)=1,2,3, •••8.  X ANAH£U>(, CAUKOKr-S'.A CHART HO. Ol  ISO.ODD  CAi.3f'Cr??tf?A COMPUTER PRODUCTS, SMC.  ANAHSSSvl, CALIFORNIA  CHART MO. 0 1  1BCL OQO  HAOE IH U.S.  i-ULL I  —U  U| ! Cr-f jr.  „1_.  : I Fig.7.2 —continued  T  DENSITY,  R=/// 0 , with / = 1 . 4 3 x 10~ 3 gm/cm 3  120-000  so.ona  CAUFORr-uA COMPUTER PRODUCTS, 3WC.  140.000  ANAHEi?/}, C A U F O R ^ J A  CHART WO. 0 1  ISO.001  MADE l?l U.S.  PfEEEffifflfflSffiiE  isa.coc CAL:s~on?Jj; COu.'.-UTHH PFXIU'CTS,  lilC.  PtlAHZV:., C - U r O R i H A  CHART  KO.  '  . 1," i- .1,  d  Fig.7.3  Computer  profiles,  0.1  Method  of c h a r a c t e r i s t i c s  Energy  input  Peak  photon  Time,  at f i x e d  light  ph/520A  , w i t h A t = 0 . 2 0 (isec.  from w i n d o w ,  plotted  time  for  intervals.  source.  flux, F 0 = 1 . 1 6 x 1 0 2 2  t=(N-l)At  Distance Curves  from B o g e n  atm.  x=0.0249 X  /cm/.  (N/10)=1,2,3,  •••8.  cm2sec,  a.onn  5B.QOO  products,  a^akf.?'^, c a u f g r ^ s a  c h a : ; t no. 03  MrtDi-: ITi U.S.A  64,000  72.000  I t  rrr  r  frhi'  .L  ) i , l-l t I-J  I . I- Jl.-rt-1-t—t- Ir-;' -!—+—( -.-I • 4 .)--  1 11 Tznir -P: i-rrIt hHirti-h rhFR-fh-mi  rnrc - L i - J -  -1. • l„,-l-l, .t-.t-.l-l. .1 -1 -1-1 - !.  TTTTT  fflftff  Fig.7.3 —continued DENSITY,  R=///"0 , with  43 x. 10~ 4 gm/cm 3  0D0  coiwutnrc - r c r r j c T S , ;nc.  akahe>", c,AL:FDT?rcjA  csjart mo. oz  mmmua  72.000  -t .i  l-i-l-ul- L U4-.U-U1-.UI.. 1 1  1  I 1  1 1  1 I  J  .1  I 1  1  1  i 1  1  (  -  —!—— -  1  1  !  -  i  i  -  0=sy^l.0  „  -  -  -  • -  -  •• -  -  -  -  -  _  -  -  -  -  -  -  -  1  1 -  -  -  _  --  -  !  -  -  - -  1 -  1  1 •  1  1  1 -  -  _ i  !T;:Ltn i-1 hj-f-H-h  -  8,000  is,qoq  24,000  -  -  X  cc.'.t,?j.n;-:r; p-.-.c^-jzts, sr;c.  1  —1— i - l —H - — i i 1 ! —V- -  1  1 1  !  -  1  _ 1  -  j 1  -  1  -  !  | I  _  1  --  -  -  -  t  -  -  -  _j -  j 1 1 !_,  -  -  -  -  -  -  -  _ l  -  -  - -  — -  1 1  -  — -—  T 1  -  -  -  -  -  -  !  i  5S.Q0C!  chart no. 02 -tad!." in u.s.a  -  -  -  i  I  -  _X  T  1 1 L -  -  --  -  » 1  :  •  •64,000  1  !  1 I  - —  _  —  - -  1 - -  1  1- 1 -  1 ! 1  -  -  -  - -  —  \ -  -  -  i  X E •f 1  —  -  _  _  -  -  -- -  1 1  —  -  }  !  -  L !  1 L  — -  —  1  i  -  1  -  r.  48,000  -  -  1  —  40-qdq  -  t  .L !  — -  -  -  1 -  -  r I .J  -  - -  !  _  -  —  i -  FF  -  1  -  —  -  -  -  1  -  -  -1  -  - -  1 -  -  1  -  -  -1-  i  -  -  i 1  1  1 I i 1 1. r  32.000  -  -  j i  --  -  i  -  -  1 1  -  -  - -  -  1  - -  1 1  1  -  -  -  1  !  t  -  -  -  -  -  -  -  -  1  1  — -  - -  -  -  I 1 I  -  - - -  1  -  -  -  -  -  1 -  --  -  1  -  -  -  -  1  I  _ L_  i  L L 1  -  -  -  -  -  -  1  1  -  1  1  1  -  -  —  „ - _ -- _ -  -  1 !  1-  -  _  1 1  j  T  1  i  1  —  1  •1  -  -  -  -  1 1  -  1  -  h -  -  -  i  i  1  -  1  f=y , with  -  i  t  J—J  1 1 1 r 1  -  1  L!  I•  1  1  + H  i  i i  -  1  1  1  j i  -  -  1 1  -  1  1 1 I  -  _  i •  1  ! 1  |  72.qdg  v[r ih -Jr u-La.-ci-.il ;.i. l.»fc-U.LU.L,LL  ENERGY  INPUT,  £„=1.94  8* ODQ  Q=(M/af0/'0c0)q  xI09ergs/gm,  ID-DDQ  vCiipuTeii  M=53.3  2«4,.Q0Q  -:;c.  x  with  x  10  -18  2  10~24gm  32.000  X  a=14.94  yo.Doa  a?:a;-;e:;vi, c^vson'mp  cj:asj  us,qdo  oj  5S>ooa  614, aaa-  72.000  ,000  8, god  ib.000  24:,qoo  32.qq0  X  yo-oao  48,000  5b.0qq  M.'iut I'l U S  614,000  72.000  s-jweat^Lsfs slsStesasB  doq  a. ooo  16.000  24,000  32.ooa  X  Mu.ooa  45,000  chatvt rc. CH  58.000 MADE IN U.S.n  64:, OOO  72.000  119 7,3.2  Structure_of_.dlssociatlo^  T h e r e s u l t s of C h a p t e r 3 i n d i c a t e Fc  = 4,72 X  required  1022  ph/cm2sec  that  a s t e a d y photon, f l u x  (with an e n e r g y of  to p r o d u c e a w e a k D - t y p e r a d i a t i o n  b y a M a c h 3 s h o c k at an i n i t i a l p r e s s u r e We applied evolution ence  the m e t h o d  of f i n i t e  of the f l o w for  i n t e r v a l s and  t i o n s are g i v e n Again  the c o m p u t e r  in A p p e n d i x  it w a s  indicates  at t h e w i n d o w  maximum  it a l r e a d y Chapter  the  differcalcula-  reached.  the  cal-  H o w e v e r , the  plot  7»4{.  is 7.2 t i m e s  the d e g r e e of  the  dissociation is 4 0 0 0 °K  and  : i n i t i a l d e n s i t y ) is 0 . 3 S 0  is  1.4 c Q  is in the  460 m/see.  initial  s t a g e s of  some of t h e p r o p e r t i e s  The  Although  development,  predicted  in  • r e s u l t s m u s t be c o n s i d e r e d  s i n c e cons:Lderab1e the c a l c u l a t i o n s  difficulty  from  going  In f a c t ,  in the c o m p u t e r  such  preliminary  into o s c i l l a t i o n s to o b t a i n  oscillations  profiles  as  was encountered  a s 15 i t e r a t i o n s w e r e n e c e s s a r y values.  to c a r r y on w i t h  the t e m p e r a t u r e  (final  velocity  exhibits  These  this point  ratio  front  3„  to c a l c u l a t e  E.  impracticable  is o n l y 3 6 % ,  particle  the r a d i a t i o n  oxygen„  a s a f u n c t i o n of d i s t a n c e X , see F i g .  At  the c o m p r e s s i o n  is  preceded  The constants,  that w i t h i n 15 jj.sec the p r e s s u r e  initial value.  front  p r o g r a m m e for t h e s e  culations until steady state W a s of the p r e s s u r e  8.8 e V )  of 0 . 0 1 atm  differences  this case.  —  0  shown  are  in F i g ,  in  preventing  and as  many  self-ccmsistant  already 7.4,  in  evidence  Fig.7.4 Computer Method  profiles,  of f i n i t e  differences,  for a w e a k D - t y p e Mach 3  shock.  Photon  flux,  Time,  t=NAt  Distance Curves  0 . 0 1 a tin.  radiation  Fq=4.72 x  applied  front  1022ph/cm2sec.  plotted  for  x=0. 249 X  (N/10)=l,2,  Cond  preceded  , w i t h A t = 0 . 3 0 3 jisec .  from w i n d o w ,  to  [cml . -5.  by  4  I rl~r ( LL -t"or J  it  "I,  I I !  —A!^, J—i!  -.t_t_L i i i -1  L J_  —!—i—LL T-I_ X  —continued  PRESSURE,  -.000  r  A—  TT  Fig.7.4  m m .  r Pi  tt  -r r  I  P=p/pQ  , with  p o =0.01 atm,  s.aao  1.EDO  7 >200  7-200  Isass^izakEesES ksrisaissas:  II  b  GOD  j  000  .bcd  .,500  ZAGO  200  X  y.oao  u.boo  5,600  o.yao  7.200  120  C-H A F T E R SUMMARY  The  object  theoretically radiation of  and e x p e r i m e n t a l l y  window  for  of t h i s t h e s i s w a s to i n v e s t i g a t e  or d i s s o c i a t i n g  Five  ofccur  CONCLUSIONS  radiation}  into a t u b e c o n t a i n i n g  different  t y p e s of  of h i g h is l i t t l e  particle  there  front,  at the o t h e r e x t r e m e particle  and a shock front The  passing  densities  motion  structure within  of  known  it are  iation front  m a y be c a l c u l a t e d  the r a d i a t i o n  front  either  yield  important  may  particle with  and a l l  is  dominant front.  thermodyndetailed  and m e c h a n i s m s  occuring the  a measurement  information  the  intensities  if the  Conversely  mechanisms.  about  rad-  of  this  these  '  It w a s s h o w n t h a t for or c o l l i s i o n a l  low  or if the t e m p e r a t u r e b e h i n d  is a s s u m e d .  temperature would  and. low  of the r a d i a t i o n  speed, of the v a r i o u s d i s c o n t i n u i t i e s  amic quantities  gas.  consideration.  the p a r t i c l e m o t i o n ahead  a tran-  fronts  associated  of r e l a t i v e l y  propagates  situation  the a b s o r b i n g  intensity  with  through  steady radiation  radiation  density  and. h i g h  realistic  the experiments,! s i t u a t i o n u n d e r  At o n e e x t r e m e  both  phenomena associated  f r o n t s for the e x p e r i m e n t a l l y  ionizing  sparent  AND  8  dissociation,  the case of no the  structure  recombination  of a  steady  121 radiation  front  be described Lagrangian  produced  density  of  that,  out  Mach  F o r an  discharge  experimental  through  structed.  angle  a discharge  voltage  10'  value. perature  This of  the s o u r c e  ber densities, N 0 . " j, i l l u s t r a t e d  which  the  this  that  to  front.  A  D-type front  polyethylene,  this "Bogen"  at 5 0 0 0  A  numpre-  to be  times larger  out  experiment  fronts  arc  coa»  was  light  source  Along than  °K.  absor-  at a low of a  the  tem-  at low and h i g h  the f o r m a t i o n  0.2)  this  black body  in i o d i n e  at  (1.9 t  6 0 , 0 0 0 °K to 1 5 0 , 0 0 0  beginning of  con-  and o p e r a t e d  c a r b o n arc.  the e f f e c t i v e  is from  front  outlined.  c o n s i s t s of an  in  of  three  were carried An  vel-  investigation, of r a d i a t i o n  sterad,  is about  Experiments  N  weak'  as a s t a n d a r d  indicates  light;  it is n e c e s s a r y  of 3 . 0 k V ) w a s m e a s u r e d  times as bright  if the  unsuccessful.  intensity  of 0 . 1  the i n t e n s i t y  be m a d e  scheme was  for a  a Barrow channel  (in a s o l i d  on  c o e f f i c i e n t , oc.  the r a d i a t i o n  source,  light  The average  only  of  front.  in g e n e r a l ;  3 s h o c k but w a s  pulsed  an i n t e n s e  in t e r m s  depends  that of  of the  could  the s t r u c t u r e of a d i s s o c i a t i o n  solution was attempted  c e d e d by a  axis  is n e a r  a l l the r e a c t i o n s w i t h i n  erical  X  front  for a s i m p l i f i e d r e a c t i o n  It w a s p o i n t e d sider  This expression  steepening  A treatment  expression  correction must  o c i t y of the r a d i a t i o n c a u s e s an a p p a r e n t  radiation  and on the a b s o r p t i o n  A simple relativistic  oxygen  monochromatic  by a simple analytical  co-ordinates.  the a b s o r b e r  in  by  density,  radiation  122 front.  Although  agreement  with  theory was quite reasonable.  gests a similar d.e.  light  the m e a s u r e m e n t s w e r e q u i t e c r u d e  t y p e of e x p e r i m e n t  the  The author  be a t t e m p t e d  with  a  high  bj' m e a n s of p i e z o e l e c t r i c  pressures  lithium  of f o r m a t i o n  w a s about  T h e speed, of p r o p a g a t i o n for  pressure  probes.  (1 a t m ) the s h o c k s f o r m e d v e r y n e a r  f l u o r i d e w i n d o w , w h i l e at  the p o i n t  strong  source.  S h o c k f r o n t s in o x y g e n at a h i g h d e n s i t y , N Q } detected  sug-  all pressures,  at  low p r e s s u r e s one cm from  of the s h o c k s w a s  least  at d i s t a n c e s  were At  the  (0.03  the  atm)  window.  364-- 8  m/sec  far from  the  Li]? w i n d o w . '•'  pliotoionijsation  A t t e m p t s to d e t e c t chamber showed  only  that p h o t o n s  o from  out to  o  electrons detect  brass or  from  ionization  fronts  dielectric  efficient  with energy  input  t i c s at c o n s t a n t differences. evolution  puter  time  These  as unsteady  treated  of a  Attempts  radiation  one-dimensional  b y the m e t h o d  of  theories were  applied  results agreed well  It w a s a l s o p o i n t e d and  flow  characteris"  i n t e r v a l s or by the m e t h o d  time w e r e a v a i l a b l e  knocking  proved- fruitless.  of  finite  to c a l c u l a t e  of the s h o c k s w h i c h w e r e o b s e r v e d  theoretical results.  and  region  in  material,.  It w a s s h o w n h o w the d e v e l o p m e n t f r o n t m a y be c o n s i d e r e d  test  wavelength  in the  1 2 0 0 A to 2 0 0 0 A w e r e e s p e c i a l l y  in the  in o x y g e n .  the The  with the experimental  out  that  if s u f f i c i e n t  a constant  energy  input  comwere  123  used, these methods could he used to obtain steady state solutions.(complete with thermodynamic quantities, velocities and the front structure) calculate previously.  which we had attempted to  It had been hoped that it would be  possible to compare the results of such a calculation with the structure obtained by the method outlined-in Chapter 4 (an attempt at which proved unsuccessful)..  Since this was  not practicable the author hopes that he has at least pointed out a possible mode of attack for future work in this field,, In conclusion, the author would like to point out that future work in this field depends upon the development of extremely intense sources of radiation both d.c. and pulsed„  The author can only dream in anticipation of a  gigawatt laser j, radiating for tens of - microseconds and. adjustable to any frequency desired.  124  A P P E N D IX  A  NUMERICAL CALCULATION OF A STEADY RADIATION FRONT  IN  OXYGEN  The programme used to calculate the structure of an idealized radiation front in oxygen for a black body spectrum F ( # } and absorption cross section oc ( 3 C 1»2) is given below.  (see section  For the calculations we use the o  o  photoabsorption cross section between 1280 A and 1800 A as given by Metzger and Cook, Fig. B.3, Appendix B and assume  A o  a frequency distribution of a black body source at 6 X 10 ~ 31 as given in eq'ns (4,10) and eq'n (D.7), Appendix D, First the total photon flux is calculated by Sira~ psOn*s rule and then standard Runga-Kutta subroutine is applied to eq r ns (3.21) and (3,24) to calculate the photon flux, F/F and co-ordinate x, at selected intervals ^ z-. The o " initial value of x = 0 is chosen arbitrarily at a point where F/F Q — follows:  „  The terminology in the programme is as  h 7-V'kT =  >: —** f(2)'>  ^  —  a  n  d  $FORTRAN C 7C C  I D E A L I Z E D R A D I A T I O N FRONT FOR BLACK C O R RE'S P O N D T N ' G USING RUNGA-KUTTA SUBROUTINE DIMENSION V ( l O ) » F i l O ) ) Q U O , )  BADY  R A D I A T I O N AT 60000 KELVIN 'rN~trXTG"Etl"~————  — — cOMMair-FU'F'WTDX^  :  ...  — — —  DATA A / 0 . 0 0 2 5 1 ' 0 . 0 0 8 7 1 > 0 . 0 2 7 4 ' 0 . 0 4 9 8 1 u 7 0 v 0 . 1 8 9 U » U . 3 0 1 » 10.416>0.548>0.672>0.784'0.894>u.963?l.u>u.985>u.894> •  2  ^TTr&47T0TZ2TV"O7t)'7^5Tt)lTr4"9"8"rcrrO"3T"3T"  T = 6.0 X ( 1 ) =8.0/T  :—:  —--DX=o".Ter7T""'"~~— 10 11  60 .  2  J  =  y,2  "  -  Y(1)=4.518876 Y (2 )j=0.0 DO i 1=1,75 G A L L RK. ( Y ' F >Q > D Z > 2 > 1 ) _ WRITE (6*60) Y ( 1 ) »'Y ( 2 ) » ' DZ=-2.0*DZ ' ~ STOP END "" " S U d R U u T I , \ L " " A U X ~ R . K X Y , F T  _ A 2  ' 21  : 22  D Q  '  :  —  :  —  :  — ™ — — :  —  : — .  ——  ~ •  . : •• •  :— ' . ..••••. , -. ' '  —  .  :  ;  -  , —  :  ~  —  —  -  —  _  •  FJ —  _  _  -  —  — ~ —  COMMON FJ»FW,DX>A<21)>X(21)'F0(21) DIMENSION Y ( 1 0 ) » F ( 1 0 ) » F Y ( 2 1 ) ' D O 21 1 = 1 » 21 F Y ( I ) = F 0 (1 ) * ( E X P ( - A ( I ) *Y-( 1 ) ) ) F Y T = F Y ( 1 ) - F Y . ( 21 )  DO' "2'2" K=I vio  :  :  F0( I ) = ( X ( I ) * X ( I ) )/ (EXP (X ( I ) ) - l . U ) FOT=FO(1)-F0(21) : — - — : D O 12 = 1 '10 ^ F0T =F0T + 4«'P*F0( 2*"K)+2.0*FO( 2*K+.l ) Fw=F0T*DX73•0. WRITE ( 6 » 6 0 T FVv : — — FORMAT(1X»10E12.4) DZ =- 0 >06  __ 12  ~  :  DO-'li- I = 1 > 21 —  '  :  :  DO 10 I=l>20 X ( I +1)= X{I)+DX  '  .  :—  FYT =FYT +4.0*FY(2'*K)+2.0*FY(2-*K+1) FZ = FYT-*DX/3®0 .... F J = F Z / F W . F ( 2 ) = 1 . 0 / ( 1 . O - F J ) RETURN  ?  —  —  —~  ~~  ~  "  — —  •  -  1—1  ~  '  '  "  ~ ~  —  :  . :  :  :  :  —:  "  "  r SENTRY  "  E X D  'Vs'' 9 ;L 6  01 ~ZL  A  _LL to 9_  8 7 C  A P P E J D  I X  B  Fig.B.1 Scale drawing of Bogen light source consisting of cylindrically symmetric electrodes separated by epoxy and strengthened with fihreglass. Figs«B„2 and B.3  Absorption cross sections for iodine  and oxygen reproduced directly from the literature.  125  Fig. B e 1  Scale drawing of Bogen light source.  126  . 4500  0000 5500 A Fig. i.—Extinction-curves of iodine vapour.  6000  Pig.£.2 Iodine absorption cross sections (Rabinowitch and Wood (1936))  PHOTON ENERGY, hf(eV)  9-54 SH8 8-B6 8-55 8-27 7-80 7-75 I6-7 14-9 130 11-2  9-29 I85±l5cm"j 7-43 5-58 372  7-51  _—_/?5 3 .!H J ' l i !  I M Y V/AVELEMGTI  -if '  1-86  1300 1350 1400 1450 1500 1550  1600 1650  Y/AVELENGTH, X (A)  FIG. 6.  0 2 absorption. A few examples of results obtained by the first and second arrangements are indicated by circles and triangles.  Fig.B.3 Oxygen absorption cross sections (Metzger and Cook (1964))  127  A P P  EQUATIONS  FOR  D  E N  C  I X  SPECIAL  REACTION  SCHEME  The reaction eq'ns (4.5) are written to indicate different groups of particles (0 atoms, molecules,  molecules).  equation for each group.  molecules, Og  We can write one conservation The equations corresponding to  eq'n (4.8) in the simplified case considered in Chapter 4 (using the subscripts i = 1, 2, 3, 4 to indicate 0, 02, °3j °2  Particles respectively) are  + ^  Jdi  ~ ^ 1  Z+l—>3  - X ; Jsi-i M, Nzri - JE,- J W  '  MA/+ Mi  1  4-l~f->'3  _ jj£ j  <£jh v-  //,*Af£  C.l  __ / gr  /zyJ^j  /T*  I  *C. 2 /-rr/  j-t  -/• ^  3+1—>3  //, /i/,  ^  ^  //, >3  y^'  C.: /  5065  +  +f3£l  Jli fa/Ji V'l  +  ^  'Mtf+tti  // //,•  — ^-Jut  //<? A/c /  The notation reaction process  is as for e q ' n  process  M is the t h i r d  particle  Corresponding through  four different  where  (4„8j, the a r r o w s i n d i c a t e  (e.g. k/ii  for the r e a c t i o n  C.4  is the r e c o m b i n a t i o n  0 2 + 0 -v M^  • — > 0 3 + M-Lf  in a t h r e e b o d y  to eq'n  rate where  collision).  (4,9) the p h o t o n flux  t y p e s of a b s o r b e r s  is the p h o t o a b s o r p t i o n  the  passing  is  cross section  for  the  x•th _p a r,t i.c , le. We may now obtain  the  individual  j t e r m s in  eq'ns  ( C . l ) to (C  .4 ) „ s i n c e The  h  individual  if,  m <?X/VrW < <P A / 3  C, 6  t e r m s are  , /  thus fc  ,  -/i-^/VJcfX C.l  N  a/  f f ^ o ^ ) r '  3  r  '  < ^  e  -dVj  C. 9  ,  - 71 /  /  /  /4{ /  {*)  C.8  0,  <3  /J  C . 10  or J  129 w h e r e we h a v e split Og m o l e c u l e s 2  ^^ ^  h ^  the p h o t o a b s c r p t i o n  according  i s  Jche  0  c r o s s  (3*).  cross section  to the end p r o d u c t of the section Their  sum  associated  with  of  reaction  the  reaction  is e q u a l to the t o t a l  cross  section  Coll Finally concentration  the d e n s i t y  of the g a s is r e l a t e d  of the v a r i o u s  to  the  t y p e s of p a r t i c l e s by the  rel-  ation  ^ ^  - i M M  + & (X,*) + 4  +  -  C.12. In o r d e r know 24  for the  photoabsorptiori  of these q u a n t i t i e s experimentally For  example,  a n d are  various  ( C , 4 ) one  collisional  a r e not  known;  lumped with  has  been  poratures),  the  intex'ir.ediate  that  processes  have been  measured  fronts would  most measured  constants.  ignored.  the  the M  (over a s m a l l r a n g e of step involving  ignored  Thus  the r e a c t i o n 0 + 0  the u s e of t h e s e c o n s t a n t s  and  m o l e c u l e s are  state molecules.  molecules has been  ving radiation  What  excited  the g r o u n d  •£^0 /a 4- M  ally excited  function  a r e c o m b i n a t i o n s of r e a c t i o n r a t e s  the v i b r a t i o n a l l y  must  Unfortunate1y  cross sections.  r e c o m b i n a t i o n r a t e c o n s t a n t s of  size  ( C . 1 ) to  (6 X 4 « 2 4 ) r e a c t i o n r a t e c o n s t a n t s a s a  of t e m p e r a t u r e four  to s o l v e e q ' n s  tein-  vibration-to  empha-  in calculation;?  invol-  be s t r i c t l y  We wish  permissible  only  if  130 the v i b r a t i o n a l  and  in equilibrium,. necessary  Furthermore  in such  (i.e.  be temperature  (4,19)),  function  cross section  case  treated  in C h a p t e r  for each g r o u p of p a r t i c l e s m a y be  eq'ns  Thus we  =  and as a  had  would  dependent).  input  from  be  cross section which  the p h o t o a b s o r p t i o n  A s for the s i m p l i f i e d the e n e r g y  were  a c a s e it w o u l d  as a f u n c t i o n of w a v e l e n g t h  temperature  directly  d e g r e e s of f r e e d o m  to u s e a p h o t o a b s o r p t i o n  been measured of  translational  ( C . 7 ) to (C.10)  (see e q ' n s  4,  obtained  (4.18)  and  obtain  £>  •  ffc -  f^ K  M  C. 13 ,fl"%r)  octM)  C. 14  /?  > r  •  I / S r-s  i  dK . C. 15 JET  (/ Jx C. 16  where  -fa-, - 0 s i n c e  o^  energy  input  the sum of t h e s e i n d i v i d u a l  is s i m p l y  (p )  = 0,  The  total rate  of  contri-  butions  £ fx, -£-) This in e q ' n  treatment  of  (4.5). i n d i c a t e s  the i n t e r m e d i a t e eral  the e n e r g y  than  if these  steps  -Ax,*) (p  *f<)  the m o r e g e n e r a l  reaction  that  it is n e c e s s a r y  in a c h a i n r e a c t i o n  flux c o n t r i b u t i n g  steps were  ignored,  C , 17  process.  the  scheme  to c o n s i d e r  to the front because  .  is  In  all gen-  larger  intermediate  131 particles absorb photons in a different part of the energy spectrum than the initial pure gas.  132  A P P E N D I X  D  METHOD OF CHARACTERISTICS AT FIXED TIME__^NT^VALS  An excellent treatment of the equations of unsteady flow with no energy input by the method of characteristics is given by Hoskins (1S64) in a form directly applicable for calculations on a computer. Here, v/e will extend Hoskin1 s treatment to include energy input„ Fig0 Del shows the typical mesh in Lagrangian space and time.  In order to determine conditions at the  i+t  Fig0 D„1  point D on the tj  Calculation of an ordinary point D,  +  -j baseline we require values of the  flow variables at A and B on the t-; baseline which may be obtained by linear or quadratic interpolation between the known values at Sj ™ j„ *  J  and  +  The equ'a-  tions to be solved as given in section 1 of Chapter ? may be written in dimensionless units by making the  133 substitutions  So 'A/O Tfrfor  the  length  convenient  reference  eq'n (3.20) . dividing  m/sec. by  time dimensions „  and  by  The  the  The  cross  velocities  speed  of  by  their  r - p/pQf The rate respect  of  h » energy  input  to the e q u a t i o n  affective  adiabatic  we use  »> 1 . 4 .  gQ  These finite along  values  q is made in w h i c h  exponent  dimensionless  difference  form.  the Mach lines are  h/hos  g  is  following  at 3 0 0  are made t  = 0.  u,  = u/cos  °K, c Q  already  c  -  and  with  Since  the  dimensionless  (7.3)  in  evaluated  written  XJ o {-£„  .t&p-Xc)  =  ^  pcj^o  330  c/coo  may be written  Eqrns (7.11 )  s  Thus.  dimensionless  it o c c u r s .  by-  dimensionless  at  equations  some  dimensionless  in o x y g e n  quantities  initial  is  as defined  are made  sound  thermodynamic  dividing  section  where o ( m  d.2  D.3  D.4  A l o n g the p a t h l i n e BD„  (ho-Q  eq'ns  ( 7 „ 5 ) and  (7„4) are  -  ep  D0 5  zz D0 6 T h e r a t e of  energy  input corresponding  to e q f n  (4„19')  <9 =  ca/4$ t-  /  ' Ji  is  y  j f  'f,: f's. e*-/ D„ 7  where CONS  ' & -O 3  and. w h e r e we h a v e u s e d EQ  h ^/kt,  is the energy flux entering  flux normalised  w h e n G(tJ  that  yJ  in  eq'n  of  the  «  J!i olJ\  =3 1 0 0 0  dissociation uated- a l o n g  equation  for 0  e q u a t i o n ) a s g i v e n by e q f n a p a t h l i n e to  (Mf* J  ^ziA M  m!  time d e p e n d e n c e  (in our c a s e ) such  The conservation  n  S  the g a s a s d e f i n e d  '(4„X8)s G ( t ) is the diinensionless energy  c/<{5 ) ~  particles  (or  ( 4 . 1 4 ) may b e eval-  be  (>y)  If) / Do8  135  where we have substituted for k df and kyz/from eq'n (4„17). In this equation M is the mass of the oxygen molecule and. F q is the photon flux entering the gas.  Substituting from  eq'n (4017j for k d ? and k / i 2 eq9 n (D. 8} becomes =  * Je^S* Cy)  ffe  •—77,  JiZ  >  "  -'SVr-r-  7-  //  > D„ 9  where the temperature T  > T  T/T  tfcl k  %  « T/300 °K and where  AONS j, BONS and DONS are  AONS  CotiS  D „ 10  .3  3. & & A /a' ^ r %  D „ 11  *  D 0 12  X/o' r0 J  The equation of state 3."elates the various thermodynamic quant it ie s l/  ~  r  jo-t  '  /  r  7  Do 13 DO 1 4  y-  where /  ~ ' ^ iiH?)-  ' J ts:.._  + rye  D„ 15  136  Finally the speed of sound and the position in Eulerian co-ordinates are  and -  Xe  D.17 The procedure for obtaining a solution at the point D is to calculate approximate values of eq'ns (D.8J and  (D 0 7)„  and  Next one calculates u„  from and p  from ,eq?n*s (D „ 1) and (D„3) (using values of the thermodynamic' quantities at points A and C which are located approximately from eq'ns (D.2) and (D.4))0 culates h p  , gp  s  £  , Tp  Next one cal-  from eq'as (D.5),  (D.15) s  (Dc13) and (D„14) respectively0 Finally a p  is obtained from eq'n (D.16J.  Using  these calculated, values the whole iterative procedure is repeated until all the quantities have converged to the desired accuracy.  Finally the position X^  is obtained  from eq'n (D017)„ The point of formation of a shock is located by calculating the point where the right flowing Mach lines first intersect,,  The shocks may be treated, according to  the procedure outlined by Hoskin (1964}„ For the case of no energy input, the choice of the difference intervals /it and Zs, sr- depends upon how well a straight line approximates the actual curved. Mach lines (i0e upon the relative change in the variables across these  intervals}.  If energy input is included then we have an  additional constraint in that the interval ^z- must be small enough such that the energy absorbed in the interval is small compared to the energy incident upon it (i.e. such that straight lines segments may be used to approximate the exponential-like decay curves, see Fig. 3.3).  From  practical experience v/e found that it is best to use = ^ z - (in dimension!ess units] and also that the product  ^ 2* CONS should not exceed 10. The procedure outlined above was used in section  • 7o30.l to calculate the development of the shock fronts observed experimentally in oxygen in section 6.2.  The cal-  culations were carried out for initial pressure of 1.0 atm and 0.1 atm. 22  The peak photon flux was assumed to be 1.16 °  9  X 10 " ph/520 A cm "'sec and the time variation was taken to simulate the shape of the Bogen light pulse. The values of the constants and difference intervals which were used for each case v/e ret  1.0 atm  O.J. atm  / 0 /M - 1.69 X 10lscnf'3  f Q / M = 1.69 X 10 18 cm~ 3  AONS w 0.0131  AONS  0.131  BON.S r*0„llS5 X 107  BONS ==0.1X95 X 10  CONS « 3 .0  CONS  DONS ^ 0 o 80  DONS «0.08  7  30.0  ^ t « 1.32—^(0.1 jisec)  A t =» 0.264 —^-(0.2 |>sec)  A 2 r* 2.64 —s*(o00657 cm)  ^sr ~ 1.056 —s-(0.0263 cm)  TOO J.G O  The computer programme used in the calculations is given on the following pages.  The symbols used corres-  pond to the terminology used above except for the following temperature , T — E £  h f /kT s J *—>• S j density s  -f—^ Rj  effective adiabatic exponents g-—t-GAMj enthalpy, h—s-KHj, time, t —  ^  t  —  co-ordinate„ x — ^ X ^  D  j Zv z — ^ d z ;  Eulerian spatial  The symbols XX, IIU and CO refer to  the posit ion j, particle velocity and speed of sound of (right flowing) Mach lines0  ADDENDUM:  In section 7.1.1 we stated that for calculations  in this thesis we would neglect the term ( 3g/ < 9 1 i n eq'n (7.11).  It was found that this resulted in the calcul  ated density being consistantly too high (violating the principle of conservation of mass).  To overcome this  difficulty we were forced to include this term. in eq'ns (D. 1) and (D. 3) we assumed that ( ^ g)s  As shown at the  midpoint of the Mach lines AD and CD was equal to ( ^g). gp  -  g J along the pathline BD (see Fig. 7.1).  SPALiSE $JOB SPAGE  MOUNT 79087 ,  ST I ME'—  TAPE H.A.BALDIS 100  "10  :  :  :  ~  ~  :  -  :  :  ~  :  :  ;  —  5IBFTC MAIN C METHOD OF C H A R A C T E R I S T I C S AT F I X E D T I M E INTERVALS G~ "" P R E S S U R E = 0 . 1 ' A T M ^ ~ C F F R O M BOG-EN L I G H T SOURCE D I M E N S I O N SAM ( 8 > 6 5 » 8 ) >DX(20)>XMIN(20> ~ D T M E N'ST'O S'(^2T) » ' P O " ( " 2 T . " l " l T L " r 2 T l ~ F X t 7 2 T T T r F 7 L ( 2 i ) » A ( 2 1)  10  11  :  12  ~~~  :  • • '  —  ;  DATA A/Q.0025l>0.00871»0.0274»0.0498'0.l070»0.1890»0.30l» 1 0 . 4 1 6 > 0 . 5 4 8 > 0 . 6 7 2 > 0 . 7 8 4 ' 0 . 8 9 4 > 0 . 9 6 3 ? 1 .-0 » 0 . 9 8 5 > 0 . 8 9 4 > 2 0 . "6 4 7 > 0 . 2 2 T » 0 . 0 7 4 - 6 ~ > " 0 T 0 4 W > " 0 T 0 3 T T / — ~ ~ ~ ~ = • — : • T W=6 . 0 • S ( 1) = 8 • 0 / T W ; : ~DS = 0-. 1 6 / T W — — — D O 10 I = 1 > 2 0 S(1+1)=S(I)+DS D O - 11 1 - 1 > 2 1 — — ; — — — : : — • F L ( I ) = ( S ( I ( I ! ) / ( E X P ( S ( I ) ) - l . o ) FO( I )=S( I ) FL ( I ) ' : FL"7'= F L ! 1 ' ) - F L ( 2 1 ) — — — r — — :— F 0 T=F 0 ( 1 ) - F O ( 2 1 ) D O 12 K=1>10 F LT = F L T + 4 • 0 * F L ~ — — — — • . FO-T = F O T + 4 . - 0 * F O ( 2 * K ) + 2 . 0 * F 0 ( - 2 * K + 1 ) FWL=FLT'*DS/3 • 0  ^  -  —  —  .  -  •  F w'=r o T * D S / 3 . 0 " ~~ — — — : : FORMAT (1X>7E12.4) WRITE(6 > 60) FW ? F W L ^ j M E i ON'" X T 9 9 > 3') "> U"( 9"9 > ' T ) ~ T T 9 " 9 ~ * T f r V T S T T S T ^ ' T ^ ^ 1 H ( 9 9 > 3 ! >Q t 9 9 > 3 ) > H H < 9 9 > 3 ) »GAM'.( 9 9 > 3 ) » X X ( 9 9 > 3 j > U U ( 9-9 » 3.) » C U ' ( 9 9 > 3 ) DATA X ( 1 > 1 ) > U ( i » l ) . » Y < 1>1) >V(1> 1 ) > R ( i , l } , p ( l > i ) , H ( 1 > 1 ) >HH(1>1) » :  60  9 • xx t rv 1") > w ( i >' n>i:ijrmTTcn.TTTT6"A>raTr)-r —: ;—~— 8 0.0>0.0>0.0>1.0>l.o>l.o>1.0>1.0>0.0>0.0>1.0>1.0>1.4/ DATA  __  Q ( 1  1  Ms.L/65 >80/ =0<Q--  -  --•• -  Ml=M—1 M2 = M - 2 A O N 5 = 0 . 131 — — BONS=0.1195E+07 CONS = 30•0  00NS=0o080 DT = 0 . 2 6 4 DZ=1,036  -  :  1  "  —  :  :  — — — — " —  —  :  —  s  ;—~—~~  —  —  1  ~  _  —  —  dza=0 . 5 2 8 do 1 i=1>m  ~  *  t  U( 1 + 1,1 )=u'( 1,1) YTi+-iTi"r=r(TVT"r~~  q ( 1 + 1 v i ) =q ( i • 1 ) v ( 1 + 1 , 1 ) =v ( i , - l ) . r ( 1 + 1 , 1 ) =r ( 1 , 1 )  P( 1 + 1,1)=P(1,1) H(1+1,1)=H(1,1) rriTrTT7 = C(T7TT  :—:  ~—r—:—" '  :  '  :  !  "  :  — — —  ~~  —  ~ —  — : — :  :  :  — —  :  r  :  :—;  —  H H ( 1 + 1 , 1 ) = H H ( I » 1 )  UU( I + l, 1) =UU( 1,1) c'j (1 + 1,1) =cu ( r , t )  gam(i+i'i)=gamci>i> x x ( i + 1 » 1 ) = x x ( i , 1) + dz  x"a+irnT^x"n~,T)+D"z  ~  1  ~~~J  :  —  ~~~ ;  -  :  ~  —  ~ :  : 1  —~—  :  :  ~~  —:———  :  d i m e n s i o n g t ( 2 9 ) » t t ( 2 9 ) ' g < 100 ) , t ( 100 ) , t 5 (100 ) d a t a g t / 0 . 0 > . 0 4 8 4 , . 0 9 6 8 r . 2 1 9 » . 387» . 5 8 0 * . 813» . 9 8 1 • 1 . 0 , .99-ov 1 . 9 6 8 v v 9 3 5 » . 9 0 3 , . 86'5 > . 8"2"6" >y739v.6~58v.583 v . 5 t 6 v . 4 ^ z , ~ 2 .426».387,. 355?.332*.313 *.200,.1096>.0452>0.0/ d a t a t t / 0 . 0 , 2 . 64 » 5 . 2 8 » 7 . 9 2 » 1 0 . 5 6 >13 ® 2 0 > 1 5 . 8 4 , 1 8 . 48 » 2 1 . 1 2 » 2 3 . 7 6 >  1  2  26V40V29TO4V3  G(1)=0.0 TCI) = 0.0  :  t5(1)=0.0  ATM = 0. l do t5( do t ci do if  .  68.64>73.92'79.20,84.48,89.76,116.16»142.56»168.96»195.36/ :  ~~  •  554 j = 2 , 2 9 ~~~ j ) =atm*tt'( j ) 557 1 = 2 , l ' )=t c i - l t + d t ~ 556 j = 2 , 2 9 ( t ( i ) . g e . t 5 ( j ) > go t o 556  G I T R G T Q  ~  C(J>2)=C(J »1 )  ( J ) - C I  U - I ) ) / 1  :  :  :  ;  :—  RS U ) - 1 5 U - I ) )  —  :  : '  :  :  )~RR5TTY^TNTTRRGI  go t o 557 g ( i ) =0.0 "continue ' lll-l/10 do 3 n n = 1 » l l l do 3 nm = 1 v t 0 ' " — n=nm+(nn-1)*10 do 4 j = 1»m gam ( j , 2 ) = gam ( j » i t ih j , 2 ) = u ( j , 1 )  •  :  :  ' :  :  1  :  :  ~  :  —  V . m ' cv  f* N  aaimri uivyd n n  *  X *  o z 0  Q 1 > I  * >  .*X n z o  +  >-  Kl CW. * >—  CQ +  Ll  +  +  — cm cm rH  *  cm 'i—i  !+ _i ^ > - I* u. cm * o•>rU. cm +O  I ~I > —~>  —  Ll  W >!< — > * O > + -0 • +— co f-h z c1z to* * >c m o -~ * fM < o E> ho S) % x to o! c ;m 3 * op * «—1 in03 * ! + u . . — i — + o tQ *£L o * ! CM ^ o 1— o i -> p > + >- Mo 1— > o •*• in u- s: o >-c.*m • • H > x > m c m * cr\ * o i— +• + p — 1 O * + < uj x -I H ll. > \ tP ~ — *u o 1— • * b o o > N * I < o — '* < •M CM b X M in1— P > C I I O C M O j CO UJ * + o o a. " * o • q m "> ~> X !• X 1— o ,J|<•p. • • J " M k. r-i — ~)' i e " + !o<T • — \ I- 1 f n rl Hi* x UJ • ~1X >—i 11r-i • h- cm b ~> — oCM i-« !® > X 0 P — J £ ft >e * ! _ j ~ v\— t- _l »-—-'>- iii CH >X CL >- UJ f- " « j. • Q. CT> * UJ CO + +! 11 _l+ >-->*>- 1 :~> 3 I ill ii I z3 * * ffl x x co d o * i * o z o o ;co s: s —i : ii — i n iZ >II O i— UJ z in m owjj uj |® » in o ; o~ u.> * >- i * 13 « rv CMrsj CM —-<>11 t\jM]f.«HX~-OOH « • n o •O rvj ,-nj II LLC.M II cm rh kj1 I I o 1 u u m « h p o ii ^ 11 i— ; n n o h — ii 'f- ll ii c 'm II • 'h- i l l ( M h ; I I > > i l l Z) P ~> il II S 1 Z ill II X i > > 3 II II x ! — i _ l h _Jh " II — T -j ii n — Z > > > < U b — O I- UJ LL. k. ^ O! o > : 0 o — : o 3 u. >- V a:.> x X u u z >- u> X X -i x a M tslP NJ rsj Q O >- ll a 3 <ii.iL.u_ Ll.q L l a. u_ m  ~>  — >—  •—  —  >— *  *  >—  >-  —  -  ^  *  *  —  •->  o o .1—1  i  co  CO  o  jco  co "00  cm _y c.i -i o  1  (M !CM 01 Q) f^ CO  C> c^ LO 'J" CO LTJ 'D  it o h _to  q3iiwn nivao w '  I q: s: < o  J Q  < i  vj O  <M  O *  E<E L3 N _J U * rH _ J  cm  az + az  r N  SI u < *  Nl nI O O  *Q 5: <  cm  QZ 1 QZ  N  N  o ic  —  ->  a  *  3 — H s : r + D co < S -1 n •O e? 1 • .-I —J ) cn I+ * * 3 CQ I QZ O D I CL + !+ a. CL + cm > + n— r, U J 1 + * - u; — o: r- u s: M OZ n < a 01 u u u u < — Q 3 n H*H Vj l* U j * I< iu I a e> * tD —1< u X. — oz •HD r _ l Q .1 3 n cm r-t a rh o 'rh o O 1 1 en r. I II rH * — + :+ + + * • ^ :~> -i ~> 3 » az -J U D CM CM az QZ h - >* Q- 33 • K N ,U N ab a * 3 * — I i s : C M I I _j In — i 11 I cm # » o * or u cl o -< az ju • r ; o u l o — CL x • IT-<< H H jH O 3 • co — u u 1—f P 1—I uj —->-1j •*r-|(ib ~* ! < — — U r ~ U cm * 3 3 p * i (M * (\! * r -> r> ~> s:! i oc h & b; •h — > — — i e> .—s. QZ cd o + pj uu cl Cl + •n>- cm :cm r\jCM a. 3 b si cm Kj jM CL p 3 rsj CM<O '_j 1 (M ~) CM K( 2 : f < I * — LQ « ' j c — INI * '11 QZ5 >+ :+- cm cm .— + :< .D p az i r h ~> + — oz * * * I 3 p -j q >(< >[< * a o u ->QL -) p-)Q u o o * a;Di _J QZ tH ~> - ) + ~ — N <<;< — — — — <3 + — :+ — — 2 — 0 a: b ** o. .— ul uNluNl CL! r.cm cm ^ —• !< az < yj az U t-Nl Nl kl <—i O CL u o: U N 3 cl * I u a: < * u I < 1 * ~> X>* jl<< + *C> i• n "v l IV i +!+ QQ Q Q + + •D in * V I OZ\— — {- • * n I I I 3 < < c u u * III I ai 5: r-i UU — X in i* r-H az~ — — — O CL CM ! E# i—i* * i-tuoz a cl OJ • QZ I + • • i•n 3—• > X -J in QiU u rH .rH I-H X — — i n ; ' « > . — u i n * + a * 0 QZ — 1 3 i+ 3 — ii o o * * — ;CNJ ^ : <£ if. < Llir\j -) * 3 3 UJ o + 1 1 c —-CM inIn —I > . > im•m• Clu uj• xl O 3 3 b O LH CL I • ^• r ^m 'oz O : • -'m m• cmr- m — II • ; « II O <Xo u j • o; cl CL « a;— II p ozoII us-• o a . d 6 < o b oz I I — ii — o (O I I C L r > a o o o I I I I II — (MOjus o cIoI cmo 11 —U i II IIii ,< II II hi X II II a ii a : ii u I I ! n u ii ii ii ii ii a a: II ——1 1 1 ~> iii oz II< :< cc < a;O ti-u u U U .JU N l u O u KD Ll. U- oz _l ll. L l — i- d. I- X > < sj .- <~>'<3 <o oz »:< < < u aza: i Q a a . u ca u O a : cl b a U Q cl a a . u o . Q . M " H O Z uj o UQ > X !< oz ~>  —.  —  +  +  —•  - )  —  ~>  h  q  Q ' Q  -  +  —  Nl  :+  '•*  ~>  vO h-  XI  — >  O CD (N 10 IO m  *  +  "GAM ("JTT) =1.0+11 •  Y i J » 2 )•) /GA  v ( j » 2 ) = 3 ' » 5 * ( gam ( j » 2 ) - 1 . 0 ) *hh i j > 2 ) ' i gam i j * 20 *p i j »2 ) ) r ( j 5 2 ) = 1 . 0 / v ( j »2 )  WtUT2~r=PT'U"r27W-(-3TZT7^T0TYTDT2TT~—;—;—;—— numg=numc+1 i f ( n u m c . l e . 2 ) go t o 1 0 1 . : : cccc=c ( j > 2 ) '— : — ———~—: c<j>2)=sqrt((gam(j>2)-1.0)*hh<j»2)/0.4) numb=numb+1 . /  — —  j > z ) r~b o i o l urr  if (abs(y(j>2)-yyyy).gl..ool^y(j>2)) go to l u u i f ( abs ( c (-j > 2 ) - c c c c ) . g i • • u u 1 # c \ j > 2) ) go io l u u •xtotz)•'='x-(-j', • • :— zom^zon z i m=z i n : —" : —:— co n t t n u e :— : ——do 86 k=1>m1 xx(k'2)= xx<k»1)+(uu(k>1>+cu(k»1))*di : dc 32 j j = 1tm — ' : —1 i f ( x ( j j » 2 ) « g t . x x ( k > 20 ) go 10 83 do 84 i = 1> m2' •  LT = M-I  :  :  uu (1_l »2 ) =uu ( l l — 1 > 2 ) cu(ll»2)=cu(ll-1'2) xx ( l l »2 ) = x x ( l l - 1 » 2 ) uu (1» 2 ) =0 • 0cul.i>2i-c(l»2)  :  :  ———7—:—  :  :  :  :  ~  XX ( 1 Tz) = 0 ® o  go t o 86 x f = x( j j > 2 ) x e = xt u j - 1 > 2 ) ~ : xfe=xf-xe uf =u(j j > 2) 0e = i j ( j j - i » 20 — ~ ufe=uf-ue c f = c c j j. >. 2 ) : : ce='c( j j - l > 2 ) c.fe=cf—ce bes=0.5*dt*(ufe+cfe)/xfe  •  1  • :  :  •  :—  --  c e s"= o 7B"*DT*"cwncrrr-pcutktitwetcet"  x x ( k > 2 ) = (xx(k.»1 ) + ' c e 5 - b e i > * x e ) / u . - w - b e s ) txx = ( x x ( k » 2 ) - x e ) / t x f - x e ) uutkj 2 )=ue+txx*ure " ~ ' ~~ cu(k > 2)=ce+txx*cfe continue  • :  :  :——:  :  — ~  ~  62'  FORMAT (1X,I6) WRITE (6,62) N . W'R I » r t I , 2 ) ' GT^f' 2 ) ' P 1 I ' 2 ) > RTT'TM 1 H ( I ? 2 ) » HH £ 1,2) » GAM i I » 2 } > AA I I , 2 ) » I = 1' M1 ) RNM=NM : - IF( RNM-lO.T7bV71V7T"'~ • •. ' •. •• : : — 71 DO 72 1 = 1 »M > Ir ^»1 )=X( I ,2 .—_ ) _ _ sSAM -AiVi((K NNT'NI ^D T F T 2 T -:  '72 70 ~32 _  SAM CNN,I>3)=PCI,2) SAM( NN»I , 4 ) = Y (I ,2) S A M C N N , I ,5)=G( I,2 ) SAM(NN,1,6)=Rt1,2) SAM(NN»I,7)=H(I,2) SAM( NN , I ,6)=GAM ( 1,2 ) CONTINUE DO 32 1 = 1 >M v(I,3)= Y(I , 1) ; Q.( I , 3 ) =Q( I > 1 ) DO 33 1=1,M  U(I»1)=U(I , 2) Y( I ,1)=Y( I ,2) ' V ( I , 1) = V (1,2) — R(I,1)=R(I,2 ) '.•'•.. P(I,1)=P(I,2) : q'('i'7i j~=U ( I", 2 )'~ H ( I , 1 ) =H(1,2) HH(I,1)=HH(1,2) -3-3GAM( I , 1T=GAM ( I , 2 ) ~ DO 34 1=1,Ml XX(1,1)=XX(1,2) : — uurrrrr=uu-cTT2n—: 34 CU(I , 1)=CU(1,2) 3 CONTINUE _ W R I T E ( 7 ) SAiv; CALL PLOTS M8=M*8  :  —  — :  ~  —  _____ ' :  ;  • ~: " _____—  ;  ________ :  : — 1 :  ~~  :  —:  ~ :  : —  :  :  :  ^—  — —  DO 200 1=2,8 CALL SCALE ( SAM ( 1»l.» I V » M8 »6 • U »AMI N 1 I ) , DA ( I ) , 1 ) CALL AXIS CO„"OVOTOV-rHX>^rr9VOVOVOVX?TrNVrr'"D^"TTT CALL AXIS(0.,0. »1H v 0 , 6 . »9U . , AM IN 1 I ) »• DA ( I ) ) DO 201 J=1 ,8  'j l ;e' '  . :  —  ~ ~ —  —  TP zi  :  20.2  c a T l pltotts/cm ( jti~» i") rsam ( j t 1 > n »+3 ) do 202 k=1>m c a l l plo'ubama j » k » 1 ) >oafh j > f s . i > > + 2>  200  call ploh12.u>u.u»-3) continue  :zoT~~coNTTNaE" ~  ~~  -^Qj- P1I0TWE , stop end  •SENTRY"-  ~—~~  ~~  • '  ~~  1  '  ;  ~ ~  — — —  139  A P P E N D  METHOD OF  IX  FINITE  E  DIFFERENCES  The method of finite differences to calculate fluid flow in one space variable is illustrated in Fig, E.l.  The  general procedure is to calculate the state of the fluid at £  1  Tn-r  1  1 1  u  —  1 1  N ..,.,„„ P  77  1  —  1 1  n-i 3-1 F  ig 0 E o1  1 1  I Sf-l  Lagrangian mesh for finite difference calculations  constant time t Q in each cell in the -2 direction with special techniques being employed at the beginning and end of the interval. at intervals of tin:3  One then repeats the procedure ^ t0  Each cycle depends upon  quantities calculated during the preceding cycle0  Thus  the state and dynamics of the gas can be calculated as a function of  s? (or x) at any time,, t.  The procedure which v/e describe below is similar to that described by Richtrnyer and Morton (1S67), Chapter 12,, (together with comments on "centering"f stability criteriap  140  etc.) and a potential user should refer to this reference before attempting to use it„  Since our equations contain  the rate of energy input q and our centering is slightly different from that of Richtmeyer and Morton and v/e present these equations in difference "form in dimensionless units. (These equations are very similar to those presented in Appendix d,j  They are;  i u ,  -  • i -f , S- _ r  <  &  (b)  e.I  ri-f, JL? - :U -t  ^  ,  )Jj 3  -> Wri- /  o  /  —•  yj  ^ ' • * ( ' - K - ^ j . J r  /  _(-•  ^  — — — ^  HTIf'^-^)  -h SortS-ft I / . ^ l i •'-I Si l/f+'k.  jV '  a  ^  Js -  f/-?* 2.  E» 5  141  The various constants found defined  in A p p e n d i x D 0  The  an average  value  o t h e r hand  the s u b s c r i p t  in the m i d d l e (see Fig„  « 0  j  -n{r  5 ( P ^  0  ^  i m p l i e s that  + P ^  d o e s not  ) j „ On  indicate  calculated  E o1}o  Calculate  Ui j  p r o c e d u r e used, is as 1  "Ti ' 1 '  and X . j  at b o t h  follows:  1  from e q ' n s  (using a s p e c i a l  (E0.1) and procedure  ends}„  n -!- 1 Calculate Vj  +  x  g r a m m e ) from e q r n + 3o  Calculate or a v e r a g e procedure  Y.  (appearing  as Y ( J  ~ 1, 2 ) in  pro-  (E„3)  1 from e q ' n  v a l u e s for Y j is n e e d e d  for  (E.4) u s i n g  , ^ „ T^f If Tt  approximate  lA .A  )d;f.  the  an  b e t w e e n j and j + 1  ( E 0 2 j for a l l v a l u e s of j  2.  are  indicates  the v a l u e of P is  of t h e i n t e r v a l  The calculation 1„  superscript  (e.g. P ^  a v e r a g e v a l u e J it  in t h e s e e q u a t i o n s  special  142 Calculate q"  •4.  culated  from  in the m i d d l e  of the time  50  C a l c u l a t e E3? + ^  6.  U s e an i t e r a t i v e p r o c e d u r e  J  H ? + -1 from  from e q 9 n  z  eq'ns  7.  Repeat  8„  If n e c e s s a r y  (E.5).  eq'n  (Note Q is  cal-  interval.)  (E.6). to c a l c u l a t e P1?  J  (E.7) and  and  z  (E.8)  s t e p s 2 to 6 for a l l v a l u e s of j„ (because of i n s t a b i l i t i e s or to  the n u m b e r of i t e r a t i o n s r e q u i r e d ) c a l c u l a t e , „ ,„n + 1 n + 1 , v a l u e s ox P.; . and. II-j . j. (i.e. a v e r a g e or  J '^  o  z  the r e s u l t s of s t e p 6 w i t h t h o s e o b t a i n e d  shorten weighted weigh  in a  previous  iteration). 9.  Check  the s e l f - c o n s i s t a n c y  if t h e r e return 10.  is i n s u f f i c i e n t  to s t e p  If s u f f i c i e n t  11.  Increase  is o b t a i n e d  and r e t u r n  not examined  our case  in d e t a i l .  increase  in the v a r i a b l e s per  ceed  about 30%  accuracy  and  z  to s t e p  1, criteria  for  h o w e v e r , v/e find that  ^t  interval should  (even in t h i s c a s e we are not  not  sure  the ex-  about  s i n c e we u s e a l a r g e v a l u e  for  the the  viscosity).  The procedure 7 . 3 . 2 in a n a t t e m p t weak D-type  the v a l u e s  c a l c u l a t e Qs"' j .<„xfor  the s t a b i l i t y  In g e n e r a l 9  of the r e s u l t s  artificial  store  and  j.  time index  V/e h a v e  accuracyy  obtained  1.  accuracy  all v a l u e s of  of the r e s u l t s  outlined  to s i m u l a t e  front preceded  tions for this case were  above was used  in  the d e v e l o p m e n t  by a M a c h 3 s h o c k .  the same  as used  section of a  The  in F i g .  steady  condi3.6  143 initial pressure  21 X  of 0 . 0 1 a tin and a p h o t o n f l u x , F  =  4.72  2  1 0 ~ photons/cra sec for w h i c h AONS »  CONS »  121.8  DONS «  0.008  DT  »  0.04  D Z  *  0„08  appearing  were  0.532  BONS ^ 0.1195 X  T h e v a l u e of a 2  the v a r i o u s c o n s t a n t s  107  8 w a s c h o s e n for t h e n u m e r i c a l  constant  in the a r t i f i c i a l viscosity,, see e q ' n (7.12}°  The computer programme used g i v e n on the f o l l o w i n g p a g e s „  for Most  these c a l c u l a t i o n s  is  of the s y m b o l s vised  in the p r o g r a m m e are v i r t u a l l y the same as used, in oAppe:o,™ dix D . T h e a r t i f i c i a l v i s c o s i t y i s Q ^ — > Q S and a — A A .  SPAUSE ' MOUNT TAPE" $JOB 79296 ROB MORRIS SPAGE , 100  S-fTM'ET"—~  ;  to--—  '  —  —  :  ~  :—:  • J  —  -  $ IBFTC MAIN C METHOD OF FINITE DIFFERENCES C — PRESS'URE = 0 . 0 r A T M ~ — — "—: > ^ C WEAK D-TYPE + MACH 3 SHOCK DIMENSION SAM ( 5*80.* 8) »DX(20> »XMlNI2U) :  —  —DTME~N~STON~"Si~2T")~>~FQ~(^l^  11  12 60  ~  :  —  —— —  ;  t 21) > A I 21) —  ;  —  ~  -  DATA A/0.0°251 »0.00871 »0,0274'U.:U498'u. iu7u»u.l89u»u. 3^1 v l0.416'Q.548'0.672>0.784>0.894»0.963'1.0v0.985»0.894>  2  :  10  :  ~~~  .  T W-6.0 S ( 1) = 8 . 0 / T W Q5 = 0 .16/ TW DO 10 1=1 >2:0 S( 1 + 1)=S(I)+DS D0  :  o.647'0'.2'2r'Ov0746rvO-.t)4:98^0vO-3-737 —;  : ;  —  —  :  FL< I )=(S( I )*S( I ) )/(EXP(S( I ) >-1.0) FO( I ) =S,( I )*FL( 1 ) : F L T = f"'L"( 1) - F!_'('21 ) — — — FOT=FO(1) —FO(21 ) DO 12 K=1>10 _ p L T = F L T + ^ ,0 * FL" ("2*K')+2T0^FLT2^K+X) ~ FOT=FOT+4.0*FO(2*K)+2.0*FO(2*K+1) FWL = FI_T#DS/3.0 ..•"-'•••'•••  FWi= F0T #D S/3. 0  :  ;  :  J — J. • 21  ^  "  — :  —  ' -•  FORMAT (1X»7E12.4) WRITE(6>60) FW »FW L - D I MEN'S I ON OT3 ")"• »"0"r9UT3 ) » Y -l 9 U »3 )'» v i 9o > 3 ) »E < 9 v » 3. )•»P't'9 v•> 3) » 1 H(90 > 3)>Q(90 > 3) »QS(90 > 3). ______ DATA X ( 1 >1 ) >U( 1., 1) > Y( l, l) ,V I 1 > 1) 1 >1 ) >Hl 1 '1) / _ . >E<1_ _>_1) _ _>P l——— DATA M'L/80 > 50/ AONS=0.532 B 0 N S = 0 . • 119 5 E+07 CONS=121.8 DONS=0.008 :—~G=T. o : — dt=o.04 DZ=0.08 ; AA=8 .0 ~~~  QS(1»1)=0.0 DO 1 1=1?M  '  "— —-  :  - ' —: —  — —  :  . —:  —  ;  — — — — -  G1 ~J CO 1 £>H o.<-  aajuwn NIVUD ~t  vj a  *  <t  a m  *  O 3i n *  £I  f)p Q rg 3 it * i — (\j — Ik Q a+ D .N. i• n 00 rH 1— — 3 3 3 > CO* * •d"rH M (M * C O * 3 m g rH rH rH O« o H+ ® ill ~ O 1 I 111 1— *x in O3 N tM I 1 — + o 1 ! >h-0 3'CMP. > C L o > l-J + + P H- O + > r0 3 rH > O * H UI + D jc\j CcM O CM CM (M CMo• —• I * —I I— irH rHr-4t-H.-H * o +S rH m hCO X ' > rH i i i 1 + • ;>-H i HH r H 3 C O ' U J : 1 + I 1 1 1 * 3 33 II _j o* ~* Oc n x 3 3 3 £ • X i I— UJ 3 ~ b — _i i—( I— 1 • t X ID t— r-t o o s > X a. 2: 3 3 in2: s: iJa.X II X > > l • >- h - r-H >x > cl b•o — i j II 1I 1IMl 11 II ~ II I ' « e > II II II — J ;2 X H x N 11 N u — > — * " " O III II Z O O <M ~ — ~ i< M* II 2: II II |M II *rg**roe CL CTv o 2 : * : ! • *O 'rH 1—1 rHP rHi—tr~'t ^ I— * tZ 2 z ; I H I I I U t C M C M C M I I X X C O o inm nn m onLU UJ • rH 3 m o >—< o »» — 3 —. — j~ 3 * ii• i — ( « • > \ z : z — ~ ~ r H ~ r . r H 1 _ e | H I ' • X O O rHill z « : « > [1—1 rHrHu rH r+H'-< rH +_|i + • f\l <M II ; rHrHrHI;0 Oe CoM 2: CM CMvOCMI >—O O III« 111- II Iu I — O H || II •+ + + 1+ + I <n co 2: - CD :<h I I I 1 2 in II II X > ! > 3: II II > 3 2 1 3 *—i — -—' (——4 _I J ' ZrHrHSI, 3 3 3 I I — U W I- t— — (— 1— LU. u I-O H H 3 U.h _J O O II — — 3 O — " — r-HCLO 3 LL.X 3 O >>>->Q > > CL X-« x O X X |3 V > hjCL x a x  t  -—  ®  ^  ^  O O  co O 00 CO .0 co  ^10  CM O  o -J CO CO  >-  *  X * z o p  CM  0  1  >  I  3 3 • 1+ r-l r- I  I— >-  X  *  z o  L 3  CO +  £  M  +  I  ^ *  I ~  -  H-  CM  >-  >+  O +  Ll *  M  cm  O  CM  P  P no P—  * ( M  O  >  C<1 P— I  I  3 >-  D£> rH I  CM I  3  CM I -3 1-1 — CO > 3 +  ->  I  <T-  — >  *  fM X * 1—I -O 3 * o• —1 • 3 K•P O •  O ®  m ir<~i  CM  — CM LL. CM p Cl * Ml <~ I 3 3 m z p cm ; — ^ vO CM * CI o P D. X O I P CO ® ft + o + + r-t P — »!<: V C M INl c m ~ > ; * — 3 < t c m c m o 6 0 i n t r * U- Ico — — 'o '• CM * a I-I CO a * I r-1 * i :l O t-. X CM P 3 p 3I 3 i• n CM * P ^r ~> i o r o z P >:< 3 I >- CMcn r-i . <X — c m ! o 3~ p w < > • > cn*u a x— ll — >I - LL! > CNJ — o ii cm 3 +1 I * Jj« r-H CM H O UO• |lO•I~> cm tm cmcm cm cm m in ® i Q 3 | » I S D * — — I ^ | r-l r-l t—i cm uj M (M UJ -h o Ll) 2: • ; • uj — -I Ift + : • O O O >: cl + c m « I I 2 5: X s 2: & r~t 5jC H I— vj"-I II Oh- II II ! II o cm I I 11 o• i ii cm .cm II II _J + vO l 5: auj cq 11 co cm — •> • ;CM >- > q. x o > - —• — >O II -• z v: v i—LL CM P L L «C M CM O 3 cm cm 3 3 2: cm cm s11 p p ;-> ll II >nftr-l« ^ ^ Z Z 3 » ^ 3 to co in o— e VH ~ ~ >- LL. >* »• Z»>—{< rsl t-r H-l < t 1 — I r—\ | | c m lu C M l l L L • > ' Z c ^ i < < , 0 3 c m r l r l | | I I u . 3 O I C M C M c. cm f 1- r-II—I SJ " - rH— ;|1 |1 n — IICM I— I I I ffl >I- I u>- I I QII III I Q 'z z z111 | \ p U I 11 n 2 -< zP ~> S 3 p p f- 2: 2: _J i—II 3 2: 1 !-. l u P u . o 3 — r 3 o ll ll o <3 > < — — 3 U h Q a. -x. Q >-<•-< Q p - > Kl O tsl NO < U. U_U_ U-O > ax z 3 P >- > a x U_ U.U- >- ZU_ P LlJ Z * O  CM  -H >-  •  S  L.  ~  O  "  O  _I  I*  <  P *  —  >-  r  Q. *  3  3  LT-  *  V  - -  O  _L  •  + —-  U-  0-  *  *  P  O  O  I  U. *  L  O  U-  A.  X  fc>-  O  ,M  «-  S>  ~  I  >"  —  0.  I>  P  O  »  1  *  P  M  Q  —  I -  *>  >-  + *  •  O  3  —  UJ  *  Z  T-  LL  I  I  >-<  —  5:  H  ]|  H  P  O  >  IL  O  P  II  II  U_  W  CM  O  -«  ~ O  II  X LU  >-  >  CM  1  •-> —  O  D  UJ  CM M  <1- c<1 O o  P  r cr<  CO  00  vO o  -  in p  -  X  2:  Q ( M > 2 ) =0 ( Ml, 2 ) """" • " " .-: " QS(M,2)=QS(Ml, 2 ) FORMAT (IX , 9 E12 . 4 ) "F0R M AT~~"( TXT2 I6T — • — — — — — —-———— — WRITE (6>62) N , NUMB WRITE .( 6 > 6 1 ) (X(.I »2 ) »U( I »2) »Y ( I-i »2 ) »V( I-l»2 } »E< I-l»2 ) »'p< -r-l»2) » : 7 I""' Hfl-i'i2 ) VQ (T-X»-2 ~~~ RNM=NM I F (RNM-10 • ) 7 0 , 71 > 71 • . " ' • • ' • ': . ' D0~~7 2" 1=1 »M ~~~ ~ ~~~ ~ " ~~ ~~ : _ SAM ( NN, I »1)=X( 1,2). SAM (NN,I,2)=U(I,2 ) , : "SAM (NNYTY3 )~= P (1,2) . ~ ~~ ~~ ~ SAM(NN,I > 4 ) = Y (I>2) SAM (NN, I * 5)=Q( 1,2) : : 5AMrN.N¥TT6T=VTr>2rr : : : — — -— SAM{NN»I>7)=H ( I > 2) SAM(NN,I>8)=E(I>2)__ •' _ \ : : GO TO 32 ~ , '" ~~~ 1 DO 9 I = 2 »M U(I,3)=U(I>2) : "p"n-i'T3")=PTr-TT2")" : • ~ . ~ : ~ " H (I—1»3)=H(I—1* ) NUMB = NUMB+1 L : : go t o ioo : ~ : : ~ ~~ ~ ~ : DO 20 I = 2 > M ". .. x(i,i)=x(i,2)  "U ( i > 1) = U ( I ? 2 ') Y(1-1,1)=Y(1-1,2) V( 1-1,1 )=V( 1-1,2). E (I-1,1 f = E ( 1-1,2) : P<I-1,1)=P(1-1,2) QS(.1-1, 1) =QS( 1-1,2 )  ' . ' ' •• , • • - • •-•• ' . ' . • : ———— ~ — - ^ " -  CONTINUE WRITE (7) SAM 7 : : 1 : CALL PLO IS - — ; M8=M*5 • • CALL SCALE(SAM(1»1»1)»M8»10.,XMIN(1),DX(1),1) DO 200 1=2,3 "" — ——— CALL SCALE(SAM(1>1»I ) >M8,6•0 »XMIN(I) ,DX( I) , 1) CALL AXIS (0.0»0.0»lHX-»-l»iO#»0.0»XMIN<-l> *DX(1)> C A LL A XTS ( 0 . * 0 V'»l:H"T""0"»"6T"»~9''0T"»-XM"rN"rr)~»"D"X_(T"r) : DO 201 J =1,5 CALL PLOT I SAM( J ,1 »1) »SAM( J ,1 » I ) ».+3 )  202 201 : 200 _  ~ "DO * C A L L P L O T ( SAM ( J » K » D » SAM ( J » K » I ) » + 2 ) CONTINUE CA'LL" ~ P L O T 1 1 2 . ~ O V O T O V - 3 1 :—: CONTINUE CALL PLOTND : .. S T O p END  SENTRY  BIBLIOGRAPHY  Ahlborn, B., Phys. Fluids: 9, 1873 (1966) Ahlborn, B. and Salvat, M., Z. Naturforschg. 22 a 9 260 (1967) — — » Axford, W„ I Phil. Trans, R„ Soc. London. A253„ 301 (1961) Blackman t V. s J. Fluid Mech. i s 61-85 (1956) Camac t M., J. Chem. Phys. 34, 448 - 459 (1961) Camac f M., and Vaughan, A. , j. Chem. Phys. 34_f 460-470 Chandrasekhar, S . " R a d i a t i v e Transfer", Dover Publica'tion y New York (1960) Courant, R., and Friedricks, K.O., "Supersonic Flow and Shock "Waves"j Interscience, New York (1948). Elton, R. C , j Plasma Phys.(J.Nuc.En.Part 0)6,401  (1964)  Goldsworthys F.A. , Phil. Trans. R. Soc. London,, A253, 277 (1961) " Hoskin, N. E., Methods in Computational Physics 3, 265 (1964) ..•'•.• Hurle, Ic R c y Reports on Prog. Phys. xxxf 149 (1957) Kahn, F. D., B.A.N. , 12, 187 (1954) Kuthe, R., and Neumann, Kl.K, 9 Ber Bunsenges Phys. Chemi jS8g 692 (1964) Landolt - Bornstein (1950) II.4, p 717 Lun 5 kin, Yu„P„f Soviet Phys. - Techn. Phys. 4, 155 • (1959) Mathews s D. L„ s Phys, Fluids 2, 170 - 178 (1959) Metzger s P. H„ 9 and, Cook„ G„ R o p J 0 Quant. Spec„ Rad. Tr.ans. 4 P 107 (1964) Mises, R. v o y "Mathematical Theory of Compressible Fluid Flow", Academic Press N.Y„ (1958)  145  mill, Mo R., and Lozier, W. W,, J. Opt. Soc. Am. 52, 1156 (1962) Oswatitsch, K. , "Gas Dynamics" , Academic Press, New York (1957) Panarella, E., and Savic, P., Can. (1968)  J. Phys. 46, 183  Porterf G. , and Smith,J., Proc. Roy. Soc. 261, 28 (1961) Rabinowitch, E.f and Wood, W. D. , Trans. Faraday Soc. 32, 540 (1936) Ramsden, S. A., and Savic, P., Nature 203, 1217 - 1219 (1964) Rich tin y erp R. D. , and Morton, I£. W. , "Difference Methods for Initial Value Problems", 2nd ed. Interscience Publishers, New York (1967) Rink, Jc P., Knight, H. T., Duff, R. E., J. Chem. Phys. 34, . 1942 - 1947 (1961) Schwarz, W. M.,"Intermediate Electromag-tfetic Theory"', John Wiley and Sons, New York, (1964) Shapiro, A. H. , "The Dynamics and. Thermodynamics of Compressible Fluid Flow", Ronald Press Co., 1 New"York (1954) Zel ! d o v i c h Y a . B. , and Raizer„ Yu„ P.,, "Physics of Shock V/aves and High-Temperature Hydrodynaraic. Phenomena.", Academic Press, New York (1966)  

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