##### Document Text Contents

Page 1

VOLUME 1

APPLIED PHYSICS AND ENGINEERING

An International Series

Relaxation in Shock Waves

Page 2

Relaxation

in Shock Waves

Yeo V. Stupochenko

S. A. Losev

and A. I. Osipov

Translated by

Scripta Technica, Inc.

Translation Editor

Richard Shao-lin Lee

College of Engineering

State University of New York

Stony Brook, New Yorh

SPRINGER-VERLAG • Berlin • Heidelberg • New York • 1967

Page 201

192 Relaxation in Shock Waves

garded as valid for electron concentrations of 1014_1017 cm -3, correct within

10% and in good agreement with experimental results. Here Ne is then deter-

mined by comparing the contour of the observed line with a number of cal-

culated contours of the same line, obtained for different values of Ne and by

selecting a contour for calculations which is closest to the experimentally

obtained contour [1, 78]. Alyamovskiy and Kitayeva [1], using a shock

tube, have obtained an accuracy of at least 15-20% in measuring Ne• McLean

and others [442] have measured Ne by the widening of lines of ionized helium

with an error of up to 50%. Griem, however, assumes that in the study of

contours of Hp lines with Ne '" 2 x 1017 cm - 3 and T~ 18,000 oK it is possible

to obtain an accuracy in measuring Ne of not less than ± 5%. Comparison

with experimental data shows that it is possible to appreciably reduce the

error by measuring the widening of a number of lines at the same time,

determining Ne and then averaging the result thus obtained.

In the case of the quadratic Stark effect the line with")' and the shift of the

maximum Ll, according to the Weiskopf-Lindholm theory, are determined by

(13-8)

where C4 is the constant of the quadratic Stark effect, v is the relative veloc-

ity of the atom and the exciting particle, and N is the number density of the

exciting particles. These formulas, however, are valid only for small v.

Mandel'shtam and Mazing [120] have found that the experimental results

obtained in argon plasma do not conform to Eqs. (13-8). The subsequent

development of the theory of the quadratic Stark effect has made it possible

to overcome these disagreements by taking into account nonelastic collisions

of electrons with atoms and the reduction of the effect of rapid traverses by

electrons by the amount of widening (within the framework of the non-

steady-state theory due to Sobel'man-Vaynshteyn [181]). In this case the

widening and shift of the lines are determined by the parameter

Z2 (2ncLlEq)tCt

f3 = 2 '

V

(13-9)

where Z is the ionic charge, LlEq is the distance from the ion under considera-

tion to the nearest exciting ion (cm -1), so that Eqs. (13-8) now contain

instead of")' and Ll the correction factors I' (f3) and 1" (f3), respectively. Spec-

ific values of f3 and C4 for a number of lines of inert gases can be found in

[120, 116, 117], and the values of functions I' (f3) and 1" (f3) are given in [181].

Satisfactory agreement with experimental data is here noted.

The setting up of experiments for study of the Stark effect in plasma be-

hind a shock front is complicated by the need of simultaneously obtaining

(after passing of the shock wave) the dependence of the radiation on the

Page 202

Nonequilibrium Phenomena in Shock Waves 193

wavelength (over the spectrum) as well as on time with a high resolving

power. The photographic method is most frequently used in experiments

[475, 1, 442, 416]. In this case either the spectral region under study is

projected on a moving film, or the spectrum is scanned in time by moving

the image of the spectrum over a stationary film, using a disk with inclined

slits which rotates in front of the input slit. The time resolution which is thus

obtained is not too high. Thus, in experiments by Petschek, et at., [475], it

comprised about 12-24 microseconds (in terms of the natural time); the

authors have found that the measured Ne (Ne~ 1016 _1017 cm -3) corresponds

to equilibrium values. This is understood, since the resolving power in time of

the arrangement under experimental conditions is comparable with the time

needed to establish ionization equilibrium (obtained from later measure-

ments by other methods [474]).

The use of photoelectric recording methods is more promising. For this

it is necessary to divide the image of each spectral line into a number of sec-

tions and to direct each of them to the photomultiplier. A number of methods

is available for successfully overcoming this problem, such as the use of

multi component slits and mirrors, hair optics for breaking up the ray, in-

clined slits, interferometer techniques, etc. [410]. Measuring the image of

one line into seven segments has made it possible to study the behavior of the

contour of a number oflines in shock waves in helium and deuterium [305].

Malyshev and others [118] suggest the use of the Fabry-Perot standard [in-

terferometer] with a monochromator and an electron-optical amplifier for

obtaining contours of spectral lines with a high time resolution.

[5] Recording the Optical Radiation

As was noted in the preceding section, the intensity of the continuous

radiation produced by free-free and free-bound electronic transitions in

quasi-neutral plasma is proportional to the square of the electron number

density. It is known that. the radiation intensity in this case depends also on

the electron temperature (see, for example [22]). However, in the region

where Ne is sufficient for observing the radiation the electron temperature

does not vary appreciably (see [25]). Hence for the change in I, the radiation

intensity, with time behind a shock front we have

1 dI 2 dNe

I dt Ne dt

(13-10)

The arrangement for measuring the distribution of the radiation intensity

behind a shock front is similar to that described in the preceding section.

Particular attention must here be given to reducing the level of light scat-

Page 402

Subject Index 393

resonance, diffusion of, 302

short-wave, 93

ultraviolet, absorption of, 143,248

Radio-frequency method, 98

Random molecule wandering, stochastic

process of, 275

Random walks, theory of, 279

Rankine-Hugoniot relationships, 211

Refraction index, study of changes in, 121

Relaxation process, kinetics of, 348

study of, 71

(see also Shock Waves)

Resonance method, 188, 189

Reynolds number, 78, 79

Ribbon lamps, tungsten, 175

Riemann invariant, 40

Rosen's variational principle, 212, 213

Rotational equilibrium, 35

relaxation, 215

Roth's experiment, 250

Rozhdestvenskiy's hooks method, 99, 131

Runge-Kutta method, 113,318

Saha's equation, 51

Sakurai's method, 2 I 3

Schrbdinger equation, 228, 229, 232, 254

steady-state, 232

Schumann-Runge bands, 141,312

region, 346

system, 143, 172,311

Semiconductor receivers, germanium, 173

Shock adiabatic, 7

Shock front, flow inhomogeneity behind, 72

study of light reflection from, 1 17

Shock-front curvature, measurements of,

83

problem of, 83

Shock tubes, 37,49

gasdynamic flows in, 61

nonequilibrium phenomena in, 72

pulse-type magnetohydrodynamic, 172

shock-wave propagation in, 72, 80

theory of, 37, 39

Shock-tube chambers, filling of, 52

operation, techniques of, 52

two-diaphragm arrangement of, 61

Shock waves, basic properties of, 3

dimensions, 64

dissipative processes in, 8

formation and acceleration of, 64

motion of, 17

gas radiation in, 162

increasing speed of, 52

method of obtaining strong, 49

nonequilibrium phenomena in, 38, 104

kinetics in, 38

study of, 99,100,306

relaxation processes in, 206

theoretical analysis of, 206

structure of strong, 211

study of formation and propagation of,

38

thermal ionization kinetics in, 189

theory of, 3, 364

velocity of, 40

measuring, 91

Single-component system, thermal disso-

ciation in, 281

Single-photon energy transfer, concept of,

246,253,260

Small disturbances, propagation of, 359

Sobel'man-Vaynshteyn unsteady-state

theory, 192

Sodium salts, 100

heating of, 100

Spectral absorption analysis, methods of,

138,161

line reversal, method of, 176

Spectrometer, time-of-flight, 200

Speed of sound, isentropic, 1 I

Stark effect, 180, 191

basic theory of, 192

in plasma, study of, 192

quadratic, 192

use of, 191

Statistical thermodynamics, expression for,

280

theorem on, 350

Steric factor, 33, 275

Structureless particles, 13

Targets, 137

chromium, 137

copper, 137

tungsten, 137

Temperature-dependent coefficients, 243

Temperature measurement, spectral

methods of, 179

Page 403

394 Relaxation in Shock Waves

Tepler's schlieren systems, 46, 94, 97, 121,

122,124,125

operating principle of, 121

Test gas, preparation of, 100

Thermal conductivity, coefficient of, 9

diffusion processes, contribution of, 182

dissociation kinetics, 271

process, 272

theory,286,325

ionization, mechanism of, 301

of air, 342

process of, 272

study of, 342

pickups, 94

Thermoanemometer, defined, 197

probe, 197

use of, 95, 96

Thermodynamic equilibrium, 349

parameters, 349

temperature, 350

Thermodynamics of irreversible processes,

methods of, 352

Thyratron arrangements, 96

systems, actuation time of, 96

Time-measuring devices, chronographic, 96

Transition probabilities, averaging value

of,236

qu?:-:tum theory of, 231

quantum-mechanical problem of cal-

culating, 224

semiclassical theory of, 228

Transitional layer, boundary of, 2

structure of, 2

Translational energy, transfer of, 23

Transport phenomena, theory of, 225

Transverse waves, effect of, 85

Trimpi-Cohen theory, 67

Tubes, 100

Ultrahigh frequencies, method of, 183

Ultrasonic waves, dispersion and absorp-

tion method of, 215

vibrations, propagation of, 245

Valency schemes, method of, 263

Van der Waals-type forces, 308

Vl'vilov's experiments, 140

Vibration excitation, process of, 115

Vibrational energy, relaxation equation

for, 257

photons, transfer of, 313

relaxation, study of, 36, 226, 263

theory of, 31

transfer, probability of, 253

Volt-ampere characteristics, 180

Water vapor, kinetics of decomposition of,

161,162

Weiskopf-Lindholm theory, 192

Widom's calculation, 218

Wien's radiation law, 175

Wigner's distribution function, energy

representation of, 242

Wray's recommendations, 326, 327

Xenon, ionization kinetics of, 188

Xenon lamp, arc-discharge, 158

X-ray radiation, use of, 136

VOLUME 1

APPLIED PHYSICS AND ENGINEERING

An International Series

Relaxation in Shock Waves

Page 2

Relaxation

in Shock Waves

Yeo V. Stupochenko

S. A. Losev

and A. I. Osipov

Translated by

Scripta Technica, Inc.

Translation Editor

Richard Shao-lin Lee

College of Engineering

State University of New York

Stony Brook, New Yorh

SPRINGER-VERLAG • Berlin • Heidelberg • New York • 1967

Page 201

192 Relaxation in Shock Waves

garded as valid for electron concentrations of 1014_1017 cm -3, correct within

10% and in good agreement with experimental results. Here Ne is then deter-

mined by comparing the contour of the observed line with a number of cal-

culated contours of the same line, obtained for different values of Ne and by

selecting a contour for calculations which is closest to the experimentally

obtained contour [1, 78]. Alyamovskiy and Kitayeva [1], using a shock

tube, have obtained an accuracy of at least 15-20% in measuring Ne• McLean

and others [442] have measured Ne by the widening of lines of ionized helium

with an error of up to 50%. Griem, however, assumes that in the study of

contours of Hp lines with Ne '" 2 x 1017 cm - 3 and T~ 18,000 oK it is possible

to obtain an accuracy in measuring Ne of not less than ± 5%. Comparison

with experimental data shows that it is possible to appreciably reduce the

error by measuring the widening of a number of lines at the same time,

determining Ne and then averaging the result thus obtained.

In the case of the quadratic Stark effect the line with")' and the shift of the

maximum Ll, according to the Weiskopf-Lindholm theory, are determined by

(13-8)

where C4 is the constant of the quadratic Stark effect, v is the relative veloc-

ity of the atom and the exciting particle, and N is the number density of the

exciting particles. These formulas, however, are valid only for small v.

Mandel'shtam and Mazing [120] have found that the experimental results

obtained in argon plasma do not conform to Eqs. (13-8). The subsequent

development of the theory of the quadratic Stark effect has made it possible

to overcome these disagreements by taking into account nonelastic collisions

of electrons with atoms and the reduction of the effect of rapid traverses by

electrons by the amount of widening (within the framework of the non-

steady-state theory due to Sobel'man-Vaynshteyn [181]). In this case the

widening and shift of the lines are determined by the parameter

Z2 (2ncLlEq)tCt

f3 = 2 '

V

(13-9)

where Z is the ionic charge, LlEq is the distance from the ion under considera-

tion to the nearest exciting ion (cm -1), so that Eqs. (13-8) now contain

instead of")' and Ll the correction factors I' (f3) and 1" (f3), respectively. Spec-

ific values of f3 and C4 for a number of lines of inert gases can be found in

[120, 116, 117], and the values of functions I' (f3) and 1" (f3) are given in [181].

Satisfactory agreement with experimental data is here noted.

The setting up of experiments for study of the Stark effect in plasma be-

hind a shock front is complicated by the need of simultaneously obtaining

(after passing of the shock wave) the dependence of the radiation on the

Page 202

Nonequilibrium Phenomena in Shock Waves 193

wavelength (over the spectrum) as well as on time with a high resolving

power. The photographic method is most frequently used in experiments

[475, 1, 442, 416]. In this case either the spectral region under study is

projected on a moving film, or the spectrum is scanned in time by moving

the image of the spectrum over a stationary film, using a disk with inclined

slits which rotates in front of the input slit. The time resolution which is thus

obtained is not too high. Thus, in experiments by Petschek, et at., [475], it

comprised about 12-24 microseconds (in terms of the natural time); the

authors have found that the measured Ne (Ne~ 1016 _1017 cm -3) corresponds

to equilibrium values. This is understood, since the resolving power in time of

the arrangement under experimental conditions is comparable with the time

needed to establish ionization equilibrium (obtained from later measure-

ments by other methods [474]).

The use of photoelectric recording methods is more promising. For this

it is necessary to divide the image of each spectral line into a number of sec-

tions and to direct each of them to the photomultiplier. A number of methods

is available for successfully overcoming this problem, such as the use of

multi component slits and mirrors, hair optics for breaking up the ray, in-

clined slits, interferometer techniques, etc. [410]. Measuring the image of

one line into seven segments has made it possible to study the behavior of the

contour of a number oflines in shock waves in helium and deuterium [305].

Malyshev and others [118] suggest the use of the Fabry-Perot standard [in-

terferometer] with a monochromator and an electron-optical amplifier for

obtaining contours of spectral lines with a high time resolution.

[5] Recording the Optical Radiation

As was noted in the preceding section, the intensity of the continuous

radiation produced by free-free and free-bound electronic transitions in

quasi-neutral plasma is proportional to the square of the electron number

density. It is known that. the radiation intensity in this case depends also on

the electron temperature (see, for example [22]). However, in the region

where Ne is sufficient for observing the radiation the electron temperature

does not vary appreciably (see [25]). Hence for the change in I, the radiation

intensity, with time behind a shock front we have

1 dI 2 dNe

I dt Ne dt

(13-10)

The arrangement for measuring the distribution of the radiation intensity

behind a shock front is similar to that described in the preceding section.

Particular attention must here be given to reducing the level of light scat-

Page 402

Subject Index 393

resonance, diffusion of, 302

short-wave, 93

ultraviolet, absorption of, 143,248

Radio-frequency method, 98

Random molecule wandering, stochastic

process of, 275

Random walks, theory of, 279

Rankine-Hugoniot relationships, 211

Refraction index, study of changes in, 121

Relaxation process, kinetics of, 348

study of, 71

(see also Shock Waves)

Resonance method, 188, 189

Reynolds number, 78, 79

Ribbon lamps, tungsten, 175

Riemann invariant, 40

Rosen's variational principle, 212, 213

Rotational equilibrium, 35

relaxation, 215

Roth's experiment, 250

Rozhdestvenskiy's hooks method, 99, 131

Runge-Kutta method, 113,318

Saha's equation, 51

Sakurai's method, 2 I 3

Schrbdinger equation, 228, 229, 232, 254

steady-state, 232

Schumann-Runge bands, 141,312

region, 346

system, 143, 172,311

Semiconductor receivers, germanium, 173

Shock adiabatic, 7

Shock front, flow inhomogeneity behind, 72

study of light reflection from, 1 17

Shock-front curvature, measurements of,

83

problem of, 83

Shock tubes, 37,49

gasdynamic flows in, 61

nonequilibrium phenomena in, 72

pulse-type magnetohydrodynamic, 172

shock-wave propagation in, 72, 80

theory of, 37, 39

Shock-tube chambers, filling of, 52

operation, techniques of, 52

two-diaphragm arrangement of, 61

Shock waves, basic properties of, 3

dimensions, 64

dissipative processes in, 8

formation and acceleration of, 64

motion of, 17

gas radiation in, 162

increasing speed of, 52

method of obtaining strong, 49

nonequilibrium phenomena in, 38, 104

kinetics in, 38

study of, 99,100,306

relaxation processes in, 206

theoretical analysis of, 206

structure of strong, 211

study of formation and propagation of,

38

thermal ionization kinetics in, 189

theory of, 3, 364

velocity of, 40

measuring, 91

Single-component system, thermal disso-

ciation in, 281

Single-photon energy transfer, concept of,

246,253,260

Small disturbances, propagation of, 359

Sobel'man-Vaynshteyn unsteady-state

theory, 192

Sodium salts, 100

heating of, 100

Spectral absorption analysis, methods of,

138,161

line reversal, method of, 176

Spectrometer, time-of-flight, 200

Speed of sound, isentropic, 1 I

Stark effect, 180, 191

basic theory of, 192

in plasma, study of, 192

quadratic, 192

use of, 191

Statistical thermodynamics, expression for,

280

theorem on, 350

Steric factor, 33, 275

Structureless particles, 13

Targets, 137

chromium, 137

copper, 137

tungsten, 137

Temperature-dependent coefficients, 243

Temperature measurement, spectral

methods of, 179

Page 403

394 Relaxation in Shock Waves

Tepler's schlieren systems, 46, 94, 97, 121,

122,124,125

operating principle of, 121

Test gas, preparation of, 100

Thermal conductivity, coefficient of, 9

diffusion processes, contribution of, 182

dissociation kinetics, 271

process, 272

theory,286,325

ionization, mechanism of, 301

of air, 342

process of, 272

study of, 342

pickups, 94

Thermoanemometer, defined, 197

probe, 197

use of, 95, 96

Thermodynamic equilibrium, 349

parameters, 349

temperature, 350

Thermodynamics of irreversible processes,

methods of, 352

Thyratron arrangements, 96

systems, actuation time of, 96

Time-measuring devices, chronographic, 96

Transition probabilities, averaging value

of,236

qu?:-:tum theory of, 231

quantum-mechanical problem of cal-

culating, 224

semiclassical theory of, 228

Transitional layer, boundary of, 2

structure of, 2

Translational energy, transfer of, 23

Transport phenomena, theory of, 225

Transverse waves, effect of, 85

Trimpi-Cohen theory, 67

Tubes, 100

Ultrahigh frequencies, method of, 183

Ultrasonic waves, dispersion and absorp-

tion method of, 215

vibrations, propagation of, 245

Valency schemes, method of, 263

Van der Waals-type forces, 308

Vl'vilov's experiments, 140

Vibration excitation, process of, 115

Vibrational energy, relaxation equation

for, 257

photons, transfer of, 313

relaxation, study of, 36, 226, 263

theory of, 31

transfer, probability of, 253

Volt-ampere characteristics, 180

Water vapor, kinetics of decomposition of,

161,162

Weiskopf-Lindholm theory, 192

Widom's calculation, 218

Wien's radiation law, 175

Wigner's distribution function, energy

representation of, 242

Wray's recommendations, 326, 327

Xenon, ionization kinetics of, 188

Xenon lamp, arc-discharge, 158

X-ray radiation, use of, 136