THE USE OF REAL GASES IN A SHOCK TUBE by Russell EarlPuf. A dissertation submitted in?.part ia.aL fulfillment of the requirements.for-the degree of Ddctor -of Philosophy. in; the University of. Michigan 1951 Committee in charge: Professor Otto Laporte, Chairman Assistant Professor Ernst Katz Professor Wilbur C. Nelson Professor Gordon B. B. M. Sutherland Professor George E. Uhlenbeck

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ACKNOWLEDGNEMETS The author wishes to express his gratitude to Dr. Otto Laporte for his valuable assistance in initiating and directing this investigation and to Mr. Robert N. Hollyer, Jr., for his cooperation in the design, construction, and operation of the shock tube and associated equipment. The financial support for this investigation was provided by the Office of Naval Research. ii

TABLE OF CONTENTS Page ACKNOW LEDGEMENTS ii LIST OF TABLES vi LIST OF FIGURES vii Iv INTRODUCTION 1 1. Historical Sketch 1 2. Brief Description of the Apparatus 4 3. Preliminary Statement of the Problem of Employing Gases Other Than Air in a Shock Tube 6 II. GAS FLOW IN THE COMPRESSION CHAMBER 11 lo Taub Equation for a Gas with Constant Specific Heat 11 2. Compression Chamber Temperature 13 3. Flow Velocity Produced by a Rarefaction Wave in an Ideal Gas with Temperature-Dependent Specific Heat 15 4. Flow Velocity Produced by a Rarefaction Wave in a Van der Waals' Gas with Constant Specific Heat 22 III. SHOCK WAVE EQUATIONS 27 l. Rankine-Hugoniot Shock Equations 27 2. Shock Equations for a Gas with Temperature-Dependent Specific Heat 30 3. Numerical Calculation of Shock Parameters for Various Real Gases 35 4. Experimental Verification of Variable Specific Heat Predictions 51 APPENDIX TO CHAPTER III 54 iii

Page IV. VIBRATIONAL RELAXATION 55 1o General Discussion of Vibrational Relaxation 55 2. The Effect of Vibrational Relaxation on Shock Waves 60 3. Approximate Calculation of the Time Dependence of the Flow Parameters Behind a Shock Wave 64 4. Position-Time Curves for the Flow in the Real Gases of a Shock Tube 73 5. Effects of Vibrational Relaxation Investigated in the Shock Tube 76 a. Variable Region Behind the Shock 76 b. Bow-Wave Curvature. 81 c. Reflected Shock Velocity in Carbon Dioxide 86 APPENDIX TO CHAPTER IV 98 1i. Reflected Shock Velocity in Nitrogen 98 2. Reflected Shock Velocity in Argon 98 3. Reflected Shock Velocity in Sulfur Hexafluoride 103 V. THE DURATION OF UNIFORM FLOW IN A SHOCK TUBE 106 1. Duration Index 106 2. The Motion of a Detached Bow Wave as an Indication of the Development of a Steady State 111 VI. FURTHER DISCUSSION OF GAS PARAMETERS AND SPECIFIC GASES FOR SHOCK TUBE WORK 118 1. Effects of Gas Parameters 118 a. Specific Heat Function- 118 b Molecular Weight 122 c. Index of Refraction 123 iv

Page d. Viscosity Coefficient 124 e. Chemical Activity 125 f. Boiling Point 125 g. Solubility 125 2. Choice of Specific Gases for Use in a Shock Tube 126 a. General Aerodynamic Investigations 126 b. Physical Investigations 127 c. Representative Photographs of Flows in Complex Gases 128 VII. CONCLUSIONS 133 BIBLIOGRAPHY 135

LIST OF TABLES Table Page I. Hydrodynamic Variables for a Rarefaction Wave in Hydrogen; Constant and Variable Specific Heat 20 II. Hydrodynamic Variables for a Rarefaction Wave in Hydrogen; Ideal and Van der Waals' Gas 25 III. Values of B for Carbon Dioxide, Nitrogen, and Sulfur Hexafluoride 36 IV. Shock Wave Characteristics in Carbon Dioxide 37 V. Shock Wave Characteristics in Nitrogen 43 VI. Shock Wave Characteristics in Sulfur Hexafluoride 46 VII. Flow Mach Number Behind Shock Waves in Carbon Dioxide 52 VIII. Initial and Final Values of Shock Velocity, Flow Velocity, and Pressure Ratio Across the Rarefaction Wave for U/aO = 2.34 in Carbon Dioxide 75 IX. Relaxation Times for Various Gases 77, 78 X. Initial and Final Reflected Shock Velocities, Temperatures, and Pressure Ratios in Carbon Dioxide 91 XIo Position-Time Data for the Normal Reflection of Shock Waves in Carbon Dioxide 95 XII. Position-Time Data for the Normal Reflection of Shock Waves in Nitrogen, Argon, and Sulfur Hexafluoride 101 XIII. Detached Bow-Wave Position as a Function of Time for a 1/16-in., 30~ Wedge at Mach Number of 1.3 114 XIV. Chemical and Physical Properties of Gases 119 vi

LIST OF FIGURES Fig. Page 1. Initial Pressure Across the Diaphragm as a Function of Shock Strength 14 2. Compression Chamber Temperature as a Function of Shock Strength 16 3. Flow Velocity Produced by a Rarefaction Wave 21 4. Shock Wave Temperature Ratio in Carbon Dioxide 38 5. Shock Wave Density Ratio in Carbon Dioxide 39 6. Shock Wave Pressure Ratio in Carbon Dioxide 40 7. Shock Wave Flow Mach Number in Carbon Dioxide 41 8. Initial Pressure Ratio for Helium and Carbon Dioxide 42 9. Shock Wave Temperature and Density Ratio in Nitrogen 44 10. Shock Wave Pressure Ratio and Flow Mach Number in Nitrogen 45 11. Shock Wave Temperature and Density Ratio in Sulfur Hexafluoride 47 12. Shock Wave Pressure Ratio and Flow Mach Number in Sulfur.Hexafluoride 48 13. Bow Wave Attached to a Wedge 52 14. "Pillbox" through a Shock Wave 61 15. Various Flow Regions in a Shock Tube 65 16a,b.Flow Variables Behind a Shock Wave 69, 70 17. X-T Diagram of the Flow in a Real Shock Tube 74 18, Length of Transition Region Behind Shock Waves in Carbon Dioxide 82 19. Streamline Through an Oblique Shock Wave 83 20. Effect of Vibrational Relaxation in an Attached Bow Wave 84 vii

Fig. Page 21. Reflected Shock Strength as a Function of Initial Shock Strength 88 22, Normal Reflection of a Shock Wave 90 23. Normal Shock Reflection in Carbon Dioxide; U/ao = 4.835 92 24. Normal Shock Reflection in Carbon Dioxide; U/ao = 3.417 93 25. Normal Shock Reflection in Carbon Dioxide; U/ao = 2.472 94 26. Normal Shock Reflection in Nitrogen; P =.0783 99 27. Normal Shock Reflection in Argon; r =.0898 100 28. Normal Shock Reflection in Sulfur Hexafluoride; U/ao = - 3590 104 29. Duration Index versus Flow Mach Number 110 30. Shock Wave Detachment in Nitrogen 112 31. Shock Wave Detachment in Sulfur Hexafluoride 113 32. Detached Bow Wave in Sulfur Hexafluoride 130 33. Subsonic Flow in Petroleum Ether About an Airfoil at an Angle of Attack 130 34. Schlieren Photograph of the Normal Reflection of a Shock Wave in Carbon Dioxide 131 355 Detached Bow Wave in Carbon Tetrachloride in Front of a Wedge at an Angle of Attack 131 36. Attached Bow Wave in Carbon Tetrachloride on a Wedge at an Angle of Attack; Early Photograph 132 37. Attached Bow Wave in Carbon Tetrachloride on a Wedge at an Angle of Attack; Late Photograph 132 viii

THE USE OF REAL GASES IN A SHOCK TUBE I. INTRODUCTION 1. Historical Sketch In spite of the fact that the shock tube was first used over fifty years ago, its full potentialities are only now being recognized. In 1899, Vieillel proved the fundamental fact that a compression wave produced by breaking a diaphragm separating regions of different pressure travels faster than sound. The theory of the ideal shock tube was developed as early as 1910 by Kobes2 in a paper discussing the hydrodynamic theory of vacuum and pressure-air brakes for railroad rolling stock. In 1932, Schardin3 referred to the work of Kobes in a discussion of the noise generated when a diaphragm separating regions of different pressure in a pipe is ruptured. These early papers were unknown to the several scientists who derived the shock tube theory independently within the last decade. 1 Vieille, Paul, "Sur les Discontinuities produites par la detent brusque de gas compremes, Comptes Rendus, 129 1228-1230 (1899). Kobes, K., "Die Durchschlagsgeschwindigkeit bei den Luftsauge- und Druckluftbremsen", Zeitschrift des Osterreichischen Ingenieur- und Architekten-Vereines, 62, 558 (1910)Schardin, H., "Bemerkungen zum Druckausgleichvorgang in einer Rohrleitung", Physikalische Zeitschrift, 33, 60 (19)2). 1

2 Apparently no experimental investigation was made of shock waves produced in the shock tube until Payman and Shepherd4 performed a series of experiments in which the processes occurring in the tube were investigated in some detail by means of a wave-speed camera. Then, at Princeton University in 1943, Reynolds5 devised a shock tube for the calibration of pressure gauges suitable for blast measurements. Several other laboratories now have shock tubes designed solely for this purpose. The fundamental equation relating initial pressure ratio across the diaphragm to the shock strength produced was derived by Taub for this report. For convenience, this relation shall be referred to as the Taub equation throughout this report. The first quantitative use of the shock tube in a hydrodynamic investigation was made by Smith6 at Princeton University in 1945. Shock reflections, both regular and Mach, were examined by observing the primary tube shock after reflection from a wall whose inclination could be varied. This investigation of Mach reflection has been continued by Bleakney7'8 and his co-workers at Princeton University using a much better shock tube and a Mach-Zehnder interferometer. Payman, W., and Shepherd, W. F. C., "Explosion Waves and Shock Waves: VI, The Disturbance Produced by Bursting Diaphragms with Compressed Air", Proceedings of the Royal Society, A-186, 293-321 (1946). 5 Reynolds, George T., A Preliminary Study of Plane Shock Waves Formed by Bursting Diaphragms in a Tube, OSRD Report No. 1519, June, 1943. 6 Smith, Lincoln G., Photographic Investigation of the Reflection of Plane Shocks in Air, QSRD Report No. 6271, November, 1945. 7 Bleakney, W., Fletcher, C. H., and Weimer, D. K., "The Density Field in Mach Reflection of Shock Waves", Physical Review, A, 323 (1949). 8 Bleakney, W., and Taub, A. H., "Interaction of Shock Waves", Reviews of Modern Physics, 21, 584 (1949).

The possibility of using a shock tube as an intermittent wind tunnel was first proposed by Mautz, Geiger, and Epstein9 at the University of Michigan in 1948. This suggestion was investigated and found promising by Mautz and and Geiger.l0'1'1l2 At the present time the investigation of the applicability of the shock tube as an instrument for aerodynamic research is being continued. During the past year the shock-tube laboratories at Princeton and Michigan have prepared numerous reports on the diffraction of shock waves around various obstacles.l3-18 Shadowgraph, schlieren, and interferometric photography were used in these investigations. This paper will discuss the advantages of using gases other than air in the shock tube to extend its range and increase its usefulness. 9 autz, C. W., Geiger, F. W., and Epstein, H. T., "On the Investigation of Supersonic Flow Patterns by Means of a Shock Tube", Physical Review, y, 1872 (1948). 10 Mautz, C. W., The Use of the Shock Tube in the Production of Uniform Fields of Transonic and Supersonic Flow, Dissertation, University of Michigan, 1949. 11 Geiger, F. W., On the Shock Tube as a Tool for the Investigation of Flow Phenomena, Dissertation, University of Michigan, 1949. 12 Geiger, F. W., and Mautz, C. W., The Shock Tube as an Instrument for the Investigation of Transonic and Supersonic Flow Patterns, Report on O0R Contract N6-OfR-232, TO IV, Engineering Research Institute, University of Michigan, 1949. 13 Bleakney, Walker, The Diffraction of Shock Waves Around Obstacles and the Transient Loading of Structures, Princeton University, Technical Report II-3, Contract N6-ONR-1051 TO II, March 16, 1950. 14 White, D. R., Weimer, D. K., and Bleakney, Walker, The Diffraction of Shock Waves Around Obstacles and the Resulting Transient Loading of Structures, Princeton University, Technical Report II-6, Contract N6-ONR-105, TO II, August 1, 1950. 15 Uhlenbeck, G. (submitting work done by R. E. Duff and R. N. Hollyer, Jr.), Diffraction of Shock Waves Around Various Obstacles, Engineering Research Institute, University of Michigan, Report No. 50-1, Contract N6-ONR-232, TO IV, March 21, 1950.

2. Brief Description of the Apparatus The procedures and techniques of shock tube research are well described in the papers of Mautz and Geiger10,11'12 However, the shock tube and auxiliary equipment originally installed by Mautz have been extensively modified for this investigation. The shock tube itself now consists of a rectangular tube 26-1/2 feet long of 2-in. x 7-in. rectangular cross section. The compression chamber is 8-1/2 feet long, and the distance from the diaphragm to the test-section window is 15 feet. The lengths of the compression and expansion chambers were chosen so that the contact surface would limit the duration of uniform flow at all supersonic Mach numbers. One outstanding feature of this tube is its 7-in. x 11-1/2-in. test-section window. This very large window makes it possible to examine the flow in the shock tube from the top to the bottom of the tube and along a considerable length of the expansion chamber. The study of shock wave diffraction about obstacles on the floor of the tube is thus facilitated. Furthermore, the investigation of flow phenomena several chord lengths in front of and behind airfoil sections is also simplified. 16 Hollyer, R. N., Jr., and Duff, R. E., The Effect of Wall Boundary Layer on the Diffraction of Shock Waves Around Cylindrical and Rectangular Obstacles, Engineering Research Institute, University of Michigan, Report No. 50-2, Contract N6-ONR-232, TO IV, June 21, 1950. 17 Duff, R. E., and Hollyer, R. N., Jr., The Diffraction of Shock Waves Through Obstacles with Various Openings in Their Front and Back Surfaces, Engineering Research institute, University of Michigan, Report No. 50-3, Contract N6-ONR-232, TO IV, November 7, 1950. 18 Hollyert R. N., Jr., and Duff, R. E., Growth of the Turbulent Region at the Leading Edge of Rectangular Obstacles in Shock Wave Diffraction, Engineering Research Institute, University of Michigan, Report No. 51-2, Contract N6-ONR-232, TO IV, January 18, 1951.

Flow pictures may be taken by shadowgraph or schlieren photography.. The basic photographic optical system now consists of a highintensity, short-duration spark light source which illuminates an f/8.0, 6-in. parabolic mirror which produces a beam of parallel light. A similar mirror is used as the focussing element when schlieren photographs-are desired. The reduction of aperture of the photographic optical system from f/2.5 to f/8.0, which resulted when the original lens was replaced by the new mirror, necessitated a ten-fold increase in spark intensity-. This increase was obtained by doubling the spark voltage and capacity and enclosing the spark gap in an atmosphere of argon at a pressure of 200 lbs per sq in, Sufficient freedom was provided in the mountings for the spark, for the shadowgraph and schlieren mirrors, and for the plate holders so that any portion of the test section could be photographed easily. The strength of the primary tube shock is determined by measuring the transit time of the shock between two light stations. The equipment used is, however, very different from that used by Mautz. The stations now consist essentially of single-lens schlieren systems* mounted on ground optical benches which, in turn, are fastened to an adjustable bench completely isolated from the tube. The greatest improvement in the shock tube instrumentation has been made in the electronic control equipment. Two new binary counters and associated gates, oscillator, and delay circuit have been designed and It was found that light scattered from the surfaces of the lens which had focussed parallel light onto the schlieren knife edge, in a doublelens schlieren system, caused a large fraction of the background noise of the photomultiplier tube. Elimination of the lens improved the signal-to-noise ratio and therefore lowered by approximately 35 per cent the minimum expansion-chamber pressure at which shock velocities could be determined,

constructed by Mr. R. N. Hollyer, Jr. This new equipment allows both the shock velocity and the time at which the photograph was taken to be determined every time the shock tube is fired. Of course, this has reduced the uncertainty in shock velocity inherent in all earlier shock tube photographs made at this laboratory. At the same time the techniques of shock tube research have been greatly simplified. A complete description of the shock tube and its associated electronic equipment will be given by Mr. R. N. Hollyer and the author in an Engineering Research Institute Report now in preparation. As the Michigan shock tube now exists, it is far from a perfect instrument; but it is definitely superior to the original tube, and it is quite adequate for the investigation of many classes of hydrodynamic phenomena. 3. Preliminary Statement of the Problem of Employing Gases Other Than Air in a Shock Tube In all the early shock tube work, air was used in both chambers of the tube. However, both Smith and Mautz have pointed out that by using helium in the compression chamber, a much stronger shock wave can be produced by a given initial pressure ratio across the diaphragm; or, conversely, when helium is in the compression chamber, a higher pressure can be used in the expansion chamber to produce a given strength- shock wave. This has the great advantage of increasing the contrast and definition of the resulting photographs of the flow. Furthermore, the use of helium in the compression chamber makes it possible to obtain shock strengths that cannot be obtained in the shock tube with air. One of the limitations of the shock tube as an instrument for aerodynamic research is the fact that the flow Mach number is limited to

a relatively low value. This limit is inherent in the method for producing flow used in the shock tube and it is a function of the specific heat ratio of the gas used in the expansion chamber. Mautz first noted that the limit of 1.89 for air could easily be surpassed by using a complex gas with a low specific heat ratio in the expansion chamber. This investigation is concerned with a detailed consideration of the advantages and disadvantages of using gases other than air in the compression and expansion chambers of a shock tube. Examination of the generalized Taub equation (cf. page 2) shows that, in order to produce strong shock waves, a light gas with a high velocity of sound should be used in the compression chamber; and fundamental shock theory (cf. page 29) reveals that high Mach number flows can be obtained by using a complex gas, with low specific heat ratio, in the expansion chamber. Therefore, when considering a wide variety of real gases for use in a shock tube, one must be prepared for possible deviations from ideal shock theory caused by (1) temperature dependence of the specific heat, (2) deviations from the ideal gas equation of state, and (3) relaxation phenomena. Only the first two perturbations need be considered as possibly important for the flow produced by the rarefaction wave in the compression chamber. In Chapter II expressions for the flow velocity produced by a rarefaction wave in an ideal gas with temperature-dependent specific heat and in a van der Waals' gas with constant specific heat are calculatedo By insertion of numerical values it is shown that the.velocity perturbations are negligible for hydrogen, The calculations were performed for this gas because the Taub equation indicates that the strongest shock waves for a given initial pressure ratio across the diaphragm can

be produced by using hydrogen in the compression chamber. Special precautions must be taken, however, to eliminate oxygen from the expansion chamber in order to prevent a hydrogen-oxygen explosion which could be initiated by the high temperatures behind the shock wave reflected from the end of the expansion chamber. In Chapter III shock theory analogous to the Rankine-Hugoniot theory is developed for a gas with temperature-dependent specific heat (deviations from the ideal gas equation of state are not important here). Flow parameters calculated from these equations show that the temperature ratio across a shock wave (for all shock velocities) is lower than predicted by the Rankine-Hugoniot theory. Also,the pressure and density behind a shock wave, as well as the flow Mach number, are higher than predicted by the simple theory. Numerical calculations are made for the hydrodynamic variables behind shock waves in carbon dioxide, nitrogen, and sulfur hexafluoride. Finally, the results of a flow calibration in carbon dioxide, in the supersonic region, are shown to be in good agreement with the more general shock theory. Relaxation phenomena are investigated in Chapter IV insofar as they influence the flow through a shock wave. In general, relaxation phenomena are caused by the destruction of thermal equilibrium between various degrees of molecular freedom by rapid temperature variations. This is one refinement of shock wave theory which as yet has not been viewed from the shock tube point of view. It is usually assumed that a shock wave is, to all intents and purposes, a surface of discontinuity separating two regions of uniform state. This tacit assumption is made in the calculations of Chapter III. However, under certain conditions this is not true; specifically, it has been found that even though the rotational degrees of freedom are usually

in equilibrium with the translational degrees, the energy associated with the vibrational modes is often not able to follow abrupt temperature changes. Under these conditions, one may say that the shock wave is followed by an extended transition region in which the vibrational energy is approaching its equilibrium value. The relative rate of change of vibrational energy may be associated with a relaxation time. A large amount of theoretical and experimental work has been done in order to determine the relations between vibrational relaxation and periodic disturbances. In particular, the effects of vibrational relaxation on sound velocity and absorption in the ultrasonic region have been investigated in great detail since 1925.19 However, the effects of vibrational relaxation on an aperiodic disturbance like a shock wave have not been thoroughly investigated as yet. In Chapter IV an approximate calculation of the flow variables in the transition region behind a shock wave is carried out. The results of earlier work by Bethe and Teller are used. Experiments are described whereby the velocity of a shock wave reflected normally from a surface is determined as a function of time, For shock waves in carbon dioxide, this velocity is found to be variable, while it is constant in nitrogen and sulfur hexafluoride. The result is that, in a shock tube, vibrational relaxation influences slightly the flow of carbon dioxide, but does not affect the flows of the other gases. The conclusion is reached that, insofar as practical hydrodynamic research in a shock tube is concerned, vibrational relaxation can be ignored. The discussion of the use of various gases in a shock tube would not be complete without an analysis of their effects on the relative 19 Richards, W. T., "Supersonic Phenomena", Reviews of Modern Physics, 11, 36-64 (1939).

10 duration of uniform flow. This problem is treated in Chapter V. If it is assumed that the rate of development of a flow field is inversely proportional to the sound speed behind a shock wave, a duration index can be defined which is proportional to the relative duration of uniform flow. This duration index is a function of the shock strength and specific heat ratio. It is shown that for a given Mach number, the duration index is practically independent of the gas used in the expansion chamber. Furthermore, experiments are described which demonstrate the validity of the assumptions made in deriving the duration index. The additional effects of a large number of chemical and physical properties of gases are discussed in the final chapter (VI). In particular, the conclusion is reached that the best combination of gases for routine aerodynamic investigation is hydrogen in the compression chamber and nitrogen in the expansion chamber. However, for investigations of the fundamental hydrodynamic principles that govern the transonic and supersonic flow of gases, sulfur hexafluoride has a unique combination of properties that makes it one of the best gases for use in the expansion chamber.

II. GAS FLOW IN THE COMPRESSION CHAMBER lo Taub Equation for a Gas with Constant Specific Heat As derived by Taub and generalized by Mautz, the basic shock tube equation which relates the pressure ratio across the diaphragm to the shock strength produced is 3 |' C ( - S Y5 where 5 is the ratio, PO/Pl, of the pressure in front of the shock to that behind it, p3 is the initial pressure in the compression chamber, aOE and aOC are the sound speeds in the expansion and compression chambers, respectively, and HE and C are the functions (+ 1)/( - l)af the specific heat ratios in the expansion and compression chambers. It is assumed that ideal gases with constant specific heats are used in both chambers of the shock tube. If the same diatomic gas is used in both chambers, this becomes simply Ph i[ ass(S ) ( b2) where a is assumed to be 1440. However, if aOE(pE - l)/aoc( MC - 1) or JC + 1 could be reduced, the pressure ratio corresponding to a given ~ would be increased. In this way the initial expansion chamber pressure can also be increased 11

12 because the pressure in the compression chamber is usually. adjusted to that of the atmosphere. As a result, the quality of flow photographs is improved because the optical effect associated with a given density ratio in the flow about a model is proportional to the absolute density of the gas in the expansion chamber. Furthermore, very strong and useful shock waves can be produced in a shock tube only by a manipulation of these constants. The most effective way to increase the initial pressure ratio across the diaphragm at constant' is to increase the velocity of sound of the gas in the compression chamber. In other words, hydrogen or helium are to be used as this gas. The velocity of sound ratio appearing in Eq 1 can easily be increased by a factor of 3 in this way. Variations in the specific heat function, I, are usually less effective because this function may only be reduced from 6 to 4 by substituting a mono atomic gas for a diatomic one in the compression chamber. As mentioned in the introduction, both Smith and Mautz suggested that helium be used; but hydrogen, which would be even more effective, was not considered for fear that an explosion of hydrogen and atmospheric oxygen might be initiated by the extremely high temperatures behind the shock wave reflected from the end of the expansion chamber, This fear was Justified because an explosion does result when hydrogen and air are used in the chambers of a shock tube to produce strong shocks. However, this difficulty can be avoided by substituting nitrogen for air in the expansion chamber. The use of nitrogen does not complicate the firing procedure, because, in order to introduce hydrogen or helium into the compression chamber without breaking the diaphragm prematurely, both chambers of the tube must be evacuated. In fact, there are several

13 additional reasons why it is desirable to use nitrogen instead of air in the expansion chamber of a shock tube. These reasons will be discussed in detail in subsequent chapters. When nitrogen and helium are used in the chambers of a shock tube, Eq 1 becomes: = I ( —.5796 (3) P/ J7 ( C +' and for nitrogen and hydrogen one obtains: 6.878 [1 -C=:[.2727 ---- (47 where " for hydrogen is assumed to be 1.410. Eqs 2, 3, and 4 are plotted in Fig. 1o It can be easily seen that hydrogen produces the strongest shock wave for a given pressure ratio across the diaphragm. In particular, the advantage of using hydrogen to produce strong shocks, small values of ~, is obvious. It can also be seen that hydrogen has a lower limit shock strength for an infinite initial pressure ratio than either helium or nitrogen. 2. Compression Chamber Temperature The temperature produced by the adiabatic expansion wave in the compression chamber can be easily calculated from the pO/P2 versus g relationship. For an adiabatic process

o 1;u k in b i b3 iD a~~~~~~~~~t T -fi ~~'~I t..1- ~I i~ t.I..i;.. t-~!:. T.I.1 IN fi.!` - i I- -1 ~ l o i. o I:~ L.1 4.4'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,.,::-i i c,'iF'.... ~.~ ~;',"!. ~: I.' -! - - 44 4 -- 1 4- H++r-t, C I -~i'~' ~ K#t,. flI I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....... i. +~~~~~~ ~i-'il --- ~ ~ ~ ~ ~ ~ ~ 4 -- ------ F Tt -4bD~~~~~~~~~~~~ F 4 -1, L - - I I I I I iT -— I-TTT-I-FTT1 Ft~~~~~~~~~~~~~~~~~~~~~ in r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 14 -~~~~~~~~~~~~~~~~~~~~~~~~F 9\ t iD~~~~~~~~~~~~~~~~~~~~~~~~~ T -T I tT tI ~~lj~tf~TrrTx~ t~ttPi~~-ttt+tttt r rrrtm~ tJ T-r~i -I~-; i tri Ci ci.E li

15 and Ps PP3p os P3 Thus, T3 [v q (LLJLl). (-) This relation is plotted in Fig. 2 for the three gas combinations discussed above. It is interesting to note that with nitrogen in the compression chamber, the temperature there falls well below the freezing point of the gas (630K) for very strong shocks.* It should be noted, however, that the optical system used to detect the passage of the shock wave has only a limited sensitivity. Therefore, when nitrogen is used in the compression chamber, it is not possible to operate the tube for shocks of strength much less than =.2. The corresponding limits for helium and hydrogen are approximately' =.05 and S =.03, respectively. The lowest temperatures actually obtained in the compression chamber are, therefore, much higher than Fig. 2 indicates at first glance. 30 Flow Velocity Produced by a Rarefaction Wave in an Ideal Gas with Temperature-Dependent Specific Heat Eq 1 has been derived under the assumption that the specific heat of the gases used in both chambers of the tube is constant, Unfortunately, this is not strictly correct for either chamber. The It has actually been suggested by Antonio Ferri that the cold flow in a shock tube be used to investigate the problem of condensation of air which is very important in the nozzles of hypersonic wind tunnels.

~ t ~ I~ ~:: I I j, t -! -1 1j -T -T-~ ~r~~~~~~~~~~~~~''.....; ~.~..~-.~-....:. ~,-..-i..~-i ~r i~i-~ i i 300 N'''i" 250 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-' —J~: - r- T' — tt: L 200 i~~~~~~~~~ri z~t- v~ li —'' T(I 150,, — ~, COMPRESSION CHAMBER 100 TEMPERATURE c- 1 1-, -r -~~~~~~- T- - 7 -'- 7-1-r -7- 50 -,1 0. 1.0 0 ~ ~~~~ ~ ~~~~~~ I.3.4.5. 7.8.. S~~~~~~

17 modifications of the fundamental shock equations brought about by the temperature dependence of the specific heat of the gas in the expansion chamber will be considered in Chapter III. The temperature changes in the compression chamber discussed in the last section are sufficient to cause an 8 per cent decrease in the specific heat of hydrogen and a 2 per cent decrease in the specific heat of nitrogen. Since helium is a monatomic gas, its specific heat is not a function of temperature. The velocity imported to a gas initially at rest, by a rarefaction wave is where dp/dp is usually calculated along an adiabatic path.(6) where dp/dp is usually calculated along an adiabatic path. A comparison of the flow velocity produced, for a given pressure ratio across the rarefaction wave, by real and ideal gases will suffice to determine the effect of specific-heat variations on the flow from the compression chamber. This velocity integral can be reduced to a quadrature in the general case of an ideal gas whose specific heat is a given function of temperature. Assume an ideal gas with the equation of state p = pRT. The specific heat may be written as Cv: f(T). For an adiabatic process, the first law of thermodynamics becomes dE = P- dp (7)

The change in internal energy can be written as dE = CVdT. However, by changing the independent variables to p and p, it can be shown immediately that p _'F(T4+R A (8) de f(T) P Since Cp - Cv = R for an ideal gas, it has been shown that the adiabatic compressibility is the isothermal compressibility multiplied by the ratio of specific heats at constant pressure and constant volume. In addition, by eliminating p from Eq 6, one obtains S o _ f(T) dT (9) P -RT By substituting Eqs 8 and 9 into the velocity equation, 6, the following general expression is obtained, T RT Between 1900 and 3000K, the specific heat of hydrogen may be represented by f(T) = C To-T)2], (11) where Cv = 10ol ergs/g; = 9.31x 10-6 degree-2; and To 3000K. The integrand of Eq 10 can be expanded by the binomial theoremo Only first-order terms need be retained.

19 R 2c,' jJ~~C2.... ut = Cv I- ~' c~,c,, _ T~ -I T After integration and simplification, the following expression for flow velocity is obtained. U _ = Ctv I n a0 i O0 T0 | 133 TO b where d = Cp'/Cv' and ao = 7RTO The first term of Eq 13 can be recognized as the flow velocity produced by a rarefaction wave in an ideal gas with constant specific heat. The flow velocity is given as a function of temperature by Eq 13. However, for shock tube applications, it would be preferable if the velocity were expressed as a function of pressure. In order to transform variables, it is necessary to obtain an equation for the adiabatic curve. If one changes the independent variables of Eq 7 to T and p, the following differential equation is obtained: RT p This equation can be integrated if Eq 11 is substituted for Cpo The resulting equation for the adiabatic curve is _( cp - ( TL) oL 2Csv f(3 Tol(j T) (14) P T 3 0

20 Eqs 13 and 14 constitute a system of parametric equations which determine the flow velocity as a function of pressure. The values of flow velocity, temperature, and pressure ratio for a rarefaction wave in hydrogen are shown in Table I. The values included were calculated for both constant and variable specific heats. TABLE I HYDRODYNAMIC VARIABLES FOR A RAREFACTION WAVE IN HYDROGEN CONSTANT AND VARIABLE SPECIFIC HEAT T0K~ *P/PO P/ PO mm/A sec constant specific variable specific heat heat 300 0 1. 1.0O 290 o1082.8900.8900 280.2183.7886.7888 270.3304.6960.6965 260.4450.6113.6125 250.5609.5340.5360 240.6801~.4641.4671 230.8011.4010.4052 220.9248.344o.3496 210 1.052.2932.3001 200 1.182.2479.2565 190 1o315.2077.2177 The change in flow velocity at a given temperature, produced by assuming variable specific heat is negligible.. The pressure ratio across a rarefaction wave in hydrogen ts plotted as a function of flow velocity in Fig. 3 for constant and for temperature-dependent specific heat. It will be seen that there is little difference between the two curves. In fact, the specific heat variations will be sufficient to cause an increase in flow Mach number of

21 FIG 3 1.0,' f'1!.! I \ I I I I I NJ II I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.9 - - \ i l_ --.8 _ _ _ _ _ _ _ 5 _ _ _ + FLOW VELOCITY PRODUCED' I I I,,' I.' ~~~~~~~~~~~~~~~~~~~~~~~~~ BY A ~' ~' _ _ _ S i _ _ RAREFACTION WAVE - e~~~~~~~~~~~~~~~~~~~~~11 fil 111 11i $1J jW401114400E.7'II rri5iI \ ~! w. __ ti o E! I I' k [1 1 I I I I j1 I I i I. it0 0 00 0 M1 il T. —-: r I t I I X I I _' i I I _ _,111. -~. i I.tI' f P/P t1 4 | 11 1 | | | | <1111>L11 l000H0H00/P TAU 3 1 1L _[ II II_ T__ __ _1Li I i,\1 1. L I I i IIl0-i -L I I i III III —T ______ LL I I' 11 i IIIL 1ILl!. ~. [E~lIt:[ tll 1 I! I \ I 1E11 i t i I ~ i -.2 l- E! L1l __I _______ I 1! i I \.! I! i; i _, t1- 1mM-SS~~~~~~~~~% 111 I I 11 t-ittl~il E pU t~ _I I_ I. -,I I I t'__T __ —-T llllllll1l11111 I I i 1!! I 1 - I' I III__I __,.L L L I I I I II I 11111 Ii' _i~~~~~~~~~~~~~ S......I-.. i I I!!

22 only approximately 1 per cent in the shock tube. It is interesting to note that the flow velocity, for a particular pressure ratio, is higher for a gas with temperature-dependent specific heat than for a gas with constant specific heat. This is the direct result of the transfer of energy from the rotational to the translational degrees of freedom, which causes the specific heat variations. It has thus been shown that the small change of specific heat with temperature in hydrogen does not appreciably perturb the flow variables of a rarefaction wave from their values as predicted by constant specific heat theory, Furthermore, this statement is true for all rarefaction waves produced in routine shock tube research because the specific heat variations in helium and nitrogen are even smaller than those of hydrogen; and they will be correspondingly less important. However, it would not be true for almost complete expansions where the temperature in the wave may approach absolute zero. 4. Flow Velocity Produced by a Rarefaction Wave in a Van der Waals' Gas with Constant Specific Heat It is possible that deviations from the ideal gas equation of state would become important at the low temperatures found in the compression chamber. This may be investigated by determining the flow velocity produced by a rarefaction wave in a van der Waals' gas. The van der Waals' equation is superior to a virial approximation because the constants, a and b, can be associated with molecular dimensions and intermolecular forces. For convenience, the Cv will now be considered constant. The van der Waals' equation of state can be written

23 p RT- a. (15) -bp For an adiabatic process the first law of thermodynamics becomes dE: P It will be convenient to make temperature and density the independent variables in this equation. It then follows as a consequence of the first and second laws of thermodynamics that Cv dT = p + (6) An equation for the adiabatic curve can be obtained by integrating this expression. R P / T I A,-be or Cv+R + )( Ff) CV V (17) Eq 17 is exact. For convenience, it can be expanded, Only first-order corrections will be retained and CV d R I Cv P 0.Q P3aFPUL\ An approximate expression for dp/dp can be obtained from Eq 18, but it is more convenient to derive an exact expression by

24 changing variables in Eq 16. After simplification and elimination of p by using Eq 17, one obtains ~Y-I JP =b t'(l, - 24g. (19) The flow velocity produced by a rarefaction wave can now be obtained from Eqs 6 and 19. I &(I-bp) e Because a and b are small, the integrand can be expanded to give p-~ I+(~"+,) bp Ut P h2 [ I+(a'+t 2)P -, P ~] dp. After integration one obtains U.'' (1 ff3 \I P3 (21) 2~ 2. _ __'. (_P_1 2'- (e)' -bp I'3 II 1 l-\ ]-,) The initial conditions for the rarefaction wave in a shock tube are usually atmospheric pressure, density, and temperature. Therefore', - O and -=

25 Eq 21 reduces to the ideal gas equation immediately when a and b are zero. The flow velocity produced by a rarefaction wave in a van der Waals' gas can now be determined as a function of pressure from Eqs 18 and 21. The values of flow velocity, density ratio, and pressure ratio for rarefaction waves in hydrogen, considered as an ideal gas and as a van der Waals' gas, are presented in Table II. The following constants were used in the calculation: a =.2444 (liter)2 (atm)/(mole)2 b =.02661 liters/mole = 1.41 aO = 1.321 x 105 cm/sec TABLE II HYDRODYNAMIC VARIABLES FOR A RAREFACTION WAVE IN HYDROGEN IDEAL AND VAN DER WAALS' GAS P/po uj mm//=see unmm//."sec....p/p.. /PO P/PO.........OP/PO ideal gas van der Waals' gas ideal gas van der Waals' as.9.1377.1378.8619.8618.8.2882.2883.73u1.7299 ~7.4543.4544.6048.6045.6.6406.6408.4866.4864.5.8536.8537.3763.3761.4 1.104 1.104.2747.2745.3 1.409 1.409.1831.1830

26 If these results were plotted as in Fig. 3, the two curves would be indistinguishable. It has thus been shown that equation-of-state variations influence the flow of hydrogen from the compression chamber even less than specific heat variations. It is true that the deviations from ideal gas behavior might be more important if a heavy gas were being used in the compression chamber in order to increase the pressure difference across the diaphragm when very weak shocks are desired. However, the effects would probably be small even here. Experimentally, the fact that the specific-heat and equation-ofstate variations are not important is shown by the experimental points in Fig. 1. Since nitrogen is used in the expansion chamber in each case, the pO/P3 versus r results are indicative of the behavior of the gas in the compression chamber. If specific — heat or equation-of-state perturbations were important, one would expect the results for nitrogen and hydrogen to be noticeably different from those for helium, because helium is a monatomic gas with very small'van der Waals' constants. However, it is apparent that all the experimental results exhibit the same trend.* From a practical point-of view the most important result of this phase of the investigation is the demonstration of the fact that hydrogen can be used in the compression chamber of a shock:tube if oxygen is excluded from the expansion chamber. Hydrogen will produce a stronger shock wave than any other gas for a given initial pressure ratio across the diaphragm. See.Chapter III, Section 4, for a discussion of two mechanical factors which cause the shock wave produced by a given initial pressure ratio to be weaker than predicted.

III. SHOCK WAVE EQU ATIONS 1. Rankine-Eugoniot Shock Equations The determination of the hydrodynamic variables behind the shock wave in the expansion chamber is the fundamental problem of shock tube theory. This problem is quite easily solved by consideration of the conservation laws for mass, momentum, and energy across a discontinuity surface. Elementary algebraic manipulation leads to the desired results. In case the specific heat of a gas may be assumed to be constant, the conservation laws for the one-dimensional steady flow through a stationary discontinuity surface become: Vo )pO, P, Conservation of mass, epV e=.V.(22) Conservation of momentum, po +. V2 (23) 27

28 Conservation of energy, 2aD -+.I = 2-, 2 (24) V0 and V1 can be eliminated from these equations; the resultant relation between p and p is the well-known Rankine-Hugoniot equation __ - =VX (25) This equation shows that a discontinuity surface can exist in an ideal compressible fluid if a particular relation between the hydrodynamic variables is satisfied. In a shock tube the gas in front of the shock wave is stationary with respect to the laboratory frame of reference; but the shock wave is not. Therefore, the velocities VO and V1 can be transformed to more significant variables by translating the frame of reference so that the gas in front of the shock wave is at rest. Thus U.-LL + +5 U where U is the shock velocity and u is the flow speed behind the shock wave. An expression for the shock velocity itself can be obtained from the Rankine-Hugoniot relation, the energy equation, and Prandtl's relation. Ub t ia giv(26) Substituting this into Eq 14 gives

29 LI.u =Q.0 R -(Z-')(' -!(27) Finally, the Mach number of the flow behind the shock wave relative to the laboratory system is Q. U. PO LD 0 P s M (28) The last equation illustrates one of the more serious limitations of the shock tube as an intermittent supersonic wind tunnel. The Mach number of the flow, even for infinitely strong shocks, is limited to a comparatively low value. oM =te29) For a diatomic gas with ~ = 1.40 this limit is 5/ /i' or 1.89. The obvious way to raise the Mach number limit in the hot flow of the shock tube is to use a gas with a lower specific heat ratio, that is, higher J, in the ~expansion chamber. This was suggested by Mautz and Geigerl'1112 but no investigation of the consequences of such a substitution was made by them.

30 2. Shock Equations for a Gas with Temperature-Dependent Specific Heat If a polyatomic gas is used in the expansion chamber of a shock tube, the assumption that the specific heat does not depend on temperature is not valid. Therefore, the basic conservation equations must be revised. Eqs 22 and 23, expressing the conservation of mass and momentum, are equations involving only the mechanical variables of the problem; and they are independent of any variation in the thermodynamic properties of the gas. The energy equation, on the other hand, must be rewritten, because the assumption of constant specific heat was fundamental to its derivation. The energy balance for adiabatic flow can be stated as follows: The increase in kinetic plus internal energy of the gas must be equal to the work done on the gas by pressure forces. dT +d E = dW The increase in kinetic energy per unit mass is udu, and the increase in internal energy for an ideal gas* is CvdT. In a one-dimensional flow the work done on the gas is P - (jdpa (L~t Ci) - dde); * The ideal gas equation of state P = RTp will be used throughout this discussion. This is valid to a very good approximation because the pressures found in the expansion chamber of the shock tube are always less than atmospheric. In the case of strong shocks, where the relative change in the state variables is large, the expansion chamber pressure is usually less than 1/50 of an atmosphere. Furthermore, molecular dissociation is also neglected because the maximum temperatures attained in the shock tube are below those at which dissoctiaion becomes important for most gases.

31 where infinitesimals of higher order have been neglected. Thus v dv + cv dT + RdT =0. This equation can be integrated from one side of a shock wave to the other. 0 ~ ~ ~ ~ ~ ~ ~22. For convenience this can be written in a form similar to Eq 24. v~ +J coT =.. +cpTdT 2 2 The f Cp dT is a function of temperature which can be represented by f Tc T = p(T) RT. The lower limit of the specific-heat integral is arbitrary. In general, the specific heat integrals or 5(T) must be determined experimentally. The usual procedure is to use infra-red spectroscopic data to assign frequencies to the various possible modes of molecular oscillation. The thermodynamic functions for the gas can then be calculated from the "zustandsumme" constructed from these frequencies. When necessary, corrections for anharmonicity can also be included. In most cases the specific heats calculated in this way are more accurate than direct thermal measurements. 20 The following solution of the conservation equations is similar to that given by Bethe, H. E.} and Teller, E., Deviations from Thermal Equilibrium in Shock Waves, Aberdeen Proving Grounds, Ballistic Research Laboratory, Report No. X-117.

32 For a diatomic gas like nitrogen, P(T) may be represented by PW + R + T e where Cpa is the specific heat at constant pressure associated with the translational and rotational degrees of freedom, and 9 is a characteristic temperature of the gas. For polyatomic molecules more than one Einstein function will usually be needed to allow for the several modes of oscillation. The three conservation equations take the form: (30) 1. 2 V + PoRTG = + PRTT e)v = D5v 2 ~ _ 2 The mass and momentum equations can be combined to give a relation involving only T and V. V + RT)' V2 (V ) R (31) Since 13 is usually a transcendental function of T, it is advisable to consider this system of equations as two equations in the four unknowns VO, V1, To, and T1. An expression analogous to the Rankine-Hugoniot equation can then be derived, and the velocities on

33 each side of a shock wave can be determined from the temperatures assumed to exist in the two uniform regions. Eqs 22 and 23 now constitute two equations in two unknowns which can be solved for the velocities. Let Vo v - V = fv,. Now solve for f by eliminating V2 from the two equations. 2 T:R(T-fT2)'V4 ( I ~~~~ ( ) (32) {f(,-f) As a result V _ f = = + - (33) where 2 I The positive square root was chosen in Eq 33 to insure that the velocity ratio be greater than zero. Substituting Eq 33 in Eq 32 gives V2 = R b + T (3. When the temperatures on each side of the shock wave are given, Eqs 33 and 34 are in convenient form for calculation of the flow velocitites. The density and pressure ratios can then be found immediately by using the continuity equation and the equation of state.

34 As expected, Eq 33 reduces to the Rankine-Hugoniot equation when P0 = 2 as shown in Appendix to Chapter III, page 54~ In order to calculate the Mach number of the flow behind the shock wave, it is necessary to change the frame of reference to one in which the gas in front of the shock is at rest. The flow velocity behind the shock becomes Lt = Vo -V2 and the local Mach number is (35) where ~ 2 is the value of K corresponding to T2. A relation between pO/p3 and the shock Mach number, U/aO, can also be derived easily. The assumption is made that the flow velocity and pressure are continuous across the contact surface which separates the gases initially in the expansion chamber from those from the compression chamber, Therefore, the velocity produced by the rarefaction wave must be equal to the flow velocity behind the shock wave, It is then possible to determine the pressure ratio across the rarefaction wave from the first part of Eq 130 2 The = i - i Then to = _ P3. (56) Po

35 3. Numerical Calculation of Shock Parameters for various Real Gases The methods outlined in the last section were used to calculate the characteristics of shock waves in carbon dioxide. The necessary values of e were obtained from enthalpy data given by Kassel.21 Table III contains values of 3 as a function of temperature for carbon dioxide, nitrogen, and sulfur hexafluoride, The values of flow velocity, temperature, pressure, density, Mach number, and required diaphragm pressure ratio for shock waves in carbon dioxide are tabulated in Table IV. The corresponding values calculated from the simple constant specific-heat equations are also included. The PO/p3 calculations were made assuming helium in the compression chamber in order to remove completely the complication of specific-heat or equation-of-state variations there. Figs. 4 - 8 show these shock wave characteristics as functions of the shock Mach number. The considerations which dictated this method of presentation of the data and the curves T1/T0, Pi/4o and P1/Po, also included in the figures, will be discussed in Chapter IV, Section 2. Table V and Figs. 9 and 10 present similar information for shock waves in nitrogen. The values of 5(T) used in this calculation were taken from the Bethe and Teller report, Table VI and Figs. 11 and 12 show shock characteristics in sulfur hexafluoride, The calculations were based on Cp values calculated by Meyer and Buell.22 21 Kassel, Louis S., "Thermodynamic Functions of Nitrous Oxide and Carbcn Dioxide", Journal of the American Chemical Society, 56, 1838 (1934), 22 Meyer, Gerald E., and Buell, C. E., "Note on the Specific Heat of Sulfur Hexafluoride", Journal of Chemical Physics, 16, 744 (1948).

36 TABLE III VALUES OF B FOR CARBON DIOXIDE, NITROGEN, AND SULFUR HEXAFLUORIDE T3 B - CO2 - N2 B SF6 300 4.411 3.493 11.47 350 11.57 400 4.450 3.499 11.79 450 12.o6 500 4.579 3.508 12.35 550 12 64 600 4.732 3.521 12.92 650 13.19 700 4.886 3.541 13.44 800 5.8034 3.564 13.89 900 5.173 3.594 14.28 1000 5.302 3.625 1100 5.421 1200 5.531 1250 3.702 1300 5.632 1400 5.725 1500 5.811 3.780 1750 5.999

TABLE IV SHOCK WAVE CHARACTERISTICS IN C02 [ Variable'Specific Hea Constant Specific Heat p_2 VO PV P1 P1 T3 VO P2 OM T22 P Pi M a0 PO PO P3 ao Po P P3 300 10013 1.035 1.045 0.037.941 5 300 1.013 1.035 1.o45 037.9410 400 1.710 2.399 3.229 0.859.195 400 1.684 2.329 3.135 0.828.204 500 2.337 3.674 6.185 1.319.0712 500 2.252 3.351 5.641 1.217.0828 600 2.876 4.699 9.493 1.615.034o 600 2.712 4.o41 8.163 1.435.0451 700 3.364 5.533 13050 1.824.o0188 700 3.102 4.596 o10.84 1.581.0272 800 3.798 6.226 16.78 1.982.0113 800 3.454 4.992 13.46 1.680.0180 900 4.204 6.814 20.68 2.109.0070 900 3.770 5.302 16.07 1.754.0126 1000 4.583 7.321 24.65 2o211.0047 1000 4.064 5,552 18.69 1.813.0091 1100 4.946 7.765 28.76 2.308.0031 1100 4.339 5.755 21,32 1.862.0069 limit Mach number 1 = 2.25

4.0 XW 0W T/ RToN-3NI;X ~'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'0 Ill,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o,HT 001 Ile,~~~~ w Z~~~~~~~~~~~~~~~~- "I 00,'IO AO' ~- - c_ I ~ w~~~~~~~~~~~~~~~~~~~. /. I~ ~~~~~~~;I,,"'" o Jv I I I 1 11 I I!100;40Tt;$04;000>XL80 I 2/< INIITlllll C02li~.Or.~ ~ ~~Tl0 l,~0~~~~~~~~~~~~~0 I. oo / -~~~~~~~~~oo-' /,1 - -- / _,_ T_____ _ I _II2IIIIIIIIII/i IIII 1111 01111111111 IIIIIII111-] <1 / / Ft::,.,. To W -.u I I I 11 I 2.0 2.5 0 3 5- 4. 4l; 5.0 TU5:: _ _____ I II 1 111IIIIIIIIIII,/.,,.:E~~ 2. 10./I- ~ SHOCK WAVE11m00 |II1- TEMERTUR RATIOTTTE~ 11 *;_ _ _ __T ___ I7_ T1 11 111 III III 1 11l III II II III III III II II III/,III' LIJ _, ______T___/_ L71 lIIIIIIII 1 11111L 111111111111111/./,o

8.0 7.0 l 04.0~~~~~~~~~~~~~~~~~~~~ ~~~~~~C')I~~~~~~~~~~~~~~~~ 3.0 100000 000,gtt~tt~tI~tTT~llllllt'l S800STIIITplllllllLT~lTIITIIITI l.1 C, Jo t1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.0~~ 05 4.0 4.5 5.0 5.5 /ao 2 1.0 1 2. 2.0 3

' ~ ~ ~ ~ ~ L'~ j! I-~iCC, ~.~C1- I — ~ - ---: ~~~.-L.i.4-]-j i..- ~. Ct. ~~. h4.:._.:: " I' I..... i l I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -~ — ~-i —~~ ~ —~, L +i-<:-~'-t-r!,.-d-f: ~.i-H iirl:':-i':-'?r~i!-rT-'~ -C-I~C-C-IiCt —i-llC ~i'~":.~.::,~ ]-i.L_,~li~ iI i__~._!~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~.-. —"':-r::i -'-*!~~- -~-',-' +1!! 30~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-. "'- Ci-~ —, ~ L-~L'-H' 25'''''''' ~ ~ ~ ~ ~ ~ ~ ~ ~.-,'-''...4-' i' T:''''''!-+'.... ~ - 25, SHOCK WAVE PRESSURE RATIO _!/i:I/ I N co,' i YI~ llr 20~~~~~~~~~~~ O[ J; i j ~ i' I II! _ i i i i'i1 i J jY...... 4-~ [I-~1'' F — ~~-i:x w 15 ~1 t.- Ii cn: -- _....., C3 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'.,,''-i i~- -i —,,-t-t ~ ~.t-r-:....,..... -C-I-i-i~~~ 1:; =-..... ~.: - J —F "~, -..-l.. —b- I —..:. —i~ —F. F I_ ll- F I i.. i —~ ~~~..:, t::j ~ ~ ~ ~;.'..,, i...' -, -~ —~ —.-i..~.. _.....l..i..... i.....~:i' i I 5 ~. ~...'.o: ~- W I L I ~ ~ ~ ~~~..... I-. —.... L.. i.. j.~ 1:i;.'i~ —;..~i,....i.............' ] so 1....... l.-...:.~ -- ~ i.i 1 t 1.0 1.5 2.0 2.5 3.0 3.5 4Q.O 4.C0 5.0 5.5 U/cQ.

,,. ~........,......i........,.,..,,. ~':T,' ~' ~i,-." "~; ~"'~'"~"'" "!j'! —! ~-i-41 —4....J...!~ ~~- ~~-~- ~~.: i! ~~ i- ~~ ~ i:!!!- - ~~~ —~..,-:.i~.. —.~.-~........''...... i'" ~..... r;; -1...... ~ "'....:' i! i' _ i ~-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-'-~ 1~.~.../.. t i~.... 2.5 I,' ~........! —! — 1~-''-1'!-.. "il".....li.. i~....!-t-'rH-t.-. —-! -I-!, i. Sil t: I/ -c- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~ r —t i r —t-'-f I~ ~~~~..]~ _....~. r't ni~~~~~~-T..~ _'-._..__. s ~ o i!. ~ i i /'p I.! i / 2- o F I i..; r- r — i~~I il IIII [ 9 I [ i, I ~..... t.~........ ~.t._,i — f"-~'.... —— f —+ —-t —- -4 - -' —H ~ -i —Ur- " __-r_..... _~,. C.......'!:- ":-:: 1.5 -o ~ ~ ~ ~ ~ r, _ ~, ~,I'llo 1.0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. T,~~~~~~~~~~~M T I I ct 1,1 I I -1 —t- i t 1 — I. i N r0l i J..... ___.Ii2 I.5;;': q:I ] l-!4 1 ~'. i.,J! II I -- -- I -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - 0. Lr,~ ~ ~ ~~~Uc!l'-~ lr.0 i i.... I!i!,!. J i' ~ ~ ~.-' II }U / LI iI~ ~~ -4 —', —- i I'.~~~ I i''I J~-— t~~~~~~ —1-~ 1-_ ~ ~ ~ ~ ~ ~ ~ ~~.1..,"t-v i I,;;~ ~ ~ ~ ~ ir I~ I~i ~,~i- i I I -:~ ~~~~~~~~~~~~iI.J_'I1!~'-.,i' r~~!-il ]~~ _.~,,,, -.-,-. —~, i. I ~. I. --.i..... JI-1 I~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~I q,,' i f!i ~~, L.: I I... f ~ ~ ~.;~i I i t i~ i).1. - i.?f";O;...;Q455 U/a~~,,, r" —'!- U... f

INITIAL PRESSURE RATIO O 0C b - o rn.P o0 ao ~ b~~~~~~ OP.~~~~~~~~~~~~~~~~~~-o _ _17ll~ r _ _.1 _ _ _ _ _.. _ _ __ _ __ _ _ __ _ _I I_,,- l l l T1- 1 11 11- lr. 1 1 1-l.I. I i.1 I l-T I i-: _I_ _ _ _ _ _ _ _ _ _ _ _ _ ll _ _ _ _ _ _ _ _ _ _ I I I CA 41],I _, _ _.l t_ _ __ I I I I I: I [ 1 I i 1 1 1 1 1 - CA~~~: Cir b i: i Hi:lll-1 —T:ll-11 o L - — 1 - Z1 _ _- _-4 -.............._ _. o~~~~~~' I~ - Il 8 L 0 -L1:1 ~ t w~~~fls1 _ 7 _ _ _ _ C 7 ~~~~~~~~~I -~T_~ 1 l I tA -I _ T t __ _ _ _.I I I l I_ ~ I I II TI 1 1 1 1 1 1 1 1 1 H I I i ATS'VYIH 0 — llll -lllllllITIIPl~ 71i IL l|Illll____^1 _-rxXvZrl ir llT11] TD llIt00 Pt UI NO L7h 1 11111111ASz ml 4~~~ 1|1 1.01I1 W -1-111tt 11 _ _ _ E & S r 1 1 1 I | | E | Z t _ __ 1l.l1 11 1II II l1 II l 11 II 111d IIIL 1111 - 1-111iF t THT~~~~l~llilil 8 9 a~

TABLE V SHOCK WAVE CHARACTERISTICS IN N2 Variable Specific Heat To = 297rK Constant Specific Heat VO P2 P2 T1 Vo P1 P1 ao a_ -- M2 — ~~~~~~~~~~U M91 _1 L L. II,,po, -PO TO.a 400 1.540 1.937 2.608 0.6421 1.328 1.512 1.882 2.50 0.6189 500 2.000 2.682 4.515 0.9684 1.774 2.104 2.818 5.00 1.019 600 2.390 3.232 6.530 1.165 2.140 2.503 3.337 7.143 1.198 700 2.736 3.658 8.622 1.300 2.623 2.952 3.813 10.00 1.345 1 800 3.049 3.998 10.768 1.401 3.043 3.295 4.108 12.50 1.429 900 3.341 4.290 12.998 1.482 3.740 3.799 4.456 16.67 1.523 1000 3.616 4.538 15.278 1.549 5.132 4.645 4.871 25.0 1.629 1250 4.239 5.0291 21.166 1.676 1500 4.798 5.4114 27.330 1.773

44 FIG 9 *ME....TUREKRATIOTEMPRATUR RAT 5.0 0 ~~~~4~~~~~~~~~~~~. -.. —- i0. I.0 4 2.0 wT 0~~~~~~~~~~~~~~~~~~~~ V Xt 4-1 J tl 32.0 6.0-X X~ W. 41,,,4 5.0 -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~yw j:Y,- irl=-H2.... 30... W 4~~~~~~~~~~J 44 SHOC WAV DENS;;|-1Wrr ITY! SH RATIO - I.0 1.0 2.0 3.0 4.0 5.0 U/ i- 4- |i r. F |.. |...1. ti.tF~t ti i t_ 4* *- * l; |'5'; J~~~~~|.. Ib|4FliilIi6|J[|t;Ti|tt-t't;f>L|t 5.0~3. P2O5.

45 FFIG 10 25~~~~~~~~~~~~~~~P 25'T T+1 2 0 I +......... 1.0 w ~ ~ ~ ~ ~~~~ESR RAATIO HUG W2 nmS 1NItttt +f-~1 t'O 44-TTT M ~ ~~~~~~~~~~~~~~~~~~~~~~~ 4+[~ t|||i||111 m trtl1 ~r C.) i.;I~etlelrr.tti.. -HITIl -:F_ I f'il:ff I... T _ |'. 0t O203.0 4.0 5.0 U/ci~V 0~~~~~~~~

TABLE VI SHOCK WAVE CHARACTERISTICS IN SF6 Variable Specific Heat To = 2980~K Constant Specific Heat V0 P2 P2 V0 Pi P1 T3 ao PO2 P 1 M2 T2 Vo p P 325 1.556 2.285 2.492.8386 325 1.538 2.244 2.447 8146 350 2.043 3.691 4.335 1.378 350 1.994 3.503 4.114 1.31 375 2.471 5.063 6.372 1.778 375 2.381 4.674 5.882 1.663 400 2.854 6.351 8.525 2.087 400oo 2.719 5.724 7.682 1932 425 3.205 7.551 10.77 2.343 425 3.022 6.663 9.504 2.144 450 3.531 8.76 13.09 2.561 450 3.297 7.500 11.33 2.319 475 3.837 9.703 15.47 2.748 475 3.553 8.252 13.16 2.466 500 4.126 10.68 17.92 2.913 500 3.790 8.931 14.99 2.59 525 4.404 11.60 20. 43 3 061 525 4.015 9.544 16.82 2.707 550 4.667 12.44 22.96 3.191 550 4.228 10.11 18.65 2.796 575 4.927 13.27 25.61 3.315 575 4.431 10o62 20.49 2.882 600oo 5.165 13.98 28.15 3.410 6oo00 4.626 10o89 21.93 2.953

47 FIG II 2.25 o SHOCK WAVE TEMPERATURE RATIONKNE HUGNIO 0~~~~~~~~~~~~~~~~~...._...... _......... E 1 0 26.00 0 SHOCK WAVE TEMP~~~~~~ERATURE RATIO LIlllllllllltlTIIII!III 1.05...... 1.0203.0 4.'0 5.0

48 ~~~~FIG 12 24....... 20~~~~~~~~~~~~~~~~~~~~~~~ t-'B~EiiiiH; 2O200 05:~~~~~~~~~~~~~~~~~~~~~~~t~~~~~~ t r l 1.0AN20E-30 40ON50 U/ct~~~~~~~~~~~~~ OJ

All the figures show that when weak shocks are considered there is very little difference between the shock wave parameters calculated from the usual Rankine-Hugoniot relations and those determined by considering temperature-dependent specific heat. However, the difference becomes very pronounced for strong shock waves. The change in specific heat that does occur is brought about by a temperature increase behind the shock wave. From a generalization of Le Chatelier's principle,- one should expect the adjustment to cause the temperature behind the shock to be lower -than predicted by simple theory. This expectation is verified by Fig. 4i. Figs. 5 and 6 show that the pressure is slightly higher. and the density is much higher for a particular shock velocity than that given by the Rankine-Hugoniot equations. The continuity equation predicts that if the density ratio across a shock increases, the velocity ratio must decrease. Thus, the flow velocity behind a stationary'shock wave goes down; but, through the transformation to the laboratory frame of reference, the flow velocity behind a shock is increased. Furthermore, the decrease in temperature coupled with the decrease of 1' insures the decrease in the speed of sound. behind the shock wave. Thus, both the increase in flow speed and the decrease in sound speed tend to cause the Mach numiber to rise-above that predicted by Eq 28. Finally, since in the laboratory frame of-reference the flow velocity behind the shock wave increases as specific heat variations are considered, the pressure ratio across the rarefaction wave must also increase to match this increased flow speed. Therefore, p0 /P3 must decrease as shown in Fig. 8. These results can be summarized as follows: For a given shock

50 1. the temperature behind a shock wave is reduced considerably, 2. the pressure ratio across a shock wave increases slightly, 5. the density ratio across a shock wave increases, 4. the flow Mach number behind a shock wave is increased, and 5, the initial pressure across the diaphragm is decreased if the temperature dependence of specific heat is8 considered. The changes in the flow parameters discussed above are not necessarily related in any way to changes produced by vibrational relaxation that actually do occur in the flow field behind a shock wave. This section has only compared two methods for calculating shock characteristics. See Chapter IV, Section 2. Perhaps a word of explanation i's'in order at this point. it was shown in Chapter II that speciffic b~-eat variations did not produce important perturbations on rarefaction waves. However, it has just been shown that shock wave parameters are definitely dependent on the specific heat behavior of the gas. This seemingly contradictory situation comes about because the temperature change across a shock wave is always much greater than the change across a rarefaction wave. Fu~rthermore, the specific-heat changes produced by the shock wave result from the excitation of vibrational degrees of freedom, whereas the rarefaction wave can only partially "freeze out" the rotational contribution to the specific heat. In many cases, there are several vibrational modes to be excited, each contributing two degrees of freedom to the specific heat. In vi ew of these facts, the importance of specific heat variations in the calcula

4. Experimental Verif'ication of' Variable Specific H~eat Predictions Because the duration of' flow in the shock tube is so shri is impossible to measure the temperature or pressure behind the shock wave precisely. The density could be determined if aMach-Zehnder interferometer were available. However, it is possible to determine the flow Mach number with fair accuracy. This was done with carbon dioxide in the expansion chamber in order to see if the results obtained agree with the predictions of' constant or variable specif'ic aeat theory. The calibration was made by measuring the angle of' a bow wave attached to a 50 wedge parallel with.-the f'low. This method was used in spite of' the fact that boundary layer on the wedge itself' perturbs the bow wave to such an extent that a calibration accurate to three or f'our signif'icant figures is impossible. In this instance, however., two significant figures are sufficient to select between the constant and varilable specif'ic heat predictions. The semi-wedge angle, bow-wave angle, ratio of' specific heats,, 2 ~ ~ ~ ~ ~ ~ ~~~~~~2 where the angles are as indicated in Fig. 115, page 52. Note that M 2 is almost independent of' for small The results of' this calibration experiment are given in Table VII and the points are plotted in Fig. 7.

52 As expected, these results agree quite well with the Mach number predicted by considering the specif ic heat to be a f'unction of' temperature. The f'act that two of' the experimental points lie above the maximum Mach-number predicted by the Rankine-Hugoniot Fig. 15 theory is particularly signif'iBow Wave Attached to a Wedge cant. TABLE VI I FLOW MACH NUMBER BEHIND SHOCK WAVES IN CO2 2.18 58.75 0 24500* 1,23 2.38 50.25 2.50 1.57 2.85 41-l75 2.50 1.58 5.20 36.50 2.50 1.77 5.83 52.50 2.50 1.96 4.22 351.25 2.50 2o 4.22 351.0 2.50 2.05 4.61 29.0 2.50 2.19 4.96 27.25 2.50 2.32 5.16 27,0 2.50 2.54 5.45 28.5 2.50 2,22 It was also possible to check the P0/P2 versus shock Mach num

55 Fig. 8. Unfortunately., there are two factors other than specific-heat. variations which influence this experiment. First, in the derivation of the Taub equation, an'ideal diaphragm is assumed to be removed ins tantaneously at the start of the flow'in the tube. This situation is never found in practice, of course. The energy required to shatter the diaphragm and accelerate the resultant fragments must be subtracted from the flow, In addition, the interface between expansion and compression chamber gases is highly turbulent. Therefore, it is not unreasonable to expect that the strength of a shock wave produced by a given'initial pressure ratio will be less than predicted. This argument has been used by all,shock tube investigators to explain the discrepancy between theoretical and experimental shock strength determinations, Second, the compression chamber of the shock tube at the University of Michigan is about two per cent narrower than the expansion chambe'r. The strength of the shock wave produced by a given initial pressure ratio is further reduced as a result'. These last two factors affecting the shock strength are apparently more important than~specific heat variations, Therefore, all that can safely be said about these experiments ils that the results fall on the right side -o h hoeial curve.

APPENDIX TO CHAPTER III PROOF OF THE EQUIVALENCE OF THE RANKINE-HUGONIOT EQUATION AND THE SHOCK EQUATION FOR A GAS WITH VARIABLE SPECIFIC HEAT IN THE LIMIT 2 The Rankine-Hugoniot equation can be written in terms of velocity and temperature ratios as follows: V 2T Th euaifo = can beq solve beomtesvlctyrto V0- (L - ) ++T40T V? 11 2 where Thus~~~~~T T~~~~ Thus ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~QED VO TO To TO~5

IV. VIBRATIONAL RELAXATION 1. General Discussion of Vibrational..Relaxation The specific heat-variations discussed in the last chapter result from the excitation of vibrational degrees of freedom within the molecule. Thils variation is described by Einstein's specific-heat theory in which the vibrational degrees of freedom are treated as quantized harmonic oscil1u her-lators. However, this is an equilibriuthoyie.a theory which predicts the energy contained in the harmonic os-cillators in equilibrium with a particular temperature, defined in terms of the average kinetic energy of translation of the molecules. When abrupt temperature changes occur, thi's equilibrium distribution will be temporarily disturbed. The remainder of this secti'on will discuss the mechanism whereby equilibrium is re-established and define a relaxation time related to the rate of readjustment. If the temperature of a volume of gas is suddenly increased, the tacit assumption is that energy has been supplied to the translational degrees of freedom of the molecules. The fundamental'problern is now to determine the probabillity for the transfer of some of this energy to the rotational and vibrational degrees of freedom-of the molecule. Landau and 24le postulated that the efficiency of a collision in exciting a degree of freedom is indicated by the ratio 7 = ___ ~~~~~~~~~(38)

56 where'7is the effective duration of the collision and -tis the natC 0 ural period for the degree of freedom. It is'intuitively obvious that this ratio should be closely related to the transition probability because it represents in a simple way the departure from adiabatic conditions during the collision. If % is of the order of unity or smaller, only a few collisions will be required to excite the degree of freedom; but if %C is much greater than unity., many hundreds of thousands of collisions will be required to effect the transfer of a quantum from the translational to some other degree of freedom. In general t'Z will be approximately a/V, where a is the radius of interaction andyV the relative molecular velocity. When molecular rotation is considered, tis the time for one 0 revolution. This time will be of the order of the distance of an atom from the center. of mass divided by the velocity of the atom about the center of mass. It follows from the equipartition principle that the velocity of the atom about'its center of mass must be of the order of its translational velocity. Furthermore, the radius of the molecule will be approximately equal to or greater than its range of'interaction. It follows, therefore., that 1 for molecular rotation. As a result, molecular collisions will fully excite a rotational degree of freedom within several mean free paths. For a vibrational degree of freedom, Vis the inverse of 0 the frequency of cscillation,~) and in this case 7., becomes V O

57 Nitrogen may be considered as an example. If the relative kinetic energy of the colliding particles is kT., then V = 7 x 10 cm/sec at room temperature; the vibrational frequency is V= 7 x 1013 sec-1; and the molecular radius is approximately a -- 1.6 x 10-8 cm. Thus for nitrogen 7x/04s = 16.10 Since X is an ord~er of magnitude larger than I, many thousands of col~lisions will be required to excite the molecular vibration; or,'in other words, the probability for the transfer of energy from translation to vibration'is very small. An analogous situation would arise in an attenrtp to excite a tuning fork by striking it with a very soft rubber mallet. The translational and rotational degrees of freedom can be con~si'dered as active degrees because, to a first approximation, they are always in equilibrium with the temperature. All other degrees of freedom, vibration, dissociation, and electronic excitation will not always be in equilibrium with rapid temperature changes. These shall be called inert degrees of freedom. When X.~> 1, Landau and Teller have shown that the probability of de-exciting a vibrational state in a collision is = C e", ~~~~~~~~(159) where C is a geometric factor assumed arbitrarily to be 1/10. 19 If this probability is averaged over a Maxwellian velocity distribution, the average de-excitation probability per collision is

58 where r and. m is the molecular mass. The d~e-excitation probability per second. is more useful than the probability per collision. If N is the number of collisions per second., k 10 The probability of excitation per second. can be obtained. immed~iately by using the statistical principle of d~etailed. balancing which states that, in equilibrium, the number of molecules that return to the ground. state from the first excited. state per second. is equal to the nuxvber raised. to the first excited. state in the same time. Thus, &RT = k01 (41) If the vibrational d~egree of freed~om is consid~ered. as a linear harmonic oscillator, the transition probability from a higher excited. state can be shown to be = ~~~~~~~~~~(42) Again, using the principle of d~etailed. balancing Only transitions between neighboring levels are allowed.. culesy0, i the round stat isn 1

59; d -t0,- 0, ~O and. the time rate of change of the number of molecules, yj, in the nth excited state'is dt Except for an unimportant zero-point contribution, the vibrational energy in the nth excited state is M h"n V and the total vibrational energy of the system i's Mm (ii4) The rate of change of vibrational energy, dt can be determined by summing Eq 43K over all possible states. The result of this summation is, where Ev' is the vibrational energy in equilibrium with the local temperature as determined from the Einstein formula. whee~li te nme fmlcls ntegs Y

6o The relaxation time for the excitation of a vibrational degree of freedom can be defined from Eq 45 by letting (I - (6) Therefore, the fundamental equation describing the time rate of change of Evibrational energy isE d E~~~~ _ _ _ _ _ _ _ ~~~~~~~(4 7 ) where T3~~~~~~ e 2. The Effect of Vibrational Relaxation on Shock Waves. A shock wave is considered to be-a surface of discontinuity in the usual mathematical treatments based on a continuum theory of an ideal fluid. For a molecular point of view, however, the thickness of a shock wave must be of the order of several mean free paths because at least one or two collisions are required to establish a new Boltzmanm distribution of translational energy behind the shock wave. In the last section it was shown that the rotational degrees of freedom of a molecule can usually be excited by several collisions; or in'this case, within the thicknes's of the shock wave or at least within a distance very

61 to excite the vibrational states of some molecules In this case, no change in the vibrational energy can occur within the thicknes of the ~~~~~nary shock wave. tions for the conservation of mass, momentum and energy must be satisfied. throughout the entire flow. 0 I Fig. 14 acie ere ffreo, h tmeatr o h fo wl ecese n the~~~~~~~ otehc aecaatrsiswllas ecagd a ontem

62 where equilibrium has been established. between vibration,, rotation, and. translation, the specific heat of the gas will assume the equilibrium value associated. with the flow temperature. The shock wave characteristics will then be the limit values as calculated. from Eqs 33 and. 34 of Chapter III. Immed~iately behind. the shock wave, for a gas with a long relaxatin tmethe conservation of energy equationcabewitns 2-~~~~ wher'e C T i's the kinetic energy of th~e active d~egrees of freed~om and. pa E is the vibrational energy which-is assumed. not to vary through the shock wave. Cp is 7/2 R for all d~iatomic and. linear molecules and. 4 f or all polyatomic molecules with three d~egrees of rotational freed~om. The three conservation equations can be written as VYO4 VI 4- + 4~~~~~~~~~~~~~~~~~~~~~ -d~~ =6 for linear molecules., and. 7 for general polyatomilc molecules. The Rankine-Hugoniot equations can be obtained. inimed~iately as d~iscussed. in Chapter III. It should. be noted. that the velocity of sound. should.

63 The surprising result now emerges that the flow velocities, temperature, pressure, and. density ratios across a shock wave are identical for all diatomic and linear molecules; the sonic velocity is different for each gas, however. A similar but numerically different relation holds for all general polyatomic molecules. It should be repeated, however, that this result is valid only when the vibrational energy can be assumed constant across a shock wave. If the relaxation time is very short, no general relations can be derived for the shock wave characteristics immediately behind the shock without a detailed consideration of the shock wave from a gas-kinetic point of view. To a first aopproxi'_ma-,tion, the relaxation time must be very much greater than 10-9 second if the analysis above is to be valid. Inspection of the relaxation times lised n TbleIXpages 77 and 78, shows that this condition is not satisfied for some polyatomic gases. The values of the temperature, density, and pressure ratios that exist just behind shock waves in carbon dioxide have been included in Figs. I. through 6. Thus, for a particular shock velocity, the flow parameters change abruptly to the values indicated by subscript 1. They then change slowly to the equilibrium values, 2. It is interesting to note that the pressure behind a shock wave in carbon dioxide changes only slightly, while the initial density and temperature ratios may differ by approximately 30 per cent from their equilibrium values. The Rankine-Hugoniot values based on room-temperature specific heat may be considered roughly an average between the ac

the shock wave,because the vibrational degree of freedom is essentially not excited at room temperature. No attempt has been made to calculate the shock wave parameters immediately behind a shock wave in sulfur hexafluoride because relaxation time data are not available.' As a rule of thumb, however, the relaxation time is very short whenever the change in specific heat is large. Therefore, it may not be safe to assume that the relaxation time for sulfur hexafluoride is very large compared to 10-9 second.The regions of flow in the ideal shock tube can be summarized by referring to the plot of pr~essure against position shown in Fig. l5. The constant initial pressure in the expansion chamber extends to the shock wave. Immediately behind the shock the pressure is P1, as calculated above. The pressure gradually rises to its equilibrium value, P2, far behind the shock wave. This limit value'is maintained-until the foot of the rarefaction wave is reached. The pressure than increases through the rarefaction wave up to the,initial compression chamber value, p. A more complete picture of the various flow regions in a shock tube may be gained by reference to the x-t diagram of shock tube flow shown on page 7k. 5. Approximate Calculation of the Time Dependence of the Flow Parameters, Behind a Shock Wave if the temperature changes that occur in the flow region behind a shock wave are assumed small, so that the relaxation time, equilibrium vibrational energy,~ and total internal energy of the flow behind the shock wave may be considered constant, Eq 47 can be'integrated immediately.

P TRANSITION REGION P. 23 /\ —'/ - o CONTACT SURFACE 2 X O 0~~~~~~~~~ "x:O VARIOUS FLOW REGIONS IN A c) SHOCK TUBE TUB

66 Thus / / ~t V (VO-FVZ)e (50) where E is the vibrational energy in equilibrium with the temperature in region ck. The total internal energy of the flow is assumed constanty so that variations'in E. are compensated-by variations In the opposite direction in the energy of the translational and rotational degrees of freedom, Ea The local temperature can be defined in terms of the energy in these latter degrees of freedom. C is the constant specific heat va associated with these degrees of freedom. T ~E Thus T ~VE. E V~ VO V2 Because there is not time for the vibrational energy of the gas to change as it passes through the shook wave Therefore, Thus, under these assumptions., the temperature behind a shock wave changes exponntiall fTro it % ws iniia to %14,4 - its fina value

67 pa ~~ C +RcT- RTR ~2C is not related in a simple way to any of the referenceuspeeds derived from the energy equation for an ideal gas with constant specific heat. The f low velocity is the root Qf the equation V2 aCV + RTRo V ~C+C -T, The flow velocity can be made an explicit func tion at the time by substituting Eq 51 for T. The density can be found from the conservation of mass equation:* The equation of state may now be used to determine the pessure variation: 72(c+ +c" R-) As expected, u.= ul, p =p1, and p =p1 when t =0; and u =u2j p= P2j and p =P2 when t =Oo. These results for the time dependence of the flow parameters

T =T~~~~~-(T1-T2)e~~~~~~~52 _ _ _ _ _ _ _ _ _ _ _(+ ~12 ~RT2(T-T) Z ___ [T2 + (T9 -Tjej ~~~~~~~~(53~) C2 Ths tegeealsatmntcn emaethtwhnth emea iniialMahu numeri 2.nra Ittemaytca be seen that veoctyhensity, tmeand pressrane approach thei tequiliritume vhaluesveynal exponentially.fo tsii Thisalcuaitsiona mayue.Thefoe blw elcnider ed nsafirt-orade coresrecto tppoatheqsimleiepoeniumal adjusimtments prdceyBthe asm aebtnd- Telimler.I isnnot cneient, howeveran for mathemgathical rearsetions, ooxpes the pressults aspracsimpleireulbimvle eynal exponentiallpuyacrecin.em Thi cauItisouldy bhe refoempaie thtte preceding analysistore corefersn to the vaiati exoneofflo parausmentes.behintda sttoay sehocae and Tle.i

cl~~~~~~~~~~~a ~~~~~~- - - - - - -—'0 w CM ui I C C4. 4~~~ (Iz s~~~~~~~~o -rc W1E F 0X N W0 g W| 04 g lg gg ggW X~~~~~~~~~~4 s~~~~~~~~~~o *0 11|li1 1|1l 11 ll 1011 I Il I IfV -j.W o If1 Wtj WW'W1 t1 44-44 tl L~~~~~~~~ it o 0 0~~~~~~ 0 ma 0 sO 0 o 0 o0 0 o 0 o 0 so 0 o0 0~~~~~~~~~~ ~~ ~~~~~~~~o,.: 5. N. 0 o ~ ~. 0. o ~ ~ 0 ON ~ ~ ~ ~ ~ ~ N 1111 -oil t1 4-11 F 4-SI 1 1L. Lle- a.i. 0~ 0 HN ff | i. t | t- Q | I I =T~~~~ 4-~~~~~~~~~~~ PI ~ ~ ~ ~ ~ ~ ~ a' A. "" it

70 FIG 16 0 5 10..:.~. - -:t-,.. t..5 20q 2'-.5r1: ~~~~~~~~~~~~~~~~~~~ bi 1.00 __T~:~~~~~~~~~~~J "I~ —t-E' _.7.. ~....20 0 ~~~~5 1.0'1; 1.52 2.5 100 —...20 0.5 1.0 t 1.5 2.0 2.5 1.00 r F.50.20.10 0.5 9.0 t 1.5 2.0 285 I: -..':...'-c"-.-' FO-W V A BEHIND A SH K W E —---------............. ~:;Z....... I L......................-L~...—:'.; 1:'0?-::;?:::~f':!-?_":-,-.~-.-!~:::'!~!.:! 0 ~ ~~~~~ 5 1.0 t, 1.5 2.025,~~~~~~~~~~~~~~~~~~~~~ -7 P2~~~~~~~~~i k.:: pt.:!'-~;:-?:~:?-i~:-.:t: — -'-.- ~..:_.m-t —!?;T,7~ P? PI~~~~~~~,::...:f~,-..~:-.,:?-.5.~m.,-.......}=_:r-:.'~f...!.:.F-~h...'.:.-..............0.. --........ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.

71 the time variable is uniquely related to distance behind the shock wave. The actual space variation of' the flow parameters behind a shock wave can be determined parametrically by integrating the velocity equation; that'is, 0 t)d This equation together with Eqs 51, 52, 53i5 and 54 determine the space dependence of' the temperature, flow velocity, density, and pressure behind a shock wave. If' curves similar to those in Figs. 16a and 16b were calculated with distance behind the shock wave as abscissa, they would resemble Figs. 16a and 16b, except that the initial rates of' change would be somewhat reduced because the initial velocity is greater than the equilibrium value., However, no f'undamental change would be produced in the character of' the curves or in the relations between the curves.' The assumptions upon which this approximate calculation rests can now be reviewed and enumerated: 1. The change in temperature in the flow f'ield behind the shock wave was assumed small enough so that the relaxations could be considered constant. 2. The change in temperature behind the shc8 a sue small enough so that Ev'. the equilibrium vibrational energy, could be considered constant. This assumption is more restrictive than the f'irst, becas tevbainal energy var'ies a~s e l/T while the dependence of' the- reaato tim is onya l/Tl3

72. energy variation resulting from work done on the flow by pressure and density changes was neglected. This assumption is more generally valid than the preceding two. The preceding calculation will be valid only for very weak, shocks where the temperature change behind the shock is of the order of 500C or less. For stronger shocks the above-,assumptions are no longer valid. The flow variables may then'be calculated from the following equations: The fundamental relation between vibrational energy, time., and temperature is given by Eq 47. dE~ El The dependence of Z on temperature and pressure must now be considered. Ev' may be determined from the Einstein equation, EV~lk The first law of thermodynamics for an adiabatic process gives a relation between the'internal energy of the gas and the work done on the gas by pressure forces: dlEv + dEd where Ev is the energy associated with the vibrational degrees of freedom, and Ea is the kinetic energy of the translational and rotational degrees of freedom. Thetz loca.ql te-m-perature may be. de-fined- in terms, of the energy

.73 In addition, the flow variables must also be solutions of the continuity equation and Euler's equation: d Pd dx d AdV d~0 This system of equations can, in principle, be solved for the time and space dependence of the flow parameters behind a strong shock wav e. It should be mentioned that this flow is not isentropic for all nontrivial cases. J4. Position-Time Carves for the Flow in the Real Gase's of a Shock Tube Perhaps the processes occurring in the real gases in a shock tube can be better understood by reference to Fig. 17, which is an x-t plot of the shock tube flow. The path, RI, of the head of the rarefaction wave is strictly a straighit line., because the rarefaction wave moves into the gas of the compression chamber with the local, constant sound speed. in the undisturbed gas. However, the velocities of the shock wave, con~tact surface, and rarefaction foot will all be influenced by vibrational relaxation in the gas of the expansion chamber. In general, the flow velocity immediately behind a shock wave is less than its equilibrium value. Therefore, during the first instant of flow., the velocity produced

74+ FIG 17 F C /~~~~~~~ 0 I PtElI^,i

75 the rarefaction wave will move more slowly in the positive x direction during the early stages of the flow. The contact surface likewise has an initial velocity somewhat lower than its final value. On the other hand, the initial pressure ratio across the shock wave is larger than expected because the rarefaction wave is not as complete, initially, as it is after the flow has reached equilibrium. The shock wave, therefore has a high initial velocity. After the flow has existed for a time of the order of the relaxation time, the velocities of the shock wave, contact surface, and foot of the rarefaction wave become constant at values which may be determined from shock theory for a gas with temperaturedependent specific heat. For example, the initial and final values of shock velocity, flow velocity, and pressure ratio across the diaphragm are given in Table VIII for U/a0 = 2.34 in carbon dioxide. TABLE VIII INITIAL AND FINAL VALUES OF SHOCK VELOCITY, FLOW VELOCITY, AND PRESSURE RATIO ACROSS THE RAREFACTION WAVE FOR U/a0 = 2.34 IN CARBON DIOXIDE Shock Velocity Flow Velocity Pressure Ratio (mnm/ ~sec ) (mm/ pec)Across l(mm/sec) (mm/Ilsec) Rarefaction Wave initial final initial final initial final. 659.631.450. 459.448. 440 It can also be seen that the gas initially in the expansion

76 the flow parameters change from their values immediately behind the shock wave to their equilibrium values; and there is region II, another region of constant state in which the flow parameters may be determined from variable specif ic heat theory as discussed in Chapter-III. Representative particle paths, A and B, in the compression and expansion chambers are indicated by dot-dash lines. The particle path in the compression chamber is similar to the path derived from ideal shocktube theory; but the world line for a gas particle'in the expansion chamber is8 no longer made up of two straight segments of a broken. line. Now the world line changes slope abruptly at the shock wave. and then continues to bend further toward the x-axis throughout region I. In region II the particle path i's a straight line parallel to the path of the contact surface, The pressure distribution corresponding to a particular time is shown in Fig. l5. Effects of Vibrational Relaxation Investi ated in the Shock Tube a.VribeReinSince there is a region of variable state behind shock waei sipratt eemn hether or not this region can seriously affect experiments conducted in a shock tube. The variable which determines if the effects of-vibrational relaxation may be important is the relaxation time. If the time for the adjustment is very long or very short compared to the duration of Uniform flow in a shock tube, vibrational relaxation, Ias such, can be ignored. According to Griffith, 25 the relaxation times for most polyatomic

:77 TABLE IX' RELAXATION TIMES FOR VARIOUS GASES Temp. Releaxation Time Gas Purity and. Foreign Gas Gas OF 1,psec Present Investigation Hydrogen 58 0.021 (rot.) 99.8%, 0.2% ~120 Nitrogen 60 (-,0.00O2 (rot.) Nitrous oxid~e 65 1.12 98-99%, 1-2% N2 Carbon d~ioxid~e 65 6.95 99.956%, 0.044% ~120 Carbon dlioxid~e 65 2.02 n% ~120 (comm.) Sulfur d'ioxid~e 68 0.37 99.988%, 0.002% ~120 Ammonia 63 0.12 99.5%, 0.005% ~120 Methane 6o o.48.8 99%,, l%-C2H6-, N2j, CO2 Methyl chlorid~e 72 0.202 99.5%, <0.008% ~120 Freon 22 70 0.10 97%, 40.005% ~120 Freon 12 67 0.090 95%, <.0.001% ~120 Ethane 57 0.003- 95%,~ 5% C2~14. C3H818 C3H16 Ethylene 6o 0.207 99.5%, 0.5% air Propylene 70 4o.oo6 99% Propane 70 (0.007 99% Ethylene oxide 86 1.23998 Butane.70 (0.018 99% Butadiene 70 <0.013 99.77%, 0.23% C4H.l0

78. TABLE IX (Cont'd.) Temp. Relaxation Time Gas ~F Sec Literature References Results reported in the literature: Oxygen 70 1000 Reference a Carbon monoxide 1800 10 b Chlorine 70 18 c Nitric oxide 60 o0.8 (el. excit.) d Hydrogen 77 0.018 (rot.) e 60 0.02 (rot.) f Deuterium 60 0.015 (rot.) f Carbon dioxide 70 10.8 f 70 10.26 g Nitrous oxide 67 0.92 c 70 1A44 g Carbonyl sulfide 70 o 86 g 70 1.67 c Carbon disulfide 70 0.70 g Sulfur dioxide 70 0.181 g Water vapor 415 0.037 h Amonia 70 0o.4 i Ethylene 70 0.238 j Methane 230 0.84 c H. 0 Kneser and V. 0. Knudsen, Ann. der Phs 21, 682 (1935). G. G. Sherratt and E. Griffiths, proc. Ro. Soc.London 147A, 292 (1934).. c A. Eucken and S. Aybar, Zeits. f. physik. Chemie B46, 195 (1940); A. Eucken and R. Becker, ibid. 27, 219 (1934). H. 0. Kneser, Ann. d. Physik. 3_ 261 (1941); J. Acous. Soc. Am. 5, 122- (1933). e E S. Stewart, Phys. Rev., 632 (1946). A. van Itterbeek and R. Vermaelen, Physica 2, 345 (1942). E. F. Fricke, J. Acous. Soc. Am. 1_2, 245 (1940), A. Kantrowitz and P. Huber, J.Ch, 275 (1947). 0. Steil, Zeits. f. phsk., Chemie B31, 343 (936). 3W.,T. Richards and J. A. Reid, J. Chem, T~.2, 193 (1934); W. T. Richards 4, 561 (1936),

79 compilation of relaxation times for various gases, is from the article by Griffith. It can be seen that carbon d~ioxid~e has a relaxation time longer than any other polyatomic gas listed.. Therefore, if the length of the transition region behind. shocks prod~uced.'in a shock tube'in carbon d~ioxid~e is found. to be small compared. to the characteristic d~imensions of an experiment, then relaxation effects as such will be unimportant for all polyatomic gases'insofar as they d~isturb the uniform flow in the shock tube. A first approximation to the thickness-, d., of the transition region can be obtained. from the equation where U and. u are the shock and. flow velocities calculated. from the Rankine-Hugoniot equation, assuming the specific heat to be constant at'its room-temperature'value. The remaind~er of this section will be d~evoted. to a calculation of the length of the transition region behind. shock waves prod~uced. in a shock tube. The d~epend~ence of t on temperature and. d~ensity d~efinitely must be consid~ered. in such a calculation. The experimental d~ata of Eucken and. his co-workers, as compiled. by Katoiz for the average number of collisions required. to excite molecular vibration in CO2 as a function of temperature, can be approximated., except at high temperatures., by C 4.04 xi T 0. 222O

where n is the number of molecules per cubic centimeter, V is the average molecular velocity, and 0 is the molecular diameter. From these equations, the relaxation time for C02 becomes simply T: (Eq = 2.q7x 1Q5 seconds. RI pf~'(T-222) Both p and T in the gas behind the shock wave can be expressed as functions of the shock strength and atmospheric temperature and density. p is Po/P3 can be found from the PO/P3 equation, and p/PO is given by the Rankine-Hugoniot equations Thus eP =~ E3 i s- ~oEO(E~}') ( ) C1 (6) The temperature behind the shock wave is ~~~~~~~~~.sion for d is obtained'

81 2.97 x0-Oo ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..... This function is plotted in Fig. 18 under the assumption that carbon diide is used in the expansion chamber and helum ber. It is interesting to note that the thickness of the transition region is almost constant, except for very strong shocks, in spite of the fact that the changes in the velocities., temperature, and density are large. The thickness of approximately 35 mm is in good agreement with the estimate of 0.1 inch made by Griffith,24 For an'initial shock Mach number of 15 in carbon dioxide, a density variation of approximately 25 per cent of the change across the shock wave is to be expected within a distance of 15.5 mm behind the shock wave. An unsuccessful attempt was made to observe this variation with shadowgraph and schlieren photography. The density gradient was simply too small to give an observable effect. However, a similar investigation has been proposed by Griffith at Princeton University. This attempt may.be more successful because a Mach-Zehnder interferometer is available for use with that shock tube. b. Bow-'wave curvature. The effect of vibrational relaxation on the flow through an oblique shock wave has been discussed by Ivey and Cline. 27 In particular., they treat the problem of the attached bow wave.

1 11l I I l l I! J_i I I I I_1 ___II I I-,-......_ __ _ IIIIIIII111 1111111111111111111111111111.11..1 nT I_ I rT I I 1 ~~~~_ --' __ I! i!lf 1111 __EIlfi77 11 t D 11111111117llll —[HII'ul 111! I 1111 I 111IIIII L IIII IIII IIIIIE E a t1 11 MTb IIi Il I' |1 illl11 IlIIIIL+F L 1113 111i 3i rr r Trr rr Ii II L f I7 1111111177 11711117r r r f I tl 10IIIlF1111 I 1 l 111I'11 1 T ll ~ t 111-1t~ 1;11rrllll111lI!1| I _1 __ __I_1_______ I I:I I IT I r I I 1 1 1 1 1 17 1 1 1 17 11 1 17 1 1 1 17 7 1 I n 1 1 1 1 17 1 r 5 20.l-l 1- Ill1lllll 1illl l1 K ~ tt 1iiilt I I!t _ I ll!it t,, C K W E 1 RO i )IE oII XI %! E II IX tXtt-H I~~~~I

83 The results of Bethe and. Teller are applied. to the component of velocity perpend~icular to the shock wave. Since an oblique shock wave can be described. as a superposition of a normal shock and. a uniform flow parallel to the wave, it follows that the flow field. behind. an oblique shock wave will not be uniform -just as the flow behind. a normal shock in a gas with temperature-dependent specific heat is not uniform. From a frame of reference in which the shock is stationary, the normal component of velocity d~ecreases-as the d~istance from the shock wave increases. This is a direct result of vibrational relaxation. However, the flow component parallel to the shock wave is constant on both sid~es of the shock. Therefore, the streamlines in the flow behind. the shock wave will not be straight. Fig. 19 illustrates this fact. RELAXATION / REIO /~~~L U01~~~~~~~ IN.~~~~~~ L-L~~~

814 The shape of the bow wave attached to a wedge in a supersonic stream can now be discussed. Immediately behind the shock wave, the f low will be as predicted by ordinary aerodynamic theory. However, the values of or ~kused in these equations should be determined by consi~dering only the active degrees of Fig. 20 ~~~freedom of the moleEffect of Vibrational Relaxation in an Attached Bow Wave cule, because insufficient time will have elapsed for the transfer of energy to the inert degrees of freedom. Thus the bow-wave angle in region I near the tip of the wedge will be as predicted by the usual oblique shock equations. However, from region 2, far from the tip of the wedge and the bow wave, the shock wave and relaxation zone appear only as a thick shock wave, The flow deflection will be determined from the flow velocity normal to the shock calculated by assuming the specific heat to be a function of temperature and from the constant tangential component. Therefore, the angle, G, must decrease below the value found at the tip of the wedge in order to satisfy the boundary condition that the flow near-the wedge be parallel to the surface. Fig. 20 illustrates the type of bow wave

1. The simple theory of the bow wave attached to a wedge neglects viscous effects and assumes the wedge to be perfect. However, these assumptions are not valid at the low supersonic Mach numbers obtained in shock tubes. Both the blunt end on a real wedge and the boundary layer which develops along its surfaces cause bow-wave curvature similar to that shown in Fig. 20. Tis curvauehsbe hn ovr in time. Therefore, it would be very difficult to observe curvature ascribable to relaxation effects because of the similar curvatures., which have not yet been discussed theoretically, caused by viscosity and the imperfect airfoils. 2. Furthermore, the pressure behind a b6* wave attached to an airfoil in a low supersonic stream obtained'in a shock tube is not more than twice the initial pressure except in extreme conditions. For instance, the pressure behind a shock wave that produces a 100 deflection at a Mach number of 1.5 in air i's only 1.7 times the initial pressure. The difference between v1 and v2 produced by such weak shocks attached to thin airfoils in air is negligible. Ivey and Cline estimate that curvature of bow waves attached to thin airfoils will not become important in air until the free-stream Mach-number reaches 20. The-effects would be greater in some polyatomic gases, to be sure, but the differences in specific-heat behavior with temperature are still insufficient to make the curvature appreciable at the Mach numbers attained in a shock tube. The conclusion that relaxation effects will not notice ably, affect bow waves attached to airfoils in a shock tube can be extended toL ote features —_ 10.I. 4 I MU-4 — of the flow. The temperature inrmet racross w shok %^

86 thin airfoils are small enough so that the vibrational energy may be assumed, constant; therefore, no change in the specific heat of the gas need. be considered. However, if shock wave diffraction studies were being conducted in a complex gas., the possibility of specific-heat variations should be carefully considered, especially in the vicinity of strong detached or reflected shock waves and large-angle expansion waves. c. Reflected shock velocit in carbon dioxide. Relaxation effects should cause the velocity of a shock wave reflected normally from a wall to vary with time. Immediately after reflection the velocity should be that predicted by considering only the active degrees of freedom, if the relaxation time is long. If the relaxation time is short, the iniltial reflected velocity will be somewhat less than such a calculation would indicate. When the distance between the shock wave and the reflecting surface is much larger than the relaxation zone behind the reflected shock, the shock velocity should be as predicted by considering the specific heat to be a' function of temperature. Before the details of the calculation of the reflected shock velocity for a gas with temperature-dependent specific heat are considered, it might be worthwhile to review the velocity calculation for a gas with constant specific heat. The change in flow speed across a shock wave is given by Eq 27. Thus for a given incident shock strength, the flow velocity behind the wave is t. a% 9:1

87 As a result, the change in flow speed. across the reflected. shock is also u. From this cond~ition the reflected. shock strengthn 5 can be d~etermined as a function of incig leads distrengctly toh. a0 (Tih L+it~i ( )H From this (9) is plotted. as a function for several values of in Fig. 21. The velocity of the reflected. shock wave can then be d~et-ermined. from Eqs 26 and. 27. That is, the reflected. shock velocity is the shock velocity calculated. from Eq 26 minus the incid~ent flow velocity u. Thus ~~~ ~~0 Substituting Eq 59 and. simplifying lead~s d~irectly to. U = a ~~~~~~~~~~~~(60) Unfortunately, when the specific heat is consid~ered. as a function of temperature, an analogous explicit expression for the reflected. shock velocity cannot be written. Nevertheless, this velocity can be obtained. through a step-by-step process similar in principle to that

88 FIG 2 1 1.0,,,I:.i' —I I I 1.9 EW E 1111111 - - - - - - - - - - W 1111|111f-1111- g Hi-X# - i - 11111111111!1111 —--- ~ I~~~~~~~~.8.6.5;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ —-.-~i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-~ -~... --- ~ I~~~ L: I -~ ~ — ~ --— ~ —--- I I L;.3 -'.,. I 0tt.2.;.4.5.6'.7.8.9 1.0 -- --- i~~~~~-e-.2,4 m t -L- iT 1L.. I7 1 I X - - -!: f I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I XXR S —-HC SR G 4 4 H | l | S g 4 | 0 0 P X - T X Tt AS A FUNCTION O INITIAL SH-OC STREN 3~,,,i: 3 — - I -X 0:: — -- 0-t i Smr I E I: 10 11~~~~~~~~~~~~~~~~~ o M S i I i I E 11 41 01E 21i i11 [41 I ~~~~~~ — ~ —I -; I ~ I~~. ~ 1W I141 O t.1. 3. 5..8.10

89 Chapter III.," These parameters are then consid~ered. as initial cond~itions for the reflected. shock wave. Eqs 55 and. 34i are used. again to d~etermine shock wave parameters und~er the assumptions that the initial temperature is that of the flow behind. the incid~ent shock and. the final temperature is less than the temperature behind. a reflected. shock calculated. by constant specific heat theory. Usually several final temperatures are assumed.. The bound~ary cond~ition that the flow velocity at the reflecting surface must vanish may now be applied. graphically. From the reflected. shock calculations, an'initial and. final flow velocity are d~etermined.. A trial reflected. velocity can be obtained. by subtracting the primary flow velocity from the initial velocity required. by the assumed. final temperature. This will be the correct reflected. velocity only if'it'is equal to the final velocity behind. the reflected. wave calculated. from Eq 54. Several attempts may be required. before this cond~ition is satisfied.. This proced~ure changes the frame of reference of the reflected. shock from a sta tionary frame of reference to the laboratory frame, and. it also imposes the bound~ary cond~ition that the flow velocity at the reflecting surface must vanish. Table X., on page 91, shows the initial and. the final values of the reflected. shock velocity, temperature, and. pressure-ratio for three shock waves in carbon d~ioxid~e. It is evid~ent that large changes in the shock characteristics occur in the vicinity of the wall d~uring the course of the reflection. Fig. 22 is an x-t plot of the processes involved. in the normal

ST T C B\ A SECTION C B A A-A L~~~ / 0~ T SEOTION tt t S ~~~~~C B A T SECTION )/ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ G C-0~~~~~~~~~~~~~~~% D.~~~~~ NORMAL REFLECTION OF A SHOCK WAVE T~~~~~~

91 TABLE X INITIAL AND FINAL REFLECTED SHOCK VELOCITIES, TEMPERATURES, AND PRESSURE RATIOS IN CO2 Imme- Initial Final Initial a Ia Initial Final Initial Final diately Reflected Reflected TemperShock Temper- Temper- Pres- PresBehind Shock Shock Mach h Soc oc ature ature sure sure Shock Velocity Velocity Number OK ~K Ratio Ratio Wave mm/ ~sec mm/Jsec 4.835.0398.475.262 3104 1771 161.5 242 3.417.0802.364.232 1661 1098 66.7 88.4 2.472.1550.301.219 971 738 26.2 32.1 shock'is narrower than behind the'incident shock. This result is an immediate consequence of the fact that both the temperature and the density are higher behind the reflected shock wave; the relaxation time is therefore shorter. A representative streamline'is also indicated. Fig. 22 also shows temperature-time variation at the Sections AA, BB, and CC of the position-time graph. At the reflecting surface the temperature rises to a high value and then drops to its final value along a path resembling an exponential. On the other hand., at CC, well removed from the wall, the temperature goes through two complete adjustment processes,. behind both the incident and reflected shocks. At an intermediate location, B h relaxation proce'ss behind the incident shock has not been completed before the reflected shock arrives. A series of experiments was conducted in which the position of the reflected shock wave in carbon dioxide was determined -as a function

i~~~~~~~ I -4~~~~~~~~~~~~~~ J:~~~~~~~~ L 25~~~~ _4'i i~~CLUAE CALCULATED~,,,~~~INA REFLCTE VEOCT..... RELCE VELOCITY... ---,~, _...2..62....... _-::;,,-,z-~.:::~~~~~~~~~~~~I! ], Tl-!_!~_ _F-I!.~. _L CLCUAE __.~-i!CLUAE PO~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ cl)~! NOR__ M..h —— J NTAL RELETDHEOCKT 4' ~z —-'!: IA REFLECTE EOION.22 a.~~~~~~ 1), t-? —i-'?I CABO DIOXIDE ~........ L......_,... i''. I I.' 4~~~~~~~~~~4-'/c 4'.I3,5, ~ ~! ~ I ~~~~~~~-..4,4; _- I..., 4~~~~~~', -'-l!-,!...,'-i ~i~"! 0 i. i,, 0 20 h0 40 5L 6,0 70... 9-0,- ~. ~i~ ~ ~ ~ ~ ~~~~~TM -(! SEC)-~LjI]Ji._ _/I.

25 0-.1 20 ~ ) I — 55 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~FNORAL SOKREFLECTE VLOITY INITIAL RELECTED VEOCITY.I6 20 CABO DIOXIDE 3.417 L.t. ooo~~~~~~r ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~00 0 O0 20 30 40 50 60 70 80 90 100 TIME (1~ SEC)

35~~.. I J~~~I! 30 i.'; I LiITT.... I - -!', I' $O~~~~~~~~~~~D, Ii, CALULTE _.:~~~~~~~ f:il_:_ z::I::~: ~:":.........;..I... I..... I - CA~~~~~~cu~NRML SHCK RFLCTO I~~~~~~~~~~~l;301 1 00, 7~I__ CARBON.....OXI-DE —'-L,O,) I0 — ~ —-— 4 ~-....~~~~~~~1 J10~~~~~~...' 0 20 40 60 80 100 120 140 160 180~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I TIM.... SEC)-: _.....

TABLE XI POSITION-TIME DATA FOR THE NORMAL REFLECTION OF SHOCK WAVES IN CARBON DIOXIDE U/a0O 4.835 U/a0O 3.4117 U/a0 =2.4172 time position time position time position5.5 1.41 3.8.8 6.6 1.1 7.7 2,0 17.2 4.2 18.5 4,0 9,8 2.8 23.1 6.2 24.4Q -r5. 11.1 3.4 33.7 8.3 56.183 12.1 5.5 55.9 8.4 58.1 13,5 12.4 3.5 12.5 5.6 51.5 12.5 62.11 14,1 15,2 5.7 61.6 14.8 62.9 14,5 14.1 4.1 72.11 17.5 75,9 17.1 16.8 5.0 74.0 17.8 85.1 18.7 22.8 6.4 79.5 19.0 99.5 22.5 55.6 9.4 86.4 21.0 109.1 25.2 45.o 15.1 95.8 25.5 114.1 26.0 5o.41 14.2 159.1 52.0? 51.1 14.5 53.6 15.4 59.9 17.0 65.6 18.5 66.2 18.1'velocities are also'includ~ed. in the figures. Fig. 54 is a schliferen photograph of' the reflection taken under conditions somewhat different than those of this investigation. The data for these curves were obtained'exclusively from shadowgraph plates. The reflecting obstacle was located so that the primary shock wave could always be seen'in the photograph. Accurate determinations of the time since reflection began were obtained from measurements of the location of the incident shock and the measured

measurements o~f shock position., both of the reflected and the incident shock., were made to well defined parts of the shadowgraph image; not necessarifly the true shock location, Where necessary, very small corrections to the observed shock positions (maximum of 1 Per cent) were made to compensate for small variations in initial shock velocity. The fact tbat the initial reflected velocity seems lower than predicted'is not surprising because this velocity persists only for an infinitesimal time; and obvious experimental difficulties interfere with attempts to determine shock velocities earlier than about five microseconds after the reflection occurs. For instance, for each of the shock velocities, the first determination is made approximately one millimeter from the reflecting surface, but this distance'is approximately equal to the thickness of the shadowgraph image of a shock wave, Examination of Fi'g. 54 reveals a severe interaction between the reflected shock wave and boundary layer on the walls and floor of the tube, This interaction will extend across the tube after approxilmately 80 microseconds, Therefore, the reflected shock positions determined for later times were disregarded in the experimental determination of the final reflected shock velocity. In Figs. 24 and 23 these late points fall above the final velocity curve as expected from the nature of the shock wave- boundary layer interaction. Because the slope of the position-time curves is not constant, the velocity of the reflected shock is a function of time. Therefore, a perturbation of the shock tube flow caused by vibrational relaxation

97 that the reflected shock velocity is constant. One can conclude, there~fore, that vibrational relaxation is not'important'in these gases. Approximately 40 or 50 microseconds after reflection,$ the velocity of the reflected shock assumes the value calculated from shock theory derived for a gas with temperature-dependent specific heat. This time of adjustment'is what one would expect from the relaxation time of carbon dioxide, These results will be discussed further in Chapter VI,~ where the advantages and disadvantages of using specific gases'in the expansion chamber of a shock tube will be considered.

APPENDIX TO CHAPTER IV 1. Reflected Shock Velocity in Nitro~gen In order to compare results with carbon dioxide a positiontime curve was determined for nitrogen. The results of this experiment are presented'in Table XII and Fig. 26, It'is apparent that the reflected velocity is constant and in good agreement with that determined from a constant specific heat calculation. In this respect nitrogen behaves differently than carbon dioxide. The apparently contradictory result'is easily explained~however, when it is noted that the vibrational relaxation time for commercial nitrogen is estimated2 to be longer than 5000 microseconds, Therefore, there simply'is not time for thermal equilibrium to be established between the various degrees of freedom of nitrogen within the duration of uniform flow in a small shock tube. Thus, the reflection results'in nitrogen serve as a verification of the simple constant specific heat theory of normal shock reflection and-also demonstrate the validity of Rankine-Hugonilot calculations for shock parameters in nitrogen. 2, "Reflected Shock Velocity.in.Argon Fig. 27 and Table XII present the results of a similar reflection experiment conducted in argon. This experiment was designed originally to demonstrate exact agreement between constant specific heat theory and experiment. However., the agreement is not quite as good as that found with nitrogen. The reflected velocity is approxi'mately two

70 60 NORMAL SHOCK REFLECTION IN I I I 11pEHTTT Ililililill~~~~~~lilililililililililililillililill~~~~~~l~rIll 50 NITROGEN.3I.1X4 =.0783 40 I 111111 tl I,-. Z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~A-.!'SEIIC A — ~-I.j i I I I I I I I < t t I I I z zg CONSTANT SPECIFIC HEAT VELOCITY =.46t1 " 1 30 f _/3 _ _ I _ _ I- 1 1 I I 1 11 1 11 1 11! 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 VARIABLE SPECIFIC HEAT VELOCITY =.4152 o 20 10 i IT I' --- t ---- 0 0 10 20 30 40 50 60 70 80 90 100 TIME ( L SEC)

I 70 ~~~~~~~~~~~~~~t.._...! - ---- -- ---- 60 50 0W __~~~~~~~~~ LL_ F TR- IlTH I 1 I 11,,, I ImmTI~_.]__L~tIftft r I v I 111 111111 j 1 i 0 X l | l l.....t 0 0 U m 0 i XT CALTED'l D IY....S 4 g ( lll 402 _!_ X I I + I 4 4. 4 _ I I I I._ _00 20 1014 0 0 W IL w,L X G 0 F 4 S S LOMAL SHOCK REFLECTIN 0 1 | IIME ()1. SECI 8 m W0 L X X L Xg0W0S m k~~~~RGO> g 5n 0 xo LXS ER MLXXM W 00oF s ~~~~~~~~-.08980| o --- --- --- [ ['I l i f e L 1 1 1 1 L.',, 0! 20 30 40 50 60 70 80 90 " TIME (eA SEC)

101 TABLE XII POSITION-TIME DATA FOR THE NORMAL REFLECTION OF SHOCK WAVES IN NITROGEN, ARGON, AND SULFUR HEXAFLUORIDE Nitrogen Argon Sulfur Hexafluoride =.0783. =.0o898 U/a - 3.590 time position time position time position 9,4 3.8 14.8 7.3 18.8 1.2 17.6 7.5 26.8 13.7 27.8 1.9 19.0 8,0 42.8 22.1 29.5 2.1 22.8 9.9 56.9 29.4 56.5 4.2 29.1 13.1 69.0 35~9 81.8 6.1 38.8 17o2 75.0 39.0 108.8 8,4 48.3 21.5 133.2 10o3 49.4 21o.9 44.3 21,3 160,9 12,4 55~6 24.7 The first few pictures obtained using argon indicated a severe discrepancy in reflected shock velocity between the pictures individually, and the pictures as a whole and constant specific heat theory. The discrepancies were an order of magnitude larger than anything that had been found in carbon dioxide or nitrogen. The boxed point at t = 44,2 microseconds is an example. It was then observed that the entire region behind both the incident and reflected shocks was luminous. A pink glow could be observed visually. Since the temperature required to excite the ground state in argon is over 130,0000K, this glow cannot be due to electronic excitation of this gas. The obvious conclusion is, therefore, that the glow must result from the excitation of energy levels in the optical region in some impurity present in the expansion chamber, Unfortunately the vacuum system, including the tube itself, piping system, and the vacuum pump, was not designed to produce a vacuum better than 1/7 nnm of

102 mercury or to keep the leak rate below 1/7 to 1/5 mm of mercury per minute. Therefore, a small air contamination will always be present in the expansion chamber. However, by careful evacuation and flushing of the expansion chamber with argon, better and better results were obtained. The results shown in Fig. 27 were obtained by flushing the expansion chamber three times with argon before firing the tube. It seems a reasonable asp sumption that if more complete elimination of impurities could be obtained better agreement with theoretical predictions would resultThe radiation of energy from the flow by ahy mechanism will have a similar effect on the flow parameters as the absorption of energy by vibrational degrees of freedom; that is, kinetic energy will be removed from the translational and rotational degrees of freedom by both mechanisms. The ultimate reservoir for the energy is unimportant. It must be noted, however, that no relaxation time in the sense used in this chapter can be associated with the radiation process. The impurity gas will continue to radiate energy so long as the mechanism for excitation remains. Therefore, one should expect that the temperature behind the reflected shock should decrease slowly until some value is reached, independent of initial shock velocity, where the excitation mechanism could no longer operate. The impurity present in by far the largest proportion is atmospheric nitrogen. The lowest excited electronic state is in the ultraviolet, and a temperature of approximately 100,0000K would be required to excite it. No mechanism available in the shock tube appears capable of exciting this level. However, oxygen is also present as an impurity to the extent of approximately 0.1 per cent. The excited electronic levels in oxygen lie much closer to the ground state than for argon or nitrogen. It is

103 therefore conceivable that some mechanism, operated by conditions found in the shock tube, could excite electronic levels of oxygen which would radiate visible light, The details of this mechanism, if it even exists, are not understood at the present time, An interesting investigation for the future might be a further study of this glow. By employing spectroscopic techniques, it should be possible to identify the radiating agent and possibly determine the excitation mechanism. 3. Reflected Shock Velocity in Sulfur Hexafluoride Finally, a similar reflection experiment was conducted in sulfur hexafluoride. It is evident from Fig. 28 and Table XII that the reflected shock velocity is constant, but it is not the value predicted by a variable specific heat calculation. There are two possible explanations for this result. The initial expansion chamber pressure was only about 7 mm of mercury. The inevitable air contamination could easily represent several per cent of this pressure. (The gas supply on hand was insufficient to allow thorough flushing of the expansion chamber before each shot,) Since the velocity of sound in air is much higher than that in sulfur hexafluoride, and since the specific heat function,, is lower for air than for the heavy gas, the reflected shock velocity should be higher than predicted for pure sulfur hexafluoride. In addition, the specific heat of this gas as a function of temperature is not as well known as for carbon dioxide or uitrogen. The only data available are contained in a short letter to the editor which does not indicate just how the specific heat was calculated, nor is any experimental verification of the calculation presented, If this prediction of the temperature dependence of the specific heat of sulfur hexafluoride should be in

~~~,,,,1,,:,lrI 0.oo ~~~~~CALCU LATEDO L _ I I1Ltit~t_~tttftS-~~~f-CSrf1 tt4t FINAL REFLECTED VELOCITY. 01 IN 5i SUN LUR HEO RAFELUOREIDE O i-t ~~~~~~~~o, 3,5 ~~~~5.90 I.poio-l:. 4....4 T 0 — ~I- i~:i:..i-.i 1.i-. l.!i l ~ II ~~~~..... j"- ir ~~ t- - 0 20 40 60 80 100 120 140 160 180 200 TIME I.. SEC)

105 error, it might help explain the discrepancy between the calculated and experimentally-determined reflected shock velocity. At any rate, the fact that the position-time curve is a straight line demonstrates that vibrational relaxation in' sulfur hexafluoride is not noticeable in shock tube flow. The relaxation time, therefore, is probably less than one microsecond - a very reasonable value for such a gas.

V. THE DURATION OF UNIFORM FLOW IN A SHOCK TUBE 1. Duration Index An important consideration in the discussion of the use of various gases in the expansion chamber of a shock tube is the duration of uniform flow, because one of the serious limitations of a shock tube as an instrument for aerodynamic research is the restricted time interval in which experiments can be performed. In this chapter, the influence of the gas parameters on this duration shall be investigated. The surprising result will emerge that the effective duration of uniform flow, as a function of flow Mach number, is practically independent of the gas used in the expansion chamber. The specific heat will be assumed constant at its room temperature value; and all velocities will be calculated from the RankineHugoniot equations. It has been shown in the preceding chapters that this assumption is not valid in general. However, the effects of variable specific heat are usually not important until the flow Mach number reaches approximately 1o4. The errors introduced by this assumption will be discussed later in this chapter. MautzlOhas discussed the general problem of the duration of uni-,form flow in a shock tube. The essential limitation is the arrival of the contact surface at the test section. The arrival of the reflected shock and of the reflected rarefaction can be postponed indefinitely by increasing the length of the expansion chamber beyond the test section and by increasing the length of the compression chamber. The limiting duration is simply 106

107 <= L( l- _ ) (61) where Lw is the distance from the diaphragm to the test section. This is just the difference between the time of arrival of the contact surface and the shock wave. Using the basic shock wave equations, this becomes o f -) (,-) (62)- Examination of Eq 62 shows that the duration of uniform flow depends strongly on the speed of sound in the expansion chamber. As the' speed goes down, the duration goes up. However, this increase is not necest sarily significant because the rate at which a stationary flow configuration will be approached will also be reduced. It is possible to define a dimensionless index depending only on the specific heat function 1 and the shock strength 5 which can be used to compare the effective flow durations in various gases. The fundamental assumption is that the rate of development of a stationary flow is directly proportional to the sound speed behind the main shock wave. In other words, it is assumed that, at a given Mach numbers the steadystate flow about a given obstacle will be attained in a time ca/al, where a depends only on the obstacle and the flow Mach number, The duration index is defined as the duration of uniform flow per unit length of expansion chamber divided by the time for a sound wave to travel a unit distance, Thus

1o8 I _ I L U I Q. or.......(y ) _ _ "'___ v?-,)(,lr-s)' Ads D - Simplifying gives D =,.,~I)2 Ad (63) Note that the duration index is independent of the sound speed. In most investigations the comparison of effective duration of flow in different gases should be made for constant flow Mach numbers instead of constant shock strengths. Therefore, the Mach number equation M _/(KOI) ( *I~~ should be used in conjunction with Eq 63. Thus, in principle, the relative duration of uniform flow in a shock tube for two different gases can be determined as follows: The values of shock strength corresponding to a particular flow Mach number are first calculated for both gases. The duration indices for these shock strengths are then calculated from Eq 63. The largest D obtained

109 represents the longest relative duration, and the ratio of effective durations is given by the ratio of the duration indices, The duration index has been plotted as a function of Mach number in Fig. 29, for f = 4, 6, and 0o. This figure shows that there is surprisingly little difference in the relative duration of uniform flow in various gases. It is now possible to discuss qualitatively the changes that will be produced by allowing the specific heat to increase with temperature. It has been shown that at a particular shock velocity the flow Mach number increases above the constant specific. heat value, Furthermore, the duration of uniform flow is decreased because the velocity of the contact surface is increased, and the speed of sound behind the shock wave is decreased. These two facts insure that the duration index will decrease. The variation in Mach number and duration index tend to compensate for each other as the slope of Fig. 29 indicates. In fact, the changes in Fig. 29, introduced by temperature-dependent specific heat, are indicated by representative examples for sulfur hexafluoride (the V of SF6 is effectively oo ) and nitrogen. The points marked A, calculated for a particular shock velocity from constant specific heat equations, become the points A' for the same shock velocity, but calculated by assuming the specific heat, to depend on temperature. It is apparent that, to a first approximation, the relation between duration index and Mach number is unchanged by considerations of the temperature dependence of the specific heat.

5~~~~~~~~~~~~~. I ~~~~~-4 5.0; 41-iA4 —----- — I-!J ll..1.1.1.-.......... i__...................SF6 LIMIT DURATION INDEX VERSUS FLOW MACH NUMBER 3.0~ ~ Llll I[IIIIi ll~lTltll11 11111 f111 111 111fill 31111111 I3 4.0 QL~~~A LrJ TT~ ~ ~~~i| I: r]: I I:- ~ ~ I - ~ ~: ~- ~ _ I ~ ~ ~ I I I flB|| I I I I I I |. 111__ 1} r l! O 2 2D At 1.0 t'i- = t ~.5 j.2.3.4.5.6.9 DURATION INDEX I l 1ill VE11111 iilllllilll llllllllliiiiiiiiiiiiiiRSUSHN Q 1 1 X l l| | ll l ll l l l l l l _ _ l l l l ll l l l ll l l l l ll! < x ll11_ llll ll- 2 1 _ 1ll11Z1l 111lll11l tll__ 1lWr 2 5\ 2 1111 11 lllll llll llll ____ IIII IIIII III IIIII I 1111'111111' FLOW1 MACH1 NUM11BIER11111111111111111111111111 1.0 ~. "11111111 l'''>: L t 1 1 1 1 1 l 0 t 5W i1|11111 ~ 111 ilt - 4 211 1 1 1 1 + m.~~~~~~~~ I OZ1. 3. 5.6 -. 8.. l ~ ~ ~ ~ ~ DRTO NE

111 2. The Motion of a Detached Bow Wave as an Indication of the Development of a Steady State It is known that at high supersonic Mach numbers, a bow wave can be attached to a pointed wedge. If the flow Mach number is decreased, the bow-wave angle (29 of Fig. 13) is increased. As the Mach number is reduced further, a value is reached at which the bow wave can just be attached to the wedge. Beyond this limit, the wave becomes a detached bow wave and assumes a stable position somewhere in front of the wedge. The distance from the wedge to the bow wave is a function of free-stream Mach number and wedge geometry. If the flow becomes subsonic, the bow wave will recede from the wedge to infinity. In the shock tube, one can determine the distance between the bow wave and the wedge as a function of the time since the flow began at the wedge, This distance increases asymptotically from zero to the steady-state value. Mautz has determined the influence of scale and Mach number on the bowwave detachment in air for two wedge geometries. The validity of the assumptions made in deriving the duration index was investigated by comparing the time of arrival of steady state for the detached bow wave in front of a given wedge at a given Mach number for two very different gases. In particular, the experiment consisted of determining the displacement of the bow wave as a function of time in front of a 1/16-in. plate with a symmetrical 300 wedge at its leading edge. The free-stream Mach number in both gases, nitrogen and sulfur hexafluoride, was 1.30o The results of these experiments are presented in Figs. 30 and 31 and in Table XgV. The arrival of steady state was estimated to be the time

3.5 3.0 as~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l.2 mm - 2.0 I.5 ler~~~~~~~~~~~~~r z W ~s~~~~~~~~~~~~~~~~~~~~~~~~BWWAEDTCMN ii i~~~~i-... ~ ~ ~~......... CO) 0 F 0 100 200 300 400 500 600 700 800 900 1000 TIME (tSEC)

3.5 3.0 - 1 I _ -0 2,0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I 1 Llr1 r Ir I tS rl_4X040LL_0 SS40012i-;4j tllll~ iiiiltllllllllllL L 1.5,Fo ~~'1. BO +0W 0 WAVE DETACHMENT!_ I I II I N I _ Ir~ ~ t -'- — Hi —~t i —-1 (.O I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -.t.Ct -.C —l- 1 -i'., M X~~~~ 5".l' ~ __iX. o W<<,XII~1111!1,1'-:-tI ~i-l- I 0 300 600 900.1200 1~00 1800 2100 2400 27-00 30 TIME ( x SEC) 2.0-t' __::...... tn~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! I -r —i ~~:1.~._i_~ i': ~ ~ ~ ~ 1 ~ ~ ~ ~ ~ ~ ~ ~t.5 __._t_,_~,: ~_ --'.........i. _ Ii;i'' 0;;-'FH-I: -, ~'.i.?..j..j "-' T'...i..I i- --- O~~~~~~~ —,_.~~ Q 300 600 900 1200 1500 1i BOW WAVE DETACHMENT TIME (r..-,

114 TABLE XIII DETACHED BOW-WAVE POSITION AS A FUNCTION OF TIME FOR A 1/16 IN., 300 WEDGE AT MACH NUMBER OF 1.3 Nitrogen Nitrogen Sulfur Hexafluoride time position time position time position 20 03 395 2.5 73.5 24.45 397 2.6 75.5 44.8 399 2.1 176 1.1 54.9 409 2.2 270 1.4 70 1.2 410 2.25 394 1.9 72 1.3 421 2.0 429 1.8 425 2.0 102 1.6 483 2,1 118 1.6 426 2.2 599 2.2 131 1.8 428 2.0 830 2.4 160 1.85 430 2.05 1043 2.6 170 2.05 430 2.5 1225 2.6 197 1.9 435 2.2 1381 2.6 207 2.0 611 2.2 1932 2.4 251 2.1 621 2.1 2390 2,6 258 2.35 809 2.5 262 2.0 265 2.1 265 2.1 269 2.0

115 when the bow wave reached a stable position. These estimates are indicated in the figures. The measurement of bow-wave detachment was made to well defined parts of the shock wave image. It was possible to measure to the white leading edge of the shadowgraph negative of the shock wave in sulfur hexafluoride; however, it was necessary to measure to the white-black interface of shock waves in nitrogen. Therefore, no significance can be ascribed to the difference in steady-state position of the bow wave in the two gases. The measurements in nitrogen were complicated by a multiplicity of the bow wave presumably caused by boundary-layer interaction and by boundary-layer growth on the -wedge itself. Measurement errors of approximately.2 mm may be expected. The three high points in Fig. 29 at approximately 400 microseconds are not the result of poor measurements, however. Boundary-layer interaction is apparently less important in sulfur hexafluoride because no bow-wave multiplicity can be detected. As a result, the position measurements are probably accurate to.1 mm. It was then possible to verify the assumption on which the derivation of the duration index rests; namely that the steady state will be reached in a time a/al where a depends only on the particular model and the Mach number. The values of a determined from this experiment for nitrogen and sulfur hexafluoride are given below. ts = a/a1 for nitrogen: =N2 325 =.5516; caN2 = 179

116 for sulfur hexafluoride: 1200 = SF6 174.1i447; ~SF6Also, the theoretical duration of uniform flow was calculated for both gases by using Eq 63. Next, the ratio of steady-state time to total duration was determined for both gases. Finally, the inverse ratio of the above ratios was compared with the ratio of the duration indices for the two gases. In other words, the ratio of the duration indices was compared with the experimental ratio of relative durations. The actual figures used are tabulated below. Nitrogen Sulfur Hexafluoride (a) time of arrival of steady state 325 1200 (b) theoretical duration of uniform flow 1670 6800 (c) a/b.195-.176. (d) duration index.209-.224 *195 Ratio of relative durations = = 1.11.176 Ratio of duration indices = 22 = 1.07.209 These results are surprising in that the values of a for nitrogen and sulfur hexafluoride agree within approximately 3 per cent and the ratio of duration indices is in good agreement with the duration ratio as determined experimentally, In fact, the agreement is better than could be expected from the type of measurements made. The conclusions that can be safely drawn are that the rate of approach of steady state depends linearly on the sound speed behind

117 the primary shock wave; and that the relative duration of uniform flow in the expansion chamber is practically independent of the gas used.

VI. FURTHER DISCUSSION OF GAS PARAMETERS AND SPECIFIC GASES FOR SHOCK TUBE WORK The preceding chapters of this paper have discussed some of the problems of using gases other than air in a shock tube. In particular, the influence of variations of specific heat with temperature were considered in detail for the compression and expansion chambers. This final chapter will discuss the additional influence of a large number of other physical and chemical properties, some of which are compiled in Table XVI for representative gases. The results obtained below are not new; they are immediate consequences of the basic shock equations and physical logic. However, the results are necessary and important to anyone interested in using complex gases in a shock tube. First, the influence of specific gas parameters will be considered, then the applicability of particular gases for shock tube research will be discussed, and finally, representative photographs of flows in several gases will be presented. 1. Effects of Gas Parameters a. Specific Heat Function (i) The specific heat function, JJ = (+ 1)/( - 1), plays a fundamental role in shock tube theory. The limiting Mach number that can be obtained in the expansion chamber is 118

119 TABLE XIV CHEMICAL AND PHYSICAL PROPERTIES OF GASES Gas Molecular Viscosity Index Chemical Boiling Weight at of Activity Point room temp. Refraction ~C (,~ poise) N2 28.02 1.40 6 176 1.0003 inert -195.8 Air 28.8 1.40 6 185 1.00024 inert 120 18.02 1.37 6.41 93 1.00026 inert 100 CO2 44.01 1.304 7.58 150 1.00045 inert - 78.5 CC14 153.84 1.13 16.4 100 1.00177 inert 76.7 n-C5H12 72.15 1.086 24.3 67 1.0017 inert 36.2 SF6 146. 06 1.096 21.9 1.00078 very inert _ 63.8 S' 07like N2 H2 2.016 1.41 5.88 87 1.00014 inert - 252.8 He 4.003 1.667 4 194 1.000036 inert - 268.9 A 39.94 1.667 4 222 1.00028 inert - 185.7 Cl2 70.91 1.355 6.63 133 1.00077 active - 33.7 Br2 159.83 1.32 7.25 155 1.00113 active 58.78 TeC14 269.4 1.00260 414 UF6 352.1 180 attacks metals 56.2 and glass

120 This limit can be raised by increasing the ~ of the gas in the expansion chamber, that is, by using a polyatomic gas. However, when this is done, the shock theory derived under the assumption of constant specific heat is no longer valid because the specific heat of complex gases usually depends strongly on the temperature. Fortunately, however, the specific heat variations always produce a further increase in flow Mach number. Thus the use of complex gases in the expansion chamber increases the flow Mach number limit in two ways. (ii) Another important parameter of the flow in the expansion chamber which depends on U is the temperature ratio across the shock wave. Simple shock theory predicts a temperature ratio of < = 1+Fg- (64) As T.-j o, T1/T0 - 1 for a given shock strength. Physically this can be interpreted by noting that gases with large j have more vibrational degrees of freedom excited than gases with e of 7 or smaller. Therefore, these internal degrees of freedom can absorb a larger relative proportion of the kinetic energy transformed into internal energy at the shock wave. This means that the translational kinetic energy of these gases is not increased nearly as much by a shock wave as in the case of monatomic or diatomic gases. Furthermore, a comparison of the temperature ratios for different gases should be made at constant flow Mach number, not at constant shock strength. Eq 28 shows that for a given Mach number, as ~ increases so does 5. By differentiating Eq 64 at constant [ one obtains

121 T 2 d TO= + e+25 r It is evident that increasing 5 decreases T/TO for a particular gas. The fact that the temperature ratio across a shock wave that produces a particular Mach number may be reduced by using a complex rather than a simple gas in the expansion chamber is important for two reasons. The first of these is illustrated by Fig. 11, which shows the shock wave parameters in sulfur hexafluoride. Even though the specific heat of this gas is strongly dependent on temperature, the difference between the Rankine-Hugoniot and variable specific heat parameters for a given Mach number is. actually less than the differences in carbon dioxide and nitrogen because the actual temperature rise and the relative increase in specific heat across the shock wave are smaller for sulfur hexafluoride than for the other gases. In other words, the effects of temperature-dependent specific heat will be noticed at a lower flow Mach number in carbon dioxide and rnitrogen than in sulfur hexafluoride. The second reason is that a large part of the work to be done in shock tubes in the future will involve the determination, by means of gauges, of pressures on various parts of obstacles. Most pressure gauges now available have such high temperature coefficients that they cannot be used for this work. The use of complex gases in the expansion chamber may reduce the temperature behind the shock sufficiently to allow crystal gauges to be used in situations where mechanical gauges must otherwise be employed. (iii) Finally, the initial pressure ratio po/p3 across the diaphragm required to produce a particular shock strength depends on ) F

122 In Eq 1, the effect of )E on a0E can, in general, be neglected with respect to its effects in other parts of the equation. Differentiation of the equation indicates that pO/p3 decreases with increasing B E at constant and b. Molecular Weight. The molecular weight of a gas enters the theory of shock waves through the sound speed. a = RT The role of the molecular weight is much more important than that of' in this equation, because Y can be increased easily by a factor of 5, but ( can be reduced by only about 30 per cent in going from a diatomic to a complex gas. (i) The velocity of sound at room temperature is important in several places in shock tube theory. Consider first the po/p3 relation discussed in the last section. The decrease in sound speed and increase in p obtained by using complex gases in the expansion chamber lead to a increased pressure ratio across the diaphragm for a given flow Mach number. For instance, the initial pressure ratio required to produce a Mach number of 1.1, using nitrogen in both chambers, is approximately 0.014; while the ratio required to produce the same Mach number with sulfur hexafluoride in the expansion chamber and nitrogen in the compression chamber is 0.162, an increase of pO by more than a factor of 10. (ii) The effect of the molecular weight and sound speed on the duration of uniform flow in a shock tube was discussed in Chapter Vo It was pointed out that the duration was inversely proportional at constant shock strength to the sound speed in the expansion chamber. It

123 follows that the time duration of uniform flow may be increased manyfold, at constant Mach number by using a complex gas. In Chapter V it was shown that this added duration does not mean that flow phenomena will develop to a higher degree, but it does mean that more time is available in which to make measurements of the hydrodynamic variables behind the shock wave. Thus, the resolution or time response required in the measuring instruments can be reduced; or conversely, the details of the flow can be examined in more detail with given instruments if the longer duration and slower flow development are available. It is obvious that this constitutes a great advantage in shock tube research because the resolution of most gauges available today is not satisfactory for shock tube pressure measurements. c.- Index of Refraction. The optical effect produced by a given density gradient in a flow is directly proportional to the index of refraction and to the absolute pressure of the gas. As Table XVI shows, the indices of refraction of complex gases are much greater than that of air. This fact, coupled with the fact that the expansion chamber pressure is usually increased by using heavy gases, results in photographs of flows in heavy gases that show much more detail and, therefore, provide more information than similar flows in air. In fact, two new classes of information can be obtained under certain conditions. If a gas is used whose index of refraction is very large (CC14 for instance), the small disturbances produced by dirt and imperfections on the windows are made visible. More precisely, the wakes produced by these disturbances can be seen. These wakes indicate

124 particle paths which become streamlines in the steady state; they are visible until wall boundary layer thickens to the point where the imperfections are completely buried in the layer. Thus, for the first time, a method is available for photographing streamlines in a supersonic flow without injecting particles of some type into the stream. Three examples of streamline photographs are shown in Figs. 35, 36, and 37. Furthermore, the index of refraction of almost all heavy gases is sufficient to make Mach lines from small imperfections in models visible. These Mach lines make it possible to determine the flow Mach number in the vicinity of airfoils. d- Viscosity Coefficient. In physical investigations of shock wave- boundary layer interactions or boundary layer growth, it would be convenient to be able to vary the Reynolds number at constant Mach number over wide ranges. This can be accomplished to a certain extent by varying the initial pressure in the expansion chamber while keeping the pressure ratio across the diaphragm fixed. However, the Reynolds number of supersonic flows can be changed conveniently only by a factor of approximately 10 in this way. Larger variations can be obtained by varying the viscosity coefficient by changing the gas in the expansion chamber. Table XVI lists the viscosity coefficients for several gases. This method of varying Reynolds number has the definite disadvantage that the specific heat ratio is usually changed also; but it should be consideredat least until the significance of Reynolds number in a compressible supersonic flow is established and the important variables in the various interactions are determined.

125 e. Chemical Activity. Several gases which would be ideal for shock tube work insofar as their specific heat, molecular weight, and index of refraction are concerned cannot be used because of their chemical properties. For instance, chlorine, bromine, uranium hexafluoride, and many acid gases would corrode the metal parts of the tube. Carbon tetrachloride undergoes some reaction* after several hundred microseconds under the conditions found behind strong shock waves. The chemical activity and stability of a particular gas must be investigated before it can be considered for shock tube work, f. Boiling Point. Many of the heavier organic and inorganic compounds are liquids at room temperature. This does not necessarily exclude them from consideration because, in many instances, the vapor pressure of the gas at room temperature is greater than the desired expansion chamber pressure. The boiling point of CC14, for instance, is 76.70C. Supersonic phenomena can be investigated with this gas; but at room temperature the vapor pressure is too small to allow transonic or subsonic flows to be obtained in the ordinary way. In general, it is desirable to use a gas with as low a boiling point as possible. g. Solubility. The final physical property of gases to be considered is solubility. Some heavy gases are difficult to use in a shock tube because they dissolve in and contaminate the oil in the vacuum pumps and manometers The temperature behind the shock is not high enough to cause thermal dissociation, and the reaction seems to be concentrated in the boundary layer. It may be that carbon tetrachloride reacts with absorbed water vapor on the windows of the tube.

126 and the rubber in various vacuum connections. However, special precautions can be taken to flush all chambers and lines completely before evacuating the system. These precautions will minimize this contaminationo 2. Choice of Specific Gases for Use in a Shock Tube a. General Aerodynamic Investigations. For general aerodynamic investigations the best gas combination is hydrogen in the compression chamber and nitrogen in the expansion chamber. Some of the advantages of nitrogen compared with air have been mentioned in the earlier chapters. These advantages and several others will now be stated and discussed. 1. Since nitrogen and hydrogen do not react, hydrogen can be used in the compression chamber if nitrogen is used in the expansion chamber. Hydrogen cannot be used with air. The advantages of using hydrogen are discussed in Chapter II and illustrated in Fig. 1. 2. The specific heat ofnitrogen varies very slowly with temperature; in fact, more slowly than air because oxygen has a lower vibrational frequency than nitrogen. 3. The vibrational relaxation time for commercial nitrogen has been estimated to be longer than 5000 microseconds. This time is so long that specific heat variations in nitrogen do not have time to take place during the interval of interest in the shock tube. Therefore, all shock wave calculations can be made with the relatively simple Rankine-Hugoniot constant specific heat equations. 41 Humidity variations are completely eliminated. The most important effect of humidity on the shock tube flow enters through the

127 velocity of sound in the shock velocity equation, 26. The velocity of sound of moist air is ard I -.320 oL P where ah is the velocity in humid air, ad is the velocity in dry air, P is the barometric pressure and Pw is the partial pressure of water vapor. This equation can be obtained from the equation a = VoRT/M e Daily humidity variations are sufficient to require correction of measured shock velocity if accurate shock strength determinations are desired. 5. The effect of water vapor on the relaxation times of air is very pronounced. For instance, on a humid day the relaxation time of atmospheric nitrogen may be reduced to approximately 100 microseconds (Kantrowitz). A similar reduction is made in the relaxation time of oxygen. Under these conditions small relaxation effects might be noticeable behind strong shock waves. These effects would change as humidity changed and they might introduce small errors into experimental results. In view of these definite advantages of nitrogen, as compared to air, for use in the shock tube for aeronautical research, it is suggested that nitrogen be used in the expansion chamber of the shock tube wherever possible. b. Physical Investigations, To the physicist, the study of the flow in heavy gases is as interesting as the flow in air. The range of problems which can be treated in a shock tube can be increased markedlyby using a heavy, polyatomic gas in the expansion chamber. It is necessary, however, to

128 consider the specific heat as a function of temperature for complex gases in the calculation of shock wave characteristics. In general, relaxation effects can be ignored because the relaxation times for most polyatomic gases are less than one microsecond. Sulfur hexafluoride is perhaps the most suitable complex gas for general experimentation. It has a low boiling point and can be compared in chemical stability to nitrogen. Its molecular weight and specific heat function are high. Its index of refraction is almost three times that of air. This value is not as high as that found in some other heavy gases; but it is sufficient for most purposes. Finally, compressed sulfur hexafluoride is available in large cylinders from gas supply houses. n-Pentane would be ideal for investigations of boundary layers because of its high index of refraction and low viscosity. Low viscosity provides a high Reynolds number for a given characteristic length. Investigations of relaxation effects can best be conducted in carbon dioxide, which has a relaxation time longer than any other common polyatomic gas. Streamlines can be photographed in some heavy gases. They have been seen in carbon tetrachloride and ethyl iodide, and, no doubt, would be visible in several other gases. c. Representative Photographs of Flows in Complex Gases. In this section, several photographs will be presented which serve to demonstrate what can be done with heavy complex gases in the shock tube. Fig. 32 shows a photograph of the detached bow wave in sulfur hexafluoride in front of a 1/16-in. plate with a 30~ angle at its

129 leading edge. Some of the details of the starting process whereby steady state is reached can be seen. Fig. 33 is an example of the subsonic flow in petroleum ether.* The starting vortex, three rarefaction waves, the early stages of a separated turbulent boundary layer, and several interesting shock wave interactions are visible. A schlieren photograph of the normal reflection of a shock wave in carbon dioxide is shown in Fig. 34. The lambda shock caused by very severe reverse flow in the boundary layer on the floor and walls of the tube is easily seen. Figs. 35, 36, and 37 are examples of flows in carbon tetrachloride in which particle paths can be seen. The definitely subsonic character of the flow between the detached bow wave and the first corner on thelower surface of the wedge in Fig. 35 is to be compared with the supersonic character of the flow behind the attached bow wave in the same region of Figs. 36 and 37. In both of these photographs, the Mach number in the various regions of the flow can be determined from measurements of Mach angles. Which particular complex gas should be used for a particularinvestigation must be determined by considering= the special[reqeuirements of the investigationand the physical and chemical properties of various gases. It is thought that some of the advantages of the use of complex gases in the investigation of hydrodynamic phenomena in a shock tube have been demonstrated by the several flow photographs shown on the following pages. * Petroleum ether is a mixture of hydrocarbons boiling between 30~ and 400Co This mixture has properties similar to pentane; but it is not suggested for shock tube use because the concentration of the various fractions in the vapor phase will vary in time.

130 FIG 32 ~~~~~~-~ ~ ~ ~ ~ ~ ~ ~ t i~~~~~~~~~~~~~~~~~~~~~~~~~~~'' DETACHED BOW WAVE I N SULFUR HEXAFLUORIDE FIG 33 SUBSONIC FLO~~~~~~~~~~~)W IN71 PETRU 4, vvr-~E T H E R

131 FIG 34 NORMAL SHOCK REFLECTION IN CARBON DIOXIDE FIG 35 DETACHED BOW WAVE IN CARBON TETRACHLORIDE IN FRONT OF A WEDGE AT AN ANGLE OF ATTACK

132 FIG 36 ATTACHED BOW WAVE IN CARBON TETRACHLORIDE ON A WEDGE AT AN ANGLE OF ATTACK EARLY PHOTOGRAPH FIG 37 ATTACHED BOW WAVE IN CARBON TETRACHLORIDE ON A WEDGE AT AN ANGLE OF ATTACK LATE PHOTOGRAPH

VII. CONCLUSIONS The strongest shock wave that can be produced by a given initial pressure ratio across the diaphragm of-a shock tube may be obtained by using hydrogen in the compression chamber. However, oxygen must be removed from the expansion chamber when hydrogen is used in the compression chamber in order to prevent an explosion in the tube, which could be initiated by the high temperature behind the shock wave reflected from the end of the expansion chamber. Expressions for the flow velocity produced by a rarefaction wave in a gas with temperature-dependent specific heat and in a van der Waals' gas were derived. It was shown that both of these departures from the ideal conditions assumed in the derivation of the Taub equation are negligible for hydrogen. However, they might be important if a polyatomic gas were used in the compression chamber for a special inve stigation. Shock wave equations analogous to the Rankine-HIugoniot equations were developed for shock waves in an ideal gas with temperaturedependent specific heat. The flow parameters calculated from these equations showed that the temperature behind a shock wave is lower than.calculated from the Rankine-Hugoniot equations. Furthermore, the pressure and density behind the shock, and the flow Mach number in the shock tube, are higher than predicted by the simple theory. The effect of vibrational relaxation on the flow parameters behind a shock wave was investigated, and an approximate calculation of the time dependence of these parameters was made. It was shown that for 133

134 weak shocks the temperature changed exponentially from its initial to its final value, while the other parameters followed a pseudo-exponential law. The velocity of a shock wave in carbon dioxide reflected normally from a wall was found to vary in time. The measured equilibrium velocity agreed with the predictions of theory assuming temperature-dependent specific heat. The variation of reflected shock velocity is the direct result of vibrational relaxation. The influence of the specific heat ratio of gases on the relax tive duration of uniform flow in a shock tube was demonstrated by the introduction of a dimensionless parameter, the duration index, which depends only on [ and A. At a given Mach number the ratio of the duration indices of two gases is also the ratio of the effective durations of uniform flow in the gases. It was shown that, to a good approximation, for a particular Mach number the duration index is independent of the gas used in the expansion chamber. The fundamental assumption made in the derivation of the duration index is that the rate of development of a steady-state flow configuration is proportional to the sound speed behind the shock wave. This assumption was verified experimentally for the special case of a detached bow wave in nitrogen and sulfur hexafluorideo Finally, the effects of a large number of chemical and physical properties of gases on the applicability of the gas for shock tube research were discussed. Nitrogen was found to be the most satisfactory gas for use in the expansion chamber for standard aeronautical. research, and sulfur hexafluoride the most satisfactory for the investigation of the physical details of hydrodynamic phenomena.

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