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Investigation of the dynamics of radiation fronts 1968
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Title | Investigation of the dynamics of radiation fronts |
Creator |
Zuzak, William W. |
Publisher | University of British Columbia |
Date Created | 2012-05-29 |
Date Issued | 2012-05-29 |
Date | 1968 |
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 [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2012-05-29 |
DOI | 10.14288/1.0085835 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/42426 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0085835/source |
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INVESTIGATION OF THE DYNAMICS OF RADIATION FRONTS • - by WILLIAM W, ZUZAK ,E. (Eng.Sc.Phys.) University of Saskatchewan, M.Sc., University of Saskatchewan, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the department of PHYSICS V/e accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1968 In presenting this thesis in partial fulfilment of the requirements for an.advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and Study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by h ils representatives. It is understood that copying or publication of this thesis for financial gain shal1 not be allowed without my written permission. Department of J?// 5 The Universjty of British Columbia Vancouver 8, Canada Da t e Z-f, 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 temp- erature may be calculated if the structure and all re- actions within the radiation front are taken into accountc An analytic expression can be obtained if part- icle 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 demon- strated 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 mod- ified to include the energy input. The agreement between the theoretical and experimental results was satisfactory. iv TABLE OF CONTENTS page Chapter 1 INTRODUCTION 1 1.1 The problem 1 1.2 An outline of the thesis 7 Chapter 2 BASIC EQUATIONS AND ASSUMPTIONS 9 2.1 The possible types of flow 9 2.2 Conservation equations for a dis- continuity in one-dimensional flow 13 2.3 Rarefaction waves in a one-dimensional flow 17 2.4 The equations of state 20 2.5 Estimation of the temperature behind the radiation front 23 Chapter 3 PROPERTIES OF STEADY RADIATION FRONTS 29 3.1 Idealized propagation of a radiation front 29 3.1.1 Case of one frequency and one absorption cross section, 30 3.1.2 Case of black body radiation F ( P ) and continuous absorption cross section j, oc( 36 3.2 Relaxation of restrictions on particle motion and recombination 42 3.2.1 The coefficient, 5- 42 3.2.2 The energy input, W/ S, v7 43 3.3 Weak R-type front 44 3.4 R-critical front 48 3.5 Weak D-type front preceded by a shock wave 52 3.5.1 General relations 53 3.5„2 Iterative procedure for calcula- 55 tions mmmmcmiv Table of Contents —Continued page 3.6 D-critical front preceded by a shock 59 3.7 M-critical front preceded by a shock 64 Chapter 4 THE STRUCTURE OF STEADY RADIATION FRONTS 69 4.1 Conservation equations of mass, monentum, and energy within the radiation front 70 4.2 Reactions within a radiation front 71 4.3 Special case of a dissociation front in oxygen 74 4.3.1 Conservation equations for absor- bing particles 75 4.3.2 The rate of energy input per unit volume, / ( x , t) q (x,t) 78 4c3c3 Calculation of the front structure 80 4.4 Concluding remarks on Chapters 2, 3 and 4 82 Chapter 5 THE BOGEN LIGHT SOURCE 85 5.1 Description of light source 86 5.2 Measurement of intensity 88 o 5.2.1 Absolute intensity at 5000 A with discharge voltage at 3.0 kV 89 5.2.2 Intensity as a function of wave- length at 3.0 kV 92 5.2.3 Intensity as a function of dis- charge voltage . 92 Chapter 6 EXPERIMENTS AND RESULTS 94 6.1 Beginning of formation of dissociation front in iodine . 95 6.2 Shock fronts in oxygen 101 6.3 Attempts to measure ionization in the test chamber 107 vi Table of Contents — Concluded page Chapter 7 UNSTEADY ONE-DIMENSIONAL FLOW WITH ENERGY INPUT .. 108 7.1 Method of characteristics 109 7.1.1 Physical characteristics in Eulerian and Lagrangian co- ordinates 111 7.1.2 State characteristics 112 7.2 Method of finite differences 114 7.3 Application of the two methods to dis- sociate fronts in oxygen 116 7„3 „ 1 Shock formation for time depend- ant radiation from Bogen source 117 7.3.2 Structure of a steady dissociation front 119 Chapter 8 SUMMARY AND CONCLUSIONS 120 Appendices A NUMERICAL CALCULATION OF A STEADY RAD- IATION 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 page 1.1 Classification of conditions encountered by radiation fronts 4 1.2 Hypothetical experimental situation 5 2.1 Schematic representation of flow veloc- ities for various values of FQ/N 11 2.2(a) Steady discontinuities in an R-critical front 14 (b) Steady discontinuities in an M-critical front 14 2.3 Propagation of a rarefaction wave 19 3.1 Radiation front travelling in + x dir- ection with velocity vr 31 3.2 Plot of the radiation equations for various values of FQ/CNQ 35 3.3 Idealized radiation front in oxygen for black body radiation F( 40 3.4 Weak R-type radiation front 47 3.5 R-critical radiation front 51 3.6 Weak D-type radiation front preceded by a shock 58 3.7 D-critical radiation front preceded by a shock 63 3.8 M-critical radiation front preceded by a shock 68 4.1 Plot of velocities versus N 0/F Q 83 5.1 Schematic representation of light source 87 5.2 Light pulse from Bogen source 89 5.3 Experimental setup for absolute intensity measurements 90 5.4 Intensity of Bogen source as function of discharge voltage 93 6.1 Schematic representation of experiment with iodine 96 6.2 Typical oscilloscope traces for measure- ments in iodine 100 viii List of Figures — Concluded page Figure 6.3 Increase in light intensity during time of light pulse 100 6.4 Schematic of experiment in oxygen 102 6.5 Oscilloscope traces of piezoelectric probe 105 6.6 Shock strength as function of d at 400 Torr oxygen 105 6.7 Velocity of shock at 400 Torr oxygen 106 7.1 Mach lines and path lines of charac- teristic net 109 7.2 Computer profiles, 1.0 atm 118 7.3 Computer profiles, 0.1 atm 118 7.4 Computer profiles, 0.01 atm (method of finite differences) 119 B.l Scale drawing of Bogen light source 125 B.2 Iodine absorption cross sections (Rabinowitch and Wood (1936)) 126 B.3 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 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 '/z c„ velocity of sound defined by c„ = ( •Pp/oV)c. fa • o o D dissociation energy E ionization energy £, internal energy per gram F • - F(xst) ~/F(^,x ft)d^ 8 photon flux (eq'n (3.16)) F Q - F(o,t) g ==. h/£ , the effective adiabatic exponent (eq'n (2.12)) G(t) S F(xst)/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 Symbols — 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 temperature u particle velocity in lab frame of reference v V particle velocity in frame of reference of the closest discontinuity velocity of radiation front (eq'ns (3.2), (3.25')) W energy flux Eulerian spatial co-ordinate y degree of dissociation Lagrangian spatial co-ordinatej (see eq'n C (3o20) x xi List of Symbols — Concluded. oC photoabsorption coefficient y' 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 (101) f density X. ionization or dissociation energy * denotes a molecule in a vibrationa11y excited state ACKNOWLEDGMENTS The author wishes to acknowledge the stimul- ating 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 espec- ially 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 I : INTRODUCTION 1.1 The Problem In most plasmas produced in the laboratory, radiation is considered an undesirable energy loss mech- anism, 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 investig- ation 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 theor- etical 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 mech- anisms 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 dis- sociated (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 bet- ween the particle density No, the- photon flux and the velocity of the radiation front,"V^ (e.g. Goldsworthy (1961)). ' _ F V f - Fo/ N o This relation assumes that each photon ionizes (or dissociates) exactly one particle and F C , wherecis- the speed of light (see section 3.1)« 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 prop- agates 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 encoun- tered 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 i s 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 D-critical weaic weak R-type M-type D-type Pig. 1.1 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 I weak critical strong j v, < O- , -zr, < ex, 1/; < a, D-type = ir^ >ftj | iT, > a, •p~, > (X, type j ĝ. >- & a •Vk = fti In this thesis, we wish to make further theoret- ical and experimental investigations of the development and propogation of radiation fronts and phenomena/assoc- iated 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 sit- uation, let us consider a tube containing N0 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 ener- gy is directed into the absorbing gas, see Fig. 1.2. -transparent /- radiation window / front Y /Ekaions) oicm sec } absorbing gas N 0 (cm ) Fig. 1.2 Hypothetica,l experimental'situation. A radiation front will form and propagate away from the window into the undisturbed gas. According to eq'n (1.1) the velocity 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 -^compared 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 F0/N0 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 poss- ible 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 occur- ring within the radiation front„ A knowledge of this temperature makes unique solutions possible. Thirdly, the experiments performed here, indicate the existance of rad- iation fronts in agreement with our theoretical investigation For this, we have modified the theory of unsteady one-dimen- sional 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 10^sec« 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 ex- pected 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 experimen- tally. The main results of the thesis are summarized in Chapter 8. C H A P T E R 2 BASIC EQUATIONS AND ASSUMPTIONS Let us now consider in more detail^ the experi- mental 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 conser- vation equations of mass and momentum may he applied. The energy equation must he .modified to include the radiant- energy 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 be- hind the radiation front) there is no difficulty in predic ting the type of flow which will occur. For F 0/N 0>^a4, the radiation front propagates so rapidly that the parti- cles 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 propa- gates 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* a4 • I»et us envisage what occurs as we decrease the radiation flux F 0 from a value at which vr = ^0/5 a4* -?he various cases are illustrated in Fig. 2.1. 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 rel- ative to the particles entering (vH = Vp+a^)while the tail travels at the speed of sound of the stationary par- ticles behind it ( vf = 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 com- pression wave) followed by a, rarefaction v/ave. velocity Fig. 2.1 Schematic representation of flow velocities for various values of P0/N0 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 = vp+a4 where vp is the particle drift velocity). 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, â ., 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 F0, 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 men- tioned 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 rad- iation 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 K 4 f, K h, If, (a) iTr^CLs F fA % & « f, pA . ^ r* = • Pz = h3 p, h, 1 1 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 a r e assumed to be identical, the velocities v2 and v3 are, of course, dif- ferent in their respective frames of reference.* If a rarefaction wave exists, the quantities behind it are la- belled 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) Compare footnote on page 4 j C _ L JL tr z Z. 2.1 2.2 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 is defined by W — < 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 discontinu- ities 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). The value of g varies between 1.06 and 1.7 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 ^ / p, ML= ! 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 dif- ferent 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 differen- tial along an isentrope c s ~ 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^. At time t = o, the tube comes to a complete stop. 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 P 5 / 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 , fJL ' r z - • 2.10 ^ V / To obtain the temperature T5 we use the relation 7V _ ^ Jl 9 n -T% - 4 AS* ' where M4 and M5 are the initial and final molecular weights of the gas in regions 4 and 5. rarefaction rarefaction I * I P+ j * • s~ | •—:—s- _ Pig. 2.3 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. £ 4 a n 20 2.4 The ..'equations-.;of state An equation of state relates the thermodynamic quantities such as the pressure, density, enthalpy, tempe- rature, 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. = -f y 2.13 since . We note that the temperature does not appear explicitly in these equations, however, the adiabatic expon- ent 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 ioni- zation respectively, n- is the particle density of the d component, T. is the temperature associated with the tran-j ( 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 ' k 2.15 y — Z- J J ' ,/jfg ^ 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 fa 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-= m^. S ti 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 temper- ature. 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 contri- bution to the internal energy is given by (see Zel1dovich and Raizer (1966),"p 181) / - V J6 M ^ L - • 2 .16 £ -tr-Lli — ^ h vy^ J ) where for oxygen hY'/k = 2228 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 /= (nm + &na> M. Combining eq'ns (2.14') and (2.15') we obtain an expression for £, in the form •L~- J 2 . 13 1 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 equa- tion 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^ = vF 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 al- ways 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 problem. In order to obtain the specifying equation, one must be able to calculate in detail the state of the par- ticles as they pass through the radiation front. It is also necessary to know the detailed structure of the discontinuity. In general, this is a formidable task since one must consider the reaction rates occurring in the front, collision and excitation cross sections, transition proba- bilities, energy transfer by means of radiation and so forth. In addition to the obvious complexity, we have a further problem in that quantitative values of many of the reaction rates, etc. are either inaccurate or unknown. Por the present, we will not attempt to obtain the structure of the radiation front but will, instead, show how to estimate or give limits for the final temper- ature, T/m behind the radiation front. A first order approximation to the temperature behind the radiation front may be obtained by considering the energy equation (2.3). If we consider a dissociation front entering a diatomic gas at room temperature , the enthalpy h-j is given by h-| = 3.5 kT-j/M, where M is the mass of the molecule. The enthalpy behind the front is given by h4 = (5kT4 + D)/M where D is the dissociation energy and where we assume com- plete dissociation but no ionization. In section 3.2 we will show that the term ¥ / / 1 v 1 may be written J H = 5<^> 3 '29' 7? if, " — ? Thus eq'n '(2.3) becomes 3.5 kl-i/M + h ^ 2 + 5 < h ^ / M = (5k T 4 + D)/M + £ v 4 2 which may he solved for the end temperature behind a dissociation front * " ~ 5 - 2.18 The corresponding temperature for an ionization front is 2-s^A >7 5'<" -•£• + -j -m*. z) • ~ ~ : : i 2.18- where E is the ionization energy and m„ is the mass of ci the atoms. From-eq'ns (1.1) and (2.18) or (2.19') it is evident that values of f and v^ are required to calculate the final temperature. One can expect the temperature to be between rela- tively narrow limits. For a dissociation front the equi- l i b r i u m temperature behind the discontinuity must be larger that some minimum temperature.,' T m i , at which virtually all the particles are dissociated (for example, we arbit- rarily assume that"virtually all " is 99.8$). For an ioni- zation front the equilibrium temperature* behind the discontinuity must be at least such that a. 11 the particles are still ionized after passing through the rarefaction wave. In fact, even if radiation losses from the plasma behind the discontinuity are ignored, an equilibrium temperature cannot be reached since there is a finite probability of particles recomblning even at extremely high tempera- tures. On the other hand, if radiation losses are con- sidered, then it is not possible for the radiation front and the flow associated with it to reach a steady state unless (perhaps) the radiation flux increases in time in some special way. Nevertheless, the concept of an' equilibrium temperature is necessary in the "steady state" approximation of radiation fronts. For a dissociation front in oxygen at atmospheric pressure, we choose* T m i n = 6000 °K at which temperature only 0.2# of the gas is in molecular form and 0.005^ is ionized (Landolt- BOrnstein II.4, p 717). On the other hand, we can define another tempera- ture, This temperature is that value of T 4 v/hich one obtains from eq'n (2.18) for a weak R-type front with v 1 = v 4 and f. = 1. As an example, let us consider a dis- sociation front produced by 1420 A (8.8 eV) photons enter- ing a cloud of oxygen molecules which have a dissociation energy of 5.1 eV. In this case, T _ ^ 8 9 0 0 °K. max If the photon flux JP is extremely high ( such that the weak R~type front travels so rapidly that there is little compression (v^ = v^) and there is little time for the hot dissociated atoms to dissociate other molecules In. collisions. Also, at a temperature of 8900 °K, one could expect little recombination. Thus one photon dissociates only one molecule,, such that ^ = 1 . In this case, we expect the temperature behind the radiation front to be T N = 8900 max If the intensity of the 8.8 eV" radiation is rela- tively low ( such that 3 ? 0 / N 0 « a^) we shall obtain a weak D-type front preceded by a shock. In this case, although there is a large expansion (such that v ^ > > v^) the term o p (v 3 - v^ ) is small compared v/ith ?<h'Z//-D in eq'n (2.18), Also, there is plenty of time for the hot 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 T4__slmin = 6000 For intermediate values of (such that Po/.No^-a4) for which we obtain D, M or R-critical fronts, the situation is more uncertain. In this case, the term (v.,2- v, 2) or 2 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, then1>1 and we expect T. „ . If collis-4 max ional dissociation is predominant over recombination, then j < 1 and we expect T . < 1 A • T , . Unfortunately, there ' uiJ-Xi _ [flci.A seems to be no criteria by which one could predict the relative importance of collisional dissociation and three body recombination within the front. As another idealized example, consider radiation o at 912 A entering a 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 if ^ J i T ^ k , f = / 4 max T4 = Tmin i f ,• S ^ or (ii)- <(h^>~~?f > f o / N < , « ° L < t T%<i T4 < \>ax ' i f <*>*> I , - ^ , S < T4> Tmax if , 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 prop- erties and the flow velocities of the gas for each of the five types o Por this calculation,, we must make two assum- ptions o 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). Secondly, we assume the temperature» 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 NQ 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 vf to approach a steady state. Let us con- sider such a steady front for the case where the radiation Fq 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 Casej3i one fr e que nc an d one absorpt ion cross section,, . — 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. In the 3 lab frame of reference Fc/c ™ c/V0 photons/cm travelling with the speed of light, c, enter a stationary gas of NQ 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 M B SYSTEM o o cm^sec Ir C 6 0 w /absorber^ ~x V2=c v ^ O FRONT SYSTEM —to— / — 0 ( <2/0 absorbers F oVcm^sec —Jso— -X' V - C vj=0 Fig, 3<>1 Radiation front travelling in tx direction with velocity v F. \ entering the front from the left is F^ y(l - v^/c)WQ (photons/cm2sec ) , where X s (X » v^2/c2) * . Equating the photon flux to the particle flux, we obtain ? which may be solved, for vF 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 (vp~>c). For low intensities such that F0/Nq c, we obtain the expected non relativistic relation, vF = ^ 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•left to zero at the extreme right. Similarily, the absorber density X ) varies from N at the extreme right to zero at the extreme left. 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 I 3.5 PCX,*). = 'Ac exF , /-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 - so, in general, we must include the first order correctionf F' - (1 -w/c )F. Thus instead of the usual differential equation for exponential decay * Since we are not particularly interested in the relativ- istic 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 S f - A - - - ^ . 3.7 This differential equation is readily solved by separation of variables using the transformation u = (F/F0) and then y = u - ! • f ^ = ^M, {/-f- fc/K i-?/-?*) - ( / />/«£-//„ ) x4 or, retransforming / 3.8 where the proportionality constant yQ(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 yQ(t) may be evaluated. A solution which satisfies the wave equation and the boundary conditions d is FK.-t) F0 ' f(Xj-t) / fof X - , zfr ~ > L t for • K = o > ° > for- 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/F0 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 FQ. 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 N0. The front speed is given by the ratio F0/N0. These dependencies are illustrated in Fig. 3.2 in whibh F/F0 and N/No of eq'ns (3.10) and (3.11) are plotted versus «<N0* for F0/cN0=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 Nq = 10 cm , F q= 1020cm 2 sec"^ and photoionization cross section for hydrogen at 912 A, <*„=. 6.3 X 10~18 cm2. 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 108 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 1 q density extreme. Here, typical values are N = 10 cm , F 0 = 10 22 cm 2 sec~^ and the photodissociation cross sec- —1 ° 2 tion o(Q2 - 15 X 10 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 spec- trum. Eq'n (3.12) indicates that the width of the radiation front is inversely proportional to the absorption cross sec- tion. 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 is 0.01 o^ tho thickness will be 100 cm. 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 ^ x ) J r .̂ 14 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+F0/cN0) from eq'n (3.2) v/e obtain -Vl [h fW*,*) jr) = -{<+ tH/) N M . 3.18 The following procedure proves convenient in obtain- ing 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(») <= N0) . 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 7£ ZfX,*) •== No + f ^/rrJ*] 3.20 and where is the maximum absorption cross section in the range ̂ t o ^ - Also, we have written o, t) — o ) since the incident radiation is constant in time. 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 det- ermine 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 pas- sing 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<„ and fox' negative x A / -* /j ) 3.24 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') app- roaches but never reaches £—>0. We now have F(2-)/FQ as a function of from eq'n (3.21) and x as a function of from eq'ns (3.24). We may thus plot F(x)/F0 as a function of x to obtain the overall structure of the radiation front. The individual frequencies FC^x)/F0 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 temp- erature 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/F0 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 va- lid 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 ion- ization or dissociation are negligible permitted us to ob- tain 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 N0 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*/*)F0 = j X 3.25 or ac = — — - — — ~ F / + ^ / 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 ) - f j 1 ^ 3 . 2 6 whei'e F < T h = /, h//F(^)d^/ is the energy flux and O / ' I (1 -fp/c) is the usual relativistic correction. If the par- ticles ahead of the front are stationary with resjject to the source then v = . The density may be written as -/f M <w0 -t . --••/; 3.27 where the subscript 0 refers to the absorbers (of density N ) and the index j refers to the dissociated or ionized particles and to any impurity particles which may be pre- sent 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. Thus we may write "^f^T ~ " - w . + )Ar/= > 3.28 which with the help of eq'n(3.25') becomes -wlc, I M, / f 3.29 v/here we have written V]_ - vF. 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. The hot gas is com- pressed with a compression ratio between 1 and 2 (1 /4//1 < 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_ = v4 (assuming 5 - 1, for (b^-J- > kTrnin . otherwise for <{h i /)>-'X ^ kTmin> T4 Tmin^ a n d t h e 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. FD/J Nq > 10a4) we can make the further approx- imation 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 •A- ̂ / * -tr^i) ̂ * £ • - • « frg) i M • • • 3.30 where K = * ~f \ V1 • Substituting for W/ "?f vj from eq'n (3.29') we obtain A ~ , ̂ fklz'll^f! h fe^lfe*-') • vf [_ Mors J z '[ 3.30' where v/e have let = NQM and =» as defined in eq'n (3.25 For the non relativistic case (vA = F0/5N0) we obtain to first order which illustrates the relationship between the compression ratio and F0, N0 and % . The particle velocity behind the front (v = vi~v4) is related to the density through the relation v p 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 inver- sely proportional to the velocity of the radiation front, Vfz . The coefficient J and the function g4 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 p̂ ^ = N 0 kTi and = nQ M. This ratio should check with the value obtained from the equations of state. Obtaining numerical solutions from the above equa- tions 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 ^ (3kT4 +ZJ/M where X is the ionization or dissociation energy. The effe- ctive adiabatic exponent is then g4 = h 4/£ 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 calcula- tion 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_=2.69x1025ph/cm2sec F -2.69x10 ph/cm sec .Fig. 5-,4 Weak R-type radiation front formulae above or directly from eq'n (2.5} and the con- servation eq'ns (2.1) to (2.3J. An example of a weak R-type radiation front is shown in Fig. 3.4. For this and other examples (Figs 3.4 to 3.8 inclusive) we use oxygen at a pressure of 0.01 1 HI Q atm (N0 = 2.69 X 10 particles/cm )and a temperature of Tj = 300 °K. We work out the examples for two final tem- peratures T m i n ~ 6000 °K and T m a x » 8900 °K. For Train we use h 4 = 2.31 X 1011 ergs/gui, g4 « 1.146 and a 4 = 1.835 X 10 5 cm/sec. For T m a x > h4 -=5: 2.78 X 1011 ergs/gm, g 4 = 1.228 and a 4 = 2.517 X 105 cm/sec. The dissociating photons have an average energy of 8.8 e K and a value of 5.08 e V 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 var- ious 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 = 10s cm/sec. At T m a x we find vx = 1.17 X 106 cm/sec ( 5 FQ/vA N 0 - 0.855) and at T m i n, vx = 1.33X106 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 rela- tive to the undisturbed gas ahead of the front is still super- sonic. This is an R-critical front. It corresponds to the Chapman - Jouguet point and can be considered the high den- sity 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 v4 = (g4 f -f ̂) s= a4 (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 where we note that px/ -fxvl ^ 1 f o r a n R-critical front (M 1 2 » 1). The particle velocity behind the front is given bv ft • Vr ^ ^ _ f ^ 1 2 which to first order in p ^ may be written as 3 3.36 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-̂ , -Z7̂ and W. Neglecting terms of p ^ / / ^ we obtain • ^ (z^-')^,) or, solving for and substituting for v^ from eq'n (3.33), r " * ~ • 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 m./^nj. A value f J 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 g4 is obtained. We treat the rarefaction wave following the front as an isen- tropic expansion in a manner outlined in section 2.3. For a f i r s t approximation we assume g5• « g4. (Quantities behind the tail of the rarefaction carry the subscript 5). Once the temperature T &, pressure p5, density -f5 and enthalpy h 5 are approximately known, a more accurate value of g 5 may be ob- tained and the final properties of the gas calculated more accurately. An example of an R-critical radiation front calcul- ated 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 1022 photons/cm 2 sec and the front- travels at a velocity vx = 4.55 X 105 cm/sec ( S -0.777). For T • F = 6.25 X 1022 photons/cm2sec and Vj = 3.41 X 105 • 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 by_a_shoc^va^ Weak D-type 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 ^ ___ — -Tn r - r - ' l i W y c * 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 vx 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 fol- lowing relationship for the particle velocity between the shock and radiation fronts: o .42 where the velocities are defined as in Fig. 2.2; the sub- scripts s and. p refer to shock and front respectively. The final pressure, obtained from a momentum equa- tion 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 p4 must fall in the limits £P 2 <^^4 ^ p2' The equations in this weak D-type case do not lend themselves to approximate solutions as easily as in the weak R-type and R-critical cases. We, therefore, fol- low a numerical method of attack. As usual, we assume the 55 final temperature T4. We then assume a reasonable shock velocity v^, calculate the thermodynamic quantities behind the shock front, calculate the velocity of and thermodyna- mic properties behind the radiation front and finally, cal- culate the radiation intensity required to produce the vel- ocity. An iterative procedure is required to obtain exact solutions. For potential users of this technique, we will out- line the iteration procedure in more detail in the next sec- tion. 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: We choose a shock velocity, v-j,, and calculate p2, f2, T 2, h2, g2, and the velocities v 2 and Vp. (For oxygen we use curves of p2//?1, T2/"rl v e r s u s Mach number given by Gaydon and Hur.le/page 52). We will see below that the final pressure p4 must fall between £ P 2 < P4_<P2 and can thus calculate a value of h 4 within 5% (using assumed T4) and. g4 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 / ^ V ^ )Z ^ _I±. • 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 v4 = v p + v3 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 v3 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 and v 3 m i n usually agree within a factor two. 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 v3 and -f^. (The solutions of v^ tend to oscillate about the final value and it is thus best to aver- age the initial value of v3 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 v3.) We then calculate v4 and P 4 from eq'ns (3.42)and (3.43). 57 With a relatively accurate value of P4 we can obtain accurate values of h^ and g^. Similarily, one can calculate accurate values of V3, v^, 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 F0, 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 1021 photons/cm2sec, enters the shocked gas with a velocity v3 = 0.4.17 X 104 cm/sec < S = Fo A / v 3 ^2 No - 0.719). For T m i n, F 0 = 4.62 X 1021 photons/cm2sec and v3 = 0.491 X 104 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 5 - 1 x( 1 Cr c m ) — x(l05cm) ]?o=4.62x1021ph/cm2sec 1^=4..72x1021ph/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 N0. This behavior is markedly different from the weak R-type (super- sonic) radiation fronts. 3.6 D-critical front preceded by a shock We will now discuss the low density limit of sub- sonic (weak D-type) radiation fronts. This limiting solution is called the D-critical front. The appearance of a D-cri- tical radiation front is exactly the same as for a weak D- type 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 v4 = a4 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 P2//2 v3 2 ̂ 1 (°r M32-£<1) for a D-critical front. Eq'n (3.48) may be rearranged and solved for vo 60 ^r - - I 1 — , T,u 2 where we have used the definition a^ « 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 us an extra relation involving the energy flux W. We solve this equation for v g to obtain •v* - sv -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 pres- sure drop across the radiation front are the same as for the weak D-type case (see eq'ns (3.42) and (3.43)). From these equations and eq'n (3.48) we may obtain the pressure ratio 3.52 If typical numerical values are inserted we find P 4 ^ g- P 2 - From this equation and equation (3.51) we obtain the pressure behind the radiation front + [_ - ^ / J ~ 3 . 5 3 With a lengthy calculation we can obtain the shock front velocity associated with a D-critical front. For this purpose we use the relation v p = a 4 - v 3 to rewrite eq'n (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 v p - ( * / * * • ! ) - / ^ J / vlf v/here M-̂ 3 v^/a^ is the Mach number of the shock. For strong shocks the term in brackets approaches unity. We may write a 2 2 = a x 2 (TaAj.) , where • • o Substituting these equations into eq'n (3.54) we obtain a quadratic equation,the solution of which is \ O t 0 0 62 with the correction terms J f c j ^ r i ^ / 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 calculation with the correction terms included. The critical density may be calculated by using the pressure ratio where again we assume, g 2 = g±. From the definition a 1 2 - g x w e obtain f<t*+i\ %rJL- Jfps) =Y-T). ft'tixr) > 3 / 5 8 where the initial pressure P 2 ( ^ ) and. velocity v x (PC) are defined in eq'ns (3.51) and (3.55) respectively. We note that in the first approximation depends linearly upon the photon flux F c. Since the D-critical case is a singular solution separating the weak D-type and M-critical cases eq'n (3.58) is useful in predicting which front shall occur for a given set of 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 K t-o 0 O T-=6000 °K 1 x(10 cm)• P0=2.32x1022ph/cm2sec 10 0 T,=8900 °K 4 0 I c- x(10 cm) • P0=3 • 96x1022ph/cm2sec 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 105 cm/sec) precedes the radiation front. The radiation flux, 22 2 F Q = 3.96 X 10 photons/cm sec enters the shocked gas with a velocity, v3 = 0.183 X 105 cm/sec ( F Q /?1/v3 / 2 NQ = 1.02). For T m i n ;a Mach 6.02 shock (vi =1.98 X 105 cm/sec) precedes the front. F Q ^ 2.32 X 1022 photons/cm2sec and v ^ 0.162 X 105 cm/sec ( 0.893). We notice that the «J appearance of the flow is very similar to the weak D-tyBe case| however, the shocked region is narrower and the pres- sure 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, NQ and Ta it is found that the high density limit /j of the supersonic radiation fronts is still considerably below the low density limit /]_ (0C) of the subsonic radiation fronts (see Fig 1.1). The region between (*<r) and ^ (^r) corresponds to the M~type conditions of Kahn. 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, >aA! must have a rarefaction wave following it. 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 travel- ling at the same velocity. This can be confirmed by num- erical calculations. 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 v̂ . - a^ (g/.p̂ / 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 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, vg ** a n d from eq'n 59) we obtain 3.59 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 g4 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 ĝ p̂ //̂ .-. 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 des- cribed 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„ 3.4.) 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 10 5 cm/sec ( $ 0 . 9 6 8 ) . For T m i n , F 0 = 4 0 8 7 x 1 0 2 2 ph/cm2sec and v 3 0 . 2 7 7 x 1 0 5 cm/sec ( S = 0 . 7 3 5 ) . 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 treated radiation fronts as ideal discontinuities behind which the gas was all dissociated,, It was necessary to assume a final tempera- ture in order to calculate the flow associated with the front. The previous results would be unique, if the final temperature could be calculated. We feel that it is one major contribution of this thesis to realize that this temperature can (at least, in principle) be obtained if all details of the rates of ionization, dossociation and recombination processes within the front are considered s and a stepwise integration across the front is carried out. It is the aim of.this chapter to outline such a detailed calculation. As an example we will discuss a dissociation front in oxygen. Again, we consider the radiation front as a one-dimensional steady state discontinuity with energy input. . Unfortunately, it turns out that most of the required rate coefficients are not yet known, and also that the numerical integration is quite difficult (and was actually finally not successful). Therefore, the 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. (Zel'dovich and Raizer (1966)f Chapter VII), we will include viscous forces and heat transfer in our discussion of the radiation front structure. The conservation equations may be written • Similar to the treatment of shock front structures f (X) Vfx) = 4.1 PCX) -h ffx) ~ e/jrfxl J X z 4.2 JTCx) c/ X •f \ i dirod\ ' ^ "ITx/ - 7T>. 71 The terms on the right hand side of these equations are con- stants of integration, expressed in terms of the initial values of the flow variables, distinguished by the subscript "o". vQ 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 mol- ecules are found in various stages of electronic, vibrat- ional and rotational exitation. Collisions between hot particles within the front tend to cause further dissocia- tions . 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 free- free or free-bound two body collisions or radiational de- excitations. "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 sec at atmospheric pres- sure ) . It takes a much longer time to reach equilibrium between the vibrational and translational degrees of freedom„ Blackmail (1956) estimates that 2.5X10 7 collisions at 19 o 300 ° K (9.6/ftiec at a standard density of 2.69 X 10 cm u ) and 1.6 X 103 collisions per particle at 3000 °K (0.083^sec at a density of 2.69 X 1019cm~3) are required to reach equil- ibrium 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 col- lisions 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 dis- sociation 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 are- dominant : / /& OO ^ ^ — ^ & * & At s M ^ +Af O £>x ^ • — + M . & + ^ ^ 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 con- cepts 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 Conservation equations for absoxMng ;part- icles. 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 __ - m)* + ( f l , ) 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 photons and is defined by eq'ns (3.18 and (3.16). We assume that the photon flux is sufficiently small so that the term (1 + F0/cN0) may be neglected.* The solution of eq'n (3.18) may be written as • „ fix * L tj.x&Jr - i ' ̂ e * ' 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 con- sideration is ; 4.11,, where M is the mass of the 0 2 molecule. The degree of dis- sociation y is defined as 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-;/) M 4 ' 1 2 ) -1 In * Actually the term (1 » vF/c) » (I + F 0 A N Q ) „ :LS„ 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/Vat + 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 recombina- tion 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 temper- ature 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 equili- brium 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 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 temperature the values of k̂ . obtained from eq'ns (4.17) may be significantly in error. Nevertheless we shall use these values for calculations In the thesis. /( X; t) % ( X, — 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 obtain 4.17 large as (-3) and small as (- -t). Consquently, at room 4,3.2 The rajte of energy input per unit volurae 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 resul- ting 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 develop- ment 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 , / ' sy st • ~ U- -tr 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 deriva- tive 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 f ij^) u j^ji 2- (t-^)- sjr^ . 9n m I 3 X / A/ cPX /yf ^ , 4,<1U where -tr- u - ŷ 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 v2/2 from eq'n (4.3). Unfortunately, the iterative procedure did not converge negative values of the density and imaginary values of the velocity always occurred. Perhaps this is hardly surprising since the v2/2 term is about 10" times smaller than the Vq 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 vP and vs as a function of N0/F0, The diagrams in Fig. 4.1 show such a plot. 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 pres- ented in Fig 4.1 unique in terms of the final temperature. Pig.4 .1 Plot of velocities versus Nq/F0 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 through- out 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 5 THE BOGEN LIGHT 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 ex- perimental 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 ef- fective 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 kKis 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 polye- thylene 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 con- sists 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 prem- aturely. 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 inlet for helium polyethylene dump chambers window baffles test gas \ test chamber 25AB\20kV capacitor bank over clamped Fig. 5.1 Schematic representation of light source 88 separated by epoxy strengthened with fibreglass , A 3/4" diameter threaded polyethylene rod is screwed into the epoxy such that the 2 - 4 mm diameter hole serves as the axis of the cylindrically symmetric apparatus. Although this design was quite satisfactory the polyethylene tends to crack after many shots especially at relatively high discharge voltages ; Also, the ringing frequency of the bank decreases as the discharge channel in the polyethylene increases in diameter. Con- sequently, the light intensity was not strictly reproduc- ible from shot to shot and the peak intensity tended to become delayed after many shots were fired„ The dump chambers consisted of 6 inch diameter aluminum tubing of various lengths (2 inches to 12 inches) sealed with 0 - rings. The LiT? and quartz windows were 1/4" thick by 1" diameterj the actual aperture for the radiation entering the test chamber was 1.7 cm diameter. A mechanical shutter consisting of sheet metal, was installed to stop the light from e n t e r i n g t h e t e s t chamber . It w a s operated from outside the chamber by means of a magnet. 5.2 Measurement of intensity A typical oscilloscope trace of the light pulse is The peak intensity of the light pulse shown in Fig. 5.2, 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) The experimental setup is indicated in Fig. 5.3. 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 mirror aperture 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. The optimum size (at a discharge voltage of 3.0kV 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 direc- tion 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 Intensity as a function of wavelength at :Ll£J£L. T o measure' the intensity in the wavelength o region from 2500 A a procedure similar to that described above was used. However, no lenses were used and the neutral density filters were replaced by a set of frosted quartz windows (the transmission of the various, combinations was measured as a function of wavelength prior to intensity measurements), The measurements show that the intensity of the Bogen source gradually increases with wavelength to a value of 4.2 X I0 6 watts/ (cm2ster |i) at 2500 A (i.e. about 10 times larger than at 5000 A). Unfortunately the i.nten- o sity of the carbon arc is very small at 2500 A and accurate measurements are difficult. Neverthe less bs? comparison with Planck black body radiation the values of the intensities o at 2500 A indicate an effective black body temperature of the order of 40,000 °K. 5,2.3 Intensity as function pf_discharge voltage. ---. At low voltages the intensity of the Bogen light source inc- reases quite linearly. However, at higher voltages it tends to increase more slowly indicating a saturation level is being reached„ at around 6 kV. Also this saturation level seems to be larger for larger diameters of the channel in the polyethylene insert. A typical curve of intensity versus discharge voltage is shown in Fig, 5,4. From this curve it 93 discharge voltage (kV) Fige 5.4 Intensity of Bogen light source as a function of discharge voltage. appears that the optinura discharge voltage is around 5 to 6 kV. Unfortunately, unless the window is very far from the light source, it gets badly chipped at these voltagesj consequently, it was preferable to use lower - discharge vol tages and place the window closer to the light source. 94 have ated dow. a window which transmits photons of energy below the ion- ization 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 wave- o 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 C H A P T E R 6 EXPERIMENTS AND RESULTS• Throughout our theoretical investigations we considered steady radiation fronts which are gener- in a semi infinite tube sealed by a transparent win- For experiments with ionization fronts one must use 95 investigations are described below. 6.1 Beginning of formation of dissociation front. in iodine An experiment to observe the beginning of the for- mation of a radiation front requires a test chamber of finite length (see Fig.6.1). If v/e choose a test gas which absorl 3S in the visible wavelength region (where v/e may use conventional monochrornators, filters and photomultipliers), it is possible to measure the amount of light passing through the test chamber. An increase with time in the amount of radiation passing through the test chamber rel- ative to the radiation incident on it (i.e. an increase in transmission) indicates the development of a-dissociation front. Furthermore p if the conditions correspond to a weak R-type case (see Chapter 3) or if the predicted, width of a steady radiation front corresponding to such condit- ions is wide (such that the pressure gradient is small), then there will be little motion of the particles during the 10/M sec light pulse and, therefore, the interpretation of the results is simplified. To carry out such an experiment iodine was chosen as the test gas since it is photodissociated by radiation o ° in the region 4600 A to 5000 A. It has a photoabaorption cross section of 2.4 X 1 0 ~ 1 8 c m 2 at 4995 A (see results 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~30 Cm6/moIecuIe2sec 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 noise ratio ( ̂ trans- mission: transmission) was large. Secondly, 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 . This • dictated the use of a short, test chamber. 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 re- combination 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 1017 particles/ cm3. 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 con- trol 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 jfl̂ j =s (1,12 t 0.03) X 10 cm S8 The experimental setup, is shown in Fig.6.1. It has an optical system similar to the arrangement for measur- ing the absolute intensity of the Bogen source (see Fig. 5.3) A 2.5" focal length lens was used to seal the dumping cham- ber of the Bogen source and also to focus the radiation into the test chamber (with an area magnification of 5.3). It was necessary to clean this lens after every four shots. The transmitted light F passed into a monochromator and was measured by photomultiplxer A plane glass plate was inserted to divert a small fraction of the incident radiation F 0 , into a second monochromator and photomultiplier which served as a monitor, A D~ 4.0 neutral density gelatin fil- ter could be placed either in front of or behind the cell. Since the difference between the signals F and F Q is small and, furthermore, not reproducible, it was necessary to use a differential technique. The experimental procedure is as follows: The neutral density filter is placed in front of the iodine cell and the amount of light, F , entering ° phototube ^"2 (at 4S95 A) is adjusted to reasonable levels and equalized to F entering phototube by means of addit- ional neutral density filters (not shown in Fig, 6.1). The difference.of those two signals is displayed on an oscillo- scope, (ideally this difference should be zero, but in practise this never occurs) and recorded on polaroid, film. The 4,0 N.D, filter is then placed behind the iodine cell and the procedure repeated. If there is substantial depietio: CQ of the 1-2 molecules ia the cell during the time of the light pulse 5the signal from phototube should be larger than the signal of the monitor. The difference in these two signals is a measure of the development of the radiation front. Typical oscilloscope traces are shown in Fig. 6.2. Unfortunately the signals are not reproducible and in order to obtain a meaningful measurement it was necessary to aver- age measurements over 12 shots. The results are shown in indicate the standard deviations. Fig. 6.3.The solid error bars/ The dashed error bars indicate the results obtained with no iodine vapor in the cell (accom- plished by keeping the cell at liquid nitrogen temperatures) for which we should obtain a straight line along the horiz- ontal axis. The deviation from the expected result and the large error bars are testimony of the difficulty in detecting the radiation front in this experiment. The two solid curves in Fig. 6,3 give upper and lower limits for the expected theoretical results;1 Here we used a photon flux F Q = 1.44 X 1G22 ph/300 A cm2sec) and a photon flux half this value (corresponding to the results of section 5,2 using solid angles ox 0.256 steradians and 0.128 stera- dians). The calculations are carried, out as outlined in Chapter 7. (Drift motion and diffusion of the particles as well as wavelength dependence of the absorption cross section were neglected. Also, the radiation v/as assumed to be parallel, ) D e s p i t e the obvious shortcomings of tho Rather than plotting the original and the increased flux which differ only by about 5%, we gave the expected d e - ferences of'both signals in Fig. b.4. . 100 monitor, 2.0 v/div ) 4 > Q f i l t e r CP - Fc), 0.5 v/div) ± n fr°nt °f C e l 1 monitor, 2.0 v/div ) , ' j N.D. 4.0 filter (F - F0), 0.5 v/div) behind cell Fig. 6.2 Typical oscilloscope traces for measurements in iodine. 2 5 0 - 200 - 150 - 100 - 8 10 t(/<sec)- solid error bars -- with iodine vapor in_cell dashed error bars' theoretical curves -- no iodine vapor in cell / S "" I* =1 .44x10^ph/cm^sec & ^ o ' " ' ' ~ ' 0 Fig.6.3 Increase in light intensity during of light pulse 101 measurements and theoretical curves there is general agree- ment 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 develop- ment 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 par- ticle motion over short periods of time. Oxygen was chosen as test gas for this purpose. 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 lcT18cm2 Similar shocks were reported by Elton, (1964). 102 o . - at 1420 A is six times larger than the value for iodine O at 4-995 A. Also the particle densities used were in the region of 10 1 8cm ° to 2,69 X 1 0 1 9 cm" 3 s substantially higher than for the case of iodine. Consequently s a radia- tion front tends to be much narrower than in the case of iodine and the pressure gradients much larger. Fig. 6.4 Schematic of experiment in oxygen. The experimental setup is illustrated in Fig. 6.4. The test chamber consisted of a 2" diameter pyrex T-junc- tion filled, with oxygen at the desired pressure. The rad- iation passes through the 1.7 cm diameter opening in the lithium flouride (LiF) window, is absorbed in the oxygen gas and produces a shock which travels in the direction • S^nce there are no containing walls it also tends to d:l perse outwards in the radial direction, however, this Tseems to have no effect on the axial propagation of -cue shock since a 1.8 cm I.D. tube inserteo co prevent xa?.s diffusion resulted in no detectable dxfierencc xn w e strength of the shock. 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 calculate the speed and initial 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 max- imum 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 0o5 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. We compare these results with theory in Chapter 7. 105 d a 3.0 cm d = 5.0 cm Pressure «= 400 Torr Oxygen 0.05 v/div 50 |isec/div 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 elect- rodes 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 con- sidered 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 in- cident 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 obser- ved 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 al- ong 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 character- istics is given by Shapiro (1954), Chapters 23-25 and Oswa- titsch (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 + cgij any disturbance travel- ling to the left from point 2 will propagate with the speed u2 - c s 2 (where u is the particle drift velocity and cs 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 ther- modynamic 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. We then find the intersection points of 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 forma- tion 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 characteris- tics can be drawn as straight linesj however, we can sat- isfy 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 co- ordinates 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 UA , = m 7-„" / 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 /is the mass density, cs is the speed of sound and fQ is a constant reference density (e.g. the density at t « 0 when (x) is constant). 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, is the density and p is the pressure. 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)) ju , i _ . ^ + 7f ^ ; 7.6 and J^ D f . 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 Chapter 4.3 If we multiply eq?n (7.7) by the speed of sound c„ and use the equation of state in the form of eq'n (2.12) s , - ̂ >'F h ~ 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 Thus eq'n (7.9) becomes 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 = ± / C g / ^ 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 cg was introduced ad hoc and has not yet been defined. The problem here is that for a system of particles not in ther- modynamic 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. We pointed out in Chapter 2 that replacing 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 cg « gp/y7 such that the term containing {a-f/Jt)z in eq'n (7.11) is zero. 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 radia- tion 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 thermody- namic 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 fa is the velocity change over the space interval between lattice points. This viscosity appears in the momentum and energy equations (see eq'ns ( 7 . 6 ) and ( 7 . 5 ) ) . One simply makes the substit- ution p 5>p + Qc, . 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 in- put 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 prohi- bitively 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 indi- cate 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 inter- vals. (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, (A computing time of 10 minutes was used.) 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 p0 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 pG within 3.2 (isec, is 3.3 pQ at 8 jisec and 3.2 pQ 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 1022 ph/cm2sec) which'we used in our calculations was too high. On the other hand the velocity measurements were taken relat- ively 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 com- pression 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 suf- ficient 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, F0=1.16 x 1022.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 1BCL OQO CAi.3f'Cr??tf?A COMPUTER PRODUCTS, SMC. ANAHSSSvl, CALIFORNIA CHART MO. 0 1 HAOE IH U.S.A. j r . U| ! Cr-f : I T i-ULL I —U „1_. Fig.7.2 —continued DENSITY, R=///0 , with /=1.43 x 10~3gm/cm3 so.ona 120-000 140.000 ISO.001 CAUFORr-uA COMPUTER PRODUCTS, 3WC. ANAHEi?/}, CAUFOR^JA CHART WO. 0 1 MADE l?l U.S.A PfEEEffifflfflSffiiE isa.coc CAL:s~on?Jj; COu.'.-UTHH PFXIU'CTS, lilC. PtlAHZV:., C - U r O R i H A C H A R T KO. ' . 1," i- .1, d • Fig.7.3 Computer profiles, 0.1 atm. Method of characteristics at fixed time intervals. Energy input from Bogen light source. Peak photon flux, F 0=1.16 x 1 0 2 2 ph/520A cm 2sec, Time, t=(N-l)At , with At=0.20 (isec. Distance from window, x=0.0249 X /cm/. Curves plotted for (N/10)=1,2,3, •••8. a.onn 5B.QOO 64,000 72.000 p r o d u c t s , a ^ a k f . ? ' ^ , c a u f g r ^ s a c h a : ; t n o . 0 3 MrtDi-: ITi U.S.A I t r frhi' rrr -P: . L ) i , l-l t I-J I . I- Jl.-rt-1-t—t- Ir-;' -!—+—( -1 1 1 Tznir i-rrIt hHirti-h rhF R-fh-mi .-I • 4 .)-- -1. • l„,-l-l, .t-.t-.l-l. .1 -1 -1-1 - !. TTTTT fflftff rnrc - L i - J - Fig.7.3 —continued DENSITY, R=///"0 , with 43 x. 10~4gm/cm3 0D0 72.000 coiwutnrc - r c r r j c T S , ;nc. akahe>", c,AL:FDT?rcjA csjart mo. oz m m m u a -t .i l-i-l-ul- L U4-.U-U1-.UI.. !T;:Ltn i-1 hj-f-H-h J—J 1 1 1 .1 1 I j i 1 + H 1 - 1 1 - -I 1 1 1 ( 1 1 1 1 i 1 r - •1 1 I i i L! - 1 T 1 j J 1 - i — ! — — - - - 1 1 I • 1 1 1 1 1 ! i i 1 1 ! f=y , with 0=sy^l.0 i - 1 1 _ L_ t - - - - i 1 - - - -1 i - - - 1 - 1 - h - „ - - - — - - - - _ - - - - - - - - - i 1 L L 1 - 1 FF i - - - — - 1 — - -- - - - - - • - - •• - - - - - _ - - - - - „ - _ - - _ - _ 1 - - - - 1 - - - -- -1 -1- - 1 r I - 1 - - - - - 1 - - - - - . J - _ ! 1 - 1 - - - t -- - - - - i 1 1 - . L ! - -1 1 i 1 I - - t - - - 1 • - - 1 - - - -1 - - - 1 - - - - — - - - - — - - - - 1 -1 - - -! 1 1 - - - - - - - - L ! 1 - - - - - - - _ 1 L T - - _ — - - - - -1 \ 1 - - I 1 1 _ j - l - - - — 1 1 1 - - - - - - - -I - 1 _ - — -1 - - - - -1 ! 1 1 ! 1 t i 1 — — — - - - - - 1 _ - - i - - - _ - - - T 1 X j - - - -1 i _ ! i 1 ! 1 1 - 1 1 1 1 1 1 - - — I 1 1 L - — - — - 1 X E •f 1 - _ i _ 1 I 1 j - - — — - - - -i 1 1 - j - -i 1. r 1 1 ! 1 } » 1 —1— i-l— i H - — 1 - - | 1 !_, ! - 1 1 - 1 - _ : • - I - - i i - - - 1 - 1 i i I - 1 1 I | ! 1 1 ! - - — ! • —V- r . 1 1 1 8,000 is,qoq 24,000 32.000 X 40-qdq 48,000 5S.Q0C! •64,000 72.qdg cc.'.t,?j.n;-:r; p-.-.c^-jzts, sr;c. chart no. 02 -tad!." in u.s.a v[r ih -Jr u-La.-ci-.il ;.i. l.»fc-U.LU.L,LL ENERGY INPUT, Q=(M/af0/'0c0)q with a=14.94 x 10 £„=1.94 x I 0 9 e r g s / g m , M=53.3 x 1 0 ~ 2 4 g m - 1 8 2 8* ODQ ID-DDQ 2«4,.Q0Q 32.000 X yo.Doa us,qdo 5S>ooa 614, aaa- 72.000 vCiipuTeii -:;c. a?:a;-;e:;vi, c^vson'mp cj:asj oj ,000 8, god ib.000 24:,qoo 32.qq0 X yo-oao 48,000 5b.0qq 614,000 72.000 M.'iut I'l U S s-jweat̂Lsfs slsStesasBrJj doq a. ooo 16.000 24,000 32.ooa X Mu.ooa 45,000 58.000 64:, OOO 72.000 chatvt rc. CH MADE IN U.S.n 119 7,3.2 S t r u c t u r e _ o f _ . d l s s o c i a t l o ^ 0 — The results of Chapter 3 indicate that a steady photon, flux F c = 4,72 X 1 0 2 2 ph/cm 2sec (with an energy of 8.8 eV) is required to produce a weak D-type radiation front preceded by a Mach 3 shock at an initial pressure of 0.01 atm oxygen„ We applied the method of finite differences to calculate the evolution of the flow for this case. The constants, differ- ence intervals and the computer programme for these calcula- tions are given in Appendix E. Again it was impracticable to carry on with the cal- culations until steady state W a s reached. However , the plot of the pressure as a function of distance X, see Fig. 7»4{. indicates that within 15 jj.sec the pressure is 7.2 times the initial value. At this point the degree of dissociation at the window is only 36%, the temperature is 4000 °K and the compression ratio (final : initial density) is 0.3S 0 The maximum particle velocity is 1.4 c Q 460 m/see. Although the radiation front is in the initial stages of development, it already exhibits some of the properties predicted in Chapter 3„ • These results must be considered as preliminary since cons:Lderab1e difficulty was encountered in preventing the calculations from going into oscillations and as many as 15 iterations were necessary to obtain self-ccmsistant values. In fact, such oscillations are already in evidence in the computer profiles shown in Fig, 7.4, Fig.7.4 Computer profiles, 0.01 a tin. Method of finite differences, applied to Cond for a weak D-type radiation front preceded by Mach 3 shock. Photon flux, F q=4.72 x 10 2 2ph/cm 2sec. Time, t=NAt , with At=0.303 jisec . Distance from window, x=0. 249 X [cml . Curves plotted for (N/10)=l,2, -5. 4 -.t_t_L i i i -1 ( LL I rl~r L J_ I m m . r Jit I I ! "I, -t"or —A!̂ , J—i! t t - r r —!—i—T-I LL _ X r TT Pi A— Fig.7.4 — c o n t i n u e d PRESSURE, P=p/p Q , with po=0.01 atm, -.000 s.aao 1.EDO 7 >200 7-200 IsasŝizakEesES ksrisaissas: I I GOD b 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 8 SUMMARY AND CONCLUSIONS The object of this thesis was to investigate both theoretically and experimentally phenomena associated with radiation fronts for the experimentally realistic situation of ionizing or dissociating radiation } passing through a tran- sparent window into a tube containing the absorbing gas. Five different types of steady radiation fronts may ofccur for the experiments,! situation under consideration. At one extreme of high radiation intensity and. low particle density there is little particle motion associated with the front, at the other extreme of relatively low intensities and. high particle densities the particle motion is dominant and a shock front propagates ahead of the radiation front. The speed, of the various discontinuities and all thermodyn- amic quantities may be calculated either if the detailed structure of the radiation front and mechanisms occuring within it are known or if the temperature behind the rad- iation front is assumed. Conversely a measurement of this temperature would yield important information about these mechanisms. ' It was shown that for the case of no recombination or collisional dissociation, the structure of a steady 121 radiation front produced by monochromatic radiation could be described by a simple analytical expression in terms of Lagrangian co-ordinates. This expression depends only on the absorber density and on the absorption coefficient, oc. A simple relativistic correction must be made if the vel- ocity of the radiation front is near that of light; this causes an apparent steepening of the front. A treatment of the structure of a dissociation front in oxygen for a simplified reaction scheme was outlined. It was pointed out that, in general ; it is necessary to con- sider all the reactions within the radiation front. A num- erical solution was attempted for a weak' D-type front pre- ceded by a Mach 3 shock but was unsuccessful. For an experimental investigation, of radiation fronts an intense pulsed light source, which consists of an arc discharge through a Barrow channel in polyethylene, was coa» structed. The average intensity of this "Bogen" light source (in a solid angle of 0.1 sterad, at 5000 A and operated at a discharge voltage of 3.0 kV) was measured to be (1.9 t 0.2) X 10' times as bright as a standard carbon arc. Along the axis the intensity is about three times larger than this value. This indicates that the effective black body tem- perature of the source is from 60,000 °K to 150,000 °K. Experiments were carried out at low and high absor- ber densities, N0. " An experiment in iodine at a low density, N j, illustrated the beginning of the formation of a radiation 122 front. Although the measurements were quite crude the agreement with theory was quite reasonable. The author sug- gests a similar type of experiment be attempted with a strong d.e. light source. Shock fronts in oxygen at a high density, N Q } were detected bj' means of piezoelectric pressure probes. At high pressures (1 atm) the shocks formed very near the lithium fluoride window, while at low pressures (0.03 atm) the point of formation was about one cm from the window. The speed, of propagation of the shocks was 364-- 8 m/sec for all pressures, at least at distances far from the Li]? window. '•' Attempts to detect pliotoionijsation in the test chamber showed only that photons in the wavelength region o o from 1200 A to 2000 A were especially efficient in knocking out electrons from brass or dielectric material,. Attempts to detect ionization fronts proved- fruitless. It was shown how the development of a radiation front may be considered as unsteady one-dimensional flow with energy input and treated by the method of characteris" tics at constant time intervals or by the method of finite differences. These theories were applied to calculate the evolution of the shocks which were observed in oxygen. The theoretical results agreed well with the experimental results. It was also pointed out that if sufficient com- puter time were available and a constant energy input were 123 used, these methods could he used to obtain steady state solutions.(complete with thermodynamic quantities, veloci- ties and the front structure) which we had attempted to calculate previously. 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. ad- justable to any frequency desired. 124 A P P E N D I X 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 spec- trum F(#} and absorption cross section oc ( (see section 3C1»2) is given below. 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 eqrns (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/FQ — „ The terminology in the programme is as follows: h 7-V'kT = >: —** f(2)'> ^ — a n d $ F O R T R A N C I D E A L I Z E D R A D I A T I O N F R O N T F O R B L A C K B A D Y R A D I A T I O N A T 6 0 0 0 0 K E L V I N 7 C C O R RE 'S P O N D T N ' G ' r N ~ t r X T G " E t l " ~ — — — — C U S I N G R U N G A - K U T T A S U B R O U T I N E D I M E N S I O N V ( l O ) » F i l O ) ) Q U O , ) — — cOMMair-FU'F'WTDX^ ... : — — — D A T A 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 » 1 0 . 4 1 6 > 0 . 5 4 8 > 0 . 6 7 2 > 0 . 7 8 4 ' 0 . 8 9 4 > u . 9 6 3 ? l . u > u . 9 8 5 > u . 8 9 4 > • 2 T̂Tr&47T0TZ2TV"O7t)'7̂ 5Tt)lTr4"9"8"rcrrO"3T"3T" . ' — ™ — — : T = 6 . 0 X ( 1 ) = 8 . 0 / T —--DX=o".Ter7T""'"~~— :—: : : : — : — — ~ D O 10 I = l > 2 0 1 0 ' X ( I + 1 ) = X { I ) + D X • DO-'li- I = 1 > 21 — : : : — :— . : •• • 1 1 F 0 ( I ) = ( X ( I ) * X ( I ) ) / ( E X P ( X ( I ) ) - l . U ) F O T = F O ( 1 ) - F 0 ( 2 1 ) __ D O 12 = 1 ' 1 0 : — - — : ^ . ' . ..••••. , . 1 2 F 0 T = F 0 T + 4 « ' P * F 0 ( 2 * " K ) + 2 . 0 * F O ( 2 * K + . l ) -. ' ' F w = F 0 T * D X 7 3 • 0 . W R I T E ( 6 » 6 0 T FVv : — — — — ; : - 6 0 F O R M A T ( 1 X » 1 0 E 1 2 . 4 ) D Z = - 0 > 0 6 , . D Q 2 J = y,2 " - ' : : — ~ — — - — _ Y ( 1 ) = 4 . 5 1 8 8 7 6 Y ( 2 ) j = 0 . 0 - • D O i 1 = 1 , 7 5 G A L L RK. ( Y ' F >Q > D Z > 2 > 1 ) _ A _ W R I T E ( 6 * 6 0 ) Y ( 1 ) » 'Y ( 2 ) » F J _ _ 2 ' D Z = - 2 . 0 * D Z ' ~ - — - — — — ~ • - S T O P E N D "" " S U d R U u T I , \ L " " A U X ~ R . K X Y , F T — ~ — ? — ~ ~ 1 — 1 " C O M M O N F J » F W , D X > A < 2 1 ) > X ( 2 1 ) ' F 0 ( 2 1 ) ' D I M E N S I O N Y ( 1 0 ) » F ( 1 0 ) » F Y ( 2 1 ) ' D O 21 1 = 1 » 21 ~ " ~ ' ' ~ ~ 2 1 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 : — :— — — — : 2 2 F Y T = F Y T + 4 . 0 * F Y ( 2 ' * K ) + 2 . 0 * F Y ( 2 - * K + 1 ) . F Z = F Y T - * D X / 3 ® 0 .... F J = F Z / F W : : : : — : " " . F ( 2 ) = 1 . 0 / ( 1 . O - F J ) R E T U R N r " E X D S E N T R Y '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 e1 Scale drawing of Bogen light source. 126 . 4500 0000 5500 A 6000 Fig. i.—Extinction-curves of iodine vapour. Pig.£.2 Iodine absorption cross sections (Rabinowitch and Wood (1936)) 9-54 SH8 PHOTON ENERGY, hf(eV) 8-B6 8-55 8-27 7-80 7-75 7-51 I6-7 14-9 130 11-2 9-29 7-43 5-58 372 1-86 _—_/?5 3 .!H J ' l i ! I M Y V/AVELEMGTI -if ' I85±l5cm"j 1300 1350 1400 1450 1500 1550 Y/AVELENGTH, X (A) 1600 1650 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 E N D I X C EQUATIONS FOR SPECIAL REACTION SCHEME The reaction eq'ns (4.5) are written to indicate different groups of particles (0 atoms, molecules, Og molecules, molecules). We can write one conservation equation for each group. 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 - X ; Jsi-i //,*Af£ ~ ^ M, Nzri - JE,- J W MA/+ Mi 1 Z+l—>3 4-l~f->'3 1 C.l <£jh v- _ jj£ j __ / gr ' /zyJ^j /T* I j-t /-rr/ -/• ̂ //, /i/, ^ ^ //, ŷ ' C.: 3+1—>3 > 3 / *C. 2 5065 + +f3£l + Jli fa/Ji V- ^ // //,• 'l 'Mtf+tti — ^-Jut //<? A/c / C.4 The notation is as for eq'n (4„8j, the arrows indicate the reaction process (e.g. k/ii is the recombination rate process for the reaction 0 2 + 0 -v M^ •—> 0 3 + M -L f where M is the third particle in a three body collision). Corresponding to eq'n (4,9) the photon flux passing through four different types of absorbers is where is the photoabsorption cross section for the •th _ , . , x particle. We may now obtain the individual j terms in eq'ns (C.l) to (C .4 ) „ since h if, fc C, 6 The individual terms are thus , / , -/i-^/VJcfX N 3 f f ^ o ^ ) e -dVj m <?X/VrW C.l < <P A / 3 r ' C.8 C. 9 a/ r ' < ^ , / / - 71 / J / /4{ / {*) 0, <3 or J C . 10 129 where we have split the photoabscrption cross section of Og molecules according to the end product of the reaction 2 ^ ̂ ^ i s J c h e c r o s s section associated with the reaction h ^ 0 (3*). Their sum is equal to the total cross section C o l l Finally the density of the gas is related to the concentration of the various types of particles by the rel- ation ^ ^ - i M M + & (X,*) + 4 + - C.12. In order to solve eq'ns (C.1) to (C,4) one must know 24 (6 X 4 « 24) reaction rate constants as a function of temperature for the various collisional processes and four photoabsorptiori cross sections. Unfortunate1y most of these quantities are not known; What have been measured experimentally are combinations of reaction rates constants. For example, the vibrationally excited molecules are ignored and are lumped with the ground state molecules. Thus the recombination rate constants of the reaction 0 + 0 M •£^0 4- M has been measured (over a small range of tein-/a poratures), the intex'ir.ediate step involving the vibration-- ally excited molecules has been ignored. We wish to empha- size that the use of these constants in calculation;? invol- ving radiation fronts would be strictly permissible only if 130 the vibrational and translational degrees of freedom were in equilibrium,. Furthermore in such a case it would be necessary to use a photoabsorption cross section which had been measured as a function of wavelength and as a function of temperature (i.e. the photoabsorption cross section would be temperature dependent). As for the simplified case treated in Chapter 4, the energy input for each group of particles may be obtained directly from eq'ns (C.7) to (C.10) (see eq'ns (4.18) and (4,19)), Thus we obtain = £> • C. 13 ffc - f^ K M octM) , f l"%r ) C. 14 /? > r • I / S r-s i dK . C. 15 JET (/ Jx C. 16 where -fa-, - 0 since o ^ ( p ) = 0, The total rate of energy input is simply the sum of these individual contri- butions £ fx, -£-) - -Ax,*) (p * f<) . C , 17 This treatment of the more general reaction scheme in eq'n (4.5). indicates that it is necessary to consider all the intermediate steps in a chain reaction process. In gen- eral the energy flux contributing to the front is larger than if these steps were ignored, because the 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 un- steady flow with no energy input by the method of char- acteristics 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 Calculation of an ordinary point D, point D on the tj + -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„ and + The equ'a-* J 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 Tfr- for the length and time dimensions „ where o ( m is some convenient reference cross section as defined following eq'n (3.20) . The velocities are made dimensionless by- dividing by the speed of sound in oxygen at 300 °K, c Q s 330 m/sec. The thermodynamic quantities are made dimensionless by dividing by their initial values at t = 0. Thus. r - p/pQf h » h / h o s u, = u / c o s c - c/coo The rate of energy input q is made dimensionless with respect to the equation in which it occurs. Since the affective adiabatic exponent g is already dimensionless we use g Q »> 1.4. These dimensionless equations may be written in finite difference form. Eqrns (7.11 ) and (7.3) evaluated along the Mach lines are written XJ o {-£„ = p c j ^ o d . 2 . t & p - X c ) ^ D.3 D.4 Along the pathline BD„ eq'ns ( 7 „ 5) and (7„4) are (ho-Q - ep zz D 0 5 D 0 6 The rate of energy input corresponding to eq fn (4„19') is = ca/4$ t- / <9 ' Ji y j f 'f,: f- 's. e*-/ D„ 7 where CONS - 3 ' & O and. where we have used h ^ / k t , c/<{5 ) ~ S m! E Q is the energy flux entering the gas as defined in eq'n '(4„X8) s G(t) is the diinensionless time dependence of the energy flux normalised (in our case) such that « yJ J!i olJ\ when G(tJ =3 1 0 0 0 The conservation equation for 0 particles (or dissociation equation) as given by eq fn (4.14) may be eval- uated- along a pathline to be (>y) (Mf* J n ^ziA If) / M D o 8 135 where we have substituted for kdf and kyz/from eq'n (4„17). In this equation M is the mass of the oxygen molecule and. Fq 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) •—77, JiZ > " ffe -'SVr-r-7- // > D„ 9 where the temperature T > T T/T « T/300 °K and where AONS j, BONS and DONS are AONS tfcl k % CotiS D „ 10 3. & & A /a' .3 ^ r % D „ 11 * X/o' r0 J D012 The equation of state 3."elates the various thermody- namic quant it ie s l / ~ r jo-t ' / r 7 Do 13 y- DO 14 where ' J ts:.._ / ~ ' ^ iiH?)- + 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 and from eq'ns (D.8J and (D07)„ Next one calculates u„ and p from ,eq?n*s (D „ 1) and (D„3) (using values of the thermody- namic' quantities at points A and C which are located approximately from eq'ns (D.2) and (D.4))0 Next one cal- culates h p , g p s £ , Tp from eq'as (D.5), (D.15) s (Dc13) and (D„14) respectively0 Finally ap 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 ẑ- 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. The peak photon flux was assumed to be 1.16 22 ° 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 inter- vals 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 1018cm~3 AONS w 0.0131 AONS 0.131 BON.S r*0„llS5 X 107 BONS ==0.1X95 X 107 CONS « 3 .0 CONS 30.0 DONS ^ 0o 80 DONS «0.08 ^ t « 1.32—^(0.1 jisec) At =» 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 — ^ R j effective adiabatic exponents g-—t-GAMj enthalpy, h—s-KHj, time, t — ^ t — D j Zv z — ^ d z ; Eulerian spatial co-ordinate„ x—^X^ 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. As shown in eq'ns (D. 1) and (D. 3) we assumed that ( ^ g)s 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). S P A L i S E M O U N T T A P E : ~ ~ : : : ~ : ~ ~ ~ $ J O B 7 9 0 8 7 H . A . B A L D I S S P A G E , 1 0 0 ST I M E ' — "10 : : : - : ; — 5 I B F T C M A I N C M E T H O D O F C H A R A C T E R I S T I C S A T F I X E D T I M E I N T E R V A L S G~ "" P R E S S U R E = 0 . 1 ' A T M ^ ~ • • ' C F F R O M B O G - E N L I G H T S O U R C E D I M E N S I O N S A M ( 8 > 6 5 » 8 ) > D X ( 2 0 ) > X M I N ( 2 0 > ~ D T M E N'ST'O S ' ( ^ 2 T ) » ' 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 ) — ; D A T A A / Q . 0 0 2 5 l > 0 . 0 0 8 7 1 » 0 . 0 2 7 4 » 0 . 0 4 9 8 ' 0 . l 0 7 0 » 0 . 1 8 9 0 » 0 . 3 0 l » 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 ~ D S = 0 - . 1 6 / T W — ; : — — D O 10 I = 1 > 2 0 1 0 S ( 1 + 1 ) = S ( I ) + D S D O - 11 1 - 1 > 2 1 — — ; — — — : : — - — — • F L ( I ) = ( S ( I ( I ! ) / ( E X P ( S ( I ) ) - l . o ) 11 F O ( I ) = S ( I ) F L ( I ) ' FL"7'= F L ! 1 ' ) - F L ( 21 ) — — : — r — — : — . F 0 T = F 0 ( 1 ) - F O ( 2 1 ) D O 12 K = 1 > 1 0 : F L T = F L T + 4 • 0 * F L ~ — — — — • - 1 2 . FO -T = F O T + 4 . - 0 * F O ( 2 * K ) + 2 . 0 * F 0 ( - 2 * K + 1 ) F W L = F L T ' * D S / 3 • 0 - • F w'=r o T * D S / 3 . 0 " ~ ~ — — — : : 6 0 F O R M A T ( 1 X > 7 E 1 2 . 4 ) W R I T E ( 6 > 6 0 ) 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 ) D A T A X ( 1 > 1 ) > U ( i » l ) . » Y < 1 > 1 ) > V ( 1 > 1 ) > R ( i , l } , p ( l > i ) , H ( 1 > 1 ) > H H ( 1 > 1 ) » 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/ D A T A M s . L / 6 5 > 8 0 / __ Q ( 1 = 0 < Q - - - --•• - - : : — — _ M l = M — 1 M2 = M - 2 1 A O N 5 = 0 . 1 3 1 — — 1 " — — — — — " — : — s 1 — B O N S = 0 . 1 1 9 5 E + 0 7 C O N S = 3 0 • 0 00NS=0o080 : ~ — — D T = 0 . 2 6 4 D Z = 1 , 0 3 6 dza=0 .528 ~ t * :—: : ~ do 1 i=1>m U( 1 + 1,1 )=u'( 1,1) YTi+-iTi"r=r(TVT"r~~ ~ — r — : — " ~~ — : — — q ( 1 + 1 v i ) =q ( i • 1 ) v ( 1 + 1 ,1 ) =v ( i , - l ) . r ( 1 + 1 ,1 ) =r ( 1 , 1 ) ' : ' " : : : — — r P( 1 + 1,1)=P(1,1) H(1+1,1)=H(1,1) rriTrTT7 = C(T7TT : ! — — — : — :—; : — H H ( 1 + 1 , 1 ) = H H ( I » 1 ) UU( I + l, 1) =UU( 1,1) c'j (1 + 1,1) =cu ( r , t ) ~ 1 ~~~ — - ~ : 1 : ~~ g a m ( i + i ' i ) = g a m c i > i > x x ( i + 1 » 1 ) = x x ( i , 1) + dz x"a+irnT̂ x"n~,T)+D"z ~~~J : ; ~ : — : — ~ — : — : — — — : dimension g t (29 ) » t t ( 2 9 ) ' g < 100 ) , t ( 100 ) , t 5 (100 ) data g t / 0 . 0 > . 0 4 8 4 , . 0 9 6 8 r .219 » . 387» .580* . 813» .981 • 1.0 , .99-ov 1 .968vv935 » . 903, . 86'5 > . 8"2"6" >y739v.6~58v.583 v . 5 t 6 v . 4 ^ z , ~ 2 . 4 2 6 » . 3 8 7 , . 355? .332* .313 * . 2 0 0 , . 1 0 9 6 > . 0 4 5 2 > 0 . 0 / data t t / 0 . 0 , 2 . 64 » 5 .28 » 7.92 »10.56 >13 ® 2 0 > 1 5 . 8 4 , 1 8 . 48 »21.12 » 2 3.76 > 1 26V40V29TO4V3 . 2 6 8 . 6 4 > 7 3 . 9 2 ' 7 9 . 2 0 , 8 4 . 4 8 , 8 9 . 7 6 , 1 1 6 . 1 6 » 1 4 2 . 5 6 » 1 6 8 . 9 6 » 1 9 5 . 3 6 / G(1)=0.0 TCI) = 0.0 : ~~ : : t 5 ( 1 ) = 0 . 0 ATM = 0. l • • do 554 j = 2 ,29 ~~~ ~ : : ; t5 ( j ) =atm*tt'( j ) do 557 1 = 2 , l ' t ci )=t c i - l t + d t ~ : : — do 556 j = 2 , 2 9 i f ( t ( i ) . g e . t 5 ( j ) > go to 556 G I T R G T Q ) ~ R R 5 T T Y ^ T N T T R R G I ( J ) - C I U - I ) ) / 1 R S U ) - 1 5 U - I ) ) go to 557 g ( i ) =0 .0 "cont inue ' : : : — l l l - l / 1 0 do 3 n n = 1 » l l l do 3 nm = 1vt0'" — : ' : — n=nm+(nn-1)*10 do 4 j = 1»m gam ( j , 2 ) = gam ( j» i t ' : : : 1 : ~ : ih j , 2 ) = u ( j , 1 ) C(J>2)=C(J »1 ) V . m ' cv f* N CH > ii X CL I il rv CM P ~> rsj CM m co f-h to * * E> h-x * o t--> p > > x +• + p — •M CM b X CO UJ "> ~> X !• X — \ I- 1 X 0 P — >- UJ f- " « j. • Q. CT> z3 * * ffl x x co d o * i z in m owjj uj |® » in o —-<>11 t\jM]f.«HX~-OOH « • u u m « h p o ii ^ 11 i— ;n n o h il I S 1 Z il I X i > >3 I I < U b — O I- UJ LL. >-* o to J £ «—1 in 03 — + •— 1— o i 1— > * o > > - * •H * + O 1 b 1— — • * > CM in 1— P * + > - j 1— o ,J|< •p. X • fn > - ft >-!® — ^ — * UJ I >-+ Kl CW. * >- — + — cm cm rH I I ~> ~> — — > > + + — - cm fM - ~> — > - > - — — * •*• in m cm o o I I o o • • J"M rl Hi* * o z o o + +! n o — ii a:.> x X u u z >- u > X X -i x a o o i .1—1 k. ̂ O M tsl CO ;co s: s —i ; o ~ •O rvj ,-nj 'f- ll ii 'cm ! Z :0 o — :o P NJ rsj Q o! i * ! < uj x I < o — '* < a. " * x ~ UJ • 1 * !_j ~ : ii II • ' x !—i O >- V <ii.iL. -I H >- N - k. >—i 11 I LL. h- il _l h->- > u_ Ll. aaimri uivyd n n + * cm 'i—i !+ _i ̂ >- I* u. cm * r-o > • U. cm + O co o CO jco co "00 cm * CM >- ll. * O o • r-i <T " + r-i \— 1 _l iZ >-u. CM I (M h-_J o >- q L l tP ;cm 3 !+ u. ^ o * • .cm cr\ > \ q o — ~)' i e ! • h- cm b ~> — o vt- _l »-—-'>- iii + >-->*>- 1 : I O i > * cm rh kj * X * o z 0 Q 1 > * > . X * n z o CQ + Ll — L l W >!< * O -0 • z c1z o - ~ < o — S) % * op .— * L Q £ + >- Mo u- s: * o ~ u — * CM O * m o o i— o ; I >-h" I > a. u_ — in >- i * I o il _y c.i -i o (M !CM 01 Q) f̂ LO 'J" CO T i n — -j >- ll a CM i-« * e ~> 3 — UJ 13 « 1 •-> Z) — 3 u. 3 C> ĉ 1 CO LTJ 'D it _ o h to J Q vj O O * I - u; u < I < •H r-t D I HHVlU * j * j :~> -i ~> az IT- < < cm * i e> .—s. •n>- cm .— - + — !< az V I i* r-H ;CNJ I • ̂ p oz 1 — oz II »:< < !< oz ju !< :cm r\j oz h ~> + ~ < yj az OZ jl + \— < * i-t oz in ;'«>. — • r ̂ o u • I s- o U i II II < :< cc < u az I I - ) Q- 3 I I I rH 3 » 3 — i s: - < H H jH O CM #» 3 CM — INI -j q — N U t-C> < * u az u * m 'oz « a; o II ii ,< a: isj. U Q 3 3 p * a. 3 b si cm Kj — L- Q « 'jc * * * I 3 p <<;< — — — Nl Nl kl <—i O CL q Q ' Q - + : + I I I 3 < <c ~ — — — O CL rH .rH I-H X — — ^ : <£ if. 3 3 b O LH — — I • ; « a. d 6 < o b I I hi X I I < < '< < oz a; Q- ~> 3 o a a. I q: s: < o < i <M 5: < * QZ CL + a * oz a * az az * • o Nl O Q * nI O a • co i QZ cd ir a o — <3 * I in " • CM ! E + a < Lli CL I oz aO ti- + !+ ~> n — — o: r- u 01 u u u i u I a — + + rh o 'rh o • K N ,U N * or u — u u — U r~ U (M * (\! * 1 (M ~) CM (M n s: < e> r + :+ + a b a i 11 I u -> + — u a: v l # i—i in * r\j -) c —-O <X I — a: ii u u u ca QL -) -) :+ — — u o: IV i + u Qi * in CM in r Kj h Nl p Q U -!+ Q u * U -J u '•* * In -> r> ~> CL p 3 >(< >[< * u u u Nl Nl Nl Q Q Q III I EE < L3 q3iiwn nivao 'w o u ii — u I U U .J u O a: o ii j • o; (O II !n u U N l U Q —I > . > i CL r> a ii ii ii u O u cl b a vO h- rsj CM 3 cl + + U U a cl * m m • • o o ii ii a a. cm r- ;o -> j • r-— -1 s:! i < '_j o O 2: f-+ :< u o o 2 — 0 < * •D in — N _J U * rH _J az + az SI u < * QZ 1 QZ N N — - > s: rH < S -1 e? 1 * —J 3 ) O D a. CL + + 1 OZ M Q 3 — . — * tD —1 < u _l Q. 1 3 O 1 1 * + —• — -J U D CM CM QZ * * 3 _j In — u l o * b ~ |(i * oc h & + pj u < I- * .D p az * a; a: b * u I <1-* {- • ai 5: r-i OJ i+ 3 u uj Cl • I • QZ a KD Ll. u o. I 0 QZ xl 3 O cl I a: U- oz cl b; u cl * '1 Di _J * o. + • • —1 3 3 UJ O : • CL I — P •h 5 Cl + QZ QZ CL cm .— ! r. * n in —• • 3 o 1 -'m (M j s O ~> iii o ic 3 — + D n • • .-I cn I cm r N cm * co O + CQ n - cm > r , U J * X. — n cm en r. • ^ h- >-— I cl o x • uj 1—f — CL 1—I — ~> — * + — > — — > :+ + - cm cm tH ~> - ) cm ^ —• * ~> X u _l ll. L l — i- d. Q . M " H O Z XI — X > X — ii * — m cm • o ~> II — I- X > X > * in o o + r- m I o co cm —1 1 > < uj o — > 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 to 101. cccc=c ( j > 2 ) '— : — : : — — — ~ — : — — c < j > 2 ) = s q r t ( ( g a m ( j > 2 ) - 1 . 0 ) * h h < j » 2 ) / 0 . 4 ) numb=numb+1 . / j > z ) r~b o i o l urr i f ( a b s ( y ( j > 2 ) - y y y y ) . g l . . o o l ^ y ( j > 2 ) ) go to luu i f ( abs ( c (-j > 2 ) -cccc ) . g i • • u u 1 # c \ j > 2) ) go io l uu •xtotz)•'='x-(-j- ', • • :— zom^zon z i m = z i n co n t t n u e :— :—" :—:— : — — - do 86 k=1>m1 x x ( k ' 2 ) = x x < k » 1 ) + ( u u ( k > 1 > + c u ( k » 1 ) ) * d i 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 : : : : — — — 7 — : — uu (1_l »2 ) =uu ( l l— 1 > 2 ) c u ( l l » 2 ) = c u ( l l - 1 ' 2 ) xx ( l l »2 ) = x x ( l l - 1 » 2 ) : : : : ~ uu (1» 2 ) =0 • 0- c u l . i > 2 i - c ( l » 2 ) XX ( 1 Tz) = 0 ® o go to 86 xf = x( j j > 2 ) x e = xt u j - 1 > 2 ) ~ : 1 • • x f e = x f - x e u f = u ( j j > 2 ) • 0e = ij ( j j - i » 20 — ~ : : : u f e = u f - u e c f = c c j j. >. 2 ) ce='c( j j - l > 2 ) : : : • :— : — — : c.fe=cf—ce b e s = 0 . 5 * d t * ( u f e + c f e ) / x f e -- c e s"= o 7B"*DT*"cwncrrr-pcutktitwetcet" : — xx(k>2) = (xx(k.»1 )+ 'ce5 -be i>*xe) /u . -w -bes ) txx = ( x x ( k » 2 ) - x e ) / t x f - x e ) u u t k j 2 )=ue+txx*ure " ~ ' ~~ ~ ~ cu(k > 2 )=ce+txx*c fe cont inue 62' FORMAT (1X,I6) . 'j l WRITE (6,62) N . ;e' ' - W'R I » r t I , 2 ) ' GT̂ f' 2 ) ' P 1 I ' 2 ) > RTT'TM : ~ ~ — ' 6 1 H ( I ? 2 ) » HH £ 1,2) » GAM i I » 2 } > AA I I , 2 ) » I = 1' M1 ) ' <n • RNM=NM TP -IF( RNM-lO.T7bV71V7T"'~ • •. ' •. •• : : : — — — :zi 71 DO 72 1 = 1 »M SAM ( N'N > I »1 )=X( I ,2 ) _ _ s-AiVi( KNTI^-r^DTFT2T-- .—_ : 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) '72 SAM( NN , I ,6)=GAM ( 1,2 ) — — _____ 7 0 CONTINUE DO 32 1 = 1 >M ' ~ - v(I,3)= Y(I , 1) ; : ~ ; : 3 2 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 )'~ : 1 : ~~ ~ H ( I , 1 ) =H(1,2) HH(I,1)=HH(1,2) -3-3- GAM( 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 ITE( 7 ) SAiv; — CALL PLOTS M8=M*8 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'"D̂ "TTT CALL AXIS(0.,0. »1H v 0 , 6 . »9U . , AM IN 1 I ) »• DA ( I ) ) DO 201 J=1 ,8 c aT l pltotts/cm ( jti~» i") rsam ( j t 1 > n »+3 ) do 202 k=1>m 20.2 c a l l plo'ubama j»k»1) >oafh j>fs. i > > + 2> :zoT~~coNTTNaE" ~ ~~ ~~ ; ~ ~ c a l l p l o h 1 2 . u > u . u » - 3 ) 200 cont inue -^Qj-P1I0TWE, • ' ' — — stop end •SENTRY"- ~—~~ ~~ 1 — 139 A P P E N D I X E METHOD OF FINITE 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 £ Tn-r 77 n-i 1 1 1 — 1 1 u N .,.,„ P 1 1 1 1 — 1 I 1 3-1 Sf-l Fig0 E o1 Lagrangian mesh for finite difference calculations constant time tQ in each cell in the -2 direction with special techniques being employed at the beginning and end of the interval. One then repeats the procedure at intervals of tin:3 ^ t 0 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 , - & e . I (b) ri-f, , JL? - :U -t -> ) Jj o 3 Wri- / / r • i -f S-, _ —• yj ^ ' • * ( ' - K - ^ j . J ^ — — — ^ r / _(-• -h SortS-ft I / . ^ l i •'-I Si l/f+'k. jV ' HTIf'^-^) ^ Js - 2 . a f/-?* E» 5 141 The various constants found in these equations are defined in Appendix D 0 The superscript - n { r indicates an average value (e.g. P ^ « 0 0 5 ( P ^ + P ^ ) j „ On the other hand the subscript j ^ does not indicate an average value J it implies that the value of P is calculated in the middle of the interval between j and j + 1 (see Fig„ E o1}o The calculation procedure used, is as follows: "Ti ' 1 1 1 1„ Calculate U i ' and X. from eq'ns (E0.1) and j j (E02j for all values of j (using a special procedure at both ends}„ n -!- 1 2. Calculate Vj + x (appearing as Y(J ~ 1, 2) in pro- gramme ) from eq rn (E„3) + 1 3o Calculate Y. from eq'n (E.4) using approximate or average values for Y j , ^ „ T̂ f If lA .A special procedure is needed for Tt )d;f. 142 •4. Calculate q" from eq'n (E.5). (Note Q is cal- culated in the middle of the time interval.) 5 0 Calculate E3? + ^ from eq 9n (E.6). J z 6. Use an iterative procedure to calculate P1? and J z H ? + -1 from eq'ns (E.7) and (E.8) 7. Repeat steps 2 to 6 for all values of j„ 8„ If necessary (because of instabilities or to shorten the number of iterations required) calculate weighted , „ ,„n + 1 n + 1 , values ox P.; . and. II-j . j. (i.e. average or weigh J ' ̂ o z the results of step 6 with those obtained in a previous iteration). 9. Check the self-consistancy of the results obtained and if there is insufficient accuracyy store the values and return to step 1. 10. If sufficient accuracy is obtained calculate Qs"' .<„xfor j z all values of j. 11. Increase time index and return to step 1, V/e have not examined the stability criteria for our case in detail. In general 9 however, v/e find that the increase in the variables per ^ t interval should not ex- ceed about 30% (even in this case we are not sure about the accuracy of the results since we use a large value for the artificial viscosity). The procedure outlined above was used in section 7.3.2 in an attempt to simulate the development of a steady weak D-type front preceded by a Mach 3 shock. The condi- tions for this case were the same as used in Fig. 3.6 143 initial pressure of 0.01 a tin and a photon flux, F = 4.72 21 2 X 10 ~ photons/cra sec for which the various constants were AONS » 0.532 BONS ^ 0.1195 X 1 0 7 CONS » 121.8 DONS « 0.008 DT » 0.04 D Z * 0„08 The value of a 2 8 was chosen for the numerical constant appearing in the artificial viscosity,, see eq'n (7.12}° The computer programme used for these calculations is given on the following pages„ Most of the symbols vised in the programme are virtually the same as used, in Appe:o,™ o dix D. The artificial viscosity is 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 = 0.0r A T M ~ — — "—: > ^ : — ; C WEAK D-TYPE + MACH 3 SHOCK DIMENSION SAM ( 5*80.* 8) »DX(20> »XMlNI2U) — :—DTME~N~STON~"Si~2T")~>~FQ~(̂ l̂ 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 o.647'0'.2'2r'Ov0746rvO-.t)4:98̂ 0vO-3-737 :— ^ ~ T W-6.0 S ( 1) = 8 . 0 / T W Q5 = 0 .16/ TW —; : ; — ; :— DO 10 1=1 >2:0 10 S( 1 + 1)=S(I)+DS . D 0 J — J. • 21 : : — FL< I )=(S( I )*S( I ) )/(EXP(S( I ) >-1.0) 11 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) ~ 12 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 " 60 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) >E<1 > 1) >P l 1 >1 ) >Hl 1 '1) / _ . _ _ _ _ _ ——— - 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-o H aajuwn NIVUD ~t l-J UI + irH rH r-4 t-H .-H ;>-H i > > liJ a. j I I 1Ml 1 'rH 1—1 rH P rH »» [1—1 rH rH u rH •+ + + 1+ + *—i — U |3 V > hj CL HH I ~ b — X I X I ~ I r~t i—t ' ̂ I—t i—( «•> rH rH + '-< + -—' (—4 — — x a x _j o _i i—( o rH * i i i 3 3 3 vj a * <t a m * i 3 n f) 3 i — 3 jc\j I I n r-t o o s > X a. 2: I ' « e > I I I O II I Z O O <M ~ — ~ <M H Z 2 z ; I I Ut CM CM CM I \ z: z — ~ ~ rH - ~ r. rH _|i + • f\l <M I ; rHrHrHI I <n co 2: - CD :<h I I I 2 in _J ' ZrHrHSI, 3 3 3 I _J O O I — — 3 O — " — r-H O O O i*in 1 _ 1 p it t ill N ! P D 3 O * Q rg * -— (\j — I 3 3 - * 3 m + ® ~ O tM I 3 CM ' 3 rH — I + S rH 3 ~ * 3 in £ I k Q D .N. CM g P. 1+ 2: s: (M rH I a + 111 > CL + + CM CM (M CM x 1 1 3 1 1 3 3 in • 00 rH CO * * o H-1— x * o 1-o > £ x > cl b — J ;2 X H x N " " * I 2: I I |M I rg * m — 3 —. — j~ 3 * in e CM 2: CM CM CM >—I ' • ;0 O o CL LL. 3 3 21 X 3 vO I 3 O — Q > * * ro m on • e |-H O O II I I X (— 1— Lu > CL X o • —• I CO X ' CO 'UJ : • X i I— 1 • t •o — i e CL CTv X X CO o LU UJ • X « 1 > U. I--« x H- > * I— > * I— X N o 1— •d" rH > O * rH rH 1 O « O 3 in 1— - — + > - h-0 O + > 0 O CM * O rH ^ c ® * rH m h- • 1 + UJ 3 O ID ̂ c t— • > - h- r-H > - O O rH 1- I I ! > 3: O H H O X X rH 3 ill z u — 3 U. co O 00 CO .0 co ^10 u — > — * 2: * :!• * O m o >—< o « : « > -O H || I I I > 3 h W I- t— >>>->- CM O o -J CO CO > + CM * in • O L O INl * < X CM fc>- , M *> .cm O I I >-< 1- r-'z z z !-. O - « ~ Kl O tsl N I A . X M (M UJ O «- — & r~t 5jC Z T --• z rsl t-H II SJ I— 11 I r-l < CM II X o LL U O LU > - > O < U. U_ > CNJ U - ~ I > " — 0 . -I I > - U_ >-I u. I— I - 1 1 U_ U- + I ^ I * ~ CM - H- >- * LL. CM * - H O > - • L . CM * + O ~ I * V CM * LT-CM _ L * >- CM U . S - * > O U- O » * -h o ft + : • H I— vj" I _J + v: v i— LL. >-CM l LL CM I— I _J i— O >- > O U_ U. > - * X * p z o 0 1 > I— > - > X * P z o CO + M Ll * p * z O " < _ I — 3 * U- I-* a O + • —-cn r-i P I Q 3 * — I - > - + -I II O >- — 5 : LL CM P * »• Z W H ] | >- I u I 3 2: •-> — D U- >- Z CM I 3 O 3 • 1+ r-l r- I L 3 £ 'I M P Q . * * — 3 <t 1 CM I 3 CO <T- I 3 >- * (M O + O CM CM I - 3 1-1 — I > 3 + — O > ® 3 U - 0 - * * P o z . O cn u — * P M 3 Q • — * O UJ h- I I > - —• — LL CM CM >{< «n ft r-l »—t 1—I r—\ || >- I I Q 1 n 2 3 — -r- 3 U_ P LlJ Z — CM Cl S V - - > - — »!<: 60 in I-I CO I r-3 I w < >- I > I LL! O |l • t—i cm — * H vO !l O 3 « ̂ cm lu I I -< z > < UJ P < ~ I p cm ; cm ~> ; 1 * i ~> i — cm ! > •> + 1 I O I • ~> CI o o o cm uj r 'o r pp cl + I o 5: a CM CM M <1-O c<1 o cm cm 3 ^ Z r-l r-l || I I Q P ~> S — — 3 a x z Ll) 2: cm uj cq 11 3 2: Z 3 •-> ' Z ĉi m ir<~i Ml 3 3 — ̂ vO D. X O + + r-t cm cm o I :l O 3 3 P a x — — o * Jj« r-H m in ® • ; • uj O O O I 1 • co cm cm s » ̂ 3 C<1 P I— P no P — cm D - £> rH I fM X * -> 1—I - O 3 o * • —1 • 3 K-• P O • P CO I ® ft tr co —• —' t-. X ^ r 3 ~ ii cm CM m o P P P u. o U h Q r cr< I I 3 p a. -x. lu- ll 3 CM H -|» I SD i—I ^ | >: cm « o i ii cm cm — •> 11 p p ;-> to co in — < < ,0 3 ffl " - rH ;|| |\ p o ll ll o <3 Q >-<•-< Q p CM P P a 3 >:< tm cm cm cm cm I 2 5: X CO vO o • ;CM >- > P O > II II o e VH ~ ~ 3 O I CM CM — 1 n p f- 2: 2: I L O P - -3 P >- > in p r-l r-l s 2: q. x l I cm cm c. ft X 2 : a x 00 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) » I""' Hfl-i'i2 ) VQ (T-X»-2 ~ ~ ~ : 7 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 ~ , ' " : ~~~ : DO 9 I = 2 »M 1 U(I,3)=U(I>2) "p"n-i'T3")=PTr-TT2")" : • ~ . ~ : : ~ " H (I—1»3)=H(I—1* ) NUMB = NUMB+1 go to ioo L : : : ~ : : ~ - ~~ ~ ~ : 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 CALL PLO IS - - 7— : : 1 ; : 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 ) ~ " D O * 2 0 2 C A L L P L O T ( S A M ( J » K » D » S A M ( J » K » I ) » + 2 ) 2 0 1 C O N T I N U E : C A ' L L " ~ P L O T 1 1 2 . ~ O V O T O V - 3 1 : — : 2 0 0 C O N T I N U E C A L L P L O T N D _ : S T O p .. E N D S E N T R Y BIBLIOGRAPHY Ahlborn, B., Phys. Fluids: 9, 1873 (1966) Ahlborn, B. and Salvat, M., Z. Naturforschg. 22 a9 260 (1967) — — » Axford, W„ I Phil. Trans, R„ Soc. London. A253„ 301 (1961) Blackmant V.s J. Fluid Mech. is 61-85 (1956) Camact M., J. Chem. Phys. 34, 448 - 459 (1961) Camac f M., and Vaughan, A. , j. Chem. Phys. 34_f 460-470 Chandrasekhar, S."Radiative 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 Rc 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 Lun5kin, Yu„P„f Soviet Phys. - Techn. Phys. 4, 155 • (1959) Mathewss D. L„s Phys, Fluids 2, 170 - 178 (1959) Metzgers P. H„ 9 and, Cook„ G„ R o p J0 Quant. Spec„ Rad. Tr.ans. 4P 107 (1964) Mises, R. voy "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. J. Phys. 46, 183 (1968) 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., New"York (1954) 1 Zel! dovichYa. B. , and Raizer„ Yu„ P.,, "Physics of Shock V/aves and High-Temperature Hydrodynaraic. Phenomena.", Academic Press, New York (1966)
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Beijing | 3 | 0 |
Unknown | 2 | 0 |
Tokyo | 1 | 0 |
Sunnyvale | 1 | 0 |
Ashburn | 1 | 0 |
{[{ mDataHeader[type] }]} | {[{ month[type] }]} | {[{ tData[type] }]} |
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