• 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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Investigation of the dynamics of radiation fronts Zuzak, William W. 1968
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Title | Investigation of the dynamics of radiation fronts |
Creator |
Zuzak, William W. |
Publisher | University of British Columbia |
Date | 1968 |
Date Issued | 2012-05-29 |
Description | 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 radiation 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 account. 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 experiment 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 10 μsec 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. |
Subject |
Radiation |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2012-05-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085835 |
URI | http://hdl.handle.net/2429/42426 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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