Nonequilibrium expansion flows of dissociated oxygen and ionized argon around a corner

3 NONEQUILIBRIUM EXPANSION FLOWS OF DISSOCIATED OXYGEN AND IONIZED ARGON AROUND A CORNER

I. I. GLASS Institute for Aerospace Studies, University of Toronto, Toronto, Canada

and A. TAKANO* Department

of Aeronautics,

University of Tokyo, Tokyo, Japan

Abstract. Detailed studies are presented of nonequilibrium, inviscid, two-dimensional expansion flows of dissociated oxygen and ionized argon around a corner. The numerical-graphical method of characteristics was applied to calculate the flow quantities. The computations were performed on an IBM-7090 computer for a number of cases with different free-stream conditions and wall angles. Of special interest are the flow conditions that will be tested in the UTIAS 4 in. x 7 in. hypersonic shock tube to measure recombination rate constants and other properties of the flow. It was assumed that, for dissociating oxygen, in the temperature range considered here the vibrational excitation was in equilibrium with the translational and rotational modes, except along the wave head and at the very comer, where the degree of dissociation and the vibrational excitation were frozen at their free-stream values. Consequently, it was found that the partially frozen characteristics played the dominant role in the present analysis. For ionizing argon, in the temperature range encountered the electronic excitation was always assumed at its ground state and the electron temperature was assumed to be equal to the atom and ion temperature. As a consequence, it was shown that in this case the frozen characteristics must be used throughout. It was found that, in every case, the flow produced by the comer consisted of seven basic flow areas, some of which were quite complex. In the case of dissociating oxygen, a recombination shock wave was found to exist along the expansion wave tail. However, for ionizing argon, no discontinuous feature was verified to exist, even up to large wall angles and even when using a small characteristic mesh size. The existence of a recombination shock wave in the case of oxygen was found to be the mathematical consequence of the assumed model in which the vibrational mode is instantly de-excited. Had a vibrational rate equation been used for gradual addition of energy it is doubtful if a shock wave would have appeared. Therefore, the term “de-excitation shock wave” would be more appropriate for such a disturbance rather than the term “recombination shock wave”. * On sabbatical leave at the Institute for Aerospace Studies (1962-3). 163

164

I. I. GLASS AND A. TAKANO

In every case quasi-similarities in the variations of the flow quantities along the wave head and along the wall surface were found to exist when their appropriately normalized values were plotted as functions of a relevant distance. It is shown that experimentally determined pressure distributions (using piezo gauges) and density distributions (using interferometry) along the wall surface of a corner expansion can be used to evaluate the recombination rate constants. 1. INTRODUCTION

At hypervelocities, temperatures of many thousands of degrees can easily be attained in the flow around a body as the kinetic energy of a re-entering craft is dissipated by the atmospheric gas through shock compression and viscous heating. The gas molecules, atoms, and other species, which absorb this kinetic energy, can possess modes of excitation such as vibration, electronic excitation, dissociation and ionization. These internal modes of excitation play important roles in the analysis of the thermodynamic, aerodynamic, and electromagnetic properties of a reacting gas mixture at the re-entry temperatures. Many theoretical studies were made during the past decade on the thermodynamics of dissociating and ionizing gas flows. (1-4) In the case of a dissociating pure diatomic gas, a simplification was obtained by assuming the so-called “ideal dissociating gas “.c5) It can be found’@ that with this assumption the vibrational degree of freedom is excited to half of its classical (fully excited) value. As a consequence, it is possible to calculate, in a simplified manner, all the thermodynamic properties of a pure, symmetrical, diatom& dissociating gas such as oxygen. (G-Q The assumption of an “ideal dissociating gas” can be verified to be a good approximation for the above gas in the temperature range of about 3000°K to 5000°K at a pressure of about 1 atm. Furthermore, in the temperature range, where dissociation is dominant, the electronic excitation is always assumed to be at its ground energy level.(1+2) For an ionizing monatomic gas, the electron translational temperature is usually assumed to be equal to the atom and ion temperature.t3p gt lo) This assumption can be verified to be approximately valid for the argon case in the temperature range of about 8000°K to 12,OOO”Kat a pressure of about 1 atm. In addition, in this temperature range it is also assumed that the electronic excitation is always at its ground state. The neglection of higher energy levels of electronic excitation can be foundol) to introduce an error of only a few per cent in the thermodynamic properties at low temperatures (below 15,OOO”Kfor argon). By using these assumptions, simplified thermodynamic equations can be derived for an ionizingmonatomicgas such as argon(anidealionizing gas).(gJo) Through the above assumptions, many studies were made of nonequilibrium flows of dissociating and ionizing gases. In the calculation of expansion and compression flows of a chemical reacting gas, the method of characteristics can be successfully applied,c6-* 912-17)and the important concepts of frozen, partially frozen and equilibrium sound speeds have been demonstrated.

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These sound speeds specify the characteristic directions(12* 13) along which disturbances propagate with respect to the moving gas. Many studies were made of nonequilibrium, two-dimensional supersonic flows of dissociating and ionizing gases based on the method of characteristics.(6-8p 14--17) Napolitano(@ summarized the thermodynamic equations for a dissociating pure diatomic gas and formulated the method of characteristics for an ideal dissociating gas proposed by Lighthill. c5) Numerical computations of nonequilibrium expansion flows of such a gas were performed by Cleaver(7) and Appleton.@) However, in a more realistic model for a dissociating gas, it was assumedo 16)that the vibrational excitation was always in equilibrium with the translational and rotational modes, except along the frozen wave head and at the very corner, where the vibrational mode as well as the degree of dissociation were frozen at their free-stream values (a partially excited dissociating gas). As a consequence, the partially frozen sound speed was shown(l@ to specify the characteristic directions in a nonequilibrium flow of this gas, except along the wave head and at the very corner, where the frozen characteristics could be used. It was also shown(16) that the so-called “frozen sound speed” of an ideal dissociating gas,t7y 8, which specified the characteristic directions in this gas flow, was a special case of the above partially frozen sound speed. Calculations of frozen, partially frozen and equilibrium Prandtl-Meyer expansion flows of dissociating oxygen were performed by Glass and Kawadao5) for a number of cases with different free-stream conditions and flow deflection angles. Nonequilibrium expansion flows of partially excited dissociating oxygen around a corner were calculated by Glass and Takano(t@ for several free-stream conditions with typical wall angles. By using the assumptions of an ideal ionizing monatomic gas, like the case of an ideal dissociating gas, all the thermodynamic properties of ionizing gases can be calculated in a simplified manner. t91lo) It was showno 17)that, in the entire flow region of an ideal ionizing gas, only the frozen sound speed specified the characteristic directions. Based on these characteristic properties, numerical computations of frozen and equilibrium Prandtl-Meyer expansion flows were made by Glass and Kawada(15) and nonequilibrium expansion flow around a corner were calculated by Glass and Takano.‘17) In the present paper, the previous studies cl69i7) of nonequilibrium expansion flows of partially excited dissociating oxygen and ideal ionizing argon around a corner are summarized. A comparison of the gas models is included in order to indicate clearly the subsequent differences in the thermodynamic equations of each gas. Based on the numerical-graphical method of characteristics, calculations were performed on an IBM-7090 computer at the Institute of Computer Science, University of Toronto. The digital computer programme is detailed in ref. 18. Of special interest are the flow conditions that will be experimentally investigated in the UTIAS 4 in. x 7 in. hypersonic shock tube.

166

1. 1. GLASS

AND A. TAKANO

The IBM calculations showed that, in every case, the overall flow pattern was qualitatively similar to that obtained by Cleaver”) and Appleton@) for an ideal dissociating gas. The expansion flow around a corner consists of a frozen flow at the very corner, an equilibrium flow at infinity (mathematical) and a transitional nonequilibrium flow occupying the remainder of the field. The nonequilibrium expansion flow starts with the frozen wave head, which decays almost exponentially. Only at infinity can the equilibrium wave head exist. The tail of the disturbance wave originating from the corner is a curved nonequilibrium characteristic line (frozen in the case of argon or partially frozen in the case of oxygen) and again the equilibrium expansion wave tail can exist only at infinity. However, in the case of partially excited dissociating oxygen, (16)a recombination or more accurately a de-excitation shock wave, which was anticipated by Feldman,(19) was found to exist along the expansion wave tail. The existence of such a discontinuous disturbance was proved by the occurrence of the intersection of the expansion wave tail and the following characteristic line. It was shown that the presence of such an intersection was a mathematical consequence of the assumption of a partially excited dissociating gas. Owing to the discontinuous change of the characteristic directions at the very corner from frozen to partially frozen values, a flow expands to a larger deflection angle than the prescribed wall angle behind the expansion wave tail. This over-expansion gives rise to the intersection of the wave tail and the following characteristic line. Discontinuous jumps of the flow derivatives are also found to take place along the expansion wave tail near the corner. However, in the case of an ideal ionizing gas, where only the frozen sound speed specifies the characteristic directions in the entire flow region (compared with the ideal dissociating gas case), the over-expansion of a flow at the wave tail cannot exist because there is no discontinuous feature of the characteristic direction at the corner. Therefore, it is verified that, in the ideal ionizing argon case (as well as in the ideal dissociating oxygen case) the intersection of characteristic lines belonging to the same family does not take place. Such a condition indicates the non-existence of a de-excitation shock wave in the above gases. Similarities in the relaxation processes were found to exist along the wave head and along the wall surface. In every case, the ratios of the angular derivatives of the flow quantities to their corner values decay nearly exponentially with increasing distance from the corner for all free-stream conditions. The flow quantities along the wall surface exhibit quasi-similarities when the appropriately normalized values are plotted as the relevant distance from the corner. The recombination rate parameter is assumed to be constant for dissociating oxygen,(15* 16) whereas it is considered as a function of temperature for ionizing argon.(15, 17)These values used in the present calculations must be verified experimentally. Experimental considerations are also given in the

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present paper for the determination of the recombination rate constants as well as other properties of the flow. Experimental studies of nonequilibrium expansion flows of dissociated oxygen and ionized argon around a corner will be made in the UTIAS 4 in. x 7 in. hypersonic shock tube in order to determine more realistic values of the recombination rate constants of both gases, to check the existence of a de-excitation shock wave and to measure the flow properties, which will be compared with the theoretical predictions. 2. THERMODYNAMIC

EQUATIONS

In this section definitions and equations of thermodynamic quantities for dissociating and ionizing gases will be summarized. The detailed formulations of the thermodynamic equations can be found in textbooks.(20~21~22) Particularly, the theory of an ideal dissociating gas proposed by Lighthillt5) was treated at some length by Napolitano(@ and the detailed derivations of the equations for a partially excited dissociating gas were presented in ref. 16. The equations of an ideal ionizing gas were given in refs. 9 and 10 and their derivations were outlined in ref. 17. In general, a gas like dissociating and ionizing air is composed of many species such as molecules, atoms, molecular ions, atomic ions and electrons. However, for a pure diatomic gas like oxygen where the molecular dissociation energy is much less than the atomic ionization energy,(l) *l) ionization can be considered to become appreciable only after dissociation is practically completed. Therefore, dissociating oxygen is assumed to be composed of molecules and atoms whereas ionizing argon is composed only of atoms, ions and electrons. That is, for practical purposes, pure oxygen (in the temperature range of about 2500°K to 4000°K at a pressure of about 1 atm) and pure argon (in the temperature range of about 7000°K to 13,OOO”Kat a pressure of about 1 atm) are chosen in the present work for the investigation of dissociating and ionizing gases, respectively. The temperature and pressure ranges noted above are encountered in nonequilibrium expansion flows treated in the present paper as well as in experiments planned for the UTIAS 4 in. x 7 in. hypersonic shock tube. The state of a gas mixture noted previously is uniquely described by three independent state parameters, pressure p, temperature T and dissociation degree CIor ionization degree x. The other thermodynamic quantities, i.e. density p, specific internal energy e, specific enthalpy h and specific entropy S can be given as functions of the above three independent quantities. In a frozen state the degree of dissociation or ionization remains constant and in an equilibrium state these degrees can be given as functions of pressure and temperature. It should be noted that, in the case of an ionizing gas, the electron temperature T, (translational) is in general different from the atom (or ion) temperature T and can be related to T and x. (9’lo) In an equilibrium state

168

I. 1. GLASS

AND A. TAKANO

T, = T. However, as will be noted in the subsequent subsection, it is assumed that T, = Tin the temperature and pressure ranges considered in the present

work. The following subscripts are used in the present paper for denoting quantities referred to each species of the mixture: A2 (diatomic molecule), A (atom), A+ (ion), e (electron). 2.1. Assumptions In writing the basic thermodynamic equations, the following assumptions will be made in accordance with thermodynamic theory for reacting gas mixtures : (i) The effects of viscosity, heat conduction and diffusion are neglected. Since no electron diffusion is assumed to occur, the mixture of an ionizing gas is electrically neutral. Also the Coulomb force between ions and electrons is neglected. (ii) The deviations from thermodynamic equilibrium are such that the fundamental Gibbs equation giving the chemical potentials for each species still holds. (iii) Although the total system of the mixture may not be in equilibrium, each component of the mixture is in thermal equilibrium, so that the thermodynamic properties of each component can be derived from the corresponding partition functions. (iv) Dalton’s law of partial pressures can be applied and the translational temperature is the same for molecules, atoms and ions. (v) The other quantities such as density, entropy, internal energy and enthalpy are given as the weighted sums of the corresponding properties for the single system. For a dissociating diatomic gas such as oxygen, the following additional assumptions will be made regarding the relaxations of translational, rotational and vibrational modes and dissociation. It is known(4) that it requires only a few collisions to achieve translational and rotational equilibrium, but many orders of magnitude more for other degrees such as vibration and dissociation. Furthermore, the relaxation time for each mode decreases with increasing temperature and pressure. For example, the rotational relaxation time for oxygen is of the order of 1O-9set at about 300°K and the vibrational relaxation time is of the order of 1O-6 set at about 3000°K~ 4OOO”K, whereas the dissociation relaxation time varies from 1O-4 to 10-s set at about 2500°K N 4000°K. (23)Therefore, (vi) The rotational modes can always be considered in equilibrium with the translational modes. Furthermore, it was shown that the vibrational relaxation time TVmultiplied by the pressure p is a function of temperature only(24125) (~7, = constant x

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expT-1/3). According to this theoretical prediction, the relaxation time for vibration was shown(l@ to be of the order of lo--’ set for oxygen in the temperature range of about 3000°K to 4000°K at a pressure of about 1 atm, which was smaller than the relaxation time 7D for dissociation (nearly 1O-6set for oxygen at the same conditions). The experimental or semi-experimental datac4 9B) also showed smaller values of TVthan T,, at the temperature range below 4000°K. Therefore, in the temperature range of about 2500°K to 4000°K at a pressure of about 1 atm encountered in the present problem of nonequilibrium expansion flows of dissociating oxygen around a corner, the following assumption will be made : (vii) The vibrational mode is always in thermal equilibrium with the translational and rotational modes, except along the frozen wave head and at the very corner, where the vibrational excitation as well as the degree of dissociation are frozen at their free-stream values. With the above assumptions, dissociating oxygen is termed a “partially excited dissociating gas “. (16) It should be noted that, in an ideal dissociating gas,(5v6) the vibrational mode is always assumed to be excited to half of its classical value (fully excited) in the temperature and pressure ranges noted above. At temperatures higher than about 4000”K, as pointed out by Treanor and Marrone,(27) the coupling effects of vibrational excitation with dissociation must be taken into account, although the vibrational relaxation time may be smaller than the relaxation time for dissociation. They showed that, because of the very effective removal of vibrational energy by dissociation, the vibrational temperature was less than the translational and rotational temperature during the entire dissociation relaxation time even when the vibrational excitation had already reached equilibrium. Therefore, at such a high temperature, each quantum state of vibrational excitation must be considered, from which the dissociation of molecules takes place. For an ionizing monatomic gas such as argon, the following additional assumptions will be made regarding the internal mode of electronic excitation and the electron translational temperature. (viii) The internal mode of electronic excitation is always assumed to be at its ground state. This assumption can also be applied for a dissociating gas, because, in the temperature range where dissociation is important, electronic excitation can be neglected. For an ionizing gas, the neglection of higher energy levels of electronic excitation in the summation of the internal partition function may be reasonable in the temperature range where the degree of ionization will not be too large. For example, according to Pike,(i’) the error in Saha’s equation (see equation (2.20.b) or (2.21.b)) giving the equilibrium ionization

I.

170

I. GLASS AND A. TAKANO

degree is less than a few per cent for argon at a lower temperature ( < 15,0OO”K, see Fig. lb). The error in the energy equation (and also in the ionization energy) can be estimated to be less than one per cent below about 20,000”K.(9’*l) Furthermore, for oxygen the ground state energy of electronic excitation of molecules, co,+ is chosen as a reference energy level for a dissociating gas and for argon the similar energy cOAof atoms is used. As pointed 1.

0

.6 Eq. (2.26) for i lssociating oxygen .4

‘* ’(b)Ax

I

1

I

91 = 182, 650’K .6-

.6

I

pI = 5.6443

at p = 1

x lo7 atms.

atms.

Ideal Ionizing Argon, Equation (2.21.’ .4

Results obtained by. Pike (Ref. 11) with Higher Energy Levelsof Electronic Excita-

T(‘K)

FIG. 1. Degrees of dissociation and ionization as functions of temperature at p = 1 atm for oxygen and argon. (a) Oxygen; (b) Argon.

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out by Rudin and Ragent, (22)the accuracy limit imposed by the uncertainty of the value of the ground state energy level may be most important at lower temperatures but at higher temperatures, where dissociation and ionization become significant, the error may be less than one per cent. Therefore, at higher temperatures encountered in the present work, ?? OAland cOAare usually taken as zero in a dissociating gas and an ionizing gas, respectively. The electron translational temperature T, is in general different from the atom (or ion) temperature T and can be determined as a function of T and x.(9.10) In an equilibrium state T, = T. The difference between T and T, depends strongly on the ionization degree x and slightly on the pressure p or the density p. However, in order to simplify the problem the following assumption will be made in the present work : (ix) The electron translational temperature T, is equal to the atom (or ion) temperature T. This relation is valid in an equilibrium state but is approximately correct only for small deviations from equilibrium. According to Bray and Wilson,(‘O) it can be found”@ that the above assumption is a good approximation in the temperature range of about 7000°K to 13,000”K at pressures below about 1 atm, where the degree of ionization varies from about 0.05 to O-25. The above two assumptions can be applied for the present ionizing gas flows at a temperature of about 10,OOO”Kat a pressure of about 1 atm, where a moderate ionization degree of approximately O-05 to 0.25 can be expected. Therefore, with these assumptions, ionizing argon can be treated as an “ideal ionizing gas “. (g1lo) A comparison of the assumed gas models of dissociating oxygen and ionizing argon is shown in Table 1, including a list of equations to be used subsequently for each gas. As shown in Table 1 simplified thermodynamic equations can be obtained for an ideal dissociating gas as well as for an ideal ionizing gas. Also it can be found that, as discussed later, the equations for an ideal dissociating gas can be easily derived from the corresponding equations for a partially excited dissociating gas as a special case of the latter (for example, e, = $R2T and fl= i: for the former). Numerical values of physical constants, which are used in the present calculations, are listed in Table 2. 2.2. Thermodynamic Properties According to the assumptions given in the preceding section, the specific properties, i.e. the properties per unit mass of a partially excited dissociating gas and an ideal ionizing gas can be derived from the partition functions of each species. The detailed derivations of the equations for the thermodynamic properties were given in refs. 16 and 17. An analysis of an ideal dissociating gas was summarized in ref. 6. Therefore, detailed derivations of the equations will not be included in the present paper.

I. J. GLASS AND A. TAKANO

I

-

-

NONEQUILIBRIUM EXPANSION FLOWS

N

dsqlh t

II 2

,

&&

i

173

I. I. GLASS AND A. TAKANO

174

TABLE 2 PHYSICAL CONSTANTS

k = 1.380x lo-16

Boltzmann constant

= 8616x 10-S

h = = NA = mq e=

Planck constant Avogadro number Mass of electron

For a dissociating

6.624 X Wz7 4.135 x 10-15 6.023 X 102’ 9.1066 x lo-28

gas, the density p of the mixture P = PA~+PA

erg/“K

eV/“K erg set eV set per mole g

is given by

(2.1.a)

of molecules and atoms, respectively.

where p,& and PA are the densities

However, in the mixture of an ionizing monatomic gas the mass of an electron is negligibly small compared with the mass of an atom (see Tables 2 and 3) and then the mass of an ion can be considered approximately equal to the mass of an atom. Therefore, the density of an ionizing gas is given by P = PA+PA+

(2.1.b)

where PAand PA+are the densities of atoms and ions, respectively. The mass fraction of atoms (degree of dissociation) and the mass fraction of ions (degree of ionization) in each gas are defined by a

=

E! P

x

=

PA+ P

(2.2.a) (2.2.b)

respectively. By using (2.1) and (2.2), the number densities of each species are given as follows : for a dissociating gas nA _ G-alp 2

2m

(2.3.a)

for an ionizing gas nA

-

(l-x)p m

(2.3.b)

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where m is the mass of an atom and m = 2.656 x 1O-23g for oxygen m = 6.632 x 1O-23g for argon Since it is considered that one electron is produced out of one atom to yield one ion, the number density of electrons is equal to that of ions; that is (2.4) Therefore, the total number density of the mixture is given as follows: for a dissociating gas u+4p n = nA2+nA = ~. 2m

(2.5.a)

TABLE 3 CONSTANTSFOROXYGEN AND ARMIN

-

T

Argon

Oxygen

Electronic excitation

‘molecules cOAZ= 0 atoms COA = 5% /-

Statistical weight of the ground state energy level Mass of atoms per unit mole Mass of an atom I

‘moleculesgo,, = 3 atoms gOA = g ,---

m, = mNA = 16 g/mole m = 2.656 x lo-23g

-0 atoms EOA ions c‘,A+ = kti, ‘_

atoms goA = 1 (ions go,+ = 6 m, = mNA = 39*944g/mole m = 6.632 x lo-23g RI = i = 0.2081 x 107erg/g”K

Gas constant

R1 = &

Characteristic temperature

an = 59,390”K

6, = 182,850 “K

Characteristic density

pD = 151.25g/cm3

pI = 150.27g/cms

Characteristic pressure

pc = 2.3042 x 107atm

~ pI = 56443 x 107atm

Characteristic velocity

1/(R2SD) = 3.929 x lOscm/sec

1 , ~/(RI&) = 6.169x lOscm/sec /

Characteristic distance Recombination rate constant

= 0.2598 x lO’erg/g”K

-

L = 6.5626 x lo-‘zcm

L = 1.8421 x lo-rlcm

km = k;PK(T) ksn = 0.67 x 101~cm6/mole2sec

I. I. GLASS AND A. TAKANO

176 for an ionizing gas

I?. = l?~+Iz~++II,

(l+x-)P = -.

(2.5.b)

in

According to assumption (iv), the pressure is defined by P = C Pj

(2.6)

where pi is the partial pressure of the j-species of the mixture and given by pi = njkT

(2.7)

Here, by using assumption (ix) the electron temperature T, has been assumed to be equal to T. A substitution of (2.5) into (2.6) gives the equation of state p = nkT p = (1 + a) pR2 T

for a dissociating gas

(2.8.a)

p = (1 +x) pR1 T

for an ionizing gas

(2.8.b)

where R2 and RI are the gas constants referred to the diatomic gas and the monatomic gas, respectively, and defined by (see Table 3). R2 = &

= O-2598 lo7 erg/g “K for oxygen

RI = k = 0.2081 IO7erg/g “K for argon

(2.9.a) (2.9.b)

The specific entropy S, the specific internal energy e and the specific enthalpy h can be derived from the relevant definitions by using the partition functions of each species as follows : for a dissociating gas

A-log(l-e-3

e= i+$o~+~,q+(l-c~)--&

1

(2.10.a)

1

R2T

(2.11.a)

R2T

(2.12.a)

h = e+F

h =

;+;m+a%+(l--a)&

I

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for an ionizing gas

e =

i(l+x)+x$

h = e+”

h =

1

(2.11.b)

1

(2.12.b)

RrT

P

&+x)+x;

RIT

where T, is a reference temperature and S,, and S,, are reference entropies for each gas.(16s17)The characteristic densities pD and pI for dissociation and ionization, respectively, have been defined in refs. 16 and 17 and these numerical values for oxygen and argon are shown in Table 3. According to assumption (viii), for a dissociating gas, the reference energy level is assumed to be the ground state of electronic excitation for molecules and then Eo‘42= 0

(2.13.a)

For atoms, co,_,is equal to half of the dissociation energy and therefore the characteristic temperature for dissociation is introduced by the relation EOA= +k8,

(2.14.a)

Similarly, for an ionizing monatomic gas $,A = 0

(2.13.b)

EOA+ = k8,

(2.14.b)

where 8, is the characteristic temperature for ionization. The values of the ground state energies of each species, ?? oj, and the characteristic temperatures 8, and 9, are listed in Table 3. The statistical weights goi of the energies coi are also shown in Table 3. These values have been used in deriving eqns. (2.10), (2.11) and(2.12). The terms (1 -e-3 andz/(e”- 1) in(2.10.a) to (2.12.a) are the contributions from the vibrational excitation, which is assumed in equilibrium with the translational and rotational modes (see assumption (vii)). Particularly, the last term in the right-hand side of (2.11.a) or (2.12.a) is associated with the equilibrium vibrational energy of diatomic molecules given by e0 = &RzT

(2.15)

where z=- 0” T

(2.16)

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I. I.

GLASS AND A. TAKANO

and 9; is the characteristic temperature for vibration (aU= 2273°K for oxygen). As discussed in the subsequent subsection, for an ideal dissociating gas the vibrational energy is found to be given by e, = $R2T

(2.15’)

which is just half of the classical (fully excited) value of the vibrational energy. For comparison purposes, the vibrational energies given by (2.15) and (2.15’) for a partially excited dissociating gas and for an ideal dissociating gas are shown in Table 1. 2.3. Reaction Processes and Equilibrium Equation For dissociation of molecules only binary collisions are considered and therefore dissociation and recombination of a pure diatomic gas are described by the process ko A,+X

s

2A+X

(2.16.a)

km

where AZ, A and X are a diatomic molecule, an atom and a third body (assumed to be a molecule or an atom), respectively, and kD and kRD are the appropriate dissociation and recombination rate constants, respectively (see ref. 28 for a detailed discussion). However, ionization and recombination of a monatomic gas are much more complicated processes, because atoms can be ionized in many ways and ions and electrons can recombine by many processes. A detailed discussion of the mechanisms of ionization and recombination can be found in refs. 3, 9, 10 and 21. According to these studies, it was shown(t7) that, in a nonequilibrium flow behind an equilibrium free stream of an ionized gas, the process described by the reaction equation kr

A+e;

A++e+e

(2.16.b)

knr

is dominant, where A, A+ and e are an atom, an ion and an electron, respectively, and kI and kRI are the ionization and recombination rate constants, respectively. In the following calculations, the values of the recombination rate constants kRD and kRI are required. However, definitive values of these constants are not yet available. Byron(29) and Matthews(30) carried out interferometric studies on the relaxation zone behind a strong shock wave in pure oxygen and oxygen-argon mixtures. On the basis of Matthew’s experimental results, Bloom and Steiger t3r) derived an equation of the recombination rate constant kRD of dissociated oxygen, which was proportional to the inverse square of temperature. Alternative relationships and experimental data have been published by many authors(26l *CQ and a comparison of the various values of kRD was made by Glass and Kawada. (Is) However, owing to probable experi-

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179

mental errors in the values of kRD, a constant value of the recombination rate constant for dissociating oxygen is conveniently taken aso 916) kRD = 0.67 x lOi cm6/mole2 set

(2.17.a)

in the temperature range of about 2500°K to 4000°K encountered in the present calculations. Unlike the dissociating oxygen case, the recombination rate constant kRI for ionized argon cannot be considered to be constant but is dependent on temperature.‘15) By assuming that atoms were first raised to their first energy level of electronic excitation by collisions with electrons and all of the atoms excited in this manner were later ionized, Bray and Wilson(9y lo) derived an ionization and recombination rate equation (2.16.b). Based on their rate equation, Glass and Kawada (15)obtained the expression for the recombination rate constant kRI for ionized argon. However, the above assumption made by Bray and Wilson’9 9lo) is in apparent contradiction with assumption (viii) that the electronic excitation is always at its ground state. If electrons are excited to higher energy levels, the additional terms associated with the electronic excitation must be taken into account in the relevant thermodynamic equations. However, since it can be shown(11s21) that, at lower temperatures, the contribution from higher order states may be neglected by comparison with the ground state, only the latter has been taken into account in the present analysis. Nevertheless, in order to obtain an equation for the recombination rate constant, it is necessary to consider the first energy level of electronic excitation, in lieu of the absence of a theoretical analysis that will provide a rate equation for any energy level. Accordingly, the following equation of the recombination rate constant(15) is applied in the present analysis : kRI = k:, K(T) (2.17.b) where m2 k;p = 3.349 x 10i6--” P2r

K(T)

= 2.366 x 1015cm6/mole2 set, for argon

(2.18)

= (++2)exp(!!tZ$E)

(2.19)

Here, m, is the mass of atoms per unit mole (m, = mN, = 39.944 g/mole, for argon) and T,,, is the characteristic temperature of excitation of the first energy level (T,,, = 134,OOO”Kfor argon) (see Table 3). When the mixture is in chemical equilibrium, the Gibbs free energy has its minimum value for all possible changes in the composition of a system at a given temperature and at a given pressure. From this theorem, it can be shown that, in equilibrium, the chemical potentials of each particle on both

180

I. I.

GLASS AND

A. TAKAND

sides of (2.16.a) and (2.16.b) must be balanced. The chemical potentials of each species were already given in refs. 16 and 17 and therefore are not presented here. By using the relations for the chemical potentials, the following equilibrium equations can be obtained: for a partially excited dissociating

gas 2 __

%

=

l--a,

2:

g 1’2(l-e-‘)exp(-B,/T) 0 v

(2.20.a)

exp(- 4/T)

(2.20.b)

for an ideal ionizing gas

By rearranging the above equations with the aid of the equation of state, (2.8), and the expressions for pD and pr given in refs. 16 and 17, GL,and x, are now written in terms of temperature and pressure as follows : for a partially excited dissociating gas -l/2

a,

=

4T)&exp(~dT)+

I

1

where

(2.21.a)

(2.22.a) and for an ideal ionizing gas

1

-l/2

xe =

where

A(T)

s2

exp (a#)

+ 1

(2.21.b)

In equation (2.22.a) 9, is the characteristic temperature for rotation of diatomic molecules (8, = 2.08% for oxygen). Equations (2.21.a) and (2.21.b)t4) are usually used for obtaining the equilibrium mass concentrations cc, and x,. Particularly, the equilibrium equation (2.21.b) for an ideal ionizing gas is known as Saha’s equation. (32)The values of CY,and X, obtained from (2.21 .a) and (2.21.b) for oxygen and argon at a pressure of 1 atm are shown in Fig. 1. It should be noted that, in deriving the equilibrium equation (2.20.b) or (2.21.b) of an ideal ionizing gas, the internal mode of electronic excitation was assumed to be at its ground state and the electron temperature was considered to be equal to the atom and ion temperature (see assumptions (viii) and (ix)). As discussed in sub-section 2.1, the assumption that T, = T is correct in chemical equilibrium. c9,lo) By considering higher energy levels of electronic excitation, Pike(“) gave an estimate of the error incurred in Saha’s equation

NONEQUILIBRIUM

EXPANSION

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181

(2.20.b) or (2.21.b) if the partition function for this mode was replaced by the ground state. According to his results, the above error is about 3 N 4 per cent in the temperature range of 8000°K to 15,OOO”Kand becomes larger with increasing temperature. This result is illustrated in Fig. lb, where the values of x, obtained by using Pike’s correction values (the ratios of values with higher energy levels to values with the ground state energy) are also plotted. In the equilibrium equation (2.21 .a) for a partially excited dissociating gas, the factor 1T 1 --~ (2.23) 2iYrl-ee-’ is identical with the partition function associated with the rotational and vibrational modes. Therefore, this term indicates the contributions of these internal modes to the equilibrium mass concentration u,. Here, the vibrational excitation has been assumed to be in equilibrium with the translational and rotational (also equilibrium) modes (see assumptions (vi) and (vii)). The factor Q is the “symmetry factor”, which appears only for symmetric molecules such as O2 and N1?.(I**) It should be noted that Napolitano(@ derived an equilibrium equation similar to (2.21.a) together with (2.22.a) but without the symmetry factor 4. Hence, Napolitano’s equation is only valid for an asymmetric molecule and is shown (16)to underestimate the values of CL,if it is applied for a symmetric molecule. Here, it is worthwhile to discuss the assumption of an “ideal dissociating gas” proposed by Lighthill.f5) The factor 2p~ ‘5 “*(l-e-_3 0 ”

(2.24)

in the equilibrium equation (2.20.a) is a function of temperature. However, in the temperature range of about 3000°K to 5OOO”K,the values of (2.24) change very slightlyC5)(160 N 144 g/cm3 for oxygen, compared with pD = 151.25 g/cm3 for oxygen, see Table 3). Therefore, for practical engineering purposes, Lighthill assumed a constant value of the factor (2.24) and defined the characteristic density pDL for dissociation by

1’2(1-e-z) = 150g/cm3

(2.25)

With this assumption, a dissociating pure diatomic (symmetrical) gas is called an “ideal dissociating gas”. By using (2.25), equation (2.20.a) can be written as 2

__% l-a,

=

p~eXp(-&,/T)

(2.26)

182

I. I. GLASS AND A. TAKANO

which is called “Lighthill’s equilibrium equation” for an “ideal dissociating gas”. For comparison purposes, the equilibrium equations (2.20.a) (2.20.b) and (2.26) for each gas are listed in Table 1. A comparison of the values of Q, obtained from (2.20.a) and (2.26) for oxygen at a pressure of 1 atm is shown in Fig. la. It can be found that the assumption of an ideal dissociating gas is a good approximation for dissociating oxygen in the temperature range of about 3000°K to 5000°K. It should be noted here that, as seen from (2.25), Lighthill’s assumption of a constant value of pDL is valid only when the term (I- e-l) is equal or proportional to d(GJT). Consequently, for an ideal dissociating gas, the partition function for vibration (see (2.23)) turns out to be given by (2.27) where c is a constant and from equation (2.25), c = 2pD/pDLz 2. If the partition function (2.27) for vibrational excitation is employed instead of that for the equilibrium vibrational mode cf, = l/(1 -e-3, see (2.33)), the vibrational energy for an ideal dissociating gas becomes e, = $R2T

(2.28)

which is compared with (2.15) for a partially excited dissociating gas (Table 1). For a fully excited vibration, e, = R2T. t6) Therefore, the vibrational energy of an ideal dissociating gas is just half of its fully excited (classical) value and is usually termed “half-excited “. Furthermore, the thermodynamic equations for an ideal dissociating gas can be derived only by replacing the terms (associated with the vibrational mode) (1 -e-“), e, and /3 by l/cd(aJT), 4R2 T and 4, respectively, in the relevant equations for a partially excited dissociating gas (see (2.27) and (2.28) and also Table 1). Therefore, an ideal dissociating gas can be considered to be a particular case of a partially excited dissociating gas. 2.4. Specific Heats and Sound Speed It is known that the sound speeds play dominant roles in the analysis of reacting gas flows. Detailed derivations of the equations for the sound speeds and the specific heats of a reacting gas can be found in refs. 4, 12, 13,21 and 22. Particularly, for a partially excited dissociating gas and an ideal ionizing gas, the above equations were derived and presented in refs. 16 and 17. Therefore, in this section only a remark regarding the derivations and definitions of these quantities will be made. The relevant equations are listed in Table 1. The specific heats C, and C,, at constant volume and at constant pressure, respectively, are defined as the relevant derivatives of the specific internal energy e and the specific enthalpy h with respect to temperature T. Here, the degrees of dissociation and ionization, TVand X, are constant in a frozen flow,

NONEQUILIBRIUM

EXPANSION

FLOWS

183

whereas in an equilibrium flow they are functions of temperature and pressure (see (2.21.a) and (2.21.b)). In the case of a partially excited dissociating gas, a partially frozen flow can be considered mathematically, where the vibrational excitation is in equilibrium with the translational and rotational modes but the degree of dissociation is frozen. The equations for C, and C, in each case are listed in Table 1. The last terms in the equations for C,, and C, of an equilibrium flow arise from the derivatives of a (or x) with respect to temperature T and therefore vanish in a frozen flow as well as in a partially frozen flow (only for a partially excited dissociating gas). The factor /3 for a partially excited dissociating gas arises from the derivative of the equilibrium vibrational energy e, with respect to temperature T and therefore also vanishes in a frozen dissociating gas flow. The sound speed and the isentropic index of a chemical reacting gas mixture can be obtained by following the general method developed by Chu(12) and Wood and Kirkwood.(13) For the present case the sound speed a is in general given by a = d[r( 1 + a) R2 T]

for a dissociating gas

(2.29.a)

a = d/[r(l +x) R, T]

for an ionizing gas

(2.29.b)

where r is the isentropic index and the expressions for various flows of each gas are listed in Table 1. The values of C, and C, in these relations must be obtained from the relevant equations in Table 1 for each case. The factors 2/[(1 +x)(2-cr)] and 2/[(1 +x)(2-x)] in y* arise from the derivatives of cc and x with respect to pressure p and therefore become 1 in a frozen flow as well as in a partially frozen flow. (16*17)It should be noted that the isentropic index r’ in a frozen flow of an ideal ionizing gas is the same as the specific heat ratio of a monatomic perfect gas. For comparison purposes, the equations of the isentropic indices and the specific heats for an ideal dissociating gast6* 16) are also shown in Table 1. It should be noticed here that, in this case, the frozen flow cannot exist, because the vibrational energy is always assumed to be half-excited (see subsection 2.3). However, the term “frozen” was incorrectly used in the analysis of an ideal dissociating gas.‘6$7l8, It can be found(t@ (see Table 1) that the so-called “frozen sound speed” in an ideal dissociating gas is a special case (jI = 4) of the partially frozen sound speed of a partially excited dissociating gas. The significant roles of the sound speeds in each flow will be discussed in Section 4 based on characteristic theory of a nonequilibrium flow of a reacting gas mixture. It will be shown that, in an equilibrium flow only at infinity (mathematical), the equilibrium sound speed a, specifies the characteristic directions. In a nonequilibrium flow of a partially excited dissociating gas, the partially frozen sound speed a~ will be shown to specify the characteristic

I. 1. GLASS AND A. TAKANO

184

directions, except those along the wave head and at the very corner, where the frozen characteristics must be used. For the ionizing gas case, the frozen sound speed af always specifies the characteristic directions in the entire nonequilibrium and frozen flows. Similarly, for an ideal dissociating gas the half-excited characteristics of a partially excited dissociating gas (the so-called “frozen characteristics”) can be used throughout the entire flow.

3. BASIC

EQUATIONS

OF

MOTION

A steady, inviscid, two-dimensional flow will be considered. Particularly, for an ionizing gas, the mixture is assumed to be electrically neutral. With these assumptions, the overall continuity equation and the two momentum equations remain unchanged from those for a perfect gas flow. However, the energy equation must be rewritten in a different form from that for a perfect gas flow, because of the heat exchange due to chemical reaction. In a steady, adiabatic flow, where all streamlines originate from a uniform flow, the total enthalpy must be conserved. From this theorem, the relevant energy equation can be obtained. Y

Streamline Streamline

I 0

FIG. 2. Physical coordinate system.

In writing the basic equations of motion, the natural coordinates along a streamline (s) and normal to it (n) are employed (see Fig. 2) and the magnitude of the velocity vector, u, and the flow deflection angle 6’(6’= 0 in the free stream) are used (also see Fig. 2). However, in order to determine the coordinates of characteristic mesh points in the calculation of nonequilibrium expansion flows, the coordinates parallel to the free stream (x) and normal to it (y) with the origin at the corner are conveniently used (see Figs. 2 and 4). Furthermore, for convenience in the following numerical calculations, dimensionless variables are introduced cl6917) by using the characteristic quantities for dissociation and ionization as the nondimensionalizing parameters.

NONEQUILIBRIUM

EXPANSION

185

FLOWS

The following dimensionless quantities p’, T’, p’, h’, d, s’ and n’are defined: for a dissociating gas P = P*P’* h = R2QDh’,

T = 8,T’, d(R2 6~)

U =

P =

PDP’

I

24’

(3.1.a)

for an ionizing gas P = PIPI, h = R18,h’,

T = 8,T’,

P = PIP!

l/(R1 6,)d

U =

1

(3.1.b)

and for the physical coordinates s = Ls’,

x = Lx’,

n = Ln’,

y = Ly’

(3.2)

where 8, and pD are the characteristic temperature and density for dissociation, respectively, and 19,and pI are similar quantities for ionization. The parameters pD and pI are the characteristic pressure for dissociation and ionization, respectively, and in order to obtain the system of dimensionless equations, they are defined by (3.3.a) for a dissociating gas PD = &PD~D PI = RI

PI%

for an ionizing gas

(3.3.b)

Furthermore, in order to obtain a nondimensional form of the mass production rate equation, (3.20), the characteristic distance L is defined by L

=

mk&G8D) kRDPi

L = m%(R1W 7kR P:

for a dissociating gas

(3.4.a)

for an ionizing gas

(3.4.b)

where m, is the mass of atoms per unit mole (m, = 16 g/mole for oxygen and m, = 39.944 g/mole for argon) and k RD and k;P are given by (2.17.a) and (2.18) for oxygen and argon, respectively. The numerical values of all the characteristic quantities used above are listed in Table 3. By using (3.1) to (3.3), the equations of the overall mass conservation, momenta, and energy can be written, in the same form in each case, as follows: ae aP’ Ufa,‘ +pr~+ph’ =0 ~ ur g

ad

as U’2-+__

7

; l ap’ = 0 pl ad

(3.6)

i apt = ()

(3.7)

ad

plan’

ah’

i apf pl asI

ai

(3.5)

o (3.8)

186

I. I. GLASS AND A. TAKANO

and the equations of state, (2.8) become p’ = (1+a)p’7-’

for a dissociating gas

(3.9.a)

p’ = (l+.v)p’T’

for an ionizing gas

(3.9.b)

The eqn. (3.8) is a general form of the energy equation, where h’ is given by (2.12) together with (3.1). However, it may be convenient for practical purposes of calculation to obtain a particular form in terms of T’, p’ and ccor x for each case of equilibrium, frozen and nonequilibrium flows. In a nonequilibrium flow, the degree of dissociation or ionization can be considered as one of the independent quantities (function only of the physical coordinates). Therefore, by substituting (2.12) into (3.8), the energy equation for a nonequilibrium flow can be written, with the aid of (3.1) and (3.2), as follows : for a dissociating gas (3.10.a)

where (3.11.a)

I

and 8

z=z=

T

(3.12.a)

T'

B =

79 1 0.038272 _II_ 8,T’ = ___ T’

for oxygen

(3.13) (3.14)

and for an ionizing gas /$$+,ax_lar’

= 0

a,4 pf asI

(3.10.b)

where A = $(1+x)

(3.11.b) (3.12.b)

It can be found that the dimensionless energy equations for both cases have a similar form, except for the expressions of A and B. The fact that the same form of the energy equation can be applied in each case facilitates the subsequent numerical calculations of nonequilibrium flows, particularly, on a IBM-7090 computer.

NONEQUILIBRIUM

EXPANSION

FLOWS

187

In a frozen flow CLand x are constant and the vibrational energy of a dissociating gas is also frozen (j3 = 0). Therefore, the energy equations (3.10.a) and (3.10.b) can be written as follows : for a dissociating frozen flow ‘-+-_ 7 3 c1 ?T’_.!Y = 0 ad plasI 2 2

(

1

(3.13.a)

for an ionizing frozen flow

;(l+x)g-$$ = 0

(3.13.b)

Since p’ is given by the state equation (3.9) as a function of p’, T’ and ccor x, where a and x are constant, the above energy equations can be integrated to give the following isentropic relation (dimensional form) : (3.14) where pr and T, are a reference pressure and a reference temperature, respectively, and I” is the frozen isentropic index given in Table 1 for each gas. The subscriptf denotes a quantity in a frozen flow. By using the equation of state, (3.9), the above equation gives (3.15) where pr is a reference density. Furthermore, integration of the s-momentum equation (3.6), together with the energy equation (3.13), gives

(3.16)

where Mf is the frozen Mach number defined by (3.17) and af is the frozen sound speed given in Table 1 for each gas and Mfr is a reference frozen Mach number corresponding to a reference velocity and a reference temperature. Equations (3.14), (3.15) and (3.16) are the same as the isentropic relations of a perfect gas only with the exception of the value of the isentropic index I”. Particularly, rf for an ionizing gas is the same as the specific heat ratio of a monatomic gas. A formal proof of isentropy for a

188

1. 1. GLASS

AND

A. TAKANO

frozen flow will be given subsequently. Consequently, with the aid of the remaining equations (3.5) and (3.7) which can be transformed to characteristic form, an analytical solution can be obtained in the same way as in the case of a perfect gas flow (see subsection 4.2). For a partially excited dissociating gas, a partially frozen flow can be considered mathematically, where the dissociation degree a remains constant but the vibrational excitation is in equilibrium with the translational and rotational modes (see subsection 2.4). In this case aa/%’ = 0 (also isentropic, see (3.26.a)) but /3#0. Therefore, the energy equation (3.10.a) can be written as (3.13.a’) The factor /3 is the contribution of the equilibrium vibrational energy and is a function of temperature. However, in the temperature range of 2500°K to 4000°K considered here, j? varies from about 0.94 to 0.97. Therefore, for practical purposes, /3 can be assumed to be constant @l= O*96).(*5y16) Consequently, the energy equation (3.13.a’) is also linear and then, in a manner similar to the case of a frozen flow, the analytical isentropic relations such as (3.14), (3.15) and (3.16) can be found with the exception only of the definition of the isentropic index (see Table 1). In an equilibrium flow, although it is also isentropic as shown later, the isentropic relations such as (3.14), (3.15) and (3.16) for a frozen flow cannot be found, because c( or x is a function of pressure and temperature and then the system of equations is no longer linear. The degrees of dissociation and ionization, c( and X, are given by (2.21.a) and (2.21.b), respectively, which are substituted into the relevant energy equations (3.10.a) and (3.10.b) to give the following equations : for a dissociating equilibrium flow (3.18.a) for an ionizing equilibrium flow C,aT; Fz-2

1

1

l+;(l-X,); eI

3

= 0

(3.18.b)

where C, is the specific heat at constant pressure given in Table 1 for an equilibrium flow of each gas and B is defined by (3.12.a) and (3.12.b) for each gas. The subscript e denotes a quantity in an equilibrium flow. It can be seen that the system of equations (3.5), (3.6), (3.7) and (3.18) is not linear and therefore these equations must usually be solved numerically with the aid of the state equation (3.9) and the equilibrium equation (2.21) (see subsection 4.3).

NONEQUILIBRIUM

EXPANSION

FLOWS

189

In order to specify completely the problem of a nonequilibrium flow, it is necessary to have the continuity equation for each species, namely, the mass production rate equation. For the chemical reaction described by (2.16.a) or (2.16.b), the mass production rate of atoms or ions is given by the following equations : (15,16,17)

for dissociation

G4

-_

=

(3.19.a)

__.-.

dt

for ionization _dCox) = k pLx [x-(1 dt R’mi l-x,

-x).-x2]

(3.19.b)

where d/dt denotes differentiation with respect to time t in the coordinates following a particle path (Lagrangian coordinates) and a, and x, are the local equilibrium values of the dissociation and ionization degrees determined by (2.21.a) and (2.21.b), respectively, for the local temperature and pressure. By using the continuity equation (3.5) equations (3.19.a) and (3.19.b) give the following mass production rate equation along a streamline (dimensionless, in Eulerian streamline coordinates) : for a dissociating gas , aa u,=w’

(3.20.a)

where w’ = (I+a)p’2

[f$(l-a)-a2]

and for an ionizing gas , ax u G = w’

(3.20.b)

where w’ = K(T’) p’2 x

(3.21.b) (3.22)

and (3.23)

(see Table 3 and eqns. (2.17.b), (2.18) and (2.19)). For deriving the dimensionless form of (3.20), the relations (3.1), (3.2) and (3.4) have been used.

1. 1. GLASS

190

AND A. TAKANO

For convenience of the following calculations, the equilibrium equations (2.21.a) and (2.21.b) giving CL,and x, and the eqns. (2.29.a) and (2.29.b) for the sound speeds are written in dimensionless forms by using the relations (3.1) and (3.3). With the aid of the definition for pu given in ref. 16, the equilibrium equation (2.21 .a) for a dissociating gas becomes CL, =

”

p’e”T’ +I 1-‘j2

~‘.@(i:

e-z)

(3.24.a)

where = 0.097818

for oxygen

(3.25)

(see Table 3 and eqn. (2.16)). Similarly, eqn. (2.21.b) for an ionizing gas is written as Xc = [L$+l]l’z

(3.24.b)

From (2.29.a) and (2.29.b), the sound speed is in general given by a’ = d[T(l+ a’ = d[r(l

LX) r’] +x) T’]

for a dissociating gas

(3.25.a)

for an ionizing gas

(3.25.b)

where the expressions for r are shown in Table 1 for various flows of each gas and the equations for C, and C, can also be written in the relevant dimensionless forms. It should be noted that, in a reacting gas flow the entropy equation can also be used as the energy equation instead of (3.8) or (3.10).t7) However, a knowledge of entropy change is not required in the present calculations. Nevertheless, it may be worthwhile to consider the entropy equation in order to calculate the variation of entropy due to recombination occurring through a nonequilibrium expansion flow and in order to prove the isentropy of frozen and equilibrium flows. The specific entropy S is given by (2.10), which is now differentiated with respect to s to provide the entropy change along a streamline. Here, the derivative of p is eliminated by using the state equation (2.8). If the equilibrium equation (2.20) is substituted into the resulting relation, the following (dimensionless) entropy equation can be obtained : for a dissociating gas (3.26.a) for an ionizing gas (3.26.b) It was shown(16~17)that, in a nonequilibrium flow, as/as’ was always positive as required by the second law of thermodynamics. In an equilibrium flow

NONEQUILIBRIUM

EXPANSION

FLOWS

191

CL= a, and x=x,, so that as/as’ = 0. When a flow is frozen or partially frozen, o! and x are constant, so that &S/as’= 0. Therefore, frozen, partially frozen and equilibrium flows are always isentropic. 4. CHARACTERISTIC

METHOD

OF SOLUTION

In order to solve the system of dimensionless equations (3.5), (3.6), (3.7), (3.10) and (3.20) for nonequilibrium flows of a dissociating gas (partially excited dissociating oxygen) or an ionizing gas (ideal ionizing argon), the method of characteristics can be applied. The method of characteristics for a reacting gas was developed by Chuo*) and Wood and Kirkwood.(13) They defined the sound speed in a reacting gas and found that a disturbance in a nonequilibrium flow propagated with the frozen sound speed along a frozen characteristic line specified by this sound speed. It was shown that an equilibrium flow could exist only at infinity (mathematical), where the equilibrium characteristics were used. However, it was pointed out that, although the flow quantities of the gas mixture could approach their local equilibrium values at infinity, the frozen sound speed never approached the equilibrium sound speed. A practical method of calculation based on characteristic theory was formulated by Napolitano(@ for a two-dimensional flow of an ideal dissociating gas and independent numerical computations of nonequilibrium expansion flows of such a gas around a corner were performed by Cleaver(7) and Appleton.@) In their calculation, the so-called “frozen sound speed” was always used for specifying the characteristic directions in a nonequilibrium flow. However, as pointed out in subsection 2.4, in an ideal dissociating gas a perfectly frozen sound speed cannot exist, because the vibrational mode is always assumed to be excited to half of its classical value. It can be shown”@ that the so-called “frozen sound speed” in an ideal dissociating gas is a particular case of the partially frozen sound speed in a partially excited dissociating gas (see Table 1). The detailed derivations of the characteristic equations for a partially excited dissociating gas and an ideal ionizing gas were given in ref. 15 and a practical method for calculating nonequilibrium expansion flows of these gases around a corner were detailed in refs. 16 and 17, respectively. It was found that, in a partially excited dissociating gas, the partially frozen sound speed specified the characteristic directions in a nonequilibrium expansion flow, except along the wave head and at the very corner, where the frozen characteristics could be used. It was also shown that, like the case of an ideal dissociating gas, only the frozen characteristics could be used in a nonequilibrium flow of an ideal ionizing gas. Nonequilibrium flows of a dissociating gas through a nozzle were calculated by Der, (i4) who also used the partially frozen sound speed for specifying the characteristic directions. In this section, the numerical-graphical method of characteristics developed

1. 1. GLASS

192

AND

A. TAKANO

in refs. 16 and 17 for calculating nonequilibrium expansion flows of a partially excited dissociating gas and an ideal ionizing gas will be summarized and some pertinent remarks will be made regarding the calculations of frozen, partially frozen (only for a dissociating gas) and equilibrium flows. It can be shown that the system of the present six equations has the following characteristic directions :

ad o as,=

(4.1)

ad = -~ ad '2/($2+

for a dissociating gas

(4.2.a)

ad - = ~-\/@$_ lj ad

for an ionizing gas

(4.2.b)

where Mf and A+!,are the partially frozen and frozen Mach numbers for a partially excited dissociating gas and an ideal ionizing gas, respectively, and defined by My

=

ul

for a dissociating gas

(4.3.a)

5

for an ionizing gas

(4.3.b)

af

Mf=

’

af

Here, a/ and af are the dimensionless partially frozen and frozen sound speeds for each gas (see Table 2). The first four characteristics (4.1) are identified with the streamline and the remaining two characteristics (4.2) can be shown to have the same properties as those in a two-dimensional potential flow, except for the definitions of the sound speeds and the isentropic indices. It can be found from (4.2) that, in a nonequilibrium flow of a partially excited dissociating gas, the partially frozen sound speed a/ specifies the characteristic directions, whereas in the case of an ideal ionizing gas the frozen sound speed must be used. However, in the former case the frozen sound speed ai must be used along the wave head and at the very corner, where the degree of dissociation as well as the vibrational excitation are frozen. By comparing the definitions of the sound speeds and the isentropic indices listed in Table 1, it can be found from (4.2.a) that, in an ideal dissociating gas, the characteristic directions are specified by the partially frozen sound speed of a particular form (see sub-section 2.4). It should be noted that this sound speed was incorrectly called a “frozen sound speed ”. c6,778, Furthermore, in the ideal dissociating gas case, since the vibrational mode is always assumed to be excited to half of its classical value, only partially frozen characteristics can exist in a nonequilibrium flow, including those along the wave head and at the very corner. However, in a real flow such a partially frozen state cannot exist along the wave head and at the very corner, because the vibrational mode

NONEQUILIBRIUhl

EXPANSION

193

FLOWS

as well as the degree of dissociation must be frozen there. Therefore, although the assumption of an ideal dissociating gas is usually a good approximation for practical purposes, the model of a partially excited dissociating gas is more realistic in the treatment of a nonequilibrium flow around a sharp corner. According to (4.2) two-families of characteristic lines 7 and t can be drawn at any point in the physical plane (see Fig. 3). Their directions are given by the following Mach angles :

tan/v= Q&2_*> 1

for a dissociating gas

for an ionizing gas

Streamline

X’

FIG. 3. Characteristic

directions.

The three equations. (3.6) (3.10) and (3.20), including the equation of state (3.9), are already expressed in characteristic form with respect to s’. The remaining two equations (3.5) and (3.7) can now be transformed into characteristic form with respect to .$and 7 as follows :

(4.5)

where for a dissociating gas

(4.6) 7*

194

1. I. GLASS

AND A. TAKANO

and for an ionizing gas M, and x must be used instead of Mf and a, respectively, in (4.6). In the expression for F, the quantities A, B and w’are defined by (3.1 I), (3.12) and (3.21) for each gas. Equations (4.5) indicate that pressure disturbances in a dissociated or ionized gas flow can be produced by deflection of the llow direction and mass production. The system of six characteristic equations (3.6), (3.9), (3.10), (3.20) and (4.5) is solved numerically for a nonequilibrium expansion flow around a corner (see subsection 4.4). 4.1. Frozen Flow In a frozen flow, a and x are constant and therefore from (3.20) w’= 0, so that F = 0. Furthermore, it can be easily shown later, in a frozen flow of a partially excited dissociating gas, the frozen Mach number Mf must also be used instead of the partially frozen Mach number My in (4.6) because the vibrational mode is also frozen. Owing to the absence of the mass production term F, equations (4.5) and (4.6) with constant isentropic index r,are similar to the characteristic equations for a two-dimensional potential flow and therefore a frozen flow is of the Prandtl-Meyer type. Consequently, (4.5) can be integrated to give f 8, = cos-i&-d(s)

tan-id($$:)

2/(My-

l)+const.

(4.7)

where I” is the frozen isentropic index given in Table 1 for each gas. A frozen Prandtl-Meyer expansion flow can be easily calculated by using the above characteristic equation (4.7) together with the isentropic relations (3.14), (3.15) and (3.16). Numerical computations for dissociating oxygen and ionizing argon were performed by Glass and Kawada(15) for a number of cases with various free-stream conditions and flow deflection angles. 4.2. Partially Frozen Flow In a partially excited dissociating gas, a partially frozen flow can be assumed mathematically, where the vibrational mode is assumed to be in equilibrium with the translational and rotational modes but the degree of dissociation is considered to be frozen. Therefore, like the case of a frozen flow, F = 0 in (4.5). Furthermore, it can be easily shown that, since CLis constant, a partially frozen flow is also isentropic (see (3.26.a)) and of the Prandtl-Meyer type (with the assumption that j3 = constant, see Section 3). As a consequence, for a partially frozen flow, the characteristic equations and the isentropic relations similar to those for a frozen flow can also be obtained, with the exception that the partially frozen Mach number Mj and the partially frozen isentropic index J” (assumed constant, see Section 3) must be used. Some partially frozen Prandtl-Meyer expansion flows of dissociating oxygen were also calculated by Glass and Kawada. (l 5,

NONEQUILIBRIUM

EXPANSION

FLOWS

195

4.3. Equilibrium Flow

In an equilibrium flow, a = a, and x = xe, so that from (3.21) w’= 0. Therefore, from (4.6) F= 0. Furthermore, it can be found that the equilibrium sound speed specifies the characteristic directions and therefore the equilibrium Mach number IV, = d/a: and the equilibrium isentropic index y* (a function of T’ and a, or x,, see Table 1) must be used in the expression G given by (4.6). Consequently, the characteristic equations for an equilibrium flow have the same form as those in a frozen and partially frozen flow and therefore an equilibrium flow is also of the Prandtl-Meyer type. However, a, or x, is no longer constant and is given by (3.24) as a function ofp’ and T'. Therefore, in an equilibrium flow the characteristic equations as well as the energy equation are non-linear (see (4.5) and (3.18)). The solution of those equations together with the remaining equations (3.6) and (3.9) must usually be obtained numerically. A practical method of numerical integration can be found in ref. 16 for a dissociating gas and is also applied for an ionizing gas as well. Numerical computations of equilibrium Prandtl-Meyer flows of dissociating oxygen and ionizing argon were performed on an IBM-7090 computer for several cases of free-stream conditions and flow deflection angles.(16p17J The same calculations were also performed by Glass and Kawada’15) by using a different numerical method. 4.4. Nonequilibrium Expansion Flow The characteristic equations (4.5) are similar in form to those in a rotational or axisymmetric supersonic flow of a perfect gas.(33v34)Therefore, for the present case of a nonequilibrium reacting gas flow, the graphical-numerical method of calculation developed by Ferri(33) and Cronvich’34) can be applied and computational procedures were detailed in ref. 16. The characteristic network is constructed step-by-step starting from the frozen wave head and at the corner (see Fig. 4). Along the frozen wave head the flow quantities have their free-stream values (initial values) and the frozen characteristic lines of &family are drawn from points taken on the frozen wave head with a finite interval dr>(see Fig. 4). At the corner, a frozen PrandtlMeyer expansion flow is calculated (see subsection 4.1) to provide the boundary values for the calculation of the nonequilibrium expansion flow originating from the corner. Along the wall surface the boundary condition is required that the flow deflection angle must be equal to the prescribed wall angle 0,. In the calculation of a frozen flow at the corner, by taking a finite frozen Mach number interval AM,-, the frozen Mach numbers M/ are prescribed together with the frozen free-stream Mach number i’k$i. Corresponding to MY prescribed in this manner, the flow deflection angle 8 and the temperature T’, the pressure p’ and the density p’can be calculated by using the characteristic equation (4.7) and the isentropic relations (3.14) (3.15) and (3.16), respectively (see Section 3 and subsection 4.1). The final frozen Mach number MfW

I. I. GLASS AND A. TAKANO

196

corresponding to the given angle O,Vcan be found by applying a Newton iteration. The other flow quantities such as the frozen sound speed a;, the velocity u’, the frozen characteristic direction 8, (see Fig. 3), etc., can be calculated by using the relevant equations given previously and then the frozen characteristic lines of the q-family can be drawn from the corner corresponding to the prescribed frozen Mach number (see Fig. 4). Consequently, by starting from points on the frozen wave head and from the corner, the characteristic network as shown in Fig. 4 is constructed step-by-step downstream. Behind the expansion wave tail, the characteristic

Frozen Wave H Free

flaw

Stream

at the corner

FIG. 4.

Characteristicmeshused in numericalcalculations.

lines of .$-family are reflected on the wall surface to give rise to the characteristic lines of v-family. The characteristic lines originating from the corner and the points on the wall surface are numbered for convenience as CO, Cl, c2, . ..) CW and CWI, CW2, . . .. respectively (see Fig. 4); here CO is identified as the frozen wave head and CW as the expansion wave tail. At each characteristic mesh point, the system of non-linear characteristic equations (3.6), (3.10), (3.20) and (4.5) together with the state equation (3.9) for a nonequilibrium expansion flow are solved simultaneously. These equations are reduced to a finite difference form and then, by using the known values of the flow quantities at PI and P2, the system of the finite difference equations is solved to give the values at a new point P3 (intersection of the characteristic lines drawn from PI and Pz, see Fig. 4). Here, it is necessary to use additional equations for obtaining such quantities as sound speed a’j or a;, local equilibrium dissociation degree cc, or ionization degree x,, mass production rate w’, etc. The above calculation is performed at first along the first

197

NONEQUILIBRIUM EXPANSION FLOWS

characteristic Cl starting from the mesh at the corner and then is continued to the next characteristic C2, and so on. It should be noted here that, as pointed out before, in the case of a partially excited dissociating gas the partially frozen characteristics must be used except along the frozen wave head and at the very corner, where the frozen flow quantities must be applied. Numerical calculations for dissociating oxygen and ionizing argon were run on an IBM-7090 computer for several free-stream conditions with typical wall angles shown in Table 4a and b. The digital computer programme that permitted the above calculations was described in ref. 18. TABLE 4 FREE-STREAM CONDITIONS

(a) Dissociating Oxygen Case

MS mmHg

Pi

PI atm T, “K p;xlOW

TX’ p;x 10-6

I u1

$1 Me, P/l = $1 PC1 = k?l

r, cm r: x 1012

~

Present 1

i

Cleaver 2a

Cleaver 2b

12 20 4.146 3720 0.20597 0062636 2.7411 O-85449 0.19964 2.5785 2.9100 22.82” 20+10” 1.159 0.1766

1 4250 0.04340 0.071562 033983 O-82597 0.78457 1.8176 2.1163 33.38” 28.20” 3.214 0.4990

0.1 3750 0+)04340 0.063143 0.037517 0.78994 0.83203 1.8201 2.2064 33.33” 26.95” 168.6 25.69

2.75 5.61 12.03

10.77 -

640.20 -

599.9 -

97.55 0.5 100

91.41 0.5 100

lxw cm

ew = -50 ew = -100 1 8, = -15~ Ii, x 10’2 ew = -50 ew = --loo ( 8, = --is

dr;x lOI r;x lOI

0.4186 0.8550 1.833 0.01 2.0

1.641 0.01 2.0

Appleton 3

0.21 3540 0*009114 0.059607 0.09964 0.74276 0.53463 1.9697 2.2933 30.52” 25.85” 62.48 9.520

-

-

Notes:

1. The free-stream conditions used by Cleave0 and Appleton(*) were recalculated for the present case of partially excited dissociating oxygen. 2. r; is the length of the frozen wave head taken as the initial line. Here, the number of the characteristic mesh along the frozen wave head, n,, was taken as 200 for every case. 3. The frozen Mach number interval AM, = 0.01 for every case.

198

1. I. GLASS AND

A. TAKANO

TABLE 4 (continued)

(b) Ideal Ionizing Argon Case

MS pi mmHg PI atm T, “K

p; x 10-7 T; p; x 10-6 Ul XI

Mfl NT1 Pfl

=

t&l

=

61 %I

r, cm r; x 10”

I,, cm ew = -5” ew = -10 8, = -150 i ew = -300 I;, x 10” ew = -5”

ew = -100 e, = -150 8, = -30

I

I

-

2

14 10 3.65 12,500 0.64687 0.068369 0.86705 0.62062 O-09123 1.7600 2.0849 34624 28.662” 0.995 0.5402

16 3 1.48 12,640 0.26304 OG69138 0.32908 0.73364 0.15612 2.0100 2.4021 29.836” 24.601” I.963 1.0656

2.489 4.707 9.388 76.348

6.947 12.768 24.336

1.351 2.555 5.097 41.45

-/

18 1.91 13,440 0.33820 0.073476 0.37586 0.84035 0.22463 2.1700 2.5936 27.441’ 22.679” 0.936 0.5083 3.721 6.823 12.926 -

6.93 1 -

Notes: 1. d&f, = @Ol for every case. 2. dr;= 0.2 x 1010, n/= 200, r;= 4 x 1011 except for Case 1, 8, = - 30”, where dr; = 0.2 x 109.

5. DISCUSSION

OF

THE

NUMERICAL

RESULTS

5.1. Summary of the Results The numerical computations of nonequilibrium expansion flows of dissociating oxygen and ionizing argon were performed on an IBM-7090 computer at the Institute of Computer Science, University of Toronto. The method of calculation was detailed in refs. 16 and 17 and outlined in Section 4. The numerical values of the physical constants and the characteristic quantities for oxygen and argon are listed in Tables 2 and 3, respectively. The free-stream conditions shown in Table 4a and b were chosen in accordance with the recommendations made by Glass and Kawada(15) as possible cases for interferometric experiments in the UTIAS 4 in. x 7 in. hypersonic shock tube. Particularly, the free-stream condition of Case 1 is generated by a moving

NONEQUILIBRIUM

EXPANSION

FL,OWS

199

normal shock wave at a Mach number of M, = 12, propagating into oxygen at a pressure of 20 mm Hg (see Table 4a). The degree of dissociation behind the shock wave is about 20 per cent at a temperature of about 3700°K. Also the free-stream condition of Case 1 for argon with a moderate shock Mach number of M, = 14 is an optimum case for practical experiments in the above shock tube, by comparison with Cases 2 and 3 (see Table 4b). Some experimental considerations will be given in the subsequent subsection (also see ref. 16). For the above representative cases for oxygen and argon, deflection angles

FIG. 5. Nonequilibrium

expansion flow of dissociated oxygen around a corner.

of 5”, lo” and 15” were chosen in order to provide sufficient data of nonequilibrium flow properties, particularly, along the wall surface. Computations were also performed for a wall angle of 30” for Case 1 of each gas in order to verify the existence of a de-excitation shock wave. For comparison purposes calculations were made by using the free-stream conditions of Cases 2a (0, = -So), 2b (0, = - 5’) and 3 (8, = - 100) for oxygen, and Cases 2 (0, = - 10’) and 3 (0, = - 10’) for argon. The above free-stream conditions for oxygen were employed by Cleaver (‘) and Appleton(*) in their numerical calculations of nonequilibrium expansion flows of ideal dissociating oxygen. The accuracy of the results obtained from the present numerical calculations is dependent on the characteristic mesh size employed, which is determined by the frozen Mach number interval dI$ at the corner and the distance dr;

I. I. GLASS AND A. TAKANO

200

of two neighbouring points on the frozen wave head (see subsection 4.4). As pointed out in ref. 16, the accuracy of the results can be improved by applying an iterative calculation(33, 31) at each characteristic mesh. However, in the present calculation, by using a characteristic mesh of small size, sufficient accuracy for practical purposes can be obtained. Based on the accuracy criteria given in ref. 16, the values of dMf (= 0.01 for every case) and dr> shown in

Partially

frozen

wave

he”*

7

wavehead

tbl

structure

a,

WDW

HeOds

FIG. 5. (continued)

NONEQUILIBRIUM

EXPANSION

201

FLOWS

Tables 4a and b were used for each case. It should be noted that the error introduced by a finite mesh size accumulates when the calculation proceeds downstream, particularly, behind the expansion wave tail close to the corner. This conclusion can be illustrated by the results of the calculation for Case 1 with 8, = - 30” (argon case, see Table 4b and Fig. 13b), where Ar> was taken one tenth as small as the other case.

x (C)

Variation

of

Pressure

Along

@I

Streamlines

(dl

Variation Alonq

of

Temperature

Variatton Along

of

Flow

DIrection

Streamliner

I 0

X (f)

Entropy

Chonqa

Along

Strromllncr

FIG. 5. (concluded)

The number of points, nf, on the frozen wave head, from which the characteristic network is constructed, determines the extent of a flow region to be calculated. The length of the frozen wave head is given by 5. A& which should be sufficiently large compared with the characteristic radial length r: (see (5.6) and also Table 4a and b), in order to obtain sufficient data covering a nearequilibrium flow far from the corner. The value of nf was usually taken as 200 to give a frozen wave head length of more than several times that of the characteristic radial length for each case (see Tables 4a and b). Summarizing the results of the present calculations, the overall flow patterns of nonequilibrium flows of dissociating oxygen and ionizing argon were

I. 1. GLASS

202

AND A. TAKANO

found to be qualitatively similar to that for ideal dissociating oxygen obtained by Cleaver c7) and Appleton. t8) Sketches of the predicted flow patterns are shown in Figs. 5 and 6 for the oxygen and argon cases, respectively. In these

Compression

Rsqion --’ onisenlmpic Locot Equittbrium Ftw (Constant Pressure and Flow Direction 1

(a)

Flow

Flow

Pattern

Quantity

ong

Streamline

Chorocteristic

Along

At

the

( Frozen (b)

Streamline

Corner P-M Expansion)

Structure

Line

of

Wove

b Ct

o

Distance Along Streamlines

Head

FIG.6 Nonequilibrium flow of ionizing argon around a corner,

NONEQUILIBRIUM

EXPANSION

FLOWS

203

figures, typical streamlines are drawn and some features of the variations of pressure p, temperature T, flow deflection angle 0 and entropy S along each streamline, including the wall surface, are shown (for details, see Figs. 16 and 17 and the relevant figures in refs. 16 and 17). The structure of the wave head is also illustrated in Figs. 5 and 6 for each gas, respectively.

-

0

Equilibrium Expansion of Infinity

P-M Flow Only

-Pressure Minimum Frozen P-M Exponsian Flow a1 the Corner

(c)

Variation Along

Expansion

of

----x Pressure

Streamlines

Wave

Toil

Cw

Frozen P-M Expansion Flow of the Corner

(d)

Variation Along

of

Temperature

Streamlines

FIG. 6. (continued)

1.I.GLASSAND

204

A. TAKANO

The flow at the very corner 0 is frozen and of the Prandtl-Meyer type (region 2). The values of the flow quantities in the frozen Prandtl-Meyer expansion flow can be calculated by using the characteristic equation (4.7) and the isentropic relations (3.14) (3.15) and (3.16) as described in subsection 4.1. The variations of the pressure p’ and the temperature T'for Case 1

Wove Head _/- Equlhbrlum Expansion at infinity

Wave

,,.’

0’ ‘,‘(e)

of

ot the Corner

Flow

Along

Entropy FIG.

Angle

Flow

Change 6.

Direcfion

Streomlines

rozen P-M Erporwon at the Corner

(f )

Tail

*X

Frozen P-M Expansion Flow

Variation

P-M Flow Only

Along

(concluded)

Streamlines

205

NONEQUILIBRIUM EXPANSION FLOWS

are plotted in Figs. 9 and 10 for oxygen and argon respectively, as functions of the flow deflection angle 8. The calculations of frozen flows of dissociating oxygen and ionizing argon were also made by Glass and Kawada(15) for a

2. 13”; Equilibrium

L&t.,=2.05O; Frozen 00 0

(a)

Partw.lly

Wave Frozen

Tail Wave Tail

Wave Tail I 1

2

3x

12

Characteristicdirectionsa,, +, 6, (only 6~varieswith r’along the C-lines)

FIG. 7. Variation of flow quantities along q-characteristics originating from the comer, as functions of radial distance r‘ from the comer; Case 1, 8,”= - 15”, oxygen. number of cases with various free-stream conditions and the results were shown in the relevant figures and tables in their report. The variations of the flow quantities along the characteristic lines Cl, c2 , . . ., CWoriginating from the corner are shown in Figs. 7 and 8 as functions of radial distance from the corner for oxygen and argon, respectively. It can

206

1.I.GLASS

AND A. TAKANO

be found that, owing to the presence of recombination along streamlines, the flow quantities along each characteristic line are not constant and these lines are curved (concave upstream). Therefore, a nonequilibrium expansion wave around a corner is not of the Prandtl-Meyer type and there are always two families of disturbance waves, expansive q-family and compressive t-family. In Figs. 9 and 10, the variations of the pressure p’ and the temperature T'at constant distances from the corner are plotted as functions of the flow deflec-

1

3 x 1012 I-’

(b)

Pressurep’

FIG. 7. (continued)

tion angle 8. In the case of a partially excited dissociating gas, the partially frozen characteristics are used in a nonequilibrium flow, except along the wave head and at the very corner, where the frozen sound speed specifies the characteristic directions. Consequently, as shown in Fig. 7a, in this case the slopes of the characteristic lines change discontinuously from frozen to partially frozen values at the very corner, whereas, in an ideal ionizing gas, they can vary continuously (always frozen values) as shown in Fig. 8a. Figure 9 also indicates the discontinuous change in the flow of a partially excited dissociating gas at the corner (r’ = 0) from a frozen flow to a partially frozen flow, whereas,

NONEQUILIBRIUM EXPANSION FLOWS

207

as shown in Fig. 10, in the case of an ideal ionizing gas a near frozen flow can exist in the region very close to the corner. Furthermore, in a partially excited dissociating gas, the partially frozen wave head can be defined mathematically behind the actual frozen wave head CO (see Figs. 5a and b). Across the frozen wave head, the flow quantities start to change from their free-stream values with finite values of the derivatives but the vibrational energy and the degree of dissociation remain constant. The .07

T’ 06

.05

.04 0

1

:

3 x 1012 l?’

Temperature T FIG. 7. (continued)

(c)

vibrational excitation suddenly attains its equilibrium with the translational and rotational modes at the partially frozen wave head, where the degree of dissociation remains constant (frozen at its free-stream value). As a consequence, the flow quantities also exhibit discontinuous variations across the partially frozen wave head (see Fig. 5b). As shown in Fig. 7a, the slopes of the characteristic lines just behind the frozen wave head (for example, Cl to C5) approach the partially frozen wave head angle Sfr with increasing distance from the corner (6fr = 21.933” for this case, compared with S,= 224319”). This result indicates that between the frozen and partially frozen wave heads there is no disturbance wave. However, if a continuous vibrational relaxation had

I. I.

208

GLASS AND

A. TAKANO

been taken into account, the above discontinuous variations along the wave heads as well as at the very corner would not have appeared; rather, a continuous variation of the flow quantity as illustrated in Figs. 5b and 16c would have been obtained. Furthermore, in a manner similar to the case of an ideal dissociating gas, t697I*) if only the partially frozen characteristics had been considered in the entire flow region (incorrect physically), such a discontinuous feature would also not have occurred. In summary, it can be stated that the

0

1

I!

3 x 1012 I.’

(d) Flow deflection angle 0 FIG.

7. (continued)

only wave head which would be observed (optically, for example) is the frozen wave head, since in an actual flow of dissociating oxygen along the frozen wave head the flow derivatives change discontinuously (see Fig. 5b). However, in the case of an ideal ionizing gas only the frozen sound speed specifies the characteristic directions in the entire flow region, including those along the wave head and at the very corner. Consequently, the discontinuous features behind the frozen wave head and at the very corner discussed above cannot exist in this case (see Figs. 8 and 10). This result is mathematically similar to that for an ideal dissociating gas,t7,*) where only the so-called

NONEQUILIBRIUM

EXPANSION

209

FLOWS

“frozen sound speed” (actually, partially frozen sound speed) always specifies the characteristic directions (see subsection 2.4 and Section 4). The above discussion on the assumptions for each gas model and the subsequent results will also be applied to the existence of a de-excitation shock wave, which will be considered later.

10 I I

,I

;;/ II

,/ 1

,’

I I ,’ I/ : i ,’

-

Ci

----_

de,

1’

(e) Dissociation

Local

Dissociation

Degree

Local Equilibrium Dissoclatmn Degree

degree a, cr,

FIG. 7. (concluded)

Disturbances originating from the corner propagate at first along the frozen wave head CO, across which the flow quantities start to change with finite values of their derivatives (at AI and II1 in Figs. 5a, 5b, 6a and 6b). As will be shown later in detail (see subsection 5.2), the angular derivatives of the flow quantities decay almost exponentially with increasing distance from the corner (see Figs. 1la and b). A decay distance of the wave head can be conveniently defined as the distance at which the gradients of the flow quantities have attained some prescribed non-zero values (for example, see (5.6) and (5.7)). Consequently, the frozen wave head has a finite decay distance.

1. 1. GLASS

210

AND A. TAKANO

The characteristic lines following the wave head also have increasingly larger decay distances (for example, OP1, 0P2, . . . in Fig. 6a), until the location of the equilibrium wave head is reached at infinity (mathematical), where the decay distance is also infinite. As seen from Figs. 7 and 8, the values of the flow quantities along several characteristic lines just behind the frozen wave

Frozen

C75 (CW),

Frozen

Wave

Wave Head,

Expansion

Wave

Tail

Tall

0 0

1

(a) Characteristic-line

2

3 x 1011

I

r’

slopes 6,, 8,

FIG. 8. Variation of flow quantities along v-characteristics originating from the comer, as functions of radial distance from the corner; Case 1, 0, = - 15”, for argon.

head CO approach their free-stream values with increasing distance from the corner. Therefore, for practical purposes, very far from the corner, the frozen wave head as well as the several characteristic lines following it can be considered to have vanished and the flow quantities can be supposed to begin to change when the equilibrium wave head is reached (see Figs. 5a, 5b, 6a and 6b). Along characteristic lines going downstream of the equilibrium wave head, the flow derivatives also decay but at infinity possess finite values, which are equal to those in the equilibrium Prandtl-Meyer expansion flow (region 4

NONEQUILIBRIUM

EXPANSION

211

FLOWS

in Figs. 5a and 6a). With the definition of a finite decay distance, for practical purposes, the equilibrium Prandtl-Meyer flow can be considered to exist very far from the corner but at a finite distance, Furthermore, as shown in Figs. 7e and 8e, the degree of dissociation or ionization approaches its local equilibrium

4

1, 6/

/

/ 5

/

4

/

3

/

1

2_ 0

ax,“--

1

I“

(b) Pressure p’a FIG.

8. (continued)

value and therefore the flow can be considered to be a near-equilibrium flow. Figures 9 and 10 also show the asymptotic approach of a nonequilibrium expansion flow to the equilibrium Prandtl-Meyer expansion flow very far from the corner. In the case of a partially excited dissociating gas, the intersection of the expansion wave tail C W and the following characteristic line CWl takes place for large wall angles to give rise to the formation of a recombination or

1. I. GLASS AND A. TAKANO

0. 5 c75

‘Pfw

(CW),

= 0. 04493,

Along

Frozen

Wave

Tall

(c) Temperature T’ FIG. 8. (continued)

(d) Flow deflection angle 0 FIG. 8. (continued)

NONEQUILIBRIUM

EXPANSION

213

FLOWS

de-excitation shock anticipated by Feldman(19) (see Fig. 5a and for details Fig. 12). As shown in Figs. 12a and b, for 0, = - 5”and - lo”, the above intersection is not found but, if a small size of characteristic mesh had been used in the calculation, it would have appeared, However, in the case of an ideal ionizing gas the intersection of characteristic lines belonging to the same family is not found to exist even up to large wall angles and even when using a small characteristic mesh size (see Fig. 13). As will be discussed in subsection 5.3, this result indicates the non-existence of a de-excitation shock wave in

I

2

(e) Ionization

4

x

10’1

degree x, x,

FIG. 8. (concluded)

an ideal ionizing gas, analogous to the case of an ideal dissociating gas.c7p8, It can be concluded that the existence of such a disturbance is a consequence of the mathematical model assumed in a partially excited dissociating gas and, physically, is attributed to the sudden heat addition due to the instantaneous change in the vibrational energy from a frozen to an equilibrium value at the very corner. Therefore, it is more correct to use the term “de-excitation shock wave” for this disturbance rather than the term “ recombination shock wave”. Following the expansion wave tail CW, a compression region 5 exists (see Fig. 5a and 6a), because the compressive pulses (&family) in region 3 are reflected at the wall surface as pulses of the other family (q-family) of the compression wave (C WI, CW2, . . .). These compressive characteristics

I. I.

214

GLASS AND A. TAKANO

(q-family) change into expansive characteristics (q-family) at a finite distance from the wall surface for each line. In Figs. 5c to f and 6c to f, the variations of the flow quantities along typical streamlines are shown for illustrative purposes. The variations of the pressure p’ and the flow deflection angle 0 are plotted in detail in Figs. 16 and 17 along the streamlines illustrated in

P’ I 10’ .1

“1 = 0.20597

x 10-6

.I

:quilhrlum 2.0

x 1012

rl

‘I

0.1x 1012

II = 0. 06264

-Partially

Frozen 06

.l

\

Partially

Fr

r’ = 0, Fn .a

?en_/y b)

7

05

Temperature

-.

\

, -5”

..

-10”

04

-15” 9

Fro. 9. Pressure and temperature in the corner expansion flow versus flow direction at constant distances from the comer; Case 1, oxygen. Figs. 14 and 15 for each gas, respectively. In the case of a partially excited dissociating gas, the derivatives of the flow quantities are found to change discontinuously across the expansion wave tail near the corner. Behind it, the pressure increases until its maximum value is attained (for example, at A; and B; along the streamlines 4 and b in Fig. 5c and also, for details, see Fig. 16a). Along these streamlines the flow expands to a larger angle than the prescribed wall angle at the expansion wave tail and then discontinuously increases up

NONEQUILIBRIUM EXPANSION FLOWS

215

to its maximum flow deflection angle (see Fig. 5a and, for details, Fig. 16b). Therefore, in the case of a partially excited dissociating gas, the compression region 5 is bounded by the expansion wave tail upstream and the locus of the pressure maxima downstream (see Fig. 5a). Furthermore, as seen in Fig. 5e, the temperature also changes discontinuously at the wave tail and, owing to the occurrence of recombination, it increases monotonically until it reaches its asymptotic value far downstream. I

P’

pl1 = 0.64667

x 1O-7

X 10-’ .6

Equilibrium Expansion at r’=co

Frozen

0

Prandtl-Meyer Flow

Prandtl:Meyer

Expansion

Flow

Expansion

Wave

-5”

at r’ = 0

Tail

for 0,

I -15“

_

- 1”’

-130

e

(a) Pressure versus flow deflection angle FIG. 10. Variations of flow quantities at constant distances from the comer; Case 1, argon.

GLASS AND A. TAKANO

1. I.

216

However, in the case of an ideal ionizing gas, as seen from Figs. 6a, c, d and e, the flow continues to expand behind the wave tail CW. The pressure and the temperature reach their minimum values at some distance from the corner along the streamlines and then increase (for example, the pressure has its minimum at A; along the streamline a in Fig. 6c; also, for details, see Fig. 17a). Particularly, the pressure has its maximum further downstream T’ (

-Equilibrium Expansion

Prandtl-Meyer Flow at r’ = m

.(

Frozen

Prandtl-Meyer at r’ = 0

Expansion

Flow

.(

Expansion for ew

??

Wave Tail -150

.( 0

-5”

-100

-150 0

(b) Temperature versus flow deflection angle FIG. 10. (concluded) (at A” in Fig. 6~). Therefore, in this case, unlike the oxygen case, the compression region 5 is detached from the expansion wave tail C W and is bounded by the locus of the pressure minima upstream and the locus of the pressure maxima downstream (see Fig. 6a). However, the temperature increases monotonically from its minimum value behind the expansion wave tail to its asymptotic value far downstream (see Fig. 6d). Furthermore, unlike the case of a partially excited dissociating gas, the flow deflection angle 0 exhibits a monotonic decrease behind the wave tail until it reaches the prescribed wall angle 0,,, (see Fig. 6d and, for details, Fig. 17b). The overexpansion of the

217

NONEQUILIBRIUM EXPANSION FLOWS

flow deflection angle discussed above for the case of a partially excited dissociating gas can give rise to the intersection of the expansion wave tail and the following characteristic line to form a de-excitation shock wave and is again attributed to the assumption associated with the vibrational excitation. (5. 8) and &$

(5.3.

a)

ho --C.ase2.b.

p1=0.1~tn:s.,r,=lGt;.6cm pl=O.21

atm5..

rc=62.5cm

5

0

”

1

(a)

1.0

2

Oxygen

h

I

I\

case

1,

p1=3.65

--*--

case

2,

p1 = 1.48 atms,

I‘= = 1.963

--x--

case

3.

p1= 1.91 atms,

rc =o. 936 cm.

(IJ) Argon

atms,

c

I

I

-a-

Eqs.

0

3

= ir

rc=O,

995

cln.

cm.

(5. 3) and (5.3.1,)

3

1

r;rc

FIG. 11. Decay of

angular derivatives of pressure along the frozen wave head for oxygen and argon.

It should be noted that the maxima of the pressure for every case are found to be a little larger than the equilibrium Prandtl-Meyer values, which exist behind the equilibrium Prandtl-Meyer expansion flow at infinity (region 8 in Figs. Sa and 6a) and would be attained infinitely downstream along streamlines. Downstream of region 5 and also behind the expansion wave tail far from the corner, there is an expansion region 7 (see Figs. 5a and 6a), where the 8

1. I.

218

GLASS AND A. TAKANO

pressure and the flow deflection angle decrease along each streamline. However, owing to the occurrence of recombination, the temperature increases also in this region (see Figs. 5d and 6d). Far downstream in region 6, the flow quantities approach asymptotically their ultimate values along each streamY’

CW:

Expansion

cw1:

First

.4. 956O

Wave Tail

50 Characteristic Wave) from the

IAne

cxnpression

Startmg

Point

on the

26)

T-

(a) 0,”= -4.956”

and

- 15”

(b) 0, = - 10” and - 30 FIG. 12. Intersection

of the characteristic lines of the same family (T-family or left-running family); Case 1, oxygen.

line. For every case, as seen in 5c, 5e, 6c and 6e (for details, see Figs. 16 and 17), only the ultimate values of pressure and flow deflection angle can be the same for all streamlines and equal to those behind the equilibrium Prandtl-Meyer expansion flow (region 8 only at infinity, see Figs. 5 and 6). The other asymptotic values are different for each streamline and differ from their equilibrium Prandtl-Meyer values (for example, see Figs. 5d and 6d). Furthermore, as

NONEQUILIBRIUM

EXPANSION

219

FLOWS

expected from Figs. 19f and 2Of, the degree of dissociation or ionization approaches its local equilibrium value in region 7. By using the IBM results of a and a, or x and x,, the entropy change along a streamline can be obtained numerically from (3.26) for each gas. The results are illustrated in Figs. 5f and 6f, from which it can be found that the entropy along a streamline is constant y’ Y 1011 (i)

Characteristic

L,ines

(ii)

Slopes

I .5

I 1.0

of Characteristic

Lines

A

x 1011 x’

(a) e,, = - 15”

FIG. 13. Nonintersection

of the characteristic argon.

lines of the same family; Case 1,

and decreases with distance of a streamline from the wall surface. Therefore, the flow far downstream (region 7) is in local equilibrium but non-isentropic (rotational). The isobars in the nonequilibrium expansion flow of oxygen and argon are shown in Figs. 18a and b, respectively, for Case 1 with 0, = - 15”. The overall patterns are similar in every case as well as those obtained for an ideal dissociating gas,(‘.*) with the exception that, for a partially excited dissociating gas, a significant compression region is found to exist just behind the wave tail CW near the corner,

I. 1. GLASS AND A. TAKANO

220

The above discussion regarding the variation of the flow quantities along streamlines behind the expansion wave tail CW can apply along the wall surface as well (see Figs. 5 and 6). The variations of the flow quantities are shown in Figs. 19 and 20 for case 1 of each gas. Here, the subscript w denotes the flow quantity along the wall surface and the differences of the wall values from their corner values (subscript wf, behind the frozen Prandtl-Meyer expansion flow at the corner, see subsection 4.1) are plotted for convenience.

(ii)

Slopes

Along

-15$----J

.1

of Characteristic

Lines

the Wall Surface

I .2..3

-

Results

with A rlf = 0. 2 x 109

---

Results

with Ar’,

I

I .4

I

.5

= 0. 2 x 108

I .G

I .7 x

1010

x’

(b) e, = - 30” Fm. 13. (concluded)

It can be seen that the flow quantities exhibit similar variations along the wall surface for each case, with the exception that, for the argon case, the density decreases monotonically from its corner value (see Fig. 2Oc), whereas it exhibits a complicated variation for the oxygen case (see Fig. 19c). Figures 19 and 20 also show the asymptotic approaches of the flow quantities to their local equilibrium values. For comparison purposes, calculations were performed by using the freestream conditions and wall angles (Cases 2a, 2b and 3 for oxygen) employed

NONEQUILIBRlUM

EXPANSION

FLOWS

.2

0

-_ 2

”

1.0 x 1012 x1

.4

.2

0

-.2 .5

0

1.0 x 1012 x’

-4

x

io’2

FIG. 14. Characteristic lines and streamlines; Case 1, oxygen.

221

222

1. I.

GLASS

AND A. I‘AKANO

0,

FIG. 15. +Zharacteristics

“*

-1””

and streamlines; Case 1, argon.

NONEQUILIBRIUM

EXPANSION

223

FLOWS

(c)8,=-15” FIG. 1.5.(concluded) P’ x IO’

=.20597x

.6 P’l T

1O-6

F

FP

-

\

3 -21

F:

Frozen

P

Partmlly

Wave

Head Wave

Frozen Head

a

t

-,-=.-‘_.-‘__6_

P woo=

.0759 (x’--r

1.0

x 10 co)

l.,

x llJ12 x’

FIG. 16 (a). Variation

of pressure along streamlines (Fig. 14) as a function of x’; Case 1, oxygen.

224

I. I. GLASS

AND

A.

TAKANO

by Cleaver c7) and Appleton (s) for their calculations of nonequilibrium expansion flows of ideal dissociating oxygen. It was shown in ref. 16 that their overall results for ideal dissociating oxygen were in good agreement with the present results for partially excited dissociating oxygen shown in Figs. 19a to f, inclusive. It should be noted here that, as discussed before, the Mach number for a partially excited dissociating gas changes discontinuously from FP

B

FP

(I-

-5 i’-

-11F_

__

-15 o_ .

0

FIG. 16 (b). Variation of flow direction along streamlines (Fig. 14) as a function of x’. Case 1, oxygen.

a frozen Mach number &frWto a partially frozen Mach number Mywat the corner (s; = 0, see Fig. 19e). As will be discussed in subsection 5.4, it can be shown that quasi-similarities exist for the above relaxation towards equilibrium along the wall surface for all free-stream conditions and wall angles, if appropriately normalized quantities are plotted as functions of a relevant dimensionless distance (see Figs. 21 and 22). These parametric relations are shown to be very useful for the experimental determination of the recombination rate constants and an

NONEQUILIBRIUM EXPANSION FLOWS

225

experimental consideration of this problem will be given in subsection 5.5, including the selection of the free-stream conditions as well as some remarks regarding the effects of the boundary layer and the catalycity of the wall surface.

free-stream value

o-

-

position

of partially

frozen

expected

actual variation model

streamline

head

variation

as drawn

using present (see Fig. 5b)

variation

of p’ along

partially

frozen

partially

7

wave

a streamline

wave

Kok:

The vibrational instantaneousl:y change abruptly

head

frozen characteristic lines (curved)

The values of the flow quantities constant between F and P (see -

flow

remain Fig. 5b).

mode attains equilibrium and the flow quantities from their free-stream

values.

The above figure illustrates how the c’wvcs in Figs. 16a and b were faired at the wave head. The curves shown in Figs. 5c to 5f do not show a step since the quantities even at the very corner are too small to be drawn as noticeable steps (for example, for Case 1 Ap’ = -0.00288 x 10-6, Ap’ = 0.00283 x 10-5, AT’ = -0.00023, -A9 = 0. 196 degrees are the values along a characteristic just behind the two wave heads at a distance r’ = 0. 1005 x 1011, almost at the corner).

FIG. 16~2 Variation of pressure along a streamline, oxygen.

5.2. Decay of the Frozen Wave Head As discussed before, disturbances originating from the corner propagate at first along the frozen wave head, across which the flow quantities start to change with finite values of their derivatives. Investigations, for example, by Appleton, (*) Gibson and Moore(35) and Glass and Kawada(i5) have shown that the magnitude of the finite jump of the flow derivatives across the wave head decayed almost exponentially with increasing distance from the corner. Particularly, Glass and Kawada(15) defined conveniently a decay distance at 8*

I. I.

226

GLASS AND

A. TAKANO

FIG. 17a. Variation of pressure p’ along streamlines shown in Fig. 15; Case 1, argon.

a

b

r

oi-rT-U

I

I -Expansion

_ /.--

ivave Tad for Bw = -5’

cw

FIG. 17b. Variation of flow deflection angle 0 along streamlines shown in Fig. 15; Case 1, argon.

NONEQUILIBRIUM

EXPANSION

FLOWS

227

which the flow derivatives had attained some prescribed non-zero values. Following Glass and Kawada, the decay law of the frozen wave head for a partially excited dissociating gas and an ideal ionizing gas was given in refs. 16 and 17. It should be noted that the assumptions made in the above references for simplifying the problem are equivalent to a linearization of the

FIG. 18.

Isobars in the nonequilibriumexpansion flow.

228

I. I. GLASS AND A. TAKANO

.J

1

”

i

_1

(a) Pressure p:

1. Y

(b) Temperature

T’

FIG. 19. Variations of flow quantities along the wall surface as functions of distance

S; from the comer;

Case. 1, oxygen.

NONEQUILIBRIUM

EXPANSION

FLOWS

229

problem given by Gibson and Moore. (35) In the present paper, the details of the derivations of the pertinent equations (see refs. 15 and 16) will not be included but only the results will be given, The angular derivative of pressure along the frozen wave head is given by

I[-841f=~ =

ap’

V

ap'

Air'

-_

exp(A, r’)- 1

(c) Density p:

(d) Velocity u: FIG. 19. (continued)

(5.1)

1. I. GLASS AND A. I’AKANO

230

where #I is the inclination angle of a characteristic line measured from the vertical line and [+‘/&$]l,=o is the value of the angular derivative of pressure at the corner (finite) given by

(5.2)

2

.5 0

0. 5

1.0

1.5

2.0

2.5

x

S’w

(e) Partially frozen Mach number Mt;y

(f) Mass concentration

of atoms (dissociation degree CC,,E,,,,)

FIG. 19. (concluded)

1012

NONEQUILIBRIUM

EXPANSION

2

FLOWS

3

231

4

x

1011

S’,

(b) Temperature ZYA

FIG. 20. Variations of flow quantities along the wall surface as a function of distance from the corner 8;; Case 1, argon.

I. I. GLASS AND A. TAKANO

___-___

I

J

4

x

I”..

5’w

(c)Density p:

2

(d) Velocity I& FIG. 20. (continued)

NONEQUILIBRIUM EXPANSION FLOWS

233

MI

__;;__~____*I

O,;-J” 1. 9

18 L;

1

Y

S’

w

(e) Frozen MachInumber &f,

(f) Ionization degree x,, x,, FIG. 20. (concluded)

I. I. GLASS AND A. TAKANO

234

Here, AI and A2 are functions of the flow quantities in the free stream defined by the following relations : for a dissociating gas 2 -1

I i

Al =

(5.3.a)

1

A2 = 2/(:;-

d

I-

ij

Pt

I

wf - Q, 1

__+-_-_o--

Case

2b.

case

3 , Ir,=-10”

Ow = -5’

0

-0

FIG. 21. Variations of flow quantities along the wall surface as functions of SW/fR,v, for oxygen.

NONEQUILIBRIUM

EXPANSION

235

FLOWS

for an ionizing gas (5.3.b)

where M, is the frozen Mach number in each case and r, in (5.3.a) is the 1

c

PW _ Pe. Pn

Pwf -

pe.P

atms. pwf53.286,

p,y=2.190,

pe, PM = 3.46? pe, pa = 2.462

1. ew=-150, 2a,e, = -50,

pwf=l. 414, Pwf=7.216,

pe,PM= 1.757 pe, p&f = 7.757

Case

Zb,e,

pwf z 7.256,

Pe,PM’7.769

Case

3, ew = 100,

pwf = 10. 757,

pe, PM = 12.370

h

case 1, ew= -5O, case1, I&/=-w,

m ---x---

Case Case

---+----v---

= -5’.

k

(b)

Pressure

p..,

1 Tw - Te. pii Twi

- T,.P

-a-

case

1,

8,=-j’,

-A--

Case

1,

8, =-loo,

--+--

Case

Zb, ew = -50,

case

Twf=2,

=

2b. e, -50, ew= -100,

914,

lRw lRw

= 640.2 = 599.9

cm cm

FIG. 21. (concluded)

frozen isentropic index of a partially excited dissociating gas (for an ideal ionizing gas I’,= 5/3). The function K(Y) in (5.3.b) is given by (3.22) and the functions A and B are defined by (3.11) and (3.12) and in a frozen flow for a dissociating gas

(5.4.a)

236

1. I. GLASSAND A. TAKANC) A =

;(I +x)

B

3

T’

= 2+p

1

B = “+-1. 2 T’

T’

for an ionizing gas

(5.4.b)

for a dissociating gas

(5.5.a)

for an ionizing gas

(5.5.b)

It can be verified that equations (5.1), (5.2) and (5.3.a) are the same as those obtained by Gibson and Moore f3~)based on linearized theory. Furthermore, in the case of an ideal dissociating gas, as pointed out in the previous sections,

0

0

:i

1.0

1.5

2. ” %hw

(a) Ionization degree x, FIG. 22. Variations of flow quantities along the wall surface as functions of SW/l,,, for argon. values of rj, A and B and the partially frozen Mach number Mf for this gas must be used in (5.3.a). The resulting equations can be shown to be in agreement with those obtained by Appleton.(8) As a measure of the decay distance of the frozen wave head, the characteristic radial length rL (dimensionless) or Y, (cm) is conveniently defined by 1 rc (5.6) rL = z = 21 From (5.1) for this radial distance (r’ = Y:) the angular derivative of pressure becomes 1 -aPI (5.7) a4 I[ a4 r,=O = e_l = o*5820 Therefore, the characteristic radial length ri along the frozen wave head is physically the distance from the corner at which the angular derivative of

the relevant

apl 1

237

NONEQUILKiRIUM EXPANSION FLOWS

pressure across the wave head is attenuated by the factor of O-5820. A larger value of the characteristic radial length indicates a larger distance at which the above attenuation takes place and a larger relaxation region along the p,

- pe. PM

pwf

pe. PM

1. o,,-------7--------

I

case --o-

-

1.

p1

3.65

/

atnas.

pwf=2.65 atrns., pwf=1.868 atrns,

9,=-P,

9,=-15°, case

=

pwi = 1.278

2,

p1 = 1.485

.5

atms.

pe,pM=2.&4atms.,

2.18

pe,

PM

pe

pM=l.697

=

lRw=2.4Ycm.

atms.

iRW

--4

atms.

lRw

=9.39

71 CTTI. cm.

atms.

p,f=0.i12atms.,

p,f=O.8ij

atms.,

(b) Pressure p,,, T,

- Te.PM

Twf - Te.~~

1. 0

-

--dc.

LIw=-50,

T,f

= 1”. 99ow.

T,:, PM = 12, 140°K,

1~~

Qw = - 10’.

T,f

=

9,560’K.

Te,p&q=11.755%,

li
e,=-150,

Twf =

8, 215’K.

T,Pbf

= 11,36O%,

iRw = 9 39 Cm.

case 5

2,

T1

= 2.49

rm. cm.

= 12.640’~

-9~

Bw = -10”.

T,f=

9.420’%,

T,~M

= 11. 960°K,

In&

= 12.77

-0m.m

Bw =-loo,

T,f’

8.835°K.

T,,Phf

i 12, 690°K,

&,

= 6: 82 cm:

.5

1. il

cm.

2 W’RW

(c) FIG.

Temperature T, 22. (concluded)

wave head. The values of rf and rc for each free-stream condition of each gas are listed in Tables 4a and b. It can be seen from these tables that the characteristic radial lengths rc have realistic values for shock-tube experiments, except for Cases 2a, 2b and 3 for oxygen (see subsection 5.5).

238

I. I.

GLASS AND

A. TAKANO

By substituting (5.6) into (5. l), the angular derivative of pressure is written as r’jr: aP ap r/r, a+ iij rzO = exp (r’/rA)- 1 = exp (r/r,.) - I

-A I

(5.8)

Consequently, the ratio of the angular derivative of pressure across the frozen wave head to its corner value is a function only of r/r,. The results are shown in Figs. 1la and 1lb for oxygen and argon, where the angular derivatives of pressure calculated by using the IBM data for all cases are also plotted. Figures 1la and b show that the result of the linearized theory given by (5.1) or (5.8) is in good agreement with the exact values obtained from the IBM results for each free-stream condition. It should be noted that the same relations as (5.8) can be obtained for the angular derivatives of the other flow quantities (temperature, density, velocity and flow deflection angle).“@ It can be concluded that equation (5.1) or (5.8) obtained from the linearized method can provide an adequate prediction of the decay of the frozen wave head. The characteristic radial length rr or r: defined by (5.6) is a good indicator for estimating the attenuation of the flow derivatives, i.e. the relaxation process along the frozen wave head. 5.3. Existence of a De-excitation Shock Wave In the calculation for a partially excited dissociating gas, as seen from Fig. 12, the intersection of the expansion wave tail C W and the characteristic line CWl following it was found to exist for large wall angles (0, = - 15”and - 30”).(16)Furthermore, as illustrated in Figs. 5c, d and e and Figs. 16a and b, the flow derivatives were found to change discontinuously (compressionwise) across the wave tail. The above results predicted the existence of a deexcitation shock wave, which was anticipated by Feldman.(lg) Furthermore, the distance of the above intersection decreased for larger wall angles (see Fig. 12) and for these cases the degree of the finite jump of the flow derivatives was more significant. The shock wave was shown”@ to increase in strength with increasing wall angles. However, it was also shown”@ that it was rather weak and decayed with increasing distance from the corner until it turned out into an expansion characteristic line (see Fig. 5a). Therefore, a recombination shock wave could be treated as a compression characteristic line in the numerical computations based on the method of characteristics. As discussed in ref. 16, the presence of the intersection of the characteristic lines belonging to the same family is dependent on the size of characteristic mesh employed in the numerical calculations. For example, if a small mesh size had been used in the calculations for Case 1 with 8, = - 5” and - 10” for oxygen, the intersection would have appeared for these small wall angles.

NONEQUILIBRIUM

EXPANSION

FLOWS

239

Therefore, in order to check the existence of the intersection of characteristic lines of the same family for the ionizing argon case, computations were also performed for Case 1 of this gas with 8, = - 15” and - 30” by using smaller sizes of characteristic mesh (Ar>= 0.2 x log and 0.2 x lo*, respectively, see Table 4b). Comparisons of the results with those obtained from the calculations with dr;= 0.2 x lOlo for 0, = - 15” and dri = 0.2 x log for 8,= -30” are shown in Figs. 13a and b, where the characteristic lines in the calculations with smaller size of characteristic mesh are denoted by a prime. Therefore, for example, as shown in Figs. 13a and b, C W’ corresponds to C W and C WlO’ to C Wl. Owing to the error introduced from the large size of the characteristic mesh, the slope of the characteristic line CWl is a little larger than that of the expansion wave tail CW in the region close to the corner. However, the calculations with smaller meshes show that the slope of the characteristic line CWl’ is always smaller than that of the wave tail CW’ (see Figs. 13a and b). Therefore, the characteristic line just behind the expansion wave tail C W always diverges from the wave tail and hence never intersects it. Furthermore, as mentioned in subsection 5.1, the flow can expand continuously behind the expansion wave tail (see Figs. 6 and 17), and, particularly, an over-expansion of the flow deflection angle beyond the prescribed wall angle does not take place (see Fig. 17b). Therefore, it can be concluded that, in the case of an ideal ionizing gas, the intersection of any characteristic line belonging to the same family does not appear even up to large wall angles (for example, O,,,= -30”) and even when using a small characteristic mesh size. This result indicates that a de-excitation shock wave can never exist in the present case. The existence of the intersection of the expansion wave tail and the following characteristic line to form a recombination shock wave can be considered as a mathematical consequence arising from the assumption of a partially excited dissociating gas. As described before, in this gas, the partially frozen sound speed specifies the characteristic directions, except at the very corner, where the frozen sound speed must be used instead. This discontinuous change in the characteristic direction gives rise to a discontinuous variation of a flow pattern at the corner from a frozen to a partially frozen flow (see Fig. 9) followed by an over-expansion of the flow deflection angle behind the expansion wave tail. Consequently, the intersection of the wave tail and the following characteristic line takes place to form a recombination shock wave. Physically, at the very corner the vibrational energy is considered to be frozen at its free-stream value but away from the corner it attains its equilibrium with the translational and rotational modes. Consequently, the abrupt heat addition due to the discontinuous relaxation of the vibrational excitation from a frozen to an equilibrium state would give rise to the formation of a de-excitation shock wave behind the expansion wave tail. Furthermore, it can be concluded that, if a continuous relaxation rate equation had been used for the addition

240

I. I.

GLASS AND A. TAKANO

of the vibrational energy, such a shock wave would not have appeared. However, it may still be possible that physically the rate of vibrational energy addition is rapid enough to produce such a disturbance and therefore it will be necessary to verify this conclusion experimentally. In the case of an ideal dissociating gas, only the so-called “frozen sound speed”, which was shown’16) to be a particular case of the partially frozen sound speed in a partially excited dissociating gas, always specifies the characteristic directions, including those at the very corner. As a consequence, the discontinuous feature mentioned above as well as the over-expansion of the flow deflection angle beyond the prescribed wall angle do not take place in an ideal dissociating gas. Therefore, no sign of the formation of a de-excitation shock wave was found by Cleaver(7) and Appleton(8) in their calculations of nonequilibrium expansion flow of ideal dissociating oxygen around a corner. Furthermore, it should be noted that, if only the partially frozen characteristics had been used in the entire flow region, including those at the very corner (physically incorrect, see subsection 5.1) a de-excitation shock wave would not have taken place as in the present case of a partially excited dissociating gas. Mathematically, the treatment of an ideal dissociating gas and an ideal ionizing gas is similar in that only one sound speed specifies the characteristic directions in the entire flow region. However, as noted above, in the case of an ideal dissociating gas this sound speed is a particular case of the partially frozen sound speed in a partially excited dissociating gas, whereas in the ideal ionizing gas case it is the frozen sound speed (with the condition that T, = T). As a consequence, in an ideal ionizing gas a discontinuous feature of the characteristic direction cannot take place at the corner and therefore, as discussed before, a de-excitation shock wave does not occur. However, it should be noted that, if the internal mode of electronic excitation had been taken into account, then the flow situation would have been analogous to that of a dissociating gas, where the vibrational mode had been considered, and a partially excited ionizing gas could have been defined on this basis. As a consequence, a de-excitation shock wave would probably also have occurred if it had been assumed that the latent heat of electronic excitation was instantly added to the flow as this mode attained instant equilibrium with the translational modes. Such a situation may arise at high temperatures, say above 15,OOO”K(see assumption viii, in Section 2). 5.4. Relaxation Along the Wall Surface The variations of the flow quantities along the wall surface for Case 1 with 8, = - 5”, - 10” and - 15” are shown in Figs. 19 and 20, where skis the dimensionless distance from the corner and the differences in the flow quantities (with subscript w) from their corner values (with subscript wf> just behind the frozen Prandtl-Meyer flow are plotted as functions of sh. Figures 19

NONEQUILIBRIUM

EXPANSION

FLOWS

241

and 20 show that the pressure&, the temperature Ti. and the local equilibrium dissociation degree CC,,or ionization degree x,, increase from their corner values, whereas the velocity &, the Mach number Mf;y or MfWand the degree of dissociation CL,,, or ionization x, decrease. However, in the case of an ideal ionizing gas, the density pk is found to decrease monotonically from its frozen value p&at the corner (see Fig. 2Oc), whereas it exhibits a rather complicated variation in a dissociating gas flow (see Fig. 19c). A similar variation of the density pk to that shown in Fig. 19c also was given by Appleton(*) for an ideal dissociating gas. The above results for a dissociating gas are due to the mathematical models of vibrational excitation, which is assumed to be half-excited (an ideal dissociating gas) and in thermal equilibrium (a partially excited dissociating gas). The heat added by the vibrational excitation assumed above contributes to the variations of the flow quantities through the energy equation and the mass production rate equation (including the equilibrium equation), particularly, in the region very close to the corner, where the recombination rate is significant. It will be necessary to verify the actual variation of density in a dissociating gas as well as in an ionizing gas by interferometric experiments, for example. Far downstream from the corner (s;-+ m) the flow quantities attain asymptotic values which are a little different from those behind the equilibrium Prandtl-Meyer expansion flow at infinity (with subscript e.PM). The overshoot of pressure pk and the velocity U: beyond their ultimate values are found in Figs. 19 and 20. From Figs. 19f and 20f the dissociation degree a, and the ionization degree x, can be found to approach their local equilibrium values a,, and x,, as the flow goes downstream. Therefore, the flow far downstream (region 7 in Figs. 5a and 6a) can be considered to be in local equilibrium (mathematically at infinity). The distance at which the flow can be supposed to reach near-equilibrium depends on the wall angle (approximately linearly proportional to the wall angle). As discussed in ref. 16, the pressure must be physically uniform in this region and the local equilibrium value of pressure far downstream from the corner must be equal to the equilibrium Prandtl-Meyer value P:.~~. However, owing to recombination occurring along each streamline the other flow quantities are not uniform and therefore their local equilibrium values are different from their equilibrium Prandtl-Meyer values. As described above, the flow along the wall surface is relaxed from its frozen state at the corner (s: = 0) to the ultimate local equilibrium state far downstream (sk-+w). The relaxation distance, at which the flow attains its nearequilibrium, as well as the difference in the ultimate values from the respective equilibrium Prandtl-Meyer values depends on the free-stream conditions and increases with larger wall angles (see Figs. 19 and 20). The relaxation process along the wall surface can be described rather well by using the linearized mass production rate equation.(ls* I6917)

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The relaxation distance ljRW, (dimensionless) or lR,,,(cm) along the wall surface can be introduced for convenience as follows:(16’ 17) for a dissociating gas

for an ionizing gas 1 - X,.PM ~~~. XefPM

(5.9.b)

(2 -XcPM)

where K(TL,pM) is given by (3.22) as a function of the equilibrium PrandtlMeyer temperature Ti,PM.The values of lRWand IA, for oxygen and argon for each case of the free-stream conditions and wall angles are listed in Table 4a and b. It can be found that, with the exception of Cases 2a, 2b and 3 for oxygen, the relaxation distances have values which are practical for observing the entire nonequilibrium flow (see subsection 5.5). Particularly, Case 1 with B,,,= - 15”for oxygen and argon can be considered to have the most desirable conditions for experimental purposes (Y,NN1.1 cm and lRWz 12 cm for oxygen and rc z 1.Ocm and ZRW z 10 cm for argon). As described in refs. 16 and 17, the dissociation or ionization degree, the pressure and the temperature along the wall surface can be summarized for all free-stream conditions and wall angles when the ratios %---CPM __-~~“I$ - %.PM

or

Pw-Pe.PM

Pwf,fPe.PM'

X,v-Xe.PM .Y,$ -xcPM Tw-

(5.10)

T~.PM

r,,- T~.PM

i j

are plotted as functions of s,/lRW.The results for each gas are shown in Figs. 21 and 22, respectively. It can be seen that the normalized flow quantities (5.10) vary in a similar manner for all cases and the local equilibrium states are attained at about s, = 0.8 x lRW.Particularly, as discussed before, the asymptotic values of T, and a, or x,, are found to be different for each case as well as from the equilibrium Prandtl-Meyer values Te,PMand CL,.~,+_, or X e.PM, respectively. Furthermore, the pressure pw slightly overshoots its ultimate value (p&PM) at 3, = 0’4 N O-7x l&,,. It can be concluded that the relaxation distance lRW(cm) or li, (dimensionless) is a good measure for predicting where a near-equilibrium state will be attained along the wall surface. In addition, if the normalized flow quantities (see (5.10)) are plotted as functions of s,/lRW, quasi-similarities of the flow quantities along the wall surface can be found to exist as shown in Figs. 21 and 22. However, it should be noted that, in the case of dissociating oxygen,

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the recombination rate constant kRD is assumed constant (see (2.17.a)). The normalized flow quantities are nearly independent of the value of k,, and the values of lRWand L are inversely proportional to kRD. Unlike this case the recombination rate constant kRI for ionizing argon cannot be considered to be constant but is given by (2.17.b) as a function of temperature. Throughout the calculations for argon, the value of kR, has been evaluated by introducing the function K(T) with a constant kX (see (2.18) and (2.19)). Therefore, the above remarks regarding the dependences of I,,,, and L as well as the flow quantities on the value of the recombination rate constant kRD for dissociating oxygen can also be applied for kX in the case of ionizing argon. Consequently, a change in the value of kRD or kk only stretches or shrinks the abcissae s,/lRWand s:, = s,jL in the relevant figures. The above linear dependence of the physical co-ordinates on the value of kRD or k; provides a useful means to determine the recombination rate constant. 5.5. Experimental Considerations

As discussed in subsections 5.2 and 5.4, the characteristic radial length r, along the frozen wave head and the relaxation distance lRWalong the wall surface are useful measures to predict the extent of a nonequilibrium expansion flow around a corner. These values for oxygen and argon are listed in Table 4a and b, respectively, for each free-stream condition and wall angle. In general, a higher free-stream pressure (or density) yields a smaller characteristic radial length rc and a smaller relaxation distance lRw, because the frequency of intermolecular collisions increases.(15) Such a free-stream condition can be obtained at higher shock Mach numbers and higher initial pressures in a shock tube. At the same time, if an interferometric study is to be made, then for higher initial pressures, a dimensionless fringe shift in number of fringes from the free stream to a new flow region also increases.(l@ Therefore, a free-stream condition with a higher pressure (in the same temperature range) is preferable for observing nonequilibrium expansion flows with sizable fringe shifts in an interferometric experiment. However, it should be noticed that a higher pressure will give small values of rc and lRwand therefore it is possible that the whole flow region observed in a shock tube experiment would be nearly in equilibrium, although a large fringe shift may be measured. Consequently, it is necessary to optimize the initial conditions in a given shock tube to give an adequate fringe shift and relaxation lengths. Comparing the values of r, listed in Table 4a and b, Case 1 for each gas can be found to provide preferred free-stream conditions for observing nonequilibrium flows. For example, rc = 1.16 cm for the oxygen case and rc = 1.00 cm for the argon case. Cases 2 and 3 for argon also have moderate values of rc (r, = I.96 cm and 0.94 cm, respectively, see Table 4b). Nevertheless, the freestream conditions in these cases may not be useful in practical experiments, owing to high shock Mach numbers (&I, = 16 and 18, respectively) and low

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AND A. TAKANO

channel pressure (pi = 3 mm Hg for both cases). The values of IRVdepend on the wall angles as well as on the free-stream conditions. It can be found that the case of 8, = - 15” for the free-stream conditions of Case I of each gas have the values of IRWwhich are practical for observing the entire nonequilibrium expansion flow, including the near-equilibrium flow far downstream from the corner. In addition, in the above cases, a shift of several fringes can be expected(i5) in the UTIAS hypersonic shock tube for each gas from the free stream to the flow behind the equilibrium Prandtl-Meyer expansion flow. Furthermore, in order to check the existence of a de-excitation shock wave and in order to obtain large fringe shifts and large pressure differences from the free stream to the flow considered, larger wall deflection angles are desirable (for example, 1B,,,l> 15”). From this point of view, Case 2a for oxygen with 0, = - 5” is not a useful case because the wall angle is too small, although the values of Y, and ZR, are appropriate. On the other hand, in Cases 2b and 3, sufficient data for observing a nonequilibrium expansion flow including a near-equilibrium flow cannot be expected in shock tube experiments, because not only is rc too large but also I,, is too long (estimated as 600 cm). One of the main objects of an experimental study of a nonequilibrium expansion flow is the determination of the recombination rate constant kRD or kRI for oxygen and argon, respectively. The value of kRD for oxygen was assumed to be constant, because, up to date, a definitive value of kRD has not been obtained experimentally (as discussed in Section 2). Based on the assumption of a constant value of kRD given by (2.17.a), a more relevant value of kRD can be found by comparing the experimental data of the flow quantities with the theoretical results obtained in the present calculations. One of the methods is a comparison of the angular derivatives of density along the frozen wave head and the other is a comparison of the flow quantities along the wall surface. In the former method the slopes of fringes along the frozen wave head must be measured in order to calculate the derivatives of density. In such a measurement, great accuracy cannot be expected. However, a measurement of the pressure or density distribution along the wall surface can provide a more useful method, because, in a direct pressure measurement by using a piezo-pressure gauge or in a measurement of fringe shift by using an interferometer, an accuracy of a few per cent can be expected. A practical method for the determination of a more relevant value of kRD can be found in ref. 16. However, in the case of ionizing argon, the recombination rate constant kR, cannot be considered to be constant but is given by (2.17.b) as a function of temperature, where K(T) given by (2.19) is a dimensionless function of temperature and kX is a constant defined by (2.18). For the reason discussed in subsection 5.4, in a similar manner as the case of dissociating oxygen, an appropriate value of kX can be determined by measuring the wall pressure or density. For the measured wall temperature, the function K(T) given by

NONEQUILIBRIUM EXPANSION FLOWS

245

(2.19) can be calculated. Consequently, by using (2.17.b), a relevant value of kRI can be obtained. It should be noted here that by using the above experimental method only an appropriate value of kX can be found but the function

K(T) cannot be verified. Therefore, additional studies will be required to verify the temperature-dependent relation of the recombination rate constant for ionizing argon. An experimental study of nonequilibrium expansion flows of dissociating oxygen and ionizing argon around a sharp corner will be made in the UTIAS 4 in. x 7 in. hypersonic shock tube. As discussed before, the flow conditions of Case 1 with B,,,= - 15” for each gas will be tested in the above shock tube. (M, = 12,~~ = 20 mm Hg for dissociating oxygen and M, = 14, pi = 10 mm Hg for ionizing argon, see Table 4a and b). These cases provide optimum conditions for observing nonequilibrium flows, including near-frozen flows close to the corner and near-equilibrium flows very far from the corner. An interferometer, piezo-pressure gauges and other means will be used in order to measure the recombination rate constant and other properties of the flow. Of particular interest is a check of the possible existence of a de-excitation shock wave by using optical means. For this purpose a wall angle of -30” may be preferable if necessary. Direct measurements of temperature using spectroscopic and other means would also be very valuable in an actual study. Undoubtedly, the boundary layer and the catalycity of the wall surface will affect the expansion flow around the corner. However, in view of the complex inviscid nonequilibrium external flow a theoretical estimate of these effects has not yet been made and therefore in order to predict the viscous effects appropriate corner models will be used in future experiments. An additional complexity that occurs in actual experiments in a shock tube arises from the fact that the hot flow between the attenuating shock wave and the accelerating contact front is not uniform. These effects arise from the ever-growing boundary layer on the shock tube walls and are aggravated with increasing shock Mach number and decreasing channel pressure. Broadly speaking, the initial pressure and temperature before the expansion corner will increase with time, since the arriving particles would have been processed by increasingly stronger shock waves, and this will affect the nonequilibrium expansion flow considered. The magnitude of the effects will have to be estimated in each experiment. 6.

CONCLUSIONS

A theoretical study has been made of steady, two-dimensional, inviscid, nonequilibrium expansion flows of dissociating oxygen and ionizing argon around a corner. For a dissociating gas, the vibrational excitation was assumed to be in equilibrium with the translational and rotational modes (a partially

246

I. I. GLASS

AND

A. TAKANO

excited dissociating gas), except at the very corner and along the frozen wave head, where the vibrational energy as well as the degree of dissociation were correctly considered to be frozen at their free-stream values. For an ionizing gas, it was assumed that the electronic excitation was always at its ground state and the electron translational temperature was equal to the atom and ion temperature (an idea1 ionizing gas). Based on the graphical-numerical method of characteristics, the flow quantities in the flow field were calculated on an IBM-7090 computer for several cases of free-stream conditions with typical wall angles (see Table 4a and b). Of special interest is Case 1 with a wall angle of 15”for each gas, which is an optimum case for observing nonequilibrium expansion flows in the UTIAS 4 in. x 7 in. hypersonic shock tube and will be investigated in the near future. For comparison purposes, computations were also performed for the other free-stream conditions. Particularly, the free-stream conditions of Cases 2a, 2b and 3 for oxygen were used by Cleaver”) and Appleton’8) for their calculations of nonequilibrium expansion flows of an ideal dissociating gas. With the assumptions considered, in a partially excited dissociating gas, the partially frozen sound speed was found to specify the characteristic directions in a nonequilibrium flow, except along the frozen wave head and at the very corner, where the frozen sound speed must be used instead. Analogous to the case of an idea1 dissociating gas, t69738, in an ideal ionizing gas only the frozen sound speed was shown to specify the characteristic directions in the entire nonequilibrium flow, including those along the frozen wave head and at the very corner. However, it was shown that the so-called “frozen sound speed” in an ideal dissociating gas was a special case of the partially frozen sound speed in a partially excited dissociating gas. The above differences in the assumed gas models were shown to give rise to significant differences in the flow properties. The overall flow pattern was found to be qualitatively similar for a partially excited dissociating gas and an ideal ionizing gas as well as for an ideal dissociating gas. However, in the case of a partially excited dissociating gas, the partially frozen wave head can be mathematically defined behind the actual frozen wave head. At the partially frozen wave head the vibrational mode abruptly attains its thermal equilibrium, whereas the degree of dissociation remains constant. Consequently, in this case a discontinuous change in the flow pattern was found behind the frozen wave head. In addition, the flow was also shown to change discontinuously at the very corner from a frozen to a partially frozen Prandtl-Meyer expansion flow. It was found that, owing to this discontinuous feature at the very corner, the flow expanded to a larger deflection angle than the prescribed wall angle at the expansion wave tail. However, in the case of an ideal ionizing gas, such a discontinuous feature of a flow pattern along the wave head and at the very corner as well as an over-expansion of the flow deflection angle at the wave tail were not found

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to take place. A flow pattern similar to this case was also found in an ideal dissociating gas.c7p8, For a partially excited dissociating gas, the intersection of the expansion wave tail and the following characteristic line was found to exist for large wall angles, owing to the over-expansion of the flow deflection angle at the expansion wave tail. The existence of such an intersection as well as discontinuous changes in the flow derivatives along the wave tail indicated the occurrence of a de-excitation shock wave. Physically, at the very corner the vibrational energy is frozen but away from the corner it attains its thermal equilibrium. Consequently, the abrupt heat addition due to the above discontinuous relaxation of the vibrational excitation would give rise to a de-excitation shock wave. However, in an ideal ionizing gas, it was verified that such a shock wave did not exist even up to large wall angles and even for a small characteristic mesh size. This conclusion is again analogous to the case of an ideal dissociating gas. (7*8)Therefore, it can be concluded that, if a continuous relaxation rate equation had been used for the addition of the vibrational energy, a discontinuous feature along the wave head and at the very corner as well as a shock wave would not have taken place in a dissociating gas. Furthermore, if only the partially frozen characteristics had been used in the calculation for a dissociating gas, a continuous flow pattern similar to those for an ideal dissociating gas and for an ideal ionizing gas would also have been obtained. On the other hand, for an ionizing gas, if the latent heat of electronic excitation had been assumed to be suddenly added to the flow as this mode attained instant equilibrium with the translational modes, then a de-excitation shock wave would have occurred. In any event, it will be necessary to check the possible existence of a shock wave experimentally for a dissociating gas and an ionizing gas. Similarities in the behaviour of the relaxation processes were found along the frozen wave head and along the wall surface, when the appropriately normalized flow quantities were plotted against a relevant dimensionless distance from the corner. It was found that the characteristic radial length Y, along the wave head and the relaxation distance I,, along the wall surface were good indicators for predicting the extent of the relaxation zone. It was also shown that, with the aid of the values of rc and lRw,optimum free-stream conditions and wall angles for observing entire nonequilibrium flows in shock-tube experiments can be chosen. Throughout the present calculations, the recombination rate constant for oxygen was assumed to be constant, whereas for argon it was considered to be a function of temperature. It was shown that the assumed values of the recombination rate constants can be verified experimentally and perhaps more realistic values can be obtained. However, in the case of ionizing argon, it will also be necessary to verify the temperature-dependent relation of the

248

1. I.

GLASS AND

A. TAKANO

recombination rate constant. The additional complexities arising in an actual flow as a result of the boundary layer, the wall catalycity and the nonuniformities in the flow quantities of the hot gas in shock-tube flows were not taken into account. ACKNOWLEDGEMENTS

We wish to thank Dr. G. N. Patterson for his encouragement and interest in the present work. The opportunity of a sabbatical leave for one of us (A. T.) at the Institute for Aerospace Studies offered by the University of Tokyo and the Institute for Aerospace Studies is gratefully acknowledged. The Institute of Computer Science, University of Toronto, was generous in providing us with the use of their IBM-7090 computer and we are pleased to express our appreciation. We wish to express our thanks to Mr. J. Galipeau for his assistance with the IBM computations. The work was supported by the U.S. Air Force Office of Scientific Research (AFOSR 36564) and the Canadian National Research Council and Defence Research Board. REFERENCES 1, 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

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Electric Research Laboratory, Report No. 63-RL-(3354C) (1963). 26. C. J. SCHEXNAYDER and J. S. EVANS,NASA TR R-108 (1961). 27. C. E. TREANORand P. V. MARRONE,Paper presented at the Symposium on Dynamics of Manned Lifting Planetary Entry, Philadelphia, Penn., 1962. 28. R. J. HEYMAN,The Martin Company, Research Report R-61-15 (1961). 29. S. R. BYRON,J. Chem. Phys. 30, No. 6, 1380 (1959). 30. D. L. MATTHEWS,Phys. Fluids, 2, No. 2, 170 (1959). 3 1. M. H. BLOOMand M. H. STEIGER,IAS Paper No. 60-26, presented at the 28th Annual Meeting, New York, 1960. 32. M. N. SAHA,Phil. Mug. 40,472, 808 (1920). 33. A. FERIU, NACA Report No. 841(1946). 34. L. L. CRONVICH,J. Aero. Sci. 15, No. 3, 155 (1948). 35. W. E. GI~.~oN and F. K. MOORE, Cornell Aeronautical Laboratory, Report No. HF1056-A-2 (1958).

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