I,r.,

Nonequilibriunl.Flow Bebind StrC:>Il.g l.::'.L.L"-I'v.a.

In A Dissociated J.J..L'IU..LJL~ • ';Jlc.:..c

JANUARY 2,1962

DOUGLAS AIRCRAFT COMPANY, INC.

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OOUGL/lS.~ ;;~"(}?/~

Nonequilibriurn Flow Behind Strong Shock WavesIn A Dissociated Ambient Gas

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Approved By:

J. W. HindesChief, Missiles Aero/Thermodynamics Section

JANUARY 2, 1962

DOUGLAS REPORT SM';38936

Prepared By:

G. R. IngerSpecialist - Gasdynamics Research

Missiles Aero/Thermodynamics Section

Prepared Under The Sponsorship ofThe Douglas Aircraft CompanyIndependent Research and DevelopmentProgram. Account No. 88030-025

l"

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.­.... MISSXLES AND SPACEDouglas Aircraft Company, Inc., Santa

SYSTEMS ENGXNEE&INGMonica Division, Santa Monica, California

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ABSTRACT

This report describes a theoretical study of the effects of free stream

dissociation on the nonequilibrium gas properties behind strong shock waves.

The analysis is based on a set of oblique shock relations which are gener­

alized to include an arbitrary degree of nonequilibrium vibration, dissoci­

ation or ionization ahead of and/or behind the shock front. The frozen and--~~..:,-

equilibrium post-shock gas properties in air versus the ambient atom mass, ...... -"__..... ...,.,,..• ..,....._,_........._~'.__n~~·..-'-<"....""'..,~-"'-"'~~.._,·~~~"'....,·._.....=-""...·ro.......-~.,....,.,..........-..",.",.~·"''"'_·_ ....."-.._,..__. _,..__...."'..........__~...,........-~."••

fraction and dissociation energy are presented at shock velocities rangi~g

from 15,000 to 30,000 ft/sec for normal and oblique hypersonic shock waves.

Some of the potential effects of predissociation on the intervening relax-=

ation process are also briefly examined. Comparison is made with the prop­

erties behind geometrically-similar shock waves in a perfect undissociated

ambient gas, independently of the preshock chemic~l history, for either the,

same shock velocity or the same total enthalpy. It is found that signifi-

cant changes in the post-shock density ratio, temperatUre,. dissociation and

ionization can occur if 10 percent or more of the total energy is tied up

in preshock dissociation. Sharp reductions in the usual differences between

the frozen and eqUilibrium properties associated with endothermic post-shock

relaxation are observed when 50 percent or more of the total enthalpy is in

preshock dissociation. Moreover, it is conjectured that a complete reversal

in the nature of the relaxation process to one which is exothermic (recom­

bination-dominated), and a virtual disappearance of nonequilibrium overshoot

phenomena due to the nitric oxide exchange reactions, may be possible for

shocks in very highly dissociated ambient air.

TABLE OF CONTENTS

Comparison With Shock Properties in a Perfect Ambient Gas • • 13

Density Ratio Form of the Shock Relations.

Oblique Shock Relations With a Dissociatedor Ionized Free Stream • • • • • •

Validit~ of the Hypersonic Assumptions

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• • 10

. . . 1

• • • 20

· . . . 17

· • 18

· • • • I')

• • 14

· • 15

• 16

· . . . 16

· . 17

• • • • 10

• • • • 6

· . . . 13

· . . • 5

. . . .

'.

. . .

. . .

. .

. .

. . .

Nomenclature

EqUilibrium Shock Properties

Enthalp~

Temperature •

Governing Equations • • • • •

Introduction

Pressure

Hypersonic Approximations • •

Constant Shock Velocity (CSV) • •

Constant Total Enthalpy (CTE) • • • • •

Constant Shock Mach Number (CSM)

Conservation Equations

Density Ratio •

Shock Angle-Flow Deflection Relation

Speed of Sound

Predissociation Effects on Shocked Pir

Frozen Shock Properties •

Simplified Shock Relations

Relaxation Behind the Shock Front

2.

2.1.2

Paragraph

2.1

1.

2.1.1

2.3·1

2·3·2

3.

3.1

3.1.1

3.1.2

3·1.3

3·2

3.2.1

3·2.2

3.2-3

3.2.4

303

2.1. 3

2.1.4

2.2

ii

"\I

TABLE OF CONTENTS (Cont.)

iii

44

3839

40

21

23

24

25

26

26

Page

. .

. • • • • 29

. . . . . . . . .

Minimum Normal Shock Velocity in Air for HypersonicFlow With a Dissociated Free Stream • • • • • .

Free Stream Dissociation Energy Parameter for Air -. . . 4-2

The Functions Kf:> and K., : Normal Shock • . . 43

The Functions kp and KH : Wedge at Incipient Detachment 43

The Functions Kp and Kif:0 - Wedge Shock 43Attached 30 . . .

A Short Review of Shocked Air Chemistry

LIST OF ILLUSTRATIONS

Enthalpy Parameter ~ • • • • • • • •

Thermally Significant Predissociation Effects

Predissociation Effects on Overshoot Behavior

Dissociation and Ionization

The Intervening Nonequilibrium Behavior

Shock Wave Angle Versus Density Ratio and Flow Deflection

Summary and Conclusion

Density Ratio •••

Temperature ••••

Shock Configuration and Terminology

Shock Geometry and Flow Deflection •

References

Frozen and Equilibrium Specific Heat Ratiosfor Dissociated Air • • • • • • • • • •

5.

L

7·

Figure

6A.

6B.

6c.

3·3·3

3-3·4

3.4

3.4.1

3.4.2

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2.

Paragraph

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LIST ·OF ILLUSTRATlONS (Cont.)

Predissociation Effects on Wedge Detachment 60

Predissociation Effect on Attached Shock Angle for a30° Half-Angle \-ledge • • • • 59

PredissociationEffect on Equilibrium Dissociation Level:30° - Wedge· .. .. .. .. .. .. .. .. .. .. .. .. .. .... .... .. .. • 56

r,~ f

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Page

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.. .. .. .. .. 50

Predissociation Effect on Frozen Density Ratio

Predissociation Effect on PI' /(>00V..2.. . .Predissociation Effect on Enthalpy Function KHf

Predissociation Effect on Frozen Temperature TF /Vao2.

Predissociation Effect on Equilibrium Temperature:o30 - Wedge .. .. .. .. .. .. • .. .. .. .. .. .. .. .. .. ..

Predissociation Effect on Equilibrium Dissociation Level:Normal Shock • • • • • • • • • • • • • • • • • • • 54

Predissociation Effect on Equilibrium Dissociation Level:Hedge at Incipient Detacbment • • • • • • • • • • . • • 55

Electron Mole Fraction--Predissociation Effect onEquilibrium Ionization Behind a Normal Shock 57

Predissociation Effect on Flow Deflection Angle Behinda Detached Shock Angle of 800

• • • • • • • • • • 58

Predissociation Effect on Equilibrium Temperature:Wedge at Incipient Detachment • • • . • • • •

Illustration of Comparisons Between Predissociatedand Perfect Ambient Gas Shock Waves • • •

Predissociation Effect on Equilibrium Density:\'ledge at Incipient Detacbment·. • • • • • • • • • • • • 49

Predissociation Effect on Equilibrium Density:30° - Wedge • • .. .. .. .. .. • .. .. .. .. .. .. •

Predissociation Effect on Equilibrium Temperature:Normal Shock • • • • • • • • • • • • • • • • • • • 51

Predissociation Effect on Equilibrium Density:Nor!ual Shock • • . . • • • • • • . • • • • • • • • 48

8.

9·

Figure

13A.

12.

10.

15B.

11.

15A.

17·

13B.

14A.

18.

14c.

15C.

13C.

19·

16.

14B.

iv L

NO!>1ENCLJ.l.TURE

Post-shock flow deflection angle

i-th specie mass fraction

v

Heat of formation of i-th specie

'" 2-roo Moo shock kinetic energy2. ~_ ' ambient thermal energy

2.~ooAE.' shock dynamic pressureambient static pressure

Total specific enthalpy of mixture

Undissociated mixture molecular weight

i-th specie molecular weight

Mixture frozen specific heat raUo (Equation l2B)

Total dissociation and ionization energy of mixture(Equation 6)

Shock density ratio (poolPs)- 2.h

Dissociation energy parameter---!. (Equation 15)VfIO"L

Dissociation energy parameter ho (Equation 27)hT

Specific mixture enthalpy

Specific thermal internal energy of i-th specie

Mixture specific heat ratio (Equation 12A)

Effective specific heat ratio (Equation 11)

Total atom mass fraction

l\fixture enthalpy coefficient (Equation 6)

i-th specie enthalpy coefficient (Equation 6)

Speed of sound

l'1 ' ;

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NObmNCLATURE (Cont.)

Free stre~ Mach number

Static pressure

i-th specie gas constant (Ro/Mt)

Molecular gas constant (Ro/ MM )

Universal gas constant

Mixture mass density

Wave angle

Absolute translational-rotational temperature

Velocity component normal to shock

Velocity component tangent to shock

Total :flow velocity (~ tJ.'2. + V~l)

Distance behind primary shock :front

Mixture compressibility :factor :; ~O(~ ~i~"1

SUBSCRIPrS

Denotes atom

Denotes :frozen state behind shock

i-th chemical specie

Denotes molecule

Post shock state

Denotes equilibrium state behind shock

Incipient shock det,achment on a wedge

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SUBSCRIPTS (Cont.)

Free stream conditions

Perfect ambient gas

Dissociated ambient gas

ABBREVIATIONS

Constant shock Mach number

Constant shock velocity

Constant total enthalpy

vii

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1. INTRODUCTION

It is usually assumed in computing dissociation and ionization effects on

the flow field surroupdlng a hypersonic body or following the propagating

shock in a shock tube that the free stream ahead of the shock is a perfect

gas. However, there are five practical problems in which a significantly

dissociated or ionized gas state ahead of a shock wave must be taken into

account as well as the real gas effects excited behind the shock envelope:

(1) Highly-nonequilibrium flow in the nozzle of a hypersonic testing

facility with a model in the test section immersed in a dissociated and/or

ionized gas stream (Ref. 1 - 5). (2) The calculation of gas properties

behind a shock wave reflected from the end of a shock tube (or entrance to

the nozzle on a shock tunnel) when the reflected shock propagates into the

dissociated or ionized gas created b~ the incident shock. (3) The effect

of dissociation or ionization excited in tront of an arc discharge-driven

shock wave as a result of the radiation emitted from the arc (Ref. 6, 7).(4) The use of catalytic probes to measure atom concentrations in high

velocity, dissociated gas stre~. (5) Analysis of the flow field and

wake properties associated with a hypersonic body passing through a disso­

ciated or ionized atmosphere produced by nuclear explosions or, at very

_high altitudes in the "chemosphere," by solar radiation (Ref. 8, 9).

To prOVide a general foundation for an analysis of these problems, this

report presents a theoretical study of the nonequilibrium flow behind strong

shock waves passing tbrough a dissociated and/or vibrationally-excited (but

unionized) ambient-gas ·lmich is in an arbitrary state of chemical or vibra­

tional nonequilibrium. The main objective of this investigation is to show

how "predissociation" modifies ·the post-Shock real gas behavior observed in

a perfect ambient gas at hypersoni~ velocities for preshock atom mass

2

fractions ranging from zero to unity. To this end, three aspects of the

problem are treated: (1) the derivation of oblique shock relations which

are applicable to an arbitrary degree of vibration, dissociation and ion­

ization ahead of and/or behind the primary shock front; (2) a detailed

numerical evaluation of the frozen and equilibrium shock properties in air

as a function of the degree of free stream dissociation and/or, vibrational­

excitation, for a representative set of hypersonic shock velocities, wave

angle and ambient density combinations; (3) a qualitative examination of

predissociation effects on the intervening nonequilibrium relaxation chem­

istry in shocked air.

Section 2 of this report presents the governing equations for the steady

adiabatic flow of an arbitrary reacting gas mixture across an oblique shock

front. These equations are employed to generalize the usual oblique shock

relations to account for an arbitrary degree of nonequilibrium Vibration,

dissociation and ionization ahead of and/or behind the shock. The hyper---~-'-,

sonic approximations are then introduced to obtain a single set of equa-___.:::::: ....w ~_~.._,....,..,.,....---=~'----

tions that are applicable to either detached or attached shock configura-

tions. For normal shock velocities below 25,000 ft/sec, these assumptions...............--".:.~--'"" ...._''''....-......~~-- ...._~-_.•*............"-.,.~....,...- .......-~-.-._ ..........,,.-''''~'''''...,.,..~.~,. ..........,...,.,..,.-,-

are invalid in a significantly dissociated ambient gas unless this disso--------,--_._.-~~-~=,.._--~~~-~~~ ..,""'~-_."-"",,~~.,--_.~.~-~"-ciation is in a highly-nonequilibrium state.

In Section 3, a detailed analysis of predissociated shock waves in air is

given, assuming hypersonic flow. Comparison is made with the behavior be­

hind geometrically-similar shocks in a perfect ambient gas, independently

of the upstream chemical history, for the same velocity and the same total

enthalpy. The frozen and equilibrium gas properties versus the free stream

atom mass fraction are presented ·for shock velocities ranging from 15,000 ­

30,000 ft/sec and ambient densities pertaining to altitudes of 200,000 and

250,000 feet. Three particular configurations are analyzed: (a) a normal

shock, (b) incipiently-detached flow over a wedge and (c) an attachedo

shock on a 30 half-angle wedge. The conditions under which predissociation

causes significant changes in the post-shock properties are demonstrated.

Finally, the foregoing results are applied, in conjunction with the pi-"esent

knOWledge about the nature of shocked air chemistry, to examine the potential

[

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OBLIQUE SHOCK RELATIONS I-lITH A DISSOCIATED OR IONIZED FREE STREAM

2.1 Governing Equations

2.1.1 Conservation Equations

(1)p..Voo SIN CT::::

effects of predissociation on the nonequilibrium relaxation process without

actually solving the detailed rate equations. In particular, the possibility

of predissociation-induced exothermic relaxation (which is suggested by the

aforementioned detailed calculations) is considered. Also, the probable

effects on the thermally-insignificant reactions and nitric oXide, electron

density and molecular ion band radiation nonequilibrium "overshoot"phenomena

are discussed briefly.

The following mass, momentum and energy conservation equations govern the

flow across an oblique shock wave:

Consider the steady two-dtmensional adiabatic flow of a reacting gas across

an oblique shock discontinuity (see Figure 1). The gas is assumed to be a

mixture of i. = 1, 2, ••• N chemical species which individually behave as

thermally-perfect gases at a common translational temperature T .

2.

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poe +

Voo cos V=

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=

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Ps +

plus the thermal e<d.uation of state

JJJJJJ <I

1

2.1.2 Shock Angle-Flow Deflection Relation

The following relationship between the wave angle cr, flow deflection ~~

and shock density ratio E: s :: f'oo/f's is obtained by substituting Equations

(1) and (2) into the geometrical relation Us = Ys TAN ((T - bs ):

[

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[

[

~[

( [

c:[

[

[

[

[

l[

L

(7B)

(6)

(7A)

caloric equation of state~

,

€os TAN IT ,=

::.SIN Q"'

and composition by the following

- ~~~: [e/o' -I- ~. rr ~:]= ~ RM TZ + he>h

or

the temperature

R·where the "compressibility factor" Z == 2: !XL R~ accounts for the decrease

in mixture molecular weight due to dissociation and ionization (z. 2:1) .Now the specific enthalpy of the mixture h can be expressed in terms of

1" -For dissociated, ~ioni~~_aiJr. Z and f3 are satisfactorily approximated,.., - (I - (X SO(.

by .:-__-=-~ and ~ -:'-\-l---t:_~ -~_~__~_L.!,_oc' where 7/2 ~ (3.., < 9/2 and

0( == L eXt is the total atom mass fracti;n::4TO"'~

where ~t = I ... R~~ is a nondimensional enthalpy coefficient for the i-thI.

- ~ 0(' RLcomponent, (3 :: '-' z.. R"" ~i. , and hJ> :: I:OCt h(.j. is the average heat of

formation (total dissociation and ionization energy) of the mixture. Ne~­

lectingelectronic exc~~"~~ion in the J:.~~~al energy states, .~ #0 = 5/2, and

~";---;;;ies-b~t~;~;~'-'7/2 (n;"vibrational excit;t~:;)"'~~d"~972~'(~~~;;iet;lY~x-"cite(rvlbra£Iona:r-·energyr·for··"·drat~nii~-~;l;-;;i~s.A plot of (3 versus Z

~","<-L~""_""""""-r<""'"'--'",--"""'~''''''''''''':''-'''''';''''""''''"_,,,,,,,"_'''-'''''''''~~""'~'''~'__~.·_·_",.c"",_;,,-,~ ,-~. _

for a dissociated air mixture, with ~M = 7/2, 4 (one-half the full equili"b-

rium excitation level of Vibration) and 9/2 is shown in Figure c. l It is

obvious that ~ always lies between 5/2 and 9/2, regardless of the vibra­

tion and/or dissociation rates, in the absence of electronic excitation and

ionization.

4 (L

5

(8B)

(8A)

(lOA)

(lOB)

fi. (etsvI9 , Ts )

"t'ei.

3i.(<<i.s' O(..js' Ps. Ts )

'rOC i.

:;

=

=

V. detSvuss dx

SIN rr

2.1.3 Relaxation Behind the Shock Front

where 't"e i. and 1:"« i. are pressure and tem~~~~~:p~.2.e.l~~i~~j1!!§§~'and Fi.-";"3 i. are functi~ns of composition, pressure and temperature which

-_.---~~....~..-..~ .....",,""""~I<'.-'~. ~_--,.;> ••,..~' ,-.."",.~ ......_---~""",,,-,- .....~_-~.,...----_ ..._._.--.__. -......."..~---""""_......""...""---,.,.""-..,......

Equations (1) - (6) must be supplemented by additional relations to deter­

mine f?>i. s and ()(Ls (i.e., ~ , Z. and he,). Now the sudden translational

temperature jump across a strong shock front excites a combined vibrational,

dissociative and/or ionization nonequilibrium relaxation process in the gas

flow downstream of the shock. Assuming one-dimensional flo'T, ~ i. S VI t!> and

O(LS are therefore governed by rate equations of the form

E~uation (7) is plotted in Figure 3 for values of E. s ranging from 1/4 to

1/20; also shown are the loci of the incipient detachment condition

d {T/d Ss ----. 00, given by

'1

-1 -,

:l:1--1

]

]

] ~

J ~

]-]

--1.. J

J,J

J]

JJ ~

J

vanish identically at thermodynamic equilibrium. The particular form taken

b·YEqu-;ti';;;;··'"(lOY~'··~f~o~'s~=;~'d~'p;~d;<"~'";-tI;--specific set of re actions

assumed for the medium unuer consideration (some typical reactions for

shocked air are briefly outlined in Section 3.4).

In the absence of external pressure gradients, the nonequilibrium behavior

is eVidently bounded by two limiting post-shock states at which the gas

properties can be found without the necessity of solving a set of differen­

tial equations. The first is the ';frozen" state immediately behind the

primary shock front (hereafter denoted by the subscript S : F), ~ :: 0 ,

where the gas is assumed to be in translational and rotational energy equi­

librium but has experienced no change in the vibrational energy or composi-

tion prevailing ahead of the shock. Here, (eLv'.)F :: (eL",.). ,

I3i.F : {3i.. ,0( L, :: 0'".... and ~F' = ~1lO ; the corresponding shock proper-

ties are easily computed from Equations (1) - (6), and the initial rates of

change of f3 i , 0( L ,T , etc., behind the shock follow from Equations (10).

The other limiting state is a condition of complete thermal and chemical

equilibrium (hereafter denoted by s : EQ) which is asymptotically approached

at a distance ')C:. »"t" VF Here, by definition, f LEQ :: ~ L.£.4 = 0 yield

the relations e(LE~ -= DCL,(T, p) and ~i.EQ ::. ~ i. (T) provided by classical

statistical thermodynamics (Ref. 10). A knowledge of any two thermodynamic

variables ,SUCh as rand p or hand p, is therefore sufficient to deter­

mine all of the equilibrium properties (Ref. 11 - 14).

2.1.4 Speed of Sound

The speed of sound in a dissociated or ionized gas, which is required to

define the free stream Mach number, can be written

[

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6

=

+

,~ 1..­~ ..; '11"- ¥

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pp

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LL

-Ll

(12A)

(12B)

(dh/dT),.

J(~T)/oTc(~T)/dT- I

-~

and '( is the frozen specific heat Estio~~t'l<""'~"'''''';''-'''''''''-~.P'''',"¥__.a«.~,.~~.,".,...-~""",~.~,~

where J -: I for s gas in complete equilibrium and J :: 0 otherwise.l

Here,

'0 is the mixture specific heat ratio, defined by~~'-~~--""""----->'';';;'''-'''''''~'''''''''''''',,,,,,,,~~,._'''''''''''',......=i'''''''''''''''w..--v.-_,_'''-''''''''''''_''''..,,.,..,_.,.,..~.... ,!,''-~

1\A plot of '! and 'tE.Q for dissociated air versus 0( .: Z - I , taken from

Reference 16, is given in Figure 4. It is seen that l always lies between

OM:: 9/7 and 'fA = 5/3 as long as the dissociBtion rate is finite. The1\

effective specific heat ratio 1 EQ is not bounded in this way, however,

A < <- I<O«Zsince '0EQ < 't£Q - r,.., - ¥ when - - •

2.2 Density Ratio Form of the Shock Relations

The following shock relations involving the density ratio €os can easily be

derived from Equations (1), (3), (4) and (5):

=

-1

-1 0

l]

-1

"1

-1

]

] ,~

"]

]

'-1..-'

JJJJJJ 4

J

"s . = ~40 RM TCIQ Z_ +V_ 2..

[C I - f../,)SIN2. (J + Hoo]~

(14)

[IA 2

~_R""Too Z.. + (co~oo [CI 2.) :z. H~ ]}:: - £s SI'" rr T2. {300

IThiS equation reflects the discontinuity in the definition of /sound speedin a reacting gas at the equilibrium limit, as discussed by Chu (Ref. 15).

7

and

where }{ :: 2:.ho /V.2.,. is the local mixture dissociation and ionization energy

expressed as a fraction of the free stream kinetic energy. An equation for

E:s is now obtained by combining Equations (5), (13) and (15):

[

F­

e[

[

~ [~i [.

.

b[!

[

[

[

[

[

[

(17)

(16)

~s V + ( Y_M_2. 5IN'-u-f ,]

~~s - I

~- {, +~s

=

1$~cro Zoo {T. +

V.." ~ I - ('s'-) SIN2. {T - (Ii, - H..JJ J=J" z" 2. R'" ~oo2.

(15)

~oo Zoo {I A a.

H.)l}:: Too ..... 100 Moo [(I E:.s1)SIN'4(f - (H., -~s Zs ~~-

-

Equations (13) - (17) are applicable to any real gas mixture flow with an

arbitrary degree of nonequilibrium vibrational-excitation, dissociation or

ionization ahead of and/or behind an oblique shock. They are a particularly

instructive form of the shock relations because the real gas effects are

easily perceived through the behavior reflected in the basic parameters ~ ,

J{ - -I HZ and • For example, since Zs' (f3s) and 5 - H_ obviously increase

with % for an endothermic (dissociation, ionization-dominated) relaxation

process, it follows from Equations (15) and (16) that large reductions in

both ~s and ~ can result . .According to Equations (13) apd (14),ps and

hs also increase, but to a much smaller degree. The predissociation,

or, by a quadratic solution,

i; \

L

8 l

-1 0

'-1

"1-1

']

]

] ~

J ~

]

]

]

J

ambient molecular vibrat~~~ionizationeft~cts_~ear explicitly in

_tl_le__f_r_e_e_s_t,::.e~_spe.cif~__h_e_a~J~_-<»_l., 8S a r=.~~~tiO:Of ~h;':~~~~~_-~_-_~~~"-"-­molecular weight (Zoo> I ) and, especially) _~~~~J2<~,rc:e,D.i~~<.QL<tJl~,Q.oc~<,,_ '_-.,..,_'-~~_""""~"""_",_""-~",,,""""""'_x-M_...,....,,,, ...,......:..~ ......-_.---.<.............--""'-~-""-"""'~""'''''''''''''''~' ..........,.,......_..~.",

kinetic energy contained in the heat of formation of the atoms, endLor ionsa~f~;-"Sh~k(H:'-'><» :-----'~~_'~_'__~ ~ ~__'__~_"_A_''' __'''''_~ _

. ~",_...-n"''''''''''''''~'''''_.-

For a multiccmponent reacting gas mixture, tl'~ composition-dependence of hD

c~n be expressed as a function of Z only in two special cases: (a) when

the gas is in thermodynamic equilibrium, end (b) if the gas is, or can be

approximated b~t a binar;}! mi~ture.. Now in the case of nonequilibrium dis­

sociated unionized air, where the dissociation energies of 02 and N2 differ

significantly (hfo =:: t hFH ) and the effect of nitric oxide on Z and His usually negligible, approximation (b) may be applied by assuming that 02_

dissociates complet~_~efo~e th':__~~~':,~~!-es __b:§~n _~c:_~.~~~e_.l Thisapproximation has been used in the numerical calculations for air described

in Sections 3.2 and 3.3 beCause it enables a plot of H oo versus OC_ for each

chosen V"", independently ot: the particular ambient gas reaction history, as

shown in Figure 5.2 An example involving either pure diatomic oxygen or

nitrogen is also shown in Figure 5 for comparison. It is observed that H(lOcontributes less than 20 percent to the enthalpy at shock velocities above

20,000 ft/sec in air unless there is a significant atomic nitrogen (as well

as oxygen) population ahead of the shock. However, when the free stream

dissociated nitrogen content is large ( 0(00 >> .23), ]-[00 can easily exceed

unity for VOfJ < 25,000 t:t/sec. 3

1According to Reference 3, this model can overestimate the local ,atomicnitrogen concentrat~on in anonequilibrium hypersonic nozzle flow since itdoes not take into account the catalytic effect of the nitric oxide ex­change reactions in the expansion.

3This does not imply Hs - Moo > 1; on the contrary, it follows from theenergy equation that this quantity is always less than unity and approachesunity only when the free stre~ kinetic energy is completely converted intopost -shock dissociation or ionization at "'1....-. co.

]

JJJJ "J

2 In this figure, «Ill =for DC .... > .23. a

.77, "'N =. ° for 0(00 i .23 and 0(0 =. 0, "'0 = .232.

r 9

2.3.1 Simplified Shock Relations

~. f-

[

F[

[

[

~ [. [

C[

[

[

[

[

LL

~ r 'l_

(18)

,

,

Lh) ....\n~"too-- > >/

~>es.:'-o-»/>. c ~-"'(7 > ') -f;; fi.~ £:

" ':J'

SIN2. f - K, (H$ - .H oo )

(2.f>s- I)S1N2.tJf of- Ka.(Hs - Hoe)

Vp</"- I

:t. (h'--h-A~)10

which predicts a value no greater than 1/4 when Hs ~ H... Since (1/4)2( <;, 1,

approximation (b) therefore implies that E.2. is negligible compared to unity--~~-,...-_.,~-~~".,.,-<~._-.._-._..,. ....._...,.- . J .'.__"~.,.,.~,_,~•••~~.• • ,,_,~_H<_'_""'_~'_"_" __ ' __'" • _.•-~._----_., ..~'-'-"---_._'~'~.;..._.,--..,-- .....-'.-._"-~--.'--_ .•_-_......-......~ -~.- - ,_ ••_~•._'-'_,~< •.. ,';_._._.~~_

if the post-shock relaxation is endothermic._..........- ."""'_.~,.,.-~----- ........_-~.-,......~<-, ...~-~'T..,.."-<.""~-'=''''-~.,-''~..__....." ..,.~.'"'~~''''"''.,.,."" ..,.''''''''".."''''''''''''''~........._..~"""''''''''''.-..~"'-''"-,-

The hypersonic approximations to E~uations (13) - (15), in conjunction with

E~uation (18) and the geometrical relations (8) and (9), may be shown to

yield a single set of shock relations for either detached or attached shock

configurations. In the former case, 0- is assumed to be given; in the latter,

0- is obtained from either (8A) or (9). Substituting the appropriate expres­

sion for (f into E~uation (18) and solving for €. s (neglecting €.t terms),

the results for all cases may be combined into the ~ollowingequation:

It shall be henceforth assumed that (a) the kinetic energy normal to the

shock is much gree:!~r than !fl~e the mal ene~l._9.!_,~h~~_~~E~",.,s_as(VCD~'/II'a.O:/2.RM~ooT.ZOo=iooMoo2.51t~7.cr/2.~oo=Al sIN2.(1" >:> I) and (b) the

normal momentum flux is much greater than the free stream static pressure- 'a. a -- ""- _.--_.~.. -(p.V. 51/11 U"Ip... =A,.,. 51 N2. (T' >:> I) , where A... :: 2.~oo A.. • The value of tC.,however, is not necessarily small with respect to unity. Since

5A.:S~1III ~,~, th,e f",ormer ~sumption is clearly the weaker of the two (Le.,'. !1>....'""T-'1;

will :t'ail at a higher Moo ). M a result of the letter assumption only,

E~uation (16) yields the following approximation for the hypersonic density

ratio:

2.3 Hypersonic Approximations

.....v'60' /- -Z f'1't 3'~ ];., t "'"

A =e

'~~:~":.....~,'~

- ~'t<

~\~

10

(/3). --h.:h ~ . -. I~s:z /~ I- eO" r:.. (/-~.r) ..£.pn 0-

~_ .S-~ Ji~'?c ~ cT:::' K

Js ~ (pp ~ ~U--€s)

/2' ~ 1_ /~ k; (/-/.r-/~)~Yre'v ;; . .

(~s -/) r K'i-(hj -//.,0)

= .~4-1 T*-£//l-~)-/;l4(~-~/:£.4-/ vL ~ (A::-~)

I

-z :zj; -:z -i--;; (.6 -~) .

~Yo> -/ ;I- (/~- /~ )

Jf- 1r ~-~

(iJ>s--) ~Pf =A{ITP/

where the angle q, and the appropriate constants K" I< 2. are defined in

Table 1 below. Now with AM SlNa.cr , Ae:SlN2.(T > > I and ~s" <. <. I , the cor­

responding hypersonic approximations for Ps, hs and T.s can be written

independently of £, by substituting Equation (19) and the appropriate wave

angle relation into Equations (13, (14) and (15) respectively. The results

Ts ;: Voo2. [K tt • S1 N 2. '" (Hs Hoo ) ].t R... ~sZ.s(22)

1\ 2- [KH • H..,) ]:.:z....Teo )(_ MM Sl"'~ ", (Hi -

2f>s Zs

11

(2313)

(23A)

(21)

(20)

@:4P... Voo SI~'2.~ • K p

,.,:::

=

/>$

and

are:

where

F, (ps)Sl... a.", T K3 (Ms - H oo )'Kp - F;l (~S).sIN~o/ K~(J{s - H...>+

fII

[,. (~,)SIN·.jI of- Ks(H, - H_l]KH -

FIf (f>S)SIN ~t I<.'(}{s - J{_)+-

-1

-1

--]

-1

--1

-1

']

] ~

]~

]

~]

'1--'

J,J

JJJJ .-J .~

2.3.2 Validity of the Hypersonic Assumptions

that in a perfect ambient gas at a &iven shock velocity as a result of the.....,.,..._......".~_O'·_.-...,._·"'<.:""""- ..·"""..-..._" ....,....~·..,'v.".....,~....,......'''"..-,<'...'-'''"''""..,...<,.~,....."...,•.J"'_"'- ..~~..........,..."..~" ...."",...,..,'-",_...-.._--' . ~=-":~'-"''''''~~-.."",~;.;.-.r-P''":"",,,,~,,,,,,-,,,,.-•.~,,,-,,,,.-,,,,.-<:,--~~~~....~_.~._.

FL[

[

.[

~ [

b[

[

[

[

[

[

[~ \

L

~

·f[

'fP.BLE 1

1<3' •.• "" are defined in Table 1., ...

IIJ K, Ka, Ka K.. K& K" F, F2. F3 F.,. N

DETACHED cr , 1 0 0 ~s -I 2.f>s -ISHOCKS, , , I I

'\ ,

WE.DGE. AT~s- I 2.~s 2.(~$-I)INCIPIE.HT 11/2. I 2- J/2. , 3 :t 2.~s-1 I

D~TACtfME-NT

ATiACHEDS$ -I 1.$s-1

- .... 2.~$-1 (- .,I -I -I 0 0 'f(~s-t~ 2. (35- ~ ~

SHOCI(

The parameters ~s, Zs and H, are the only unknowns appearing in Equations

(19) - (23). Therefore, by a simple adjustment of constants according to

Table 1, the change in the effect of real gas behavior with the shock con-

\! figuration is easily observed. Furthermore, the pressure and enthalpyl\ r.

\A~' /]'\ relations dif;:pl~ythepredissociationparameters explicitly; the post-shock

\)\ \~;~-;i'~~~~"~;;e~ts-~~7~-;ntainede_ntirelJin K to and K u ' Figures 6A, Band\S:., <\ ------ -----~....------~

- ;;;,~ \ \ C present the variation of K,. and K H with Z:;, and 11., - Hoo for the case

v\~ of a normal shock, an incipiently-detached wedge shock, and a 300 half-J D~

angle wedge shock, respectively. Two extreme values of {3,.,s are shown for

each Zs to illustrate that the effect of vibrational excitation is com­

paratively small. It is seen that both K p and K H are weakly influenced

by dissociation and/or ionization for detache<i shocks; however, K. H becomes

increasingly sensitive to z., and (especially)}{ s - }feo with decreasing

attached shock angles. We may'also observe that dK~/dz.s ,dK,../d(Hs-J{..,),

dKM/d Zs and dK H / d (H.. - H..)reverse sign with decreasing wave angle near

incipient-detachment (compare Figure 6B and C).

Before taking up the application of the foregoing equations, let us examine

the validity of tl.l~_..h'y_p-eJ;:.s.Qnj..g_. assumption AE, SIN2.{T >> 1 in a dissociated......._._, """'--:--""-...-.~._'---...-- -- - ......._-" .....,<;'c........,."_•."'-"""""<....-=--~~""'..,,.~· ._'-":...r'~---·"' .._~,~·.• A •••

ambient gas. Clearly, the shock I>1ach number may be considerably lower than~~.. .,."." ......."_o.=...-_"...A..-"'".-~~·-'"=:-I.~••.""""".,.,..,.."'~,,..,,.......,."-"" ......._.,.;,""""'..=.-_'_~"'...,"""'....~_..."'-•...-""'-~.....,..~__~_

12 L

1-'1 ~.

l]

'l

1]

]

] ~

]

J]

J]1

J

JJJ a

J

3·

decreased ambient molecular wc=igllt l increased specific heat and possible"'_~_'_';~k_:'~'_'-:;-_'-:'--"__'_"_'_""'.:_.__.__~~._~ . __..,..... ._'........~_....~.~~.__.w_.__~_"_"__.,................__..___".

higher ambient temperatures, depending on the upstream chemical history of_,_.__~_~'~__._~ ._~ ,_-,~."~._.•"~_"~'_'"" __ ~ __ ..,._•. _ ..••._.".~_..~~.-~_~ ._~._._•...,.,",_,, __,_~_~ """_" __'''''''_'~~''''~'_''~''_·_·_~".'4'~·_'_'._...,.,......",<. ......~._~.~ ....~......._,_.,_••_ ... .~__.. ,._.~"._._.__

the pre~is~Q,9J;Si1:;ed sboek, as well ~__th~,._,!.9.QI~lLY~.-*.Q.ci~y-,.!_ To illustrate, let~__ e'~'__""__"~~",,,,_,,__,,,,,,,, ,,,,,,,,,'''''_' '''--~- ~" -

us assume that the hypersonic assumptions are valid when Ae;SIN'2. lT ~ 10 ;

the corresponding minim\ml SQQck. velocity in air as a function of «00 for

various ambient teIl1perature~ (incluq,i:ng TCl!O~. for each DC. r is shown in

Figure 7. We observe that ne:;i.ther ,the combir -d effects of decreasing molec­

ular weight and ~qodue to predissociation 1101' the increase in p_ as a

result of ambient vibi'ation~ excitation cause any significant increase in

(Voo SIN (J" ),.. •• when"'~ (". ~TCtCo.. ~ . However, the free stream temperature his­

tory is extretllely important~ ." ~en·. theJ:'e is ).0 percent or more dissociation

in equilibrium. ahead. of the shock, the chosen criterion is never satisfied

for normal shock velocities below 25,000 ft/sec. Consequently, the hyper-".. -=--_.-sonic approximations can beinadmiss1ble for normal shock velocities at~hi~-;ppro;:U;ati~~;·ar~,· entirel~ v~i:i~~n-~~-;;;;;;~-~~;~t~>;;if>'~----_._-~-:.. ".,..".....,:-"-":,..---.""""~ ......~~....-.-._-_.."'~""'......."""...-._._-'-->........~""""" ..-...,.._"'.. " -~-"""""--"''''''''----'''''''''''''.''"'''''''''",""_.'->--:I><.""""",,.,...~........,.-,

less Too <<: T<IIl".t., • While the assumed criterion is perhaps conservative,

it is nevertheless c1.ea;r tbattb,e' toregoins hypersonic shock relations are

applicable for VIJOS,tI fT ';S, . 25,000 ft/sec in ~ dissociateq. ambient gas only

when this dissociati.:ol1 is 1n a highly nonequilibrium state.

PREDISSOCIATION EFFECTS ON SHOCKED AIR

The influence of free stre~ dissociation and/or molecular vibration on the

frozen and equilibrium shock properties behind strong hypersonic shock waves

in air, and some of the resulting implications concerning the intervening

nonequilibrium chemistry1 will now be analyzed. Since the effects of pre­

dissociation are best appreciated by a comparison with the properties behind

a similar shock in a perfect ambient gas, we shall first discuss several

types of comparisons which arise in practical problems.

3.1 Comparison With Shock Pro~rties in a Perfect Ambient Gas

There are three obvious co:p.di~ion$ under which a comparison between a shock

in a perfect ambient gas (here after denoted by the subscript ., ) and a-'" .;:. ~

geometrically-similar shock in a dissociated free stream (subsequently

I\

ttI,

13

3.1.1 Constant Shock Velocity (CSV)

i.e., the predissociated shock is accorded a greater total energy. There

is also generally a reduction in the Mach number 1

[

F[

[

[

4[~ [

C[

[

[

[

[

[

L.'0/ r

L~

(26)

(25)+,..,-

hToo = ~_R",TooZ_ +V..2.

+ "1»~ -Ve 2- (I A&;-I)

(24)

= + He + t;t

lSince we are assuming the preshock dissociation to be in a highly nonequi­librium state, the "frozen" Mach number is used (i.e., i- = 'flO ~ f,../~_-I).

indicated by oo~.) can be made: (a) equal shock velocities, (b) the same-total enthalpy and (c) equal shock Mach numbers.

the following basic property of' the comparison Voo = "., = Vt10 is evident, 2-

when both shocks are considered to be hypersonic ( A(. , Ae; > > '):"",., -2,

As schematically illustrated in Figure 8A, this condition would be used to

compare the floY field around end behind a hypervelocity body passing

through a dissociated or ionized atmosphere to that of a similar body pass­

ing through the normal atmosphere at the same flight speed. It also could

be employed as the basis of en evaluation of predissociation effects on the

shocked gas properties, relative to the assumption of a perfect ambient gas,

behind a shock propagating with a given velocity through a shock tube.

Noting that the total enthalpy may be written

14 L

3.1.2 Constant Total Enthalpy (CTE)

K.".tX.... ? ~"'~ ~ 7h, ~"'_ ='h'So a.

0 '.000 /.0411-

·2.5 .i72. .90'·50 .777 .79'1

.75 .70' .72.0

'.00 .'If'/ .b!fa

- }{r_'2.

This condition applies when one, compares the shock layer properties of a

body in hypervelocity free flight with those of a body placed in the non­

equilibrium dissociated flow of a hypersonic testing facility nozzle which

simulates the free flight total enthalpy (see Figure 8B). Furthermore, CTE

also prevails when one evaluates the effects of a nonequilibrium expansion

on the flow field around a model in the test section of a given hypersonic

nozzle. ~~_~~'::'~~~>,!!~E..{._.:thecondition hr.. ,:: hr...~ ~ ;.2. .-;ft----

requires, according to Equation (24), a smaller velocity for the predisso-'\.

ciated shock: ~... v.~, ~ ;;:--"";"1)7.-. "l.. "" l.

~1>,-_. ;;:; (/_ J,~~y~.

where}{l': .= hp / h,.; =: ~ nD /V; (<. I ):L. ' is the free stream dis-. -1 002." -'2.'

soc1ation energy energy expressed as a fraction of the given constant total

enthalpy. Correspondingly, there is a change in shock Mach number given by

where K", : J"(~f1O,.- 1)15Z_,.~_'2. ' is

shown in the accompanying Table. Although

the ratio T_:z./ T_, cannot be related to

. «""20 and/or }{-:z. until the ambient disso­

ciation history is given, it will most

likely be greater than unity in the appli­

cations mentioned above.

--1

10

l1l1l]~

J~

J]

]

J

m~ ~ce..Now in applications involving CTE predissociation, T..,. IT. < < I (Ref. 2,

:z. 111.

3, 16); hence MQ)2. can be large in spite of the' -]-{T. term in Equation-2-

(28) and, in fact, remains unchanged if the preshock nonequilibrium

1JJJ ~

J ~

(28)

15

temperature history is such that

r­.[

3.1.3 Constant Shock Mach Number (CSM)

[

F[

[

[

[

~ [

C[

[(30)+TOO2./ r ... ,

I< ~...

K M2. (J

there is also, according to Equation (24), a change. in the total enthalpy

which depends on the temperature ratio Too ./ Too~ I

Although this condition may be desired to simulate certain Mach number­

dependent phenomena for a body placed in a highly dissociated test gas

rlow, it appears to be or less practical interest than the two foregoing,-==~,~---=~:::";;..........,.... - "'--......,......,.;..._....-............---..................,,--.._..._"""n;..,.,.....~

comparisons. Furthermore, there is the following disadvantage to this~..._..._~--_---."'"'--~,

method of comparison. Since the condition MOlt : Moo : Moo imposes aI 2-

shock velocity change

sidered further.

when Moo> > I .to specify the

going methods,

shock. in terms

Consequently, the preshock chemical history must be known

energy level. Therefore, in contrast to either or the fore­

CSM does not enable a comparison with a perfect ambient gas

of ex _.. and/or h D alone and will therefore not be con-... GO 2.

4,

[

[

16

3.2 Frozen Shock Properties

The conditions ~F ~ ~oo t Z f:. ~ Z... and J{F = J{_ are now introduced into

Equations (19) - (23), asSuming either CSV or CTE. The following properties

for a dissociated and/or vibrationally-excited free stream are obtained as

a result.

rL

[

, l

L

l3.2.1 Density Ratio

(32B)

::::

, given in Table 2, is plotted as a function ofKt>oo2.K .._.

0(00 and ~M_ in Figure 10. The preshock real gas effects on PI=' / DOD re-2. 2. r

fleeted in this function are not large: a 5 - 8 percent maximum increase

due to molecular vibration and a 10 - 20 percent decrease from predissocia-

3.2.2 Pressure

tion when cr .~ (f "f.T"c..~ , and similar magnitudes of opposite sign :for attached

shock configurations. However, predissociation will cause a large reduction

of PF/p_ for CTE when tXoo2.:> .23 and V_, <. 30,000 :ft/sec (Figure 5). In

general, the ambient density f-" is determined by the preshock dissociation

According to Equation (20), the frozen pressure is related to(i)

hI" ~ 5, p. V. SIM2. tr by the following two ratios:....., I I

~ O<\'> V", ... s...:"r ( I - fL)'~. t I f~

f..PF_,-.'\-PF... I Jv. : v:

"", ""2.

Equation (19) yields

where RPI" (~-2.) ::

which is plotted versus 0(0/) and {3",,_ in Fig-Te 9. The frozen density ratio

Pr/Poo decreases from 6 (no ambient vibration) or 8 (completely excited

vibration) at 0(. .... .: 0 to a value of 4 (monatomic gas) at 01: 00 ,: I , represent­

ing a 30 - 100 percent maximum reduction due to predissociation. Also, it

is seen that a nonzero preshock dissociation leve1 exist s (0 < ~_ $. .34)

for which the opposing effects of ambient dissociation and molecular vibra-

tion on aGO cancel and E:F' = E: -! .t" F' Pf.kF. erAs - ...

l1llJJ.J ']

JJ]

JJJj

17

[

F[

[

[

4 [

~ [

C[

[

[

[

[

l.[

~ l

L

(34B)

(33B)

(33A)(I

.jl

,."-

where R" (13- ) = (:14"'~) (see Table 2) and RT are plotted versus C(dO~ 2. Meo • ~ " 2.

and {311 in Figures 11 and 12, respectively.002.

The ratio ;~""''l. (T~eo ~ VOOI2.S1~2.fT/7Rttt)' according to Equation (22), mayF'_, '

be written

According to Figure 5 and Equation (3313), the decrease in PF is less than

30 percent unless 50 percent or more of the total enthalpy is invested in

preshock dissociation.

history. However, ffoo2. may be given explicitly when the free stream mass

00,

flux per unit area foo Voo ::;m is invariant to dissociation (which is

approximately the case, for example, in nonequilibrium wind tunnel flow) ••For CSV, we see that /)t) ~ constant implies a simulation of perfect ambient

gas density (pGO, ':::::' foo,) and an approximately constant p, (within 20 per­

cent or less). On the other hand, for CTE,

3.2.3 Temperature

18

19

(35M

(35B)S'N'Z. '"

+

+

constant C ffF is defined in Table 2.

predissociation can yield a shock enthalpy

Furthermore, because the effect of 1-100 2­

decreasing normal shock velocity, the value

----RpF Rlir eMf

DETAcHE.D g6~~'- 0 I JSHOC.KS 5 2. ~ooa.-

WEOG-E. AT7 (e, - 9 f2~ il~. - :j '"INCIPIENT _I. 00\

DETACHMENT :5 \ ~OO2. 5 2. i3~2.- 5"

ATTACHE.D ~~~~..- ~ ~.2h-fJ 2.5WEDG-e. StiOCK 12 ~ - /2. ~-2.- J 3b

.... 2.

The following enthalpy ratios are obtained from Equation (21):

whereh, ~ Voo,2. SlN 4 cr and the-, 2-

According to Equation (35A), CSV

that is much greater than hi< •-,on Equation (35A) increases with

TABLE 2

of hF _ /11"00 for an oblique shock exceeds that for a normal shock wave2. I

with the same preshock dissociation. Equation (35B), as expected, predicts

a substantially smaller increase in enthalpy; indeed, there is no change

3.2.4 Enthalpy

The change in ambient gas molecular weight and P represented by RTF causes

a maximum decrease of 15 - 35 percent in r;. due to either ambient vibration

or preshock dissociation. On the other hand, the frozen temperature for CTE

can be reduced by a factor of 2 or more with 0<.00 > .23 and VClOI < 25,000 ttlsec (specific examples will be presented in Section 3.3).

JJJJ P

J

l-]

l-l-l-1 ~

J ~

]

]

J-J

J

20

for a normal shock. In spite of the constant total energy, however, a sig­

nificant increase in hr: with DeClO can occur for attached shocks. Ambient,, ~ '_"'''' '''''<''~''''_<C'="_=''~'''~'_~'><=.,._".~a"""'_"~·''<··V''~~''''_M..""••..•.......•.__ '",..••." .•~_._"._._

vibration has a negligible effect on the shocked gas enthalpy in comparison

to predissociation according to (35).

It .should be noted that the foregoing frozen shock properties were calcu­

lated independently of each other; in particular, PF and TF do not depend

on the corresponding pressure or enthalpy. In contrast, all of the thermo­

dynamic properties are completely determined by los and hs when the flow

reaches complete equilibrium behind the shock. Accordingly, the predisso­

ciation effects on fE4 , ~q and ZEq which will now be considered are

attributed to the energy parameters H<»~ or HT rather than as a result... ...~

of the changes in ambient molecular weight and specific heat.

3.3 Equilibrium Shock Properties

An iterative simultaneous solution to Equations (19) - (23), in conjunction

with the data given in Reference 14 and Figures 5 and 6 was employed to

calculate the equilibrium shock properties in air as a function of the free

stream atom mass fraction. l Both CSV and CTE conditions were assumed for

the five representative perfect ambient gas shock velocity-altitude con­

ditions shown in Table 3. Three configurations were studied: (a) a normal

shock, (b) an incipiently-detached shock on a wedge, and (c) an attached,o .

30 half-angle wedge flow. In several cases for each configuration, the

effect of an .ambient density change was evaluated by arbitrarily assuming

/-1f'oo2. poo, = 10 ,land 10. The resulting variations in equilibrium den-

sity, temperature and compressibility factor with 0<_, and a comparison

with the corresponding frozen shock properties, are presented in Figures

13 - 15 and will be discussed in detail below. The effects of ambient vi­

bration on the equilibrium properties have been omitted, since they were

negligible in every case considered•.

lMt-. D. A. Meis, of the Missiles Aerodynamics Research Group, carried outthe majority of these calculations. As a first approximation, it wasassumed that hEo. Z h F and PEQ Z PF , since h s and Ps are not toosensitive to post-shock relaxation.

.[

t[

[

[

~. [

[

[

[

[

[

[

[

L

TABLE 3

(2) Although the CSV and CTE predissociation effects on PE~ / p_ are

comparable in magnitude, there is a different cause in each case:

in the former, it is the increased post-shock enthalpy, tempera­

ture and compressibility factor; in the latter it is the decreased

value of los / {'oo (to which f£Q./P'" is directly proportional).

The CTE effect is evidently the larger of the two fora given tX_

and V00 •,

3.3.1 Density Ratio

The ratio ~: is presented as a· function of CX<lC) in figures IJA, B and C

for the normal shock, incipiently-detached and 300 half-angle wedge flows,

respectively. For purposes of comparison, the frozen density ratio p~ / p_is indicated by a· shaded region in each figure. Contours of Hoo or

2.

HT .... = constant are also shown. to illustrate the dissociation energy re­2.

quirements. The following conclusions may be drawn from these numerical

results.

l.'1 b

'-1

'1'1

J]

] '.I

] ~

]

J]

JJJJJJ ~

t:'

.J

(1)

CASE. Voo (f. p. s.) ALT1TUO~ (~t:.)•<D 3°1 °00 2.50,,()oO

® 2.5" ()()O 2..50... °00

@ 2/,000 :;tOO... 000

® 17" 00 () 2001 000

® 15) 000 2,.001 °00

Both Hco" and HToo must equal .10 or more for predissociation~,~~._...--_._.-_.~~,_.~_.__.,.._---~-_.~ .._,~-----'----'--~--"-'---" .•..~.,..--..-to have an observable effect on the density ratio. The usual__.__._. ,..,...,~--".....""",-,x~_~.~;..,.~_""" ...._._,..~_-;>a ........<....~~•.,..,.,."......aI~:"";'-~.""""""~..,..,....;,.-.,.

equilibrium shock density ratio P£Q /poo will be significantly

reduced if there is a substantial concentration of atomic nitro­

gen as well as oxygen ahead of the shock (0(00 >> .23). Indeed,

a reduction by a factor of two or more is possible if H....2"

HT"" ~ .50.2.

21

22

(3) The rate of change of equilibrium density ratio with «. increases

rapidly as the shock strength decreases; predissociation effects__....-,~~. "'''.....v .......,...y;:,~~.........~..~_..-....._

are therefore most pronounced for· attached shock configurations-- ..."'""'-=--....,.. ...__...........__.........'-...":'-."''-'..,.~~-- ..~-'--'''''-'':- .............~ .........__~......:>~,_~.,....;.""'.""' ..._."""'''_................,.>ic...~~.,..."..__•

(i.e., an effect comparable to that for a normal shock can be

realized with a much smaller fraction of the total energy in pre­

shock dissociation).

(4) A ten-fold increase or decrease in foo has a relatively weak

effect on p&Q / p.. in comparison to the effect of varying 0('00 •

Therefore, predissociation effects on the ambient density will be

negligible in practical applications.

(5) Unlike the frozen density ratio, rr.ca / f- drops sharply as OC oo

increases; consequently there is a pronounced decrease in the

usual degree of compression due to relaxation behind a perfect

ambient gas shock when H"" , HT: > .10 and OC ao > .23, particu-2. -a,

larly for CTE. Indeed, Figure 13 indicates that the sign of the

inequality Pr.~ > pv may change if O{_» .23 and VOO1 < 15,000

ft/sec (H co > 1.00 or HToo > .50), thereby implying a predisso-a. z

ciation-induced reversal in the nature of the relaxation process

to one which is exothermic (recombination-dominated). As will be

seen below, thiS possibility is also evident in the behavior of

the remaining thermodynamic properties and the thermally-signifi­

cant reaction rates.

(6) The validity of the assumption E.£q2. < < 1 progressively weakens

with increasing ()(_, particularly for CTE. Since this assumption

is questionable when E: < 1/4, the dashed portion of each curve

in Figure 13 below the line P;: :: 4 is inconsistent with the

hypersonic approximations. Furthermore, it will be recalled that

the hypersonic approximations for h , T and hence fElt are

"weaker" than the approximation for E:.F' and therefore will fail

when PEq /f.... > 4. Although the "cross-over" points rUt ~ pI:lie above the line ~.: :: 4, they do not necessarily fall within

the scope of the present hypersonic apprOXimations. For this

.1

·1[

F[

[

[

-[

[

[

[

[

[

L[

~ L~L

l]

]-

]

]

JJJ

reason, the evidence in Figures 13 - 15 is considered strongly

suggestive, but not conclusive proof, of a predissociation-1nduced;;;;;;~-~~-~h;~~~t~;'-~f--th;-'~~~~~'~~~~~;~~~~;;'~~;;;~:-"~]i~i~'-~"~-~"--"-'~-~"""""""'J.~~~""~~"";.~... e.......,.,,.."~~_""".~~~~~~"'="'·~:'o'-~P),="""""~'~"\'~""",,.

clear, of course, that the majority of any exothermic post-shock

relaxation which may exist will occur only if 50 percent or more

of the total energy is invested in predissociation and D<DO >> .23.

Moreover, this behavior cannot be analyzed on the basis of the

hypersonic approximations.

3.3.2 Temperature

The corresponding effect of preshock dissociation on the equilibrium and

frozen temperatures is shown in Figures 14A, B and C, where T£Q and T"

are nondimensionalized with respect to the values of 1';OD/ ~ .2. T_. M oo:Sltol2.tr

shown in each figure. The CSV andCTE results are quite opposite in char-

acter. In the former case, ~oo /Tr decreases gradually with CX: OO and2. 00,

is independent of shock strength, while TE~ /T~.. is increased by as-2. -00,much as 40 - 50 percent when H00 > .50 and 0( 00 >> .23 because of the en­

2-

hanced post-shock enthalpy. In contrast, CTE predissociation can result in__'-"7.~~"'''''''''''''-'''~-----_· ....""".",..,....,.~ ...........,."",...,~.",.~."",~i'''''''~'''~<V'''''''''~'',...- ....,.......,.,...._,......""""'~~-<:'".,...""'.~_ ...__,

a two or more-fold reduction in T;.oo ITF... which increases with decreasing-'....,........""-=-.....-~~--.-- ....... ...,.,.."....... "..:..-...,..,..~"'""""'"""".,..-'.,. ..""'''''''''''''..''-.......:"..'''~~~'<"'''-=-'''''''~'''~N .....'''''',..'-''"''''.,...'-'~ ....a".."~~.,.,..,:..,..''''-,,,.k .. ,,..,..,~......'C'....,·;,.....,''-·J~.~,...v_=,...,.""""''''~...,.;...l'.~~'''':'OI.:I ..r.'~~<:'~,,~;..or.''-.....,'_~''''{"y,..y..''''':"'"' ...,....'v;.-..~.~>

s~ock str~~~.ft. ~~.:r ,,,_~,~~ __~~"_~~~:~2:~!Si '!:~:=~~~~~!E.J:~M£.2~r~~~2~~~~equilibrium temperature for r:r ~ o;U"Clfbecause h£~ % constant at thesesh~~gi;;-(-;~~ig~iiic~~ti~c;;-~7i;T~'~does=;pp;-~~i~~t;;~;tt;;'he-a.~~~~ .....:..._"".....~\_,~"~-"' ......·.....,...~.""'-.....-_~l""'o':Vr..'""""""~r.("'.o::.!.>:PJ$.~u.~,,.P"...,;;. ..........."'~~cJ;P~),'~.C'.:<""""'.<.;...-;;.,.",..";:.r"'~",,:>O-"....,.. ......-:'......I'j';.........-""',,,,.."""';....""'.....;..~,,~."',:<n"'''::.~~~_''''<'''='':'~'''q~,.,.,..",-,",.",,,~, ...

shock example since h£.Q._ I hu is greater than unity in this case).~.,..."..._~~","-""",,,,,,,~*_oe,""~--.2. "'_.-=- '_ 110, =-';"-"~"'fr.-~..--.""""'~~""'---"""''''''''''''''''·"'''''"'''''''·-;~--·'''---=~-~· _'''~~'''''.~,~.~_r....,.. "~...,

In spite of the foregoing differences in behaVior, the net effect of pre-- 4>__"'---~'

dissociation in either method of comparison is a pronounced reduction (and_.___---.., .' _ ;~.....,...;o;~~-",y~~-::w,........-....."" ...""v-.......,.,...,=?-..-~_~·_· .o,._~..".-~ ...,....-..·,_·........-",,~.......".,..,..,..--=~r.:""""'~"""'''''''''''''--'_·'''Y''''''.",r-",-~.''l;'''''''''~.,"

even reversaIi~?sign) of the temperature difference TF - ~~ in a perfect~_~~...............,.")....",y~.r~"-;~O<""·~.~~""""",,,,,~~,,,~~~,,,:«<,,,,,<:,,,,,,,,,,,,,,, __.,,,,,,,,,,,,,,,~=,--~,,,;,>~"'!'"r...."",,,,,,,,,".....~_,.:,,,,,,",,,,,~c.<.w........;","''''~''''''''-''''''-_~'''''.,-\<:,.-,.,._,, •

ambient gas when HOo 2. or HT_ > .50.. Furthermore, a ten-fold change in p_..... ~ • r ,,~~_-.c.,,_..........:II':O:"---<01V ~...........~__-v:-=~~',,:,:>!:~,,;,'I?,.,'~~

has a relatively small effect on this behavior. The temperature crossover

points (<X.co 8 Tu ~ 1';) shown in Figure 14 correspond closely to the

crossover points shown in Figure 13.

23

.!

24

3.3.3 Dissociation and Ionization

The equilibrium compressibility factor ZEQ versus O{oo is plotted in Figure

15A, Band C. As a result of the post-shock enthalpy increase for CSV,

predissociation involving both atomic oxygen and nitrogen will produce a

significant increase in ZEQ (a factor of 2 in the attached shock example)1when Hoo > .50. The preshock dissociation effect is negligible, however,

~

for Hoo < .10. There is also a negligible change in ZEQ with 0(. for CTE2,. -----------.-----------.-..-••--••...• '- ----

in general when rr ~ a;U'AC.H; a significant increase in the post-shock dis-________..__._.__.~_~__~-'--_~_. ............""'_~ --.----" ._...-;...._.~~__._. ~~_......:..__~~~.~. .-''''---.o_,-~-,,-',-.''--·...--..-"'' .....__~_._"'.'"'~_''-''._'_''-~.~'''''''''''''''''._.,..;~

sociationlevel is observed in the attached shock cases only._______~.__._.. . .~_."_~ ,......~ .__""'_....~-.....".~.-" .....''''''...........--...'''''''~.;'''-......~~-.......-=--- .......,.,...,.''-''',..-''''....,.,.,,.--4.-__......,...,.~.,..:;:~ ..... ._

In agreement with the trends discussed above, CSV and CTE predissociation--- - -~--....,..-----

~~!:~_:_~~~_~_~--~~~~~-c:!-io£_J.n-_~p.~ __~:!..~~e~~~5:.:-._.~e~_~~~n _w~_~!_!'::~ Zr = ~:_~.,sU~.~h~~_~.!_:'Y._~~J~~__,!!!.__~h~__~~i~_£f. ~1i._~::_,,~£ __!Jl~L~E12~.?F' when H-2,. or

H~ >.5 and OC oo >> .23. Large changes in poo have a negligible effect....2-

on this behavior. It should be noted, however, that the slope of the CSV

curves approaches unity as o(CIO increases; this trend becomes more pronounced

as the normal shock velocity decreaSes. (A similar behavior is also ob­

served for CTE at much weaker shock strengths; for the examples shown in

Figure 15, it occurs only for the attached shock). Consequently, when the

normal shock velocity is sufficiently small (~15 - 20,000 ft / sec) ,

Z£Q (0(_) tends to merge into the line Z. = I + OC_2.as ()(.. increases. The

relation Z E.Q:: I + OCco thus becomes a good approximation to the equilibrium

dissociation behind the shock when 0(,00 > .25 - -35.2

Obviously, this limit­

ing case is of special significance in the analysis of hypersonic slender

body flow fields (Ref. 17).

It may be anticipated from the foregoing results that CSV predissociation

can. appreciably enhance the eqUilibrium ionization levels realized behind

lObserve that Z E.Q2, can exceed 2 in the stronger shock examples; hence theeffect of CSV predissociation on the equilibrium shock properties in airis not bounded by the values for a pure, unionized monatomic gas.

2This limiting behavior is to be expected, since the perturbations on thefree stream conditions weaken with decreasing normal shock velocity; thusZ £Q must approach Zoo::: f + a. at smaller £X CIt> > 0 as MaoslH (J"--. I .

.[

.[

[

E[

[

[

~ [

[

[

[

[

[

[

[

L

l.

l·"

I

1-1

l]

] ?

J]

JJJJ P

J ~

a shock wave in a perfect ambient gas (the effect of GTE predissociation,

however, should be negligible except for attached shock configuration).

Using the data in References 11, 12 and 14 in conjunction with Figures 13

and 14, the electron mole fraction %c," versus "'002. for a normal shock has

been plotted in Figure 16. A ten to fifty-fold increase in X£&. With «.~

can occur if H >.50 for GSV in air, whereas there is VirtUally no change00 . , ' _~__>_~_',_.,,__,in (x'E.L)oo for CTE. An even greater percentage increase will appear at----...."'...-.~~-.. _-==>=--...~--

smaller shock velocities and/or wave angles. Of course, it should be remem-

bered that the electron densit;y, proportional to p. Xu' does not reflect

so drastic a change because of the accomp~ying decrease in equilibrium

density, 'although an order of magnitude increase is still possible. Indeed,

CTE predissociation will actually reduce the electron density because ~E~

is relatively unchanged while ~:: decreases.

It may be recalled that nitric oxide in the free stream has been neglected.

However, because of its low ionization energy, a small amount of NO ahead

of the shock front (which can easily be the case in practice) would sub­

stantially increase the eqUilibrium ionization for either CSV or CTE, in

addition to the free stream atomic oxygen and nitrogen effects. The present

calculations are therefore conservative estimates of predissociation effects

on the post-Shock ionization level in air.

3.3.4 Shock Geometry and Flow Deflection

Predissociation effects on shock angle or flow deflection for either attached

or detached shock configurations may now be easily calculated from Figure 13and' Equations (7) - (9). As an example, 5,: and $E4 for a detached angle

of . r:r:: 8Qo are plotted versus ()(oo in Figure 17. A significant redu..E~~gn

(250

or more) in <5t:q and a corresponding decrease-of' t_~~e~~.9-2-_~!~;~ncete-S:~~~~e~~~=~~i~~~~~~~~ ..~~~;L!~~~f2£~~~?_!.,~::~~.~y~5.~~~~~~s? (~co?)below 20,000 ft/sec and «'(10 > .50.~----,-'-----......----...~,._,_ ......._---=--_....----~~ ....~~~.".-. ..,,-

Considering an attached shock configuration, a plot of fT versus 0(00 for

$s = 300 is shown in Figure 18. It is seen that purely shock-generated

dissociation, nonequilibriUm. relaxation, and GSV predissociation all have

25

26

a very small (~50) effect in the examples shown. In contrast, a 100_ 200

increase in rr with O(ClO~ is observed for CTE if HT_~> .50. The ~~e

det~~.:~~_~ave ~~:, versus tXco'1. is shown in Figure 19 and displays the

same insensitivity to real gas effects that is indicated in Figure 18 (ex-"".:.-···~'~.;~........._.."'~~-».""'·~'"""",. ..-w"e,..""'....-:;"=...."'_....""-...,.~""""" ........~.................:?_..... -~--- ..."---...,..;.,-...-..,~".,.."_":...-..~:""""""~.;""">."""",, ..........,,....~.,=--.._

cept that the sign of these effects is reversed). The correspondini deflec-'-,-'

tion az:~!:~~~E.-.:,~~~~p~~_nt_,2-~}~~~~J.,c,~~.1~Qindicated in Figure 19, are far,-_.._.._-""---'..

more sensitive to 0(00 and are similar to the deflection angle characteristics--......,.....-'•.:........,.,...,,-'-,--,..""~v_~~..".':Y_.r-"-""",,,-,.,,,,,,,,,,,,;~~

in Figure 17. These changes in bEaOETACM due to predissociation (100_ 20°

for 0(00 > .50) are well within the capability of experimental detection,

provided a sufficiently large percentage of the total enthalpy is invested

in dissociation ahead of the shock.

3.4 The Intervening Nonequilibrium Behavior

Many theoretical and experimental investigations of the nonequilibrium

relaxation chemistry in shocked air have been carried out for a perfect

ambient gas (see for example Refs. 18 - 20). However, only one rather

limited study of the relaxation process with a dissociated gas ahead of the

shock, involving atomic oxygen only, has apparently been made to date (Ref.

21). Therefore, in the following discussion we shall attempt to appraise

some of the potential effects of predissociation on the nonequilibrium

behavior of shocked air for 0 ~ tX oo < " without solving the detailed rate

equations, by combining the foregoing results and the present knowledge of

the rate chemistry.

3.4.1 ' A Short Review of Shocked Air Chemistry

Although the present understanding of the rate processes excited in shocked

air is far from complete, the combined theoretical and experimental studies

to date have clearly shown that two basically different types of reactions

may proceed simultaneously:l (1) "thermally significant" reactions and

(2). "thermally insignificant" reactions which are extremely rapid in com­

parison to (1).

lA more comprehensive review and detailed discussion may be found inReferences 18 and 20.

.!

[

E[

[

[

- [

[

[

[

[

[

[

[

L

The thermally significant-reactions possess activation energies....-o.~~~;.:""~_"""r.>"""~ :;0>:4,0: .. ' _~,..__~...,.,.~.,.,...._""~._:-cr-;>""""..-...................""w -.--"""'~~:..,.,..",..ll. ......"""""."""......,<.~ ..-:4-'i> ....""~.".,...

comparable to the shocked gas enthalpy and therefore govern the_~~_--~..",......,.,.,-""""',..v....,=~....""'~~~""",':'>-.,""""'''.'""'''''.'''''''''.,..",~,; •...,..-~-'''.........''''''''~-

distribution of the density and. temperature across the relaxation

zone behind the shock. The most important of these reactions are

the dissociation.reco~1nationreactions

Here, M is any third body catalyst. A summary of the most recent

values of the forward and reverse reaction rate parameters and the

catalytic efficiencies of the various species may be found in

References 18 and 20. There are three important properties asso­

ciated With these rate processes: (a) they describe a monotonic,

endothermic relaxation between the frozen and final eqUilibrium

states; (b) oxygen dissociation must be nearly complete before Na.

dissociation begins, and both 0 and Nionize only when N2, and 02,

dissociation is virtually complete; (c) a significant electron

-1

-1

1-1

] ~

] -

J]

1_J

JJJJ

]I

J

(1)

.."....-----..._.~.,.._~, •."y..-

lCD,2.00a. .... N\ .- 5.1 ev ~ + Nt~

I<A,

Na. + M + ,.ge~ ~ 2.N + M~

leo"

N .... a + M ~ NO + M + '.05 ev ,~

I(D3

and the ionization reactions~_'-""~...._.__..........~-.......--,.,."""",_""-

0 M 13.be~Kp..

0+ + M+ + ~ e +'t"":":"-"It ..

N + M + Iif .5e" ~ N+ + e + Nt~

I<Rs

M CJ.3et teo" +NO + + • NO T e + M

C KR"

(36A)

(3613)

(37B)

(37C)

27

~ .,. Oz. +- I ..' e Y ;;;:::.~ :(,.v0and the ionization reactions

0 3.3ev1<,.

(38A)N2,. + + ~ NO + N~2 -;<;'"""

02- + N ~ NO + 0 + , •£f ev (38B)~

(2) A second set of reactions possess comparatively small activation

energies and exothermic rate constants (KIt.), being t!leru;~~!-y

insignificant in the sense that the energy tied up in these re-~,_"~~(""'~''''---''''"'._-"......~__ ,_.~..........,..--v',-", ...,--~--.,..•".-"""•...-••_--~'-·"·-"""'·'_.__4 .....~......_~"""-_.;,~.;.•,,,,,,";,,,,,,,,,,,w.:__,'''''>'''''~'.;o...,......,.~"""....~.", .....,..............",....",.,..."",""",,,,,-,,,,,,,,",,,~,,,,,,,,,_~,,~,

actions is a relatively small fraction of the shocked gas enthalpy.

The temperature and density behind the shock are therefore negli­

gibly affected by these reactions. By far the most important of

these are the nitric oxide exchange (or Il shuffle lt) reactions

~~~",,,:,,,<,,,~,,,,,,,,,_",,,,~,,,,,,•..,-...:_,,...._...,..-_,,,,,~~4"0._-:';~~~_;,,.,;.

[

E[

[

[

~, ['

[

b[

[

[

[

[

[

[

· L

L

(39A)

(39C)

(39B)

.~~_....----""--~.-~~'"

Ko" +

N + 0 + 2..~ev NO + e< Kit

N*I<D

N+0 (5.Sev),

+ e+ + < Kit 1

0*KD > 0++ 0 4- <'.1ev) , + e ,Kit 2.

density can be produced by the nitric oxide reaction (36c) for

temperatures at which the ionization due to reactions (37A) or

(37B) is negligible. It has been customary in the past to assume

that vibrational relaxation is complete before the above chemical

reactions are appreciably excited. However, at sufficiently high

post-shock temperatures the vibrational relaxation length becomes

comparable to, or greater than, the corresponding chemical relax­

ation distances. As a result, a significant vibration-dissocia­

tion coupling effect appears and reduces the initial dissociation

rates by absorbing some of the energy otherwise entirely available

for dissociation (Ref'. 18).

28

--1

-1

1]

]~

]~

]

]

-1_J

J]

JJ

where the latter two reactions involve a very fast collisional

excitation mechanism1 resulting in electronically-excited atoms

N* and 0* which m~ easily ionize by recombination to produce

molecular ions.2 The thermally-insignificant reactions are re­

sponsible for several striking features of the nonequilibrium

behavior near the shock front which cannot be explained by reac­

tions (36) - (37) and are remarkabl~/ insensitive to the rate~ of

the latter reactions. These are: (a) unusually short dissocia­

tion and ionization relaxation times (far shorter than observed

behind a shock of comparable strength in a pure diatomic or noble

monatomic gas), (b) local values of NO concentration, electron

density and molecular ion band radiation intensity near the shock

front which are extremely high and far in excess of the correspond­

ing final equilibrium values (i.e., a decidedly nonmonotonic re­

action path involving composition and nonequilibrium radiation

It overshoots"), and (c) noticeably more atomic nitrogen near the

shock front than can be adduced to reaction (36B).

3.4.2 Thermally Significant Predissociation Effects

It has been shown in Figures 13 - 15 that predissociation can substantially

-reduce' the usual spread in the thermodynamic properties due to endothermic

re_laxation in shocked air. Moreover, there is evidence in this data which

suggests that a sufficiently high ambient dissociation level can cause the

relaxation process to become exothermic (recombination-dominated) and there­

fore an expansion flow with respect to ps. It will now be demonstrated

that the thermally-significant reactions do indeed portend a relaxation

behavior following these trends. For this Purpose, the following form of

Equation (lOB) shall be used to represent the post-shock relaxation of the

lThe exact nature of this process is not fully understood at this time(Ref. 18), although it is known to be a binary collision mechanism that isthermally-insignificant for T;:' 80000

- 10,000oK.

2The parentheses in (39B) and (39C) indicate the electronic ground stateactivation energies for these reactions.

29

and temperature exponent,

-1.5 for air), B is

1temperature. Equa-

total atom mass fractiont(s

~;r-:R':)"[.,~~_=_~_<:~~;;;:~ ..;DISSOCiATION

where .J.I\ and W are the recombination rate constant8 1015 6. 2

respectively (Ael\ Z, 0 -1 5 cm /mole -sec, W ,.,(4500 K) •

a constant (Z16) and TI> is the average dissociation

. I of- «s.'.._ ....y._-_ ...

It£CO''''&INATIOM

, (40)

30

tion (49) represents a binary "air atom-air molecule" approximation (Ref.

22) which is convenient for the following qualit,ative analysis since it

clearly displays the essential physics of the thermally-significant disso-

, t' 1 t' 2c~a ~ve re axa ~on.

In a perTect ambient gas, the initial reaction excited behind a strong

shock is a two-body dissociation rate which increases o(s and decreases

€os, ~ with J{.. (linearly for small X. ). An opposing three-body recom­

bination rate subsequently develops and slows down the net dissociation

rate and rates of increase and decrease, respectively, in lX, ('X,), €.~ and

~ ; ultimately the lagging recombination "catches up" to and balances the

forward dissociation rate when the asymptotic thermodynamic equilibrium

state is reached. The overall relaxation process is an endothermic, mono­

tonic transition between the frozen and equilibrium states (Ref. 24). Now

consider a dissociation level 0<._ ahead of (and therefore immediately behind)

the primary shock point. In contrast to the foregoing situation, Equation

(39) indicates that an atom recombination reaction is present in (dO(/d~)1:

and that fewer molecules are available to dissociate further behind the

shock. Also, since 1";. will be smaller than (1;)«.. : 0 ,the initial dis­

sociation rate exponential is reduced as well, partiCUlarly for a CTE

2A detailed discussion of the shortcomings of this binary rate equation formulticomponent air may be found in Ref. 23.

:-1

]-] ~

]

JJ

it. ' I II Ik ;

1- Jr- -Ir (~- /;j;j.,~.r7t l'~ .P.,.~?;'1-J:;-r."i

!/s /V'''''i c~vr.N . • " .4 "I_~ r.vji :-)".J~,,-_ fo, .z>'!t"",,,.};J.-..

/~-.4.- /'>....-v '-1"r d .() ;; "7 ?. v..........t- 0--' - ~ .

1comparison. Clearly, these combined effects can significantly reduce in

_._~_._• ...,-,.rA <.o<_.........--""_,_, ........"<>_..._......,,,~.,,.....,,,,,~,_" ...",...~~~~.......,..".__~............... .......".;.

the usul?.!._yal~~..Qf (~_~~~~..&..E~n:..~~n ~~~r.~~c~_~2-':'.::!.~~. For asufficiently large Gl(oo and shock velocities below 10,000 ft/sec in 1';:- , it

is possible that (d «/d%)r becomes recombination-dominated, causing an

exothermic relaxation toward equilibrium in which «$ decreases (and £.5 ,

~ increase) with %. We may note that the foregoing trends are indeed

observed in the numerical results of Reference 21, although a reversal in

the nature of the relaxation process is not in evidence since this work

considers free stream atomic oxygen only. 2

Some interesting consequences regarding vibrational relaxation may also be

observed. For example, predissociation reduces the initial vibrational~........--..----~--_.. --------.,..,..,.".,.-_....--...,....,_.........,...",............-.....-..---..,"'.,.,.....----"'_..........~"...--

excitation rate behind the shock because of the reduced 1; and fewer number_,..--.--.-or._'"-<.-..,..,~ -....-..-.-_~.,....,.""~""'"..........~<>7."'·"'_~· ..... .....,.,<;'<'..>"'...._"'''<'"'..,.. .........,.;;.._''''~A'"''' ........'O___-...". .. .....".... , .........,.."""'_,""."."..:-.-.,.._ .._.

of molecules available for excitation. This in turn slows down the disso-

ciation rate over a wider portion of the relaxation zone due to an enhanced

vibration-dissociation coupling effect. When the free stream is vibration­

ally-excited, this coupling effect is further enhanced; however, this tends

to be offset by the higher initial vibrational energy level behind the

shock. Of course, all of these effects assume a decreasing importance as

the preshock dissociation level increases.

3.4.3 Predissociation Effects on Overshoot Behavior

In a perfect ambient gas, nonequilibrium overshoot phenomena associated

with the thermally-insignificant reactions are instigated by two very fast

reaction paths acting in parallel. The first invo~ves the NO-exchange

reactions and reaction (39A): upon a slight dissociation of oxygen from

Equation (36A), these reactions rapidly drive the NO, NO+ and electron

concentrations into local equilibrium with a temperature Ts >> TSEo. ,

lIn the CTE case, this effect is somewhat opposed (but not overcome) by thecorresponding reduction in PF •

2Recall that this reversal may occur in air only when both atomic oxygen andnitrogen are present ahead of the shock with more than 50 percent of the---­total energy invested in preshock dissociation.

31

32

thereby producing a decided overshoot of the final equilibrium concentrations

near the shock front and, by (38A), a concentration of atomic nitrogen

higher than that predicted by reactiorl (36B). Simultaneously, a rapid

electronic excitation of the atoms states drives the molecular ion concen­

tration resulting from (39B) and (39C), and the radiation therefrom, into

a local equilibrium. with a temperature T.s » ~ E.Q. ; this yields a

further contribution to the electron density overshoot and an overshoot in

molecular ion band radiation intensity near the shock front. l

It is clear that preshock atomic oxygen, atomic nitro~en_~~j..Q!..,~J::!Eic

oxide will accelerate the nitric oXide exchange reactions behind the shock

~~~-d;~~bl;;'~;;:ich-'i~-t;";-~i~-'~:'o;~r'~~o~t peak;' i~NO-~;d-~"~;~'Cen-'~, ~ -_+-~ -;;::--_----' ':""wc~_''''''''''''''''''''-':----'-----_m---.... ---......-.......~.._*____

tration, and NO and Na band radiation intensity, nearer to the shock than-....-..-_~~. ..._;. ."'........__. , .......... ..........,,_-~.... ~,~._~._~_~~.........~=_~.......,..,....,.,...._='"...=~......""""....,,__.._.<;o._...,.~~=_...........~~....""'''''''''_...._._.r.~ ......''''_.~'_......._~~ _

is the case in a perfect ambient gas. Moreover, an increase in the absolute-------~------~_._._.,-_.

magnitude of these peaks would be expected. However, the decrease in T;............~---- ......~--'""-~-..,-_ ......--'-"'_..............-...---_...-.."-_-..~._. ......,....._~~~, ....-.

due to predissociation tends to reduce the initial excitation rates and

absolute magnitude of the overshoots. For air with oxygen predissociation

only, the latter effect is apparently the weaker of the two, since the

numerical results shown in Reference 21 indicate that the degree of over-....-=-""-----..----.-----.........-.~~ . .,..,. . ~.~~,................,....~...............,._---:-~......,. ...-----"""""""--x._.. .,shoot increases and moves toward the shock with increasing free stream_____-=---....__...,..._.._-...-"-"'='- ..---.-....."';""......y.-••---,-,-----~...."~.-• ..,.........--""""''"',.,~...- ..--,ooW..--.-,,.............."'.-............."..............,..-.;.,""'""'-_~>o.r"",~_=.:..."",..,.~..............~~,._.:. .•"'~",.c"._...,,,..~.....~"

oxygen atom mass fraction. However, a different trend should be observed_._ .._.-._..,...,._~,.......".__,..-.-r.....-

when there is both oxygen and nitrogen dissociation in the free stream,

since the reduction in T;. is larger and, more important, the usual differ­

ence T;: - ~q can be sharply reduced. As a result, the relative amount

of overshoot in NO, e and molecular ion radiation intensity can be drastic­

ally reduced and possibly eliminated When O{_» .23, especially for CTE.

Moreover, large reductions OfPF for CTE will also reduce the fast reaction

rates leading to overshoot, since these rates are binary and therefore

scale proportionally to Pr (Ref. 18). According to the foregoing reason­

ing, then, increasing predissociation should first lead to a slight enhance­

ment of the overshoot maxima (With these maxima moving toward the shock

lThe N~ radiation intensity, since it depends to a great extent on the ex­cess atomic nitrogen generated by reaction (38A), is rather sensitive tothe rate constants of the nitric oXi~e exchange reactions (Ref. 18).

·f[

E[

[

[

[

[

[

[

[

[

[

L[

L

4. SUMMARY AND CONCLUSION

front) and a shorter re laxation distance; however, with Q(oo >> .23 (both 0

and N ahead of shock), the trend reverses because of the sharply reducing

difference between the frozen and equilibrium shock properties and the NO,e and molecular ion radiation equilibrium overshoots decrease with increas­

ing 01.., moving downstream of the shock front and lengthening the relaxation

distance behind the shock.

A theoretical stUdy of predissociation effects on the nonequilibrium gas

properties behind the strong shOCk waves in air has been presented in this

report. The results were compared to the properties behind geometrically­

similar shocks in a perfect ambient gas for either the same shock velocity

(CSV) or the same total enthalpy (CTE). Provided that both atomic oxygen

and nitrogen are present in the free stream (cX_ >;> .23) and that 10 percent

or more of the total enthalpy is invested in preshock dissociation, signif­

icant changes in the shocked gas properties were obtained. Specifically,

the following four conclusions may be emphasized.

:1-1

]

]

:1 ~

-1 ~

]

]

:J]

]

JJJ

(1)

(2)

The equilibrium shock density ratio ~: can be reduced by a factor

of two or more when either CSV or CTE predissociation involves

50 percent or more of the total enthalpy; correspondingly, a 50 ­

100 percent increase in equilibrium temperature, dissociation and

ionization behind the shock can be observed at CSV. In contrast,

the increase in 1;.Q and LE.Q resulting from CTE predissociation

is much smal~er but is accompanied by an order of magnitude re­

duction in the shOCked gas pressure and frozen shock temperature.

The influence oLpr:~dissociationrelative to the behavior in a___•__.r...__• •• ",,"._·· -.-••• -.>- ••:.=-q_~..:....~ ~............. .~_.__._.__._~_..~ ...__~.~ ~.~~. _. -. -~---"._._. -_.

perfect ambient gas becomes more pronounced as the normal shock--~._----<~.-~.-.. --_ .. -_._._ .._".:---'_._,.-,-~,._~~---'-- .._-~-~._~-----..,---~~-.----_._~~.- ..,._..,--.~~,....-.~-----...,... ...,..----._"--',.,.....,._.....-

velocity decreases and therefore is particularly important forweakshock-~1;;-·~d7;;-~~~~;;:~d-;h~~~~--~~---~~~~ted~~~·~der bodies.

Either CSV or CTE predissociation in excess of 50 percent will

sharply reduce the usual difference between the frozen and equi-

33

34

librium shock properties that is associated with nonequilibrium

post-shock relaxation in a perfect ambient gas. Indeed, for nor­

mal shock velocities below approximately 15,000 rt/sec and ". >.50, a complete inversion of the normally endothermic nature of

this process to one which is exothermic (recombination-dominated)

may occur. Since this appears on the verge of failure of the

hypersonic shock approximations used in this investigation, how­

ever, further theoretical and (especially) experimental study is

warranted for conclusive proof.

(4) . It is predicted that a highly dissociated free stream will sub­

stantially reduce (and possibly eliminate completely) nonequilib­

rium overshoots in NO concentration, electron density and molec­

ular ion radiation intensity associated with the thermally-insig­

nificant reactions excited in shocked air. Therefore, a detailed

numerical study of the complete set of rate equations for a pre­

dissociated shock forO ~ «00 $. I , as well as experimental evi­

dence, is clearly of great interest.

In conclusion, it may be stated that although the present calculations have

been concerned with air and an unionized free stream, many of the general

qualitative trends shown should be applicable to related types of gas mix­

tures and/or preionization effects as well. Furthermore, while shock wave

behavior per se has been considered exclusively, it is clear that our results

provide a general basis for appraising the effects of free stream dissocia­

tion on hypervelocity body flow fields-

.[

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~ [

b[

[

[

[

[

LL

~L

L•.1

REFERENCE3

9. Gilbert, L. M., and S. M. Scala. Free Molecular Heat Transfer in the

Ionosphere. G.E. Space Science Lab. Rep. R61SD076, March, 1961.

8. Radcliffe, J. (Ed.) Physics of the Upper Atmosphere. New York: Academic

Press, 1960 •

4. Heims, S. P. Effect of Oxygen Recombination on One-Dimensional Flow at

High Mach Numbers. NACA TN-4l44, January, 1958.

35

"Measurements of Temperature and

and Helium Plasmas," Physical Review,

Wiese, W., H. F. Berg, and H. R. Griem.

Densities in Shock-Heated Hydrogen

November, 1960.

7. VLCLean, E. A., A. C. Kolb, and H. R. Griem. "Visible Precursor Radiation

in an Electromagnetic Shock Tube, Il Physics of Fluids, 4, No.8,

August, 1961.

3. Eschenroeder, A. Q., D. W.Boyer, and J. G. Hall. Exact Solutions for Chem­

ical Nonequilibrium Expansions of Air With Coupled Chemical Reactions.

Cornell .Aero. Lab. Rep. AF-14l3-A-l, May, 1961 (AFOSR 622, PSTIA AD-,

257 396).

2. Nagamatsu, H. T., J. B. Workman, and R. E. Sheer. "Hypersonic Nozzle Ex­

pansion of Air With Atom Recombination Present," Journal of Aero/Space

Sciences, 28, No. 11, November, 1961.

5. Whalen, R. J. Viscous and Inviscid Nonequilibrium Gas Flows. Institute of

the Aerospace Sciences Preprint 61-23, 29th Annual Meeting, Jan, 1961.

1. Bray, K. N. C. Departure From Dissociation Equilibrium in a Hypersonic

Nozzle. British A.R.C. Rept. 19-938, March 1958 (also see Journal of

Fluid Mechanics, 6, Part 1, 1959).

6.

--1-

-l ?

-'j

]

'1]

]

J '"

] ~

]

]

'J

]

JJJ.J

J "

J

.,.

36

10. Fowler, R. H., and E. A. Guggenheim. Statistical Thermodynamics. London:

Cambridge University Press, 4th Edition.

11. Treanor, C. E., and J. G. Logan. Tables of Thermodynamic Properties of

Air From 3,OOOoK to 10,000oK. Cornell Aero. Lab. Rep. AD-1052-A-2,

June, 1956 (AFOSR TN-56-343, ASTIA AD-95 219).

12. Hilsenrath, J., M. Klein, and H. W. Woolley. Tables of Thermodynamic Prop­

perties of Air Including Dissociation and Ionization From l,5000

K to

15,oOOoK. N.B.S. AEDC TR-59-20, 1959.

13. Feldman, S. Hypersonic Gasdynamic Charts for Equilibrium Air. AVCO Res.

Lab. Rep., Jan. 1957.

14. Moeckel, W. E., and K. C. Weston. Composition and Thermodynamic Properties·

of Air in Chemical Equilibrium. NACA TN-4265, April 1958.

15. Chu, B. T. Wave Propagation and Method of Characteristics in Reacting Gas

Mixtures With fl.pplications to Hypersonic Flow. Brown University WADC

TN 57-213, May 1957 (ASTIA AD-118 350).

16. Inger, G. R. One-Dimensional Flow of Dissociated Diatomic Gases. Douglas

Aircraft COIllJ?any, Inc. Report SM-38523, May, 1961 (ASTIA AD-260 027).

17. Inger, G. R. Nonequilibrium Hypersonic Similitude in a Dissociated Diatomic

Gas. Douglas Aircraft Company, Inc. Report SM-38972, October, 1961.

18. Viray, K. 1., J. D. Teare, B. Kivel, and P. Hammerling. Relaxation Processes

and Reaction Rates Behind Shock Fronts in P.ir and Component Gases, AVeO

Res. Lab. Rep. 83, December, 1959.

19. Lin, S. C. Rate of' Ionization Behind Shock Waves in l'.ir. AVCO Res. Lab.

Note 170, December, 1959.

20. Viray, K. L. Chemical Kinetics of' High Temperature Air. American Rocket

Society Preprint 1975-61, Internat. Hypersonics Conf'., M.I.T., Aug. 1961.

4/.[

[

F[

[

[

~[

~[

b[

[

[

[

[

L[

4L~ L

.- -1 .

-l '··1

'-1

l]

]

]

]

]

]

"1• .1

]

J_J

.J

Jq

].

21. Carom, J. C., B. Kivel, R.Taylor, and J. D. Teare. Absolute Intensity of'

None'iuilibrium Radiation in Air and Stagnation Heating at High .Al.tiilldes.

AVCO Res. Rep. 93, Dec. 1959.

22. Fay, J. A., and F. R. Riddell. "Theory of Stagnation Point Heat Transfer

in Dissociated Air," Journal of Aero/Space Sciences, 25, No.2 (1958).

23. Inger, G. R. Chemical None'iuilibrium Effects in the Laminar Hypersonic

Boundary La;yer, Bell Aircraft Corp. Rep. 7010-6, March, 1959 (MOOR

TN-59-237, ASTIA AD-2l2 007).

24. Bleviss, Z. 0., and G. R. luger. The Normal Shock Wave at Hypersonic

Speeds. Douglas Aircraft Company, Inc. Report SM-22624, November, 1956.

37

FIGURE .1

SHOCK CONFIGURATION AND TERMINOLOGY

~ ~.

.[[

E[

[

[

~[

4 [

C[:

[

[

[

[

[

[

-L· L

SHOCKED GASUS"

Uoo =: Veo SINO'

Us = Vs TAN (O'-8s )

PRIMARYSHOCK FRONT

" ~,""

AMBIENT GAS

a

V ~±fX"'l:::~-.~__. -::::'00_" ~_

38

'}.

l~

-j

'1ENTHALPY PARAMETER ~

39

1.0.8

FIGURE 2

.4 .6a

.2o

-2f3

",

\\

\

'" "\,,,~ ~M = 9/2 (COMPLETE VIBRATION)

'""" ~ IG'0r)': "'"

~ "Q-YI</ "'-'" '~)~ '"f3M=~ ~(. K.......

2

""- roo-, 0(NO~VIBRATION) r--...- "-....

r----. -,~r--:::

~~

~I

9

o

5

7

8

6

]

]

J~

J'" ,~

]

]

']

JJJJJj"j ~

SHOCK WAVE ANGLE VERSUS DENSITY RATIO AND FLOW DEFLECTION

.r

.[

[

t[

[

[

.; [

.[

b[

[

[.

[

[

L[

. [

" l-

6560

X POINTS DENOTE

SIN a s SIN 0$ FOR E= 10l-E .

20 35 40 45 50 55

SHOCK FLOW DEFLECTION ANGLE Os (DEGREES)

15105

FIGURE 3

O-f---+---l--+---+---+----f--+--+---+----f--r.--+---Io

30,-+------+--+---+--+--cf--l'A7'F7'Y1--+---+--4---+--I----+-----1

80+--t--+--+--+-~~---1~~...p~-+-""::::"~~4~-f-~-+---+l

20-t-----+---+-----,

70;-----jr-----t----t---t---+--r----k---t-\---!\---7ff--It---./4--/--l

10 -I------j---,

40

60G'LlJLlJCl::<.:>LlJ0

t:>LlJ 50-I<.:>z<I:LlJ>

I <I:- ~

40

_/\y, YeQ.

~ l'~ I i!~

FROZEN AND EQUILIBRIUM SPECIFICHEAT RATIOS FOR DISSOCiATED AIR

-1

lJ

]

J·] -

JJJJJJJ

1.7

1.6

1.5

1.4

1.3

1.2

1.1

Iy =5/:'

~7'./

//y~~\~ ~

y

,,0-..\\'O~ ,~

~O "" ~p ~~i~ ,,-,0

/:/ ~,«:>«:-~

¢~v

~rq\'IIij

\ )Y II\/~ VI1\ \ . /

\ \~/JVJ\\\,

t:: ~ /A ~~ /r---I--- YEQ.,,t>-1/--~ -r-- /I---r--- 1\ 'O·~-

) "

J

J ~

1.0 o .2 .4

FIGURE 4

.6 .8 1.0

41

Hoo

= 2h 0.!!2.,.

V 2""

42

5

4

3

2

FREE STREAM DISSOCIATIONENERGY PARAMETER FOR AIR

//

Ic-,' /

~'/'<. • .

~/~'/I;

~~/

$;~~/~~/

~:::>/

//

//

/;

//

//

/

Voo

= 10,000 F.P.S.

15,000 F.P.S.

20,000 F.P.S.

25,000 F.P.S.

30,000 F.P.S.

35,000 F.P.S.

[

E[

[

[

if [

,[

C[

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[

[

[

[. [~ L.

43

.7.6.5.2o .1

FIGURE 6C

-.1-.2'-.3,

FIGURE 6 B

.10-t--.......-....,.--,.....-__- ......-.....,--opo--.....- ......-.....,~_-.4 -.3 -.2 .1.2 .3 .4 .5 .6 .7

Hs - H....

FIGURE 6A

THE FUNCTIONS Kp AND KH: NORMAL SHOCK THE FUNCTIONS kp AND KH: WEDGE AT INCIPIENT DETACHMENT THE FUNCTIONS Kp AND KH: ATTACHED 30°-WEDGE SHOCK-7, -.- .

\'" 1.03.6

'I

IJ ...-...-

Z = 1.05 ...-

l .9 ...- fJM= 7/2....-...-....- 3.2 13M= 9/2

l KH

1 1.0KH .8

2.8

-l13M = 7/2

.9 13M = 9/2 2.41~

l~ Z=2.0

.5 .613M = 7/2 2.0

] fJM = 9/2

] .4 .51.6

1J

Z = 1.05 KH

J.3 .4 ....---

...- ....- 1.2

J Kp

.2 .3.8

0

J -. -.2 -.1 0 .1 .2 .3 .4 .5 .6Hs - H.... .2

.4

:"f

[

F[

[

[

-, [

~ [

b[

[

[

[

[

[

[.~ L

'L

1.0.8.6

ex""

.4

FIGURE 7

MINIMUM NORMAL SHOCK VELOCITY IN AIRFOR HYPERSONIC FLOW WITH A

DISSOCIATED FREE STREAM ...,

';-."r "7 to? --., r1..[ h~ h.?)/• <.-IT' ~ /0 ..-7 :- _.) Z I 0 ~ ,-

'AE ~ .1 ~f 1;::. ...."\.. (~npO eO ".1 7 I (I 1. -!vr ...

.2o

_13M = 7/2__ 13

M00 = 9/200

4

'\.IToo = Too IEQ.

~ 5~;'.jJl

~~T00 = 40000 K 4800 Poo==~EQ. ~ -, l'~ 4500~ 3830 _0Q-b 4150

/'"

25500/~~3350-- Too == 2000 0 K

~ lit2000

WOOoK1-------~

~

saOoKI--- .-~-1---...--l-

~

I 2000 K

~ -1--- '--

~..

5 x 10

,

z~ 2 x 104

Szin

>'i-

44

IL- ~_ L_ l_ L- i .

'----'"'-..J.\

ILLUSTRATION OF COMPARISONS BETWEENPREDISSOCIATED AWD PERFECT AMBIENT GAS SHOCK WAVES

."

Sc::amco

(A) CONSTANT SHOCK VELOCITY

TEST SECnON

(B) EQUAL TOTAL VebOC1IY

£;Y~t!/"',,~

0+--"""T'-.....,--r---r----r----r-......,r------~-""'T""-_:"1o .2 .4 .6 .8 1.0

PREDISSOCIATION EFFECT ONPr: I PoeV00

2

4

[

F[

[

[

-. [

~[

C[

[

[

[

[

[

[

~ [

~ L

1.0

INCIPIENT DETACHMENT

..8

_;;..- ATTACH ED SHOCKS­_........--

--.. ~~---':"'::::::::,,:::::,....£D ETACH ED SHOCKS

.6

........-

FIGURE 10

.4

13M"" = 7/2

13M"" = 9/2

.2

PREDISSOCIATION EFFECT ONFROZEN DENSITY RATIO

FIGURE 9

6

7

8

3

5

I'F

1.10

1.05

1.00

.95

RPF

.90

.85

.80

00

46

PREDISSOCIATION EFFECT ONENTHALPY FUNCTION RHF

13M = 7/2002

flM = 9/200

1.0

'--- f3M =7/2002

f3M = 9/2002

.8

------~INCIPIENT DETACHMENT

.6

FIGURE 11

.4

PREDISSOCIATION EFFECTON FROZEN TEMPERATURE T F / Vj

.2

............................

-............ .-'"-::::-~~--- .......... --- ............ ............

.......... ..................

""'"~INCIPIENT DETACHMENT

.90

.70

1.0

. 80

.60

O+--"""T"--r---.,.....-"""T"--"'--"""'-"""'---"--"""'--'

1.1

1.2/ ATTACHED SHOCKS

/ .

,,/,,/

/"./'

/"/

/ NORMAL SHOCK1.0--f'ClO:::-------""";---------------

-------- -- -....

J

JJJ

1l1*

]~

~]

~]

1-~

o .2 .4 .6 .8 1.0

FIGURE 12

47

PREDISSOCIATION EFFECT ON EQUILIBRIUM DENSITY: NORMAL SHOCK

CONSTANT VELOCITY CaNSTANT TOTAL ENTHALPY

.75

1.0.8.6.4

14

18

20

16

"

.~-12

.75

10

\.008

6

4

2

0.S 1.0 0 .2.6.4.2o

20

2

O+--__--r--,.---r--r--""""T---~-r--~

4

~ 16

SC::101m-W>

;Ol £/.

~~~~~.~~~~~~~~~~~~~~

IL-. :

~1

PREDISSOCIATION EFFECT ON EQUILIBRIUM DENSITY: WEDGE AT INCIPIENT DETACHMENT

CONSTANT VELOCITY CONSTANT TOTAL ENTHALPY

PEQ.

20

.75

1.0.8.6.4.2o+-.....,~.....,---.,.-....,..-...,...-.,...-.,....-.,..........,-.....,

a

1.00

1.0.8.6.4

PEQ.-20

Poo

18

16

""i5 14C::lOm-W 12CD

10

8

6

4

2

00 .2

"'"'-0

V'I0

PREDISSOCIATION EFFECT ON EQUILIBRIUM DENSITY: 30°_ WEDGE

PEQ. CONSTANT VELOCITY CONSTANT TOTAL ENTHALPYPEQ.

13Poo 13 Poo t h)Ii 'i:; I p,r \-~..._'

(i"" l I12 12 \. n 1" I {)If?

11Hoo2 = .10

11

10 10.25

9 9

:::!!QC 8 8::am .50....w(") 7

6 .75 .75 6

5 5

4 4 \ \ \\ \ \\(4'\

'/ \\\ 'Q)3 ....- 3 \\\ ~- \\, '\

Q) \\\,

2 2

1

0 .2 0 .6 .8 1.0

a·· a oo2""2h

• '"r-. r--l r-1 r--l r-1 r-1 II rJ n rn rJ II r--1 r--J r-1 rT1 II :J r-l

C>-

N ~ 0. .

N - 0

FIGURE 14A

-j" ~

-~.. -

!1

-1

!-1

I ,~

] ,~

]

]

]

1J_I

JIJ ..~I ~

J51

PREDISSOCIATION EFFECT ON EQUILIBRIUM TEMPERATURE: WEDGE AT INCIPIENT DETACHMENT

CONSTANT VELOCITY CONSTANT TOTAL ENTHALPY

1.0.4

T F /T F CURVES002 001

.2

TEO.--T

Fool

1.0

.9

.8

.7

.6

.5

.4

.3

.2

.1

00

CASE T F ,oK001~ 25,850

CD 18,470~ 9,410

HOO2 = .10

.1

00 2 .4 .6 .8 1.0

01002

• .. .. I~ Ii..

r---' l'"1 r-1 r--1 r-"i rJ r--J r-1 r--1 n-1 l1 r-1 r-1 l1 l1 rn r--1 :-'l ~

'---------J

.'

PREDISSOCIATION EFFECT ON EQUILIBRIUM TEMPERATURE: 30°- WEDGE

CONSTANT VELOCITYCONSTANT TOTAL ENTHALPY

1.0

Q)

.8.6

-.:==-}®-----Q)

.50

.4

/~\/T F /T F CURVES

002 001

TEQ.

1.2 TFool

1.1

1.0

.9

.8

.7

.6

.5

.4

.3

.2

.1

00 .2

9,8108,740

4,550

-----".~ .75

CASE

(])Q)@

.6.4

H00 = .102

.2o

.1

.2

.4

.3

00 2

U1W

PREDISSOCIATION EFFECT ON EQUILIBRIUM DISSOCIATION LEVEL: WEDGEAT INCIPIENT DETACHMENT

ZEQ.ZEQ.

2.3

2.2

2.1

2.0

1.9

1.8..,C5c::lO 1.7m-(IICP 1.6

1.5

1.4

1.3

1.2

1.1

1.0

H..2 ... 10

.2

CONSTANT VELOCITY

.4 .6a

002

2.3

1.00

.75 2.2

2. I

2.0

1.9

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

.81.0

1.0 0

,--'

CONSTANT TOTAL ENTHALPY

.4 1.0

.50

/

CONSTANT TOTAL ENTHALPY

1.7

1.1

1.9 ZEQ.

1.3

1.8

1.7

1.8

1.1

1.9

1.3

1.2

1.6 1.6

-n

Qc;iIlJ \.5 1.5m...enn

1.4 1.4

1.01.0 .0 0 .2 .4 .6 .8 1.00 aCl""2 00

2

+- • ~ :i .,r- r--J r--1 rJ r--1 r-1 r--i rJ rJ !"r1 rl r-1 rJ r-1 11 rTl r--j (I :-)

-'-Jl'

__i

'. ELECTRON MOLE FRACTION PREDISSOCIATION EFFECT ON EQUILIBRIUMIONIZATION BEHIND A NORMAL SHOCK

CD .75

--

.25.50

CONSTANT TOTAL ENTHALPY

10-2

5 x 10- J

5 x 10-2

.25

CONSTANT VELOCITY

HOO2 = .10

10- 3 10- 3

5 II 10- 4 j x 10-4

l®Q)

I@

10-4 10-4

0 .2 .4 .6 .8 LO 0 .2 .4 .6 .8 .0a oo2 a oo2

10- 1

5 x 10- 2

5 II 10- 1

."

i510- 2C

1Iam-0-

S x 10-3

1.0

\

\

.8.6

\\\

HTOO2 =.10

.21.0 0

CONSTANT TOTAL ENTHALPY

.8

PREDISSOCIATION EFFECT ON FLOW DEFLECTiONANGLE BEHIND A DETACHED SHOCK ANGLE OF 80°

CONSTANT VELOCITY

8S

65H" 2 = .10

60

55

"1150

(5c:;gm-....

40

35

30

25

20

15

100 .2 .4 .6

°""2

.:~ ...t.

PREDISSOCIATION EFFECT ON ATTACHED SHOCKANGLE FOR A 30° HALF-ANGLE WEDGE

60

q (DEGREES) ei/ q (DEGREES)

."

C5c:~m...00

50

40

30

20

.50

.4o+--~-~-~-.,..--.,....-,--.,.--.,.--.,.----.,

.6 .8 1.0 0 .2 .4 .6 1.0

PREDISSOCIATION EFFECTS ON WEDGE DETACHMENT

CONSTANT SHOCK VELOCI TY CONSTANT TOTAL ENTHALPYJ- I.· ~

.25

(J,65 (DEGREES)

HToo2 =·10.75

70

(J,65

(DEGREES)

80 HOO2 = .10

60"TI

C5C /.50::0m.... .0

-0 '\,~-;Q

'\,

/~

1:J\ '~,

40

30

1.0.8.4.21.0 0.8.6.4.2o

ii

•

i

II\,,----._.:-_---'--~ ............_-----------.......--------------------------------....--------------_...