ntrs.nasa.gov · The author wi shes to express hi s deep appreci ati on to Dr. Rudol f Hermann for...
154
• • .. ' . • UARI Research Report No. 66 INVISCID HYPERSONIC FLOW OF CHEMICALLY RElAXING AIR ABOUT POINTED CIRCULAR CONES by JOrgen This work was partially supported by the National Aeronautics and Space Administration under research grant NGL-a -002-00 University e. f Alabama Research Institute Huntsville, Alabama ' May 1969 • '; r- ., 3 • 4v ..... i ---"'(A""'Ca""'SSOo;'''''OM''MdU*'MI"EI''-) -=- (THIU) .L7__ (J,£) (.1l-//03:5L /if (NASA CI 01 fMx 01 AD MeMBER) (eA ffiiblY) .. • , II '. • https://ntrs.nasa.gov/search.jsp?R=19700020678 2020-05-18T03:49:35+00:00Z
Transcript of ntrs.nasa.gov · The author wi shes to express hi s deep appreci ati on to Dr. Rudol f Hermann for...
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UARI Research Report No. 66
INVISCID HYPERSONIC FLOW OF CHEMICALLY
RElAXING AIR ABOUT POINTED CIRCULAR CONES
by
JOrgen Thoenp~
This work was partially supported by the National Aeronautics and Space Administration
under research grant NGL-a -002-00
University e.f Alabama Research Institute Huntsville, Alabama '
May 1969
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UARI Research Report No. 66
INVISCID HYPERSONIC FLOW OF CHEMICALLY
RELAXING AIR ABOUT POINTED CIRCULAR CONES
by
JOrgen Thoenes
This work was partially supported by the National Aeronautics and Space Administration
under research grant NGL -01-002-001
University of Alabama Research Ipstitute Huntsville, Alabama
May 1969
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r
FOREWORD
The materi al contai ned in thi s report formed the a.,thor's di ssertati on
for the degree of Doctor of Philosophy in Engineering Mechanics at the
Uni versi ty of Alabama.
The author wi shes to express hi s deep appreci ati on to Dr. Rudol f Hermann
for his guidance and advice during the course of this investigation. Likewise,
the author is indebted to Dr. Jerome J. Brainerd for many stimulating discussions
concerning all phases of this work.
Sincere appreciation is also expressed to Dr. James D. Matheny, Dr.
Howard B. Wilson, Jr., Dr. Harold R. Henry, and Dr. Fran Bosnjakovic from
whose criticism the author has benefited.
The author feels much obliged to Mrs. Sylvia Heard who skillfully
programmed the nonequilibrium, as well as the equilibrium flow solutions; to Dr.
Kenneth O. Thompson who kindly furnished a subroutine for the equilibrium
composi ti on; and to Mr. Robert A. McGraw for programmi ng the frozen flow
sol uti ons.
The efforts of Mrs. Ann White who typed the manuscript are gratefully
acknowledged.
The author is indebted to the National Aeronautics and Space Administration
for partial support of thi s work under research grant NGl-0I-002-00l •
ii .,J
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ABSTRACT
Inviscid hypersonic flow of chemically relaxing air about pointed circular
cones has been calculated using Dorodnitsyn's one-strip integral method. The c:ir
model used consists of 02' N 2, 0, N, and NO, of which the relative amounts
are controlled by six reaction equations which actually represent eighteen chem-
ical reactions. Also considered are the limiting cases of nonequilibrium flow,
namely, chemically frozen flow, and the flow with chemical equilibrium. Vibra-
tional equilibrium has been assumed throughout.
The principal contribution of this work lies in the method of solution.
Specifically, it is demonstrated that the replacement of the approximate tangential
momentum equation, and of all approximate species continuity equations by their
exact forms along the body surface I eads to a system of equati ons whi ch, fi rst, can
be stably integrated, and second, yields results which compare quite well with
those obtai ned by the method of characteri sti cs. A detail ed di scussi on of the
polynomial approximations employed in the integral method, the introduction of a
characteri sti c relaxati on length defi ned in terms of the i ni tial speci es MISS frocti on
grodi ents, as well as many new resu I ts for hypersoni c flow of a: r wi th fi ni te rate
nonequil ibrium di ssociati on abou~ ci rcular cones are further contributi ons of the
work reported herei n •
iii
.-:-C '_. i
,.
The discussion c,f results includes two cases which show the very good
agreement of the results from the integral method with those from the method of
characteri sti cs. Vari ous other cases serve to illustrate the i nfl uence of free stream
vel oci ty (54oo:s; V,,,em/sec] :s; 11186), cone semi -vertex angl e (20 0 :s; 8 :s; 45 0),
and altitude (31 km to 80 km) on the variation of all flow field variables. The
anal ysi s of a cone with constant semi -vertex angl e and free stream vel oci ty at
three different altitudes is used to demonstrate the validity, and its range, ,.,f the
nonequll ibrium seal ing law Poo ' L 0= constant.
For one case, the asymptotic equilibrium state of the nonequilibrium flow is
compared with the. results of a conical equilibrium flow calculaticn. According
to theoretical considerations an agreement between the two cannot be expected,
and this is confirmed by the comparison. Although the cone surface pressures
agree very closely, it is found that the values for the asymptotic body surface
temperature are higher than those for coni cal equi I ibrium. Both, the asymptoti c
surface densi ty and the surface vel oci ty stay below thei r val ues for coni col
equilibrium.
iv
.. ;. -t-_.
TABLE OF CONTENTS
FOREWORD ii
ABSTRACT ••• III
TABLE OF CONTENTS v
LIST OF TABLES vii
LIST OF FIGURES ...
VIII
LIST OF APPENDICES ..
XII
NOMENCLA TURE ...
XIII
Chapter I. INTRODUCTION 1
Chapter II. GAS MODEL AND THERMODYNAMICS 6
201 Gas Model 6
2.2 P(.'rti ti on Functi ons 7
2.3 Thermodynami c Properti es 9 2.3.1 Thermal Equation of State 9 2,3.2 Internal Energy and Enthal py 11
2.4 Rate Processes 15 2.4.1 Finite Rate Nonequilibrium 15
; 2.4.2 Speci es Continuity Equati on 15 ~ ..
2.4.3 Rate Equations 17 2.4.4 Reaction Rate Constants 22
2.5 Equilibrium Constants 23
2.6 Equilibrium Compositicn 26
2.7 Revi ew of A ssumpti ons 29
Chapter III , FORMULATION OF THE PROBLEM 32
3.1 Basic Equations 32
3.2 Boundary Conditions 36 3.2.1 Body Surface and Shock Conditi ons 36 3.2.2 Frozen Shock Conditions 38
' .. ' 3.2.3 Equilibrium Shock Conditions 40
V
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1
Chapter IV. METHOD OF SOLUTION 43
4.1 The Method of Integral Relations 43 4.1.1 The Method in General 43 4.1.2 The First Approximation 46
4.2 Nonequilibrium Flow 47 4.2.1 General Remarks 47 4.2.2 The Standard Approximation 48 4.2.3 The Semi -exact Procedure 54
4.3 Initial Solution 59 4.3.1 Frozen Flow 59 4.3.2 Initial DerivatIves 61
4.4 Equilibrium Flow 67
4.5 Numerical Techniques 69
Chapter V. DISCUSSION OF RESULTS 72
5. 1 Frozen Flow 72
5.2 Equilibrium Flow 73
5.3 Nonequil ibri um Flow 74
5.4 Conclusions 81
REFERENCES 84
TABLES I - V 88
FIGURES 1-41 93 ..
APPENDICES A - C 133
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.-- I .. _.
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I
II
III
IV
V
i _ eia . %
LIST OF TABLES
Gas Model Data 88
Atomic and Molecular Constants 89
Oi ssaci ati on Rate Constants 90
Equilibrium and Recombinatian Rate Canstants 91
Tabulation of Cases for Chemically Relaxing Flow 92
vii
L) co .3 sc - -, :!.- Q ,~ ---..,.:=>. ...... ,,-..:.. ,..., -"",-.::;:~.==,,.,,.-,.... ' ',-
LIST OF FIGURES
Figure Title
1 Temperature [OK] and compressibility factor at the surface 93 of a circular cone for air in thermodynamic equilibrium (Data from Ref. 21).
2 Ratio of chemical to vibrational relaxation time for mole- 94 cular oxygen and nitrogen (C 0 = C N = C NO = 0).
3 Coordinate system on a pointed body with convex 95 longitudinal curvature.
4 Differential flow field geometry for the determination 96 of the metri c coeffi ci e nts.
5 Velocity diagram for locally oblique shock waves. r:r7
6 Stri p arrangement in shock layer. 98
7 Variation of shock wave angle crwith cone semi-vertex 99 angle 9 {chem. frozen, vibr. equil., Too = 250 OK).
8 V'':Iriation of angular shock layer thickness ~ with cone 100 semi -vertex angl e 9 (chem. frozen, vibr. equil., T", .,. 250 OK ) •
9 Variation of cone surface temperature with semi-vertex 101 angle 9 (chem. frozen, vibr. equil., Too = 250 oK).
10 Variation of cone surface density with semi-vertex 102 angle 9 (chem. frozen, vibr. equil., Teo=250 0 K).
11 Variati on of cone surface pressure with semi -vertex 103 angle 9 (chem. frozen, vibr. equil., Teo = 250 0 K).
12 Variation of cone surface velocity with semi-vertex 104 angle 9 (chem. frozen, vibr. equil., Teo=250 0 K).
viii
':" < ..
' .. " ... ' >.
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Surface compressibili·y factor Z~ for equilibrium cone flow (To:> = 273.16 oK, p = 1.0 am).
0
Surface tem~rature for equil ibrium cone flow (T 0:> = 273.16 oK, p = 1.0 atm).
0
Surface densi~ for equilibrium cone flow (T = 273.16 K, p = 1.0 atm).
0:> 0
Shock wave angle cr for equilibrium cone flow (T =273.16 0K, p = 1.00tm).
0:> 0
Survey of investigated nonequilibrium flow cases.
Cone surface spe -i es mass fracti ons for case 1 @ = 45. 43? V 0:> = 6638 . 0 m/sec, To:> = 273. 16 oK, Po:> = 0.01 atm).
Cone surface species mass fractions for case 2 @ = 41.07 0, V 0:> = 5974.0 m/ sec, To:> = 273. 16 oK, Pa> ,. 0.01 atm).
Cone surface temperatures for cases 1 and 2 ° (To:> = 273.16 K, Pa> = 0.01 atm).
Cone surface densities for cases 1 and 2 (To:> = 273. 16 oK, Pa> = 0.01 atm).
Shock wave angl es for cases 1 and 2 0
(Ta> = 273.16 K, Po:> = 0.01 atm).
Cone surface cPressures for cases 1 and 2 (Ta> = 273.16 K, Po:> = 0.01 atm).
Cone surface velocities for cases 1 and 2 (T = 273.16 OK, P. = 0.01 atm).
0:> 0:>
Cone surface Mach numbers for cases 1 and 2 (To:> = 273.16 OK, Po:> = 0.01 atm).
Influence of free stream velocity on cone surface species mass fractions @= 30°, altitude 80 km) •
• IX
;iido-' '- -.-."...,.--=--~ •. ='= .. ; ,... -iii=iiIiIlfiII:'lI-Cl".--_C::- ~."....> -- -- . -... _.
105
106
107
100
109
110
111
112
113
114
n5
116
117
118
---
27 Infl uence of free stream v'll oci ty on cone surface 119 temperature f- = 30°, altitude 80 km}.
28 Infl uence of free stream vel oci ty on cone surface 120 density (S c 30
0, altitude 80 km).
29 Infl uence of free stream vel oci ty on shock wave 121 angle f: = 30°, altitude 80 km).
30 Infl uence of free ~trenm velocity on cone surface 121 pressure 0"30°, altitude 80km).
31 Species mass fractions on cone surfaces for different 122 altitudes (3= 30°, V .c 9350.0 m/sec).
'" 32 Scal ed temperature on cone surfaces for di fferent 123
altitudes es= 30°, V", = 9350.0 m/sec).
33 Densi ty on cone surfaces for different al titudes 124 f. = 30°, Ven = 9350.0 m/sec).
34 Infl uence of cone semi -vertex (mgl e on surface 125 species mass fractions N", = 9350.0 m/sec, altitude 40 km).
35 Influence of cone semi-verte)' al1gle on surface 126 temperature N", '" 93:30.0 ;on/sec, altitude 40 km).
36 Infl uence of cone semi -vertex angl e on surface 127 density N", '" 9350.0 m/sec, altitude 40 km).
37 Infl uence of cone semi -vertex angl e on surface Mach 128 number N", = 9350.0 m/sec, altitude 40 km).
38 Influence of free stream velocity (and altitude) on 129 characteristic relaxation length @ = 30 0
).
39 Infl uenc" of al ti tude (and 9) on characteri sti c 130 relaxation len;Jth N", '" 9350.0 m/sec).
40 Infl uence of cone semi -vertex angl e (tlnd al titude) 131 on characteri sti c rei axati on length No:> -= 9350.0 m/sec).
x
--~. -- .. - . A e <. . 010. >.C{ - ~ .. ~ ... - --:::..:..... .....
•
.<
,;".
,.; . . .
b·--··--~' .. ---
41 Compari son of coni cal and asymptoti c equi 1 ibri um for case 10 @= 30
0, V CD = 9350.0 m/sec, 01 titude 40 km)
in a Mollier diagram (Ref. 41; T = 273.16 oK, p = 1.0 atm, p = 1.288 kg/m3~ R = 288.188 Jlkg or.). o 0
xi
. •
- = . - '~~ "-::':;;;;;,';;;.;;0;;;;;0;' ;;;% __ oii.~9"'"""'l3~C~.;;;;;;;;: =;;;;OQ~_ ;;;;;a~.O;;. '1'1 .. j .... ' ~"Qa-=. - ---:'--.. -.-
132
.... l __ r .... --. ~ - -.
LIST OF APPENDICES
A Some Expressions Related to Thermodynamics 133
B Dimensionless Functions O. for Differential Shock Relations and Initial' Derivatives 134
C Functions a .• and A. in Equations (4.30) through (4.3~
, 136
•
xii
I _. __ . __ _ ~:.-~.
NOMENCLA TURE
af
Frozen speed of sound
b Expression defined by Eq. (3.41)
C. I
Mass fraction of species i
C P
Specifi c heat (frozen chemistry)
E Total specific internal energy
E. I
Internal energy of species i
e. I
Specifi c internal energy of species
F l' F 2' F 3 Dimensionless functions defined by Eq. (4.21) through (4.23)
g. J
Degeneracy factor
H Total enthal py
h planck's constant, also static enthalpy
h1,h
2,h
3 Metri c coeffi ci ents
K longitudinal body curvature
K c
Concentration equilibrium constant
k Bol tzmann 's constant
kd,M Dissociation rate constant
k r,M Recombination rate constant
j Indicator for plane or axisymmetric flow, defined in Eq. (3.11)
l Dimensionless function defined in .Appendix A
M Molecular weight of undissociated air, also Mach number
M. I
Molecular weight of speci es i
m Particle mass
, xiii
,.----- .
m 0
Total oxygen mass fraction
N* Avogadro's number
n. I
Number of particles of species
p Stati c pressure
Q Partition function
... q Velocity vector
R Gas constant of undissociated air, also radius of curvature
R* Universol gas constant
r Radial coordinate
S Dimensionless function defined in Appendix A
T Temperature
t Time
u Velocity component in x-direction
v Velocity component in y-direction
V Volume, velocity when subscripted
W. Mass rate of formation of species per unit volume •
X. •
Concentration of species
• x1,x
2,x
3 General i zed coordinates
x,y Coordinate directions
Y. •
Molar concentration of species
y.:; V. ;tV. • • •
Mol fraction of species i
Z Compressibility factor
xiv
a
e. J
e
* e. I
e.e J
e~ I
11-
A
'V. , \).1 I I
S
p
a
Td •
• I
T~ I
cD
n
~ . . , .
. \1 ';:'
....
~ .
Indi cator, used in speci es conti nui ty equations
::: a-e
Defined by Eq. (4.45)
Defined by Eq. (4.16)
Indi cator, used in x -momentum equati on
Excitation energy
!lody surface i ncli nati on ongl e
Characteristic temperature of dissociation of species
Characteri sti c temperature of el ectroni c exci tati on, state j
Characteristic temperature of vibration of species i
Symmetry number
Characteristic relaxation length, defined by Eq. (4.68)
Stoi chiometri c coeffi ci ents
Di mensi onl ess x-coordi nate
Density
Shock wove inclination angle
Chemical relaxation time of species
Vibrational relaxation time of species
Ci rcumferenti al coordi nate
Dimensionless functions, defined in Appendix B
xv
- . i
•
.-.
t
v
r
e
d
I .... VI
b
co
s
n
t
o
tot
M
02
N2
o
N
NO
Superscri pts
Translation
Vibration
Rotation
Electronic excitatjon
Dissociation
Refers to reacti ons I .... VI
Subscripts
Body surface
Free stream
Behi nd the shock wave
Normal component
Tangential component
Species, i:= O2, N
2, 0, N, NO
Undi ssociated state
Total
Third body species
Mol ecular oxygen
Molecular nitrogen
Atomic oxygen
Atomic nitrogen
Nitri c oxyde
xvi
.-- ... --~--•• J>_ ..... ""== -.o=·i
.. "'.
CHAPTER I
INTRODUCTION
Over the past few years consi derabl e effort has been devoted to the under
standing and the theoretical prediction of the noneq"uilibrium flow field conditions
encountered in hypersonic flight. Present technology dictated the use of blunted
body shapes for reentry vehicles in order to minimize the convective "eating rate
to the body surface. Quite naturally, calculations of flow fields surrounding
hypersonic blunt bodies were of prime importance. For a survey of calculation
techniques for this class of problems the reader is referred to Hayes and Probstein
(Ref. 1). In particular, the general method of integral relations, proposed by
Dorodnitsyn (Ref. 2), proved to be quite successful in predicting shock wave shapes
and other flow field variables for certain dmple blunt bodies. Also, this method
is readily adaptable to machine calculations. The investigations of Shih, Baron,
Krupp and Towle (Ref. 3), Hermann and Thoenes (Ref. 4), and most recently,
Thompson (Ref. 5), are some examples of :nviscid real gas flow field calculations
using the integral method.
Comparatively less attention has been paid to the high temperature real gas
flow past pointed bodies, which appear to be of growing interest and importance.
Certainly the time will come when it is desirable to land on the Earth's surface
some space probes returning from for distant planets. These probes will reenter the
1
2
atmosphere at a speed far above earth parabolic speed. Both/convective and
radiative hea~;ng will occur, but because of the high speed, radiation is expected
to be the dominant source of heating. It was therefore pointed out by Allen (Ref .6)
that the shock wave sweepback provided by pointed conical entry bodies could be
used to reduce the gas luminosity grossly, and thus the total heating rote. There-
fore, be.ides being of fundamental interest, the calculation of high temperature
nonequilibrium flow past pointed cones may be of very practical importance in the
near future.
It is interesting to note that one of the first approximate theories for non-
equilibrium cone flow was developed by Chapman (reported by Stephenson, Refol)
in order to determine relaxation times of oxygen and nitrogen vibrations. Using
essentially a stream tube method, Chapman derived equations which related the
coordinates of the curved shock wave to an effective relaxation time of the flow
in the shock layer. Having obtained shock wave coordinates from experiments,
Stephenson (Ref. 7) evaluated relaxation times which agreed reasonably well with
data obtained from shock tube experiments.
If the flow field is entirely supersonic then the method of characteristics
is available for its calculation, provided that some initial data are available.
Calculations using the method of characteristics were first reported by Sedney,
South and Gerber (Ref. 8) for the case of vibrationally relaxing flow of pure nit-
ragen over a wedge. These calculations were later extended to include flows past
---j
3
circular cones (Ref. 9). The authars also reported the necessity of introducing
modifications in the method of characteristics in order to increase the accuracy of
the results. This work was followed by Capiaux anJ Washington (Ref. 10) who
treated nonequilibrium flow of an "ideal dissociating gas" (so-called Lighthill gas)
past a wedge, also using the method of characteristics.
In Ref. 11, Wood, Spri ngfi el d and Pallone i nvesti gated the flow over axi-
symmetric bodies by using the method of characteristics. They employed a multi-
component gas model and included a simple model for vibration-dissociation
coupling. Results were given for flows over blunted cones and ogival bodies.
The first application of Dorodnitsyn's integral method to the real gas flow
over pointed bodies was given by South (Ref. 12). He considered vibrationally
relaxing flow of a pure diatomic gas past a circular cone, using the one strip
approximation, and constant relaxation time. He also reported that a system of
equations containing the tangential momentum equation and the vibrational rate
equation in their exact forms could not be integrated successfully. He found it
necessary to use linear approximations in all the conservation equations. The
pressure di stributi on obtai ned from thi s set, together with a set of corrected
equati ons then served to ob tai n a corrected sol uti on. The appl i cati on of the two
and three strip integral method to the same problem is discussed in Ref. 13.
South and Newman (Ref. 14), and Newman (Ref. 15) discussed a modifj-
cati on to the one stri p integral method as appl i ed to wedge flow. Instead of usi ng
".
-- - - .. - .. 7". =-- '"-'-<-:. ~= ._ .. __ ...
4
first order polynomials to approximate certain integrands, the integrals themselves
were replaced by weighted averages of the integrands. According to the authors,
this procedure resulted in a better description of the flow within the shock layer,
but again, it was found necessary to integrate a set of corrected equations after
obtai ning an approxi mote sol uti on.
Thoenes (Ref. 16) used the one strip integral method and a simplified three
component air model, applicable in the oxygen dissociation regime, to calculate
chemically relaxing flow past cones. In particular, these calculations successfully
employed the exact tangential momentum and species continuity equation, a method
which the author labelled as the "semi -exact procedure". Resul ts from the two
strip integral method for dissociating flow were discussed in Ref. 17.
There are other methods which have been used for the calculation of non
equilibrium flow fields around pointed bodies. Lee (Ref. 18) evaluated vibrationally
relaxing flow over wedges by a perturbation method, while Dejarnette (Ref. 19)
presented an "artificial viscosity" method to evaluate wedge flows with simple
relaxation processes, including various schemes of vibration-dissociation coupling.
All these calculations with simple rate processes are of basic interest and valuable
for determining suitable numerical techniques. However, with the exception of
Ref. 11, they cannot describe adequatel y a flow fi el d i nvol vi ng several si mul
taneous relaxation processes as they occur, in air for example. Presumably, this
prompted Spurk, Gerber and Sedney (Ref. 20) to cal cui ate hypersoni c fI ow of ai r
u : 10 , _ M ."-.
5
post wedges and cones by the method of characteristics, considering a five compa-
nent ai r model i nvol vi ng numerous chemi cal reacti ons. Unfortunatel y, the appl i-
cation of the method of characteristics to nonequilibrium flow is complicated and
computation times can be very long, even on modern computers.
No cal culati ans, comporabl e to the work of Spurk, Gerber and Sedney, but
usi ng procedures other than the method of characteri sti cs, coul d be found in the
literature at the present time. It is expected that the work herein will fill this
gap by presenting a formulati on of Dorodnitsyn's integral method for hypersoni c
flow of a multicomponent chemically reacting gas about pointed bodies. The
principal advantage of the integral method is that it reduces the governing partial
differential equations to ordinary differential equations for which highly efficient
numeri cal i ntegrati on techni ques have been devel op",d .
A detailed description of the multicompanent gas model far high temperature
air, and its thermodynamic properties, is given in Chapter II. In Chapter III, the
basic partial differential equations are presented. They are cast in a suitable
coordinate system, and the discussion of applicable boundary conditions concludes
the formulation of the problem. The solution is discussed in Chapter IV. After
reviewing the method of integral relations in general, its application to the
prcbl em at hand is described in detai I. A fj nal secti on on the numeri cal techni ques
used leads to Chapter V, which is aevoted to the discussion of the results.
CHAPTER II
GAS MODEL AND THERMODYNAMICS
2. 1 Gas Model
To determine the values of the flow variables within the shock layer aoout
a body moving in the atmosphere at hypersonic speeds, it is necessary to account
for the effects af chemical reactions that take place within the layer. Such
reactions procelld at a finite rate, but for flow about infinite cones, the gas will
eventual! y reach a state of thermal and chemi cal equil ibrium. Figure 1 presents
lines of constant compressibility factor and temperature at the surface of a cone
for equilibrium flow, using data given in Ref. 21. Although this is a purely hypo-
theti cal case, the figure shows approxi matel y the condi ti ons whi ch have to be met
by the air model to be chosen. For example, if a conical vehicle with a semi-
vertex angl e of thi rty degrees enters the Earth's atmosphere at an al ti tude of 80 km
at earth parabolic speed it is to be expected that, once equilibrium is reached on
the streamline that wets the body, the surface temperature is around five thousand
degrees Kelvin, and that a sizable fraction of the nitrogen molecules must have
dissociated (Z""J.4). This example shows that, in order to calculate such flow
conditions, the gas model representin'l high temperature air must be assumed to
consist at least of a mixture of 02' N2, NO, 0, and N. It is iJrthermore
6
7
assumed that all .pecies are in translational, rotational and vibrational equilibrium
in every flow region, which implies a single temperature for oil the degrees of
freedom. Assuming weakl y interacting particles t. .e. particl es whi ch exchange
energy only by collisions, and the collision times are small compared with the time
between collisions) partition functions may be used to calculate all thermodynamic
properti es. Thus i oni zati on, vibrational nonequil ibri um, as well as vibrati on-
di$sociation coupling are neglected. The latter two assumptions will be critically
examined at the end of this chapter.
The compasition of air along with ather physical constants as used in the
present investigation are summarized in Table I.
2.2 Partition Functions
Under the assumptions stated in the previous section, all of the thermo-
dynamic. properti es of a gas may be deri ved from its parti ti on functi on. Detai Is
on the parti ti on functi on may be found in any text on stati sti cal thermodynami cs
(Ref. 22, 23, 24). In this section only those relations will be summarized which
are needed for subsequent calculations.
The partition function may be defined as
Q = ~ g. exp (- k € ~ ) • J
(2.1)
J
where €. is the common energy of several states, and g. is the degeneracy, that J J
-~ - -.;:.. -.-
8
is, the nurmer of states of the parti cI e whi ch have thi s common energy level. In
the present investigati on, the energy may be due to the translational, rotational
or vibrational motion of the particles and, in a limited way, due to the motion of
the .. Iectrons within the particle. Assuming that no coupling exists between the
difierent modes of excitation, the partition function may be written as the product
(2.2)
where the factors on the right hand si de of Eq. (2.2) are the partiti on functi ons
associated with the translational, rotational, vibrational and el ectroni c energy
levels of the particle. For diatomic molecules these factors are:
r Q "'"
QV =
Qe =
=
[ 1 - exp ( - eV / T ) ] -1
n
L: e g. exp (- e. / T)
j=O J J
(2.3)
(2.4)
(2.5)
(2.6)
r v e Here e , e , and e. are characteristic temperatures for rotation, vibration, and
J
el ectroni c exci tati on, respecti vel y. Monatomi c parti cI es have no modes of rota-
tional or vibrational excitation, therefore, the respective partition functions take
the value unity. Table II presents the atomic and molecular constants which were
used in the present calculations.
,..--._- --- ---
.'. ),
"~
.... -.- --. ------.~- - ---
9
2.3 Thermodynamic Properties
2.3. 1 _ Thermal Equati on of State
A ccordi ng to stati sti cal thermodynami cs the parti al pressure whi ch a system
of weakly interacting particles of species i exerts on its surroundings is given in
terms of the partition function as
o Pi = ni k T 0 V (I n Q)
where n. denotes the number of parti cI es of speci es I
(2.7)
contai ned in the vol ume
V, and k is Boltzmann's constant. Since, as seen from Eg. (2.3) through (2.6),
only the translational partition function depends on the volume V, one obtains
Hence, the pressure becomes
n. k T I
V
(2.8)
(2.9)
Introducing the molecular weight M., and Avogadro's number N*, Eg. (2.9) can I
be rewri tten as
= (~ Mi ) N*k Pi V N* M.
I
T (2.10)
whi ch is at once recogni zed as the eguati on of state of a thermal! y perfect gas, vi z.
(2. 11)
" ,
10
Since it is assumed that each species of the gas mixture can be treated as a
thermal I y perfect gas, the pressure of the mixture is, accordi ng to Dol ton's I ow,
the sum of its parti 01 pressures, that is
2: P.
p. = R* T .......!..... I • M.
I I
Defining the mass fraction of the ith species as
c. == I
P. I
P
the equation of state can be expressed as
p = P R* T 1: Y. • I I
where the Y. are the molar concentrations defined as I
(2. 12)
(2.13)
(2. 14)
(2.15)
It is often convenient to express the pressure of the dissociated gas mixture in terms
of the gas constant of the undi ssociated gas. Eq. (2.14) can then be wlitten as
p=pRZT (2.16)
where R = R* / M is the specific gas constant of the undissociated gas, and Z is
the compressibility factor defined by
L: i
Y. I
(2. 17) Z=M
. - ~'I
11
A gas mixture consisting of i distinct species in the dissociated state, and
of i species in the original undissociated state, requires i mass balance o 0
equati ons and 0 - i ) rei ati ons between the speci es mass fracti ons. The former o
are, first, that the sum of all mass fractions is unity,
:E C. = 1 I I
(2.18)
and, second, that the mass fraction of one element is fixed. For example, for air
(2.19)
or, alternati vel y
(2.20)
which is a direct consequence of relations (2.18) and (2.19).
2.3.2 Internal Energy and Enthalpy
Again calling on statistical thermodynamics, the internal energy of a system
of n. weakly interacting particles is given in terms of the partition function as I
(2.21 )
Introducing the molecular weight and Avogadro's number, the internal energy
becomes
E = I I (
n. M.) i N*
N*k M.
I
(2.22)
.. ~.-. ·~,-:-.. a---~~:"'·-.,,:""';;"~~- _____ --. ---i
12
Recognizing that the term (n. M. I N*) has the dimension of mass, the specific I I
internal energy, that is internal energy per unit mass, can be expressed as
= e. I
R* M.
I
2 0 T aT (In Q) (2.23)
Using Eq. (2.2) through (2.6), and (2.23), the specific internal energy for a pure
monatomi c gas is
= R* { e. M I •
I
and, for a pure diatomic gas,
:: R* { e. M I •
I
3 2
T + ~ g. 8.
e exp (- 8~ I T)
J J J J } (2.24 )
:E g. exp (- 8.e IT)
j J J
8~ ~ g.8~ exp (_8.e
IT) } I + I J J J
exp (e~ IT) -1" (-8e/T) I .t.,J g. exp .
J J J (2.25)
The last terms in the above equations, representing the contribution of electronic
excitation to the internal energy, become significant at extreme temperatures only,
and can be neglected for the range of temperatures considered in the present
i nvestigati on.
In the case of a dissociating gas mixture the dissociation ener,gy for each
molecular species must be included in the internal energy. This energy term may
be expressed as
(2.26)
, ...
;
"
where, in the present case, j ::: O2
, N2
, NO, and the subscript 0 designates
the undi ssoci ated state.
The total specific internal energy of a mixture of monatomic and diatomic
gases can now be written as
13
E = r: c. e. + L e? (2.27) I I I j J
The specific enthalpy of the mixture is defined as
(2.28)
and substiiutiol", of the pertinent relations into Eq. (2.28) yields, for the five com-
panen! gas mixture considered herein, the following expression for the enthalpy:
v YNO
e NO YN2 e~2 +----- +
e~2 IT e -1
e~O/T e -1
+ [(Y 02) 0 - Y02 ] e~2 + [(Y N2)0 - Y N2] e~2 + [(Y NO)o - Y NO]e~O }
(2.29)
Utilizing the mass balcnce equations, Eq. (2.19) and (2.20),in order to eliminate
the molar concentrations (Y 02)0 and (Y N2)0' and also considering that in the
adopted air model (Y NO)o is zero, a more convenient expression for the enthalpy
is easi I y deri ved from Eq. (2.29):
v I 7 eNO
+Y ,-T +---+ NO 2 v eNOIr
e -1
* + Y (2- T + o 2 ) + Y N ( + T + e~2 ) }
It is recognized that the specific enthalpy now has the more general form
h = r: c. h. (1') i I I
14
(2.30)
(2.31 )
For the calculations in the following chapters it is convenient to define
a frozen specific heat CIS the partial derivative of h with respect to T, that is
ah c ;; p aT (2.32)
which, according to Eq. (2.31) may be expressed as
~ ah.
c = C. -;;--T1 = L c. c
P i I 0 i I Pi (2.33)
Hence, like the specific enthalpy, the frozen specific heat of the gas mixtlJre is
simply the weighted sum of the individual specific heats. An explicit expression
for c, asderivedfromEq. (2.30), is given in Appendix A. p
-. -::. ~-~··i
15
2.4 Rate Pro.:esses
2.4.1 Finite Rate Nonequilibrium
In general, a large number of callisions among the particles is r')quired to
equilibrate the molecular vibrations, dissociation and higher modes of excitation
with the local translational temperature. This means that, once being in a state
of nonequilibrium, a finite amount of time is needed for the gas properties to
achieve thermodynamic equilibrium. The departure from equilibrium of a flowing
gas is characterized by the magnitude of this relaxation time T relative to some
translati onal time t needed by the parti cI es to change thei r state. The Ii mi ti ng
values of the ratio of T over t are a convenient means to characterize the
equilibrium state (T / t ... 0) ond the frozen state (T / t ... CD).
The purpose of thi s secti on is to present the equati ons whi ch are necessary
to account for finite rate nonequilibri um effe(:ts in the cal cui ations. Essenti all y
this is equivalent to finding an expression for the relaxation time T in terms of
the vari abl es of state.
2.4.2 Species Continuity Equation
Consider an arbitrary material volume containing particles of various species,
each species forming the partial density P. • Neglectfng diffusion, the mass rate I
of change of the ith component in the mixture must equal the rate of production
(which may be negative or positive) of the ith species, that is
....
" ' .. , "',
----- --I _"-'""'--~ _____ _
16
~t f Pi dV = f Wi d V (2.34)
V V where W. is defined as the mass rate of formation of species per unit volume
I
due to chemical reactions. Using Reynolds' transport theorem, Eq. (2.34) may be
rewritten
f [ oP. . ] aT- + 9' (Pi ~) - Wi dV = 0 (2.35)
V Since the volume considered was arbitrary, this means that
a P. I
at ...
+ 9 • (P. q) = W. I I
(2.36)
• Summi ng over all speci es i, and consi deri ng that I: P. = p, 01 so
i I I: W. = 0 i I
since total mass must be conserved, one obtains the overall mass conservation
equation
ap at
... +V"(Pq)=O
Replacing p. by p C. in Eq. (2.36), and expanding differentials yields I I
(2.37)
(2.38)
Since the first term on the left hand side vanishes according to Eq. (2.37), the
species continuity equation finally assumes the form
o C. I
at + ~ . 9C. = (W. / p) I I
It is worth noting that the quantity (p/W. ) has the dimension of time, and I
(2.39)
I
--·1
• ".
'---t_.·_ .
17
therefore may be interpreted as a relaxation time for the species i. In the
following section an explicit expression for W. will be derived. I
2.4.3 Rate Equations
Following Vincenti and Kruger (Ref. 24), a chemical reaction is considered
to have the following description:
4: v• A. I I I
~ v' A Iii
(2.40)
where the v. are the stoichiometric coefficients of the reactants, v: those of the I I
products. A M is a so-called third body species which does not participate as a
reactant. kd,M and kr,M are the respective reaction rate constants for disso
ciation and recombination.
Defi ni ng the concentrati on of the speci es A. by X., that is I I
C. X. ;:
I
I
M. o = p Y. I
(2.41 ) I
it is an experimentally observed fact that the rate of formation of anyone of the
products in a single step reaction can be expressed as
d X. Vi vM
( d tl)d = (v; -Vi)kd,M ~ Xi XM (2.42)
where the product on the right hand si de extends over all the reactants •
l i
- --I
18
Similarly, the rate of disoppearance of any species A. can be expressed I
as d X Vi V
( i) = (v. _ v.' ) k TI X. i X M d t I I r,M I I M
r
(2.43)
where again the product is taken over all of the reactants.
The net rate of producti on of the speci es /l. is then the sum of its rate of I
formation and its rate of disoppearance, vis.
d X. (d X. ) 1 _ 1
""dT - -dt d + (dd ~i ) r
(2.44 )
or, with Eq. (2.42) and (2.43):
d X. [ -.,.-_1 = (v: - v.) k
d til d,M
v. X.I - k
1 r,M
V VI] M . XM 'If Xi I
(2.45)
For thermodynamic equilibrium, the net rate of production of each and all
sped es must vani sh ("pri ncipl e of detail ed bal anci ng"), hence from Eq. (2.45):
V Vi
kd M n X. i - k M n X. i = 0 I i I r, i I
(2.46)
Eq. (2.46) can also be written in the more familiar form
n x. v:
kd,M I
'" i 1 K (T) (2.47) -
k v r, "A i
c
U X. 1 1
where K (r) is call ed the concentrati on equi I ibri um constant, and expressi on c
(2.47) represents the law of ma.s oction.
19
If it is assumed that, for gi ven temperature and reactant concentrari ons, the
rate at which a reaction proceeds in a given direction is the same whether the
system is or is not in equilibrium, Eq. (2.47) may be used to replace either k d,M
or kin Eq. (2.45). Hence let r,M
= k K r,M c
(T) (2.48)
Using this expression, and also accour.ting for the possibility of several species
acting as third bodies, the resulting rate of production is obtained from expression
(2.45) as
d Xi = (v.' _ v. ) "" k X vM [ K IT X,Vi _ U x. vi ] (2.49) d til ~ r,M M c ill 1
M Consi deri ng al so that there may be m diHerent chemi cal reacti ons of the type
,
represented by Eq. (2.40), the total rate of production of the element A. is given 1
by the sum over all reacti ons, that is
[V. I ]
K TI x. I-II X.vi c ill 1
(2.50)
It is obvious from Eq. (2.47) that K (T) is, for the mth reaction, indepenc
dent of the thi rd body speci es AM' Natural I y, for different reacti ons, the equil i
brium constants are different functions of temperature. Their eval uation is dh-
cussed in section 2.5 of this chapter.
For the air rr'odel used in this investigation the following set of chemical
reactions will be considered (Ref. 15):
---~~~================~======~~== ~--,-- ----'-- -,- ... --"-- - -----.-1
.. '. '.
m = I
II
III
IV
V
VI
I
kd,M o + M + 5.1 ev ... 2 ....
kI r,M
N + M + 9.8 ev ... 2 ....
NO + M + 6.5
NO +0 + 1.4
ev ... ....
ev ~ ....
20+ M
2 N + M
O+N+M
o +N 2
N + 0 + 3.3 ev'" NO + N 2 ....
02 + N2 + 1.9 ev: 2 NO
20
(2.51 )
(2.52)
(2.53)
(2.54)
(2.55)
(2.56)
Reactions I, II, and III are the principal dissociation reactions, and, considering
the various third body species (see Table III), actuc..lly represent fifteen different
reactions. The remaining equations represent the so called shuffle reactions.
Altogether, eighteen chemical reactions are considered bEotween the five companents
of the gas mixture.
The application of Eq. (2.50) to the above set of reactions yields the
following expressions, already simplified by introducing the molar concentrations
Yo. : I
-..:.... s:s: __
, ~ .
- -.-.. ~.
21
(2.57)
(dXN2) ;: p2{ ( y2 _y KII) [y kll +Y kIl
d t tot p N N2 c N2 r(N2) N r(N)
+ (Y 2 _ Y Y K VI ) k VI } NO 02 N2 c r
(2.58)
d X { (
NO)::: p2 (PY Y -Y KIII)(Y +Y +Y +Y +Y )kill d tON NO c 02 N2 NO 0 N r
tot
(2.59)
Proper rearrangement of terms in the corresponding expressions for the mon-
atomic species 0 and N shows that
0'
&f_ ._. ~ __ _......... 3 _;::;:: __ , . .,.;:. _ -: ~ -_. ~, -~~=====-::====-:=======-==o_" _=-000 __ "_:..c 0_._"." .. .=~_""""
--
22
(dXO ) =
CiT tot -2 (
dX02 ) _ (d XNO) d t tot d t tot
(2.60)
(dXN \ = _ 2 (dXN2) _ (axNO ) CiT Jtot d t tot d t tot
(2.61)
These equations can be obtained, i~ a simpler way, directly from the mass balance equations, Eq. (2. 19) and (2.20).
It is important to keep in mind that the expressions given by Eq. (2.57) • through (2.61) are equations representing rates of production due to chemi cal reactions taking place in a dosed system of fixed volume, assuming also a constant temperature. As painted out by Vincenti and Kruger (Ref. 24), they are customarily extended to open systems of varying volume and temperature by identifying density and temperature as the instantaneous values for the system. This implies that the internal rate processes in a maving fluid are the same as those which occur at the same state ina stati c system. The (d ~ / dt )tot is rei ated to the moss rate of formation of species i, as defined in the species continuity equation, by the relation
•
( d ~i ) d tot
(2.62) W. ::: M.
I ,
2.4.4 Reaction Rate Constants
In order to evaluate the reoction rates given in the previous section, expressions are needed for the rate constants. Although theories exist for the
;, ..
l-::--_. -..... -
-------.----
-'
calculation of dissociation and recombination rate constants (Ref. 24, 26), they
are either too complex to be useful in practice, or too simplified to produce
acceptable agreement with experimental data. However, it was recognized on
experimental grounds that, for many reactions, a plot of the logarithm of the
dissociation rate constant versus reciprocal temperature produces an essentially
straight line of negative slope, which leads to the Arrhenius equation
€:A
23
-kT kd = A e (2.63)
where €:A is the activation energy and A some constant independent of tempera
ture. A refinement of this expression is usually obtained by assuming
9* T n
kd = ATe (2.64)
where 9* is the characteristic temperature of dissociation, and the constants A
and n are determi ned by curve fi ts to the experi mental data. Expressi ons of thi s
type are numerous in the literature and can usually be traced to the papers by
Wray (Ref. 25}and lin and Teare (Ref. 26). Expressions used in the present calcu-
lations were taken from Spurk, Gerber and Sedney (Ref. 20), and are listed in
Table III.
2.5 Equilibrium Constants
The equilibrium constants, defined in Eq. (2.47), are also needed for the
evaluation of the reaction rates. Following again Vincenti and Kruger (Ref. 24),
~ .---- .----===="'--==---::==-c======:-==-=-.- -
. ,
--
24
it is found that, in terms of the partition functions, the equilibrium compasition of
a gas is given by
VI
TIi n. i I
IT v. n. I
i I
=
VI
1}Qi I
V. TIQI i
exp ( - (2.65)
where t.E is the difference in the bond energy of the reactants in Eq. (2.40), n. I
are the numbers of particles involved, and v. and v,' are the stoichicmetric I I
coefficients. The n. are easily expressed in terms of the concentrations by I
noti ci ng that, accordi ng to Eq. (2.41),
C. P. X. = ~ p = 1M.
I = M.'"
I I
n. I
VN* (2.66)
Therefore, expressing the left hand side of Eq. (2.65) in terms of the concentrations
X. , it can be shown that I
VI
IT x. i i I
V. IT x. I i I
l:V. - l:v~ I I
= (N*V)
Combining this with Eq. (2.47) and (2.65) it is seen that ~ VI
VI
IT i • n. I I
v
II i n.
I I
l:v. - £... v,' IT Q i
Kc(T) = (N*V) I I i v. exp (- ~*ET )
UQ I
I
(2.67)
(2.68)
which, when applied to the reactions given in Eq. (2.51) through (2.56), turns out
to be a function of temperature only. Eq. (2.68) shows also why the contributions
due to electronic excitation must not be neglected in the equilibrium constants •
25
Here, the ratio of the partition function is important, while in the internal energy
thei r contributi on is onl y addi ti ve •
It is clear that substitution of the proper expressions for the partition
functions in Eq. (2.68) leads to complicated functions of the temperature. In
particular, since use is often made of the equilibrium constants in comparing
dissociation and recombination rate constants, it is convenient to have them in a
simple analytic form. For this purpose Wray (Ref. 25) has fitted the equilibrium ".
constants calculated from Eq. (2.68) for the reactions given by Eq. (2.51) through
(2.56) with Arrhenius type expressions of the form
n * K (T) = A T exp (- 9 / T) c
(2.69)
These expressions, given in Table IV, are employed in the present calculations.
With respect to these curve fits a remark is in order at this time. As men-
tioned earlier, the principle of detailed balancing requires that, for a gas mixture
to be in overall equilibrium, every molecular process and its inverse must be
individually in balance. Thus it can be shown that, if the total reaction rates and
their individual contributions are set to zero, the concentration equilibrium con-
stants, for the reacti ons chosen herei n, resul t as follows:
(X2 /x ) o . 02 equil.
(2.70)
KII:= (XN2 / XN2 ) '1 c equi . (2.71 )
-. ~ - - -.. ,
., "
.,',
--'
26
K III = (XO XN / X NO ) '1 c equl . (2.72)
and it can al so be shown that
(2.73)
(2.74)
= (2.75)
Since Wray curve fitted all the K except for K VI, which was determined from c c
K IV and K V according to Eq. (2.75), it is not surprising that Wray's expressions c c
for K IV and K VI differ from K IV and K VI as given in Table IV. For the c c c c
sake of consistency, the latter expressions were obtained from K I through Kill c c
according to Eq. (2.73) through (2.75). Numerical calculations show that the
disagreement between the K given by Wray and those obtained from equations c
(2.73) through (2.75) is negligible.
2.6 Equilibrium Composition
It was mentioned at the beginning of this chapter that is is of interest to
calculate equilibrium flow even though it is, in the exact sense, a purely
.","'~'=====;;""";======-=====;;;:C:;;;"=~==:::;::::--=::-: .. ::C:,c.:::;.:~:;:,_"":,"'"";;;-~;:-::."._ ....... ~=:::: __ ':"-_-.:::-' '_-_ =.,"
b---- -
27
hypothetical case. In order to do this, the equilibrium composition of the mixture
has to be determined as a function of other flow variables.
For a five component mixture this means that five linearly independent
equations are needed to solve for all the unknowns. Equations (2.18) and (2.19),
together with Eq. (2.70) through (2.72), provide just such a set. Hence, from
Eq. (2.18) and (2.19):
~ C. = 1 I
CNO =
while from Eq. (2.70) through (2.72):
2 C = 2 p C N
N2 M K II N c
CNO =
m o
(2.76)
(2.77)
(2.78)
(2.79)
(2.80)
Substituting the expressions for the mass fracti ons CO2' C N2 and C NO into the
mass bal ance equati ons yi el ds
--~ ----...£~---====- :====~====;:5;==:O-= __ =_= ___ :-:-::: .. :::::_~. "",,": . .,-:'. = __ .. ______ _ ~ ---.-::~_- ---- - .. , - --~ ---,
•
I _ •. ___ . ~--..
·.-~-
and
C +C + 2p o N M K I
C + 2 p o M K I o e
o e
C 2 + ---,P---,,,,,, o M KIll
N e
from whi eh C N is obtai ned in terms of Co as
C ~ N
M KIll N e
pCO
[;;:, - C _ _2..:...P-=-
o 0 M KI o e
28
(2.81 )
(2.82)
C~ ] (2.83)
Using this equation to eliminate CN
from Eq. (2.81) eve ... tually result. in a
fourth order equati on for CO:
where
bO
=2 (;;:'0 M )2K I KIll o e e
b 1 =;;' 0 M 02
KeI
K eIII
KeV
[1 - -±- ] K V
e
(2.84)
29
{ 2 K~II [ 1 -4moP J- C- m ~o J-KIIIKV} b =M2 KI KV 0 P -
2 0 c M KI c M M c c o c N 0
b 3 :: PMO K~ KV [ 8
- 2 K IV - 1 J c K VI c
c
K V [ 4 c K VI
c (2.85)
Since the b's are functions of temperature and density only, Eq. (2.84) determines
the equilibrium composition for given temperature and density. It is seen from
Eq. (2.77) thai the only possible solution can be selected by requiring the atomic
oxygen mass fraction Co found from Eq. (2.84) to satisfy
o s Co s [ ~o - CO2 - (~~o ) CNOJ (2.86)
2.7 Review of Assumptions
• It is instructive to examine the validity of the assumption of vibrational
equilibrium which was employed throughout this chapter. A measure for the
validit/ of this assul""ption is glven by the ratio of the chemical over the vibrational
relaxation time of the respective species.
-'--- -----:--::- - - -.:
.. '
,.
~ ' . . ,
. . ,
"
,. I. ~. "-. I.. < \, '~ ,~
. {?:'
l
Using expressions given in Ref. 27', vibrational relaxation times for 02
and N 2 are given by
30
r ~2 P = 1.6212.10-4
exp ( ) (2.87)
r ~2 P = 1.1146· 10-6 T 1/2 exp ( 154 ) T 1/3
(2.88)
where r ~ is given in sec for the pressure in N/m2 and the temperature in oK. I
• As previousl y indicated, the quantity (p / W. ) has the dimension of time.
I
A chemi cal relaxation time can thus be defined as
rd _ i
p . W.
I
(2.89)
where the mass rate of formation, Wi ' can be evaluated for 02 and N2 from
Eq. (2.57), (2.58) and (2.62).
Ratios (rd/rv)i for i =° 2 , N2 are shown in Fig. 2 as function of
temperature, together with results for 02 taken directly from Ref. (28). It
should be noted that the values of rd / rV are independent of density.
It is seen from Fig. 2 that, at least for molecular oxygen and nitrogen,
vibrational relaxation times are always shorter than chemical relaxation times for
temperatures below 12,000 oK. Hence it is concluded that, for the t"mperature of
interest in the present investigation, vibrational relaxation will in general proceed
much faster than chemical relaxation. Therefore, the assumption of vibrational
equilibrium is a justifiable simplification.
-~~--- ----
31
However, it is also recognized, that for temperatures above 8,000 oK
vibrational and chemical relaxation times assume valL'es of the same order of
magnitude. Strictly speaking, vibrational relaxation and vibration-dissociation
coupling should therefore be considered in this temperature range. The difficulty
here is that, although vibrational relaxati on ra!es are known, an accurate model
for the coupling does not seem to be available. Conclusions drawn by investi-
gators in this particular field, Ref. 19 for example, leave the question open as to
the relative validity of the various models. Rather than to introduce equations of
whi ch the effect is not yet well understood, the vibrati onal exci tati on is assumed
to be uncoupled to the dissociation, and in equilibrium at all times.
,...---- ---- ------
,
.'
,
"
CHAPTER III
FORMULATION OF THE PROBLEM
3 . 1 Basi c Equati ons
Neglecting body forces, viscosity, diffusion, heat conduction, and radi-
ation, the basic equations for Sf6.Jdy, adiabatic continuum flow are the following:
Conservati on of mass:
... 'V'(Pq)=O
Conservation of momentum:
Conservation of energy: ...
... ... 1 + ('Vxq)xq+ - 'V P = 0
p
2 q • 'V (h + ...5.-) = 0
2
or, assuming all streamlines to originate fro~ the same reservoir,
2 h + ....9- .: H = constant
2
(3. 1)
(3.2)
(3.3)
These equations are the common equations of motion which do not depend on any
particular gas model. In order to solve these equations for the unknowns, they
must be suppl emented by a thermal and a cal ori c equati on of state whi ch do depend
on the particular gas model and were derived in Chapter II as
p=pRZT (3.4)
32
----,=======~=======--=:-.. --:;: ... ~.~-.=~ -"---_._--=---'.-.- ,. I --.-l
...
'. ,i ,.'.
-. 'I,
--., ...... <-~ •••
---_.-_._.-.. - .. -."_.-----...•. __ .-- --.---
33
and
h = L: c. h. (n i I I
(3.5)
If the gc;;s under consi derati on is chemi call y reacti ng, addi ti onal unknowns ari se
in form of the mass fractions, C., hence, additional equations for the conserI
vation of species must be added to the system. For the air model used herein
these are given by Eq. (2.39), which, for steady &Iow, reduces to
q • 'V C. = I
W. I
p (3.6)
where i = 0, N, and NO, and the mass balance ec:uations given by equations
(2.18) and (2.19).
The choi ce of the coordi nate system is largel y a matter of experi ence. For
the problem at hand, an orthogonal curvilinear coordinate system with coordinates
tangential and normal to the body surface (see Fig. 3) seems to be the most suitable.
From Ref. 2° one obtains for the gradient
.... .... ....
'VF e 1 of e2 of
+ e3 of =
~ ax 1 +r-h3 oX
3 2 oX2 (3.7)
.... where F is any scalar point function. For a vector point function F the diver-
gence is given by
and the curl by
. -. ~1
-- -~------- -
,-
34
... ... ... h1 e 1 h2 e2 h3 e3
'VxF= 1 0 0 0 (3.9)
h1h2 h 3 oX1
oX2
oX3
h 1 F 1 h2 F2 h3 F3
The metric coefficients h l' h2
, h3 follow from the differential arc length
(3.10)
Using as coordinates x, y, and <1:>, where <I:> is the circumferential coordinate,
one obtains from Fig. 4
h1 dX1
= (1 +Ky )dx
h2 dx 2 = d Y
h3 dX3
= r j d <I:> = (rb
+y cos9)j d ¢> (3.11)
where K = 1/R is the body surface curvature, and
j :;: { 0 1
for plane flow
for axisymmetric flow
Appl ying equations (3.7) through (3.11) to equations (3.1), (3.2) and (3.6) and
restricting to plane or axisymmetric flow, that is, zero angle of attack, the
resul ti ng equati ons are as foil ows •
Conservati on of mass:
o . 0 r 'J oX (pu"!)+ oyl(l+Ky) pvr J = 0 (3. 12)
. ;
~:
.;' , . -:::
--1_--... ·
x - Momentum:
u ~ + (I + Ky) v ~ + u v K + 1 ox oy P ~= 0 ox
y - Momentum:
ov (I ) ov 2 u ox + +Ky v """FY - u K + :+Ky .~ = 0 P oy
Conservati on of speci es:
o C. 0 C. W. I ( I
U ax + 1+Ky) v"""FY - (I + Ky) --.!.... = 0 P
35
(3. 13)
(3. 14)
(3. 15)
The above equations, together with the thermal equation of state, the energy
equation, and two mass balance equations, form a set of ten equations for the
unknowns u, v, p, p, T and five C.' I
Before discussing the details of the solution of the above system, it will be
convenient to have equations (3.13) through (3.15) also available in the divergence
form. After some algebraic manipulations the following equations can be obtained.
x - Momentum:
a [ 2'J 0 [ . J ox {p+Pu)rJ +ay {l+Ky)puvrJ
+ P u v K r j - j p ( 1 - Ky) sin 9 ;:: 0 (3. 16)
y - Momentum:
_ (puvrJ)+ - {l+Ky){p+pv)r J o . a [ 2'J a x oy
- (p +pu 2)K r j -j p (1+Ky)cos9 ;:: 0 (3.17)
36
Conservati on of Speci es:
o . 0 [ • • • ~(puC.rJ)+~ (I+Ky)pvC.r J ]-(l+Ky)rJ W. == 0
oX I oy I I (3.18)
The boundary conditions which are needed to solve the equations established here
are gi ven in the next secti on.
3.2 Boundary Conditions
3 .2. 1 Body Surface and Shock Condi ti ons
The body surface is assumed to be chemically inert. The condition for flow
tangency on the body surface is given by
v == 0 b (3. 19)
The condi ti ons behi nd the shock wave are obtai ned from the conservati on of mass,
momentum and energy across the shock wave.
Conservati on of mass:
P"" Y n"" == P Y s ns
(3.20)
Conservation of momentum:
Yt "" == Yts
(3.21)
y2 oJ. == Ps y2 +p P"" n "" . p"" ns s
(3.22)
Conservation of energy:
h +_I_y2 h 1 Y 2 ==
..I. __
"" 2 n"" s . 2 ns (3.23)
-.~ - -- "-.--.,
--.-- ------~-!..-~-•. ;;. ~ ...
37
where the pressure and the enthalpy are given by the equation of state, Eq. (2.16),
and the enthalpy equation, Eq. (2.30), respectively. It should be noted that the
form of the enthalpy equation implies that the molecular vibrations are in equili-
brium everywhere.
For the calculations to follow, the velocities behind the shock must be
given in the present coordinate system. The following geometric relations are
obtained from Fig. 5:
u s = V n s si n S + V t s cos S (3.24)
v s = - V n s cos S + V t s si n S (3.25)
where the velocity components as referred to the shock are given by
V = n5
P""
Ps V 00 sin C] (3.26)
V ts = V to;> = V 00 cos C] (3.27)
Fig. 4 oIso indicate .. that the relation between the shock coordinates and the
shock wave angle C] is given by
d y s
dx = (1 + Ky ) tan S
s
where, in general, i3 is a function of x.
(3.28)
.>
". ,. ;~
••
-'
38
3.2.2 Frozen Shock Conditions
For frozen and nonequilibrium flow it will be assumed that the composition
of the air does not change across the shock, that is, Z = Z • Since, in the 00 s
present calculations, the shock wave is assumed to be a discontinuity of zero
thickness, and any possible chemical reaction would need some finite thickness
in order to take place, the assumption is a logical one.
Combining the equation of state with equations (3.20) through (3.23), it
can be shown that
v '" ( RZeo T"') p:: I T = n 1 + V '" - 2(h - hJ -s RZ \,2 n s
s n '"
2 V - 2 (h -h ) n'" s eo
RZ s (3.29)
For any given free stream conditions (i .e. Teo' V , C. ) and thp .hock wave 00 100
angle a, the temperature behind the shock, T ,can now be found from equations s
(3.29) and (2.30) by solving these equations numerically. Using the continuity
equation, Eq. (3.20), and the energy equation, Eq. (3.23), the density ratio
across the shock is obtai ned as
V '" _ n
- J'"y 2 _ 2 (h _ h ) i· ., n'" 5 eo
(3.30)
Knowing the temperature and the density behind the shock, the static pressure
rati\1 across the shock becomes
T = s
Too (3.31 )
=-- - - _c __ -. __ - -,. •
.,
.--",-_ . . ':"'~~'''''-'.-'
39
which completes the calculations of the static conditions on the dow.,stream side
of the shock wave.
Unfortunately, the solution described above involves an iterative calcula-
tion of the temperature T • Since iterations are normally time consuming, il is s
impractical to calculate the shock conditions by this process at each step in the
course of a numerical calculation. Instead, the expressions for u ,v, T , p , s s S 5
and p , presented in this and in the previous section, may be expressed in diffes
rential form in terms of the shock wave angle gradient. It is then possible to
integrate for some selected key variables, say density P and temperature T , s s
while the values of the remaining variables are determined from simple algebraic
equations.
Differentiating the expressions for u , v , T ,p and p with respect to s s 5 S S
x results in the following differential shock relotions:
d T dO" s = T", 01 CfX dX (3 .32~
dp dO" s = P", °2 dx crx (3.33)
dp dO" s = R Z (p T 01 + P T 02) dx 55'" 0>5 crx (3.34)
du ~+ 5 :;:
Yo> 03 K v dx dx s
(3.35)
..•.
d v --,--:;..s =
dx - u s
40
K (3.36)
where 01
through 04
are dimensionless functions explicitly given in Appendix
8.
From Fig. 3 it may be deduced that
r = rb + y cos e 5 s
whi ch, upon di fferenti ati on, yi el ds
d r --.!.. = (1 - Ky ) sin e + (1 +Ky ) cos e tan ~ d x s s
where use was made of the relati on
d rb -- = sin e dx
which is obtainable from geometry.
3.2.3 Equilibrium Shock Conditions
(3.37)
(3.38)
(3.39)
In thi s case it will be assumed that the gas i nstantaneousl y reaches the state
of thermodynamic, that is, thermal and chemical, equilibrium behind the shock,
irrespective of the conditions in the free stream which may be arbitrary. While
for frozE!n shock conditi ons the composi ti on of the gas on the downstream si de of
the shock wave is prescribed by the free stream ccmposition, it is unknown in the
present problem. The necessary additional relations are furnished by the equations
for chemical equilibrium, derived in section 2.6 .
.... -..
41
Using the equation of state to eliminate the pressures from the equations of
conservation of mass and momentum across the shock, it can be shown that
PCI) b _~(t-)2 RZ T = s s
P 2 y2 s nCl)
(3.40)
where
R Z", T '" b = 1 + y2
n'"
(3.41)
The reci procal of Eq. (3.30), whi ch was deri ved from the conservati on of energy,
yields
(3.42)
For given shock wave angle a, equations (3.40) and (3.42) are two independent
relations for the density ratio across the shock as function of density p and s
temperature T • Hence, assuming some reasonable values for p and T ,a s s s
solution is obtained O:>y iteration on p and T ) when the difference between the s s
right hand sides of equations (3.40) and (3.42) assumes a specified minimum value.
Having determined the density p and the temperature T ,in the course s s
of which the species mass fractions C. have also been calculated, all other IS
flow variables are readily evaluated from the available equations.
Although equilibrium shock data are available in the literature (Ref. 30, 31),
these data are usually presented for selected atmospheric free stream conditions at
42
relatively low altitudes. The procedure described above, however, is applicable
for arbitrary combinations of free stream pressure, temperature and composition,
as they may occur at high altitudes or in a hypersonic wind tunnel.
,.
. ,. ; "
'It-•
, ), ;
; , t
" .~ r
CHAPTER IV
METHOD OF SOLUTION
4.1 The Method of Integral Relations
4.1.1 The Method in General
In order to appl y the method of integral relations as described by Dorodnitsyn
in Ref. 2, all the partial differential equations are cast into the divergence form,
namely of.
I + ox
oG. I
oy + H. = 0 I
(i '" 1, 2, • '" m) (4.1)
Here x and yare the independent variables, while F. , G. and H. are the I I I
known functions of the dependent variables, and m denotes the number of
equations.
Consider now the solution of the above system of m equations in the region
which is bounded by the body surface and by the shock wave y (x). Dividing s
this region into N curvilinear strips (see Fig. 6) bounded by lines "
v = y - k (x) k - ~ Ys (4.2)
where k = 0, 1, 2, ••• , N, the system (4.1) can be integrated with respect to
y across each strip. The result is a system of integral equations of the form
43
"-, , liF'-
,
',: "
.
J
i t-: -'. ~: .'
~ ,~'
t:
d y + G. k+ 1 - G. k + I, I,
H. dy I
= 0 (4.3)
where k = 0, 1, 2, ... , N - 1. Using Leibniz's rule for differentiation under
the integral sign (Ref. 32), the first integral may be rewritten and the above
equations become
d dx (
k +1 k) dys dy - N F i , k + 1 - N F i , k """'d7
+ G. k + 1 - G. k I, I , = 0 (4.4)
In order to evaluate the integrals, the integrands must be known functions of y.
Unfortunately the integrands F. and H. contain precisely those dependent I I
variables for which values are to be calculated, hence some approximations '1ave
to be made. Generally, as an approximation, any interpolation formula can be
used whi ch expresses the val ue of F. or H. at an arbitrary 0 S:y :s: y through its I I S
values on the lines y = Yk. For example, one may use polynomials of the form
N F a ~ a. (x) y" • I In
n=-O (4.5)
k H. - b in (x) yn I
,
i@/>· '.
45
where the a. (x) and b. (x) are to be evaluated in terms of the boundary values In In
at the stri p interfaces, soy
F. = F. k I I,
H. = H. k I I ,
(4.6)
Performing the necessory operations on these polynomial appn>ximations yields a
system of m. N ordinary differential equations of the following form:
N
L: n~l n=O
_ ( kN+1 k) Fi,k +1 - N Fi,k
+ ~ bin (x) ( L n+1 n=O
dy s -+G -G
dx i,k+1 i,k
= 0
where k = 0, 1, .•• , N - 1; n = 0, 1, ••• , Nand i = 1, 2, ••• , m.
(4.7)
The solution of the system (4.7), obtained when the region is divided into
N strips, is called the Nth approximation. It is obvious from the above that
with higher approximations the complexity of the problem increases.considerably.
Belorserkovskii (Ref. 37) demonstrated the convergence of the method by COi'nporing
one-, two-, and three-strip solutions for supersonic flow post a circular cylinder.
He showed that, at I east for that case, there is practi call y no difference between
the three solutions. Moreover, since previous investigations with simple rate
Lt ... .Ia."" - .----. , .. .... ..~... ' .. , ~ . --"',.-: ~
.'
processes (Ref. 12, 13) have shown that the first Oinear) approximation also agrlHlS
well with results from other methods, the linear approximation is investigated
herein.
4.1.2 The Fir~~ Approximation
For the first approximation, according to the previous section, N = 1, k =0,
n • 0, 1. Hence from Eq. (4.5) the polynOl'nial approximations have the general
form
P. • c. 0 (x) + c. 1 (x) Y I I I
(4.8)
According to (4.6), and the coordinate system, (Fig. 3), it is found that at
y=O:
(4.9)
y=y: S
P. == c' O(x)+c' l (x)y I,S I I 5
From (4.9) the coefficient functions in general are
(4.10)
(P. - P. b) 1,5 I,
Applying this procedure to F. and H., the coefficient functions are given by I I
Eq. (4.10). Equation (4.7) may now be simplified and, for the first approximation,
results Ir
,,_. ~,-....... .. . ..... -.- ....... _.,- .
,.
d -d x (F. + F. b)-
1,5 I, y 5
1 dy (F. - F. b) d x
S
I, S I,
47
+ (G. -G. b)+(H. +H. b) = 0 1,5 I, 1,5 I,
(4. 11)
The ob~ve equatJon might be called an operator equation, permitting the
transformati on of the gi ven parti al differenti al equati ons of the form (4. 1) di recti y
into ordinary differential equations for the dependent variables along the body
surface and the shock wave boundary.
4.2 Nonequilibrium Flow
4.2.1 General Remarks
As early as 1929, Busemann (Ref. 33) showed that supersonic flow of an
inviscid gas about circular cones is characterized by the fact that the pressure and
the velocity vector are constant on coaxial conical surfaces having the same vertex
as the conical body. A consequence of this is that the shock wave, forming one
boundary of this flow field, must itself be a conical surface. It will be shown in
the discussion of the results, that, in the case of nonequilibrium, chemically
relaxing flow in this case, the time dependency of the chemical reactions destroy
this self-similarity mentioned by Busemann, even though the flow is considered
to be inviscid. Therefore, no definite statement can be made, a priori, about the
shock shape or the behavior of other flow parameters. Values for all variables
"
, .
" , .. (
t ~; .. s"
must be calculated by integrating the conservation equations together with the
equation of state and the boundary conditions in a step-by-step fashion along the
I ength of the cone.
Two different sets of equations will be examined. The first set, later
referred to as the standard approximation, contains all the conservation equations,
equations (3.12) through (3.15), in approximate form. The second set, later
r~ferred to as the semi -exact procedure, uses onl y the conti nuity and the y-,
momentum equations in their approximate form, while the x-momentum and species
conti nuity equati ons are used in the exact form.
For a supersonic flow field, which will be considered here, the equations
are of quasi hyperbolic character and thus form an initial value j)rcblem. Initial
values are given by the frozen flow solution to be discussed in section 4.3, where
it will also be shown that initial gradients can be derived as functions of the
i niti 01 val ues.
4.2.2 The Standard Approximation
The application of Eq. (4.11) to the conservation equati.ons, Eq. (3.12) and
equations (3.16) through (3.18), results in the following ordinary differential
equations where use was made of the fact that Vb = 0, and y = 0, at the b.ody
surfoce.
L. ,.,&j,,~ .. • .~. --~.--"
~-.,
, )
. , , '" · , ,
· . . , , 1
: 1 ··'1 · . ,
Continuity:
1 • • dys • - (p u r J -p u rJ) -
y 5S5 bbb dX 2 •
- (1 +Ky ) p v rJ y s s s S
5
x-Momentum:
1 [ 2' .. - (p + P u ) r J -Y 5 S 5 5
5
5
dy 5
dx
2 (1 + Ky ) p u v ) - p u v K r j 55555 5555 --
+ . J
y-Momentum:
d • --..:.. (p u v r J ) dx 5 S S 5
1 --• dy
(p u v rJ)--!. 5 5 5 5 dx
49
(4.12)
(4.13)
(4.14)
" ,"
., "
l' '.I. " ~,
\ \
.... :
50
Species continuity:
= -L (p u C rsj - Q U. Y s s is 'I) I)
5
• • • •• 2 - - (1 + Ky)p v Y 555 C. r J + (1 + Ky ) r J W. + rbJ W'
I b (4.15)
IS 5 5 5 IS S
Expanding the differentials and introducing the differential shock relations
given in section 3.22, the above equations assume the following form, where 6
is der. ned as
Continuity:
x -Momentum:
dFb 2 d'b d~ _ """'dX + ub """'dX + 2 Pb ub """'dX + (1 +j flcos e>[ R Zs (ps T",O 1
+p .. Ts(2)+P",us2~ +2Ps Vcol:s 0 3 ] ~~
= -L { (p + P u 2) F - (p, + P u 2) F y 5 S 5 1 'I) b b 2 s
- P u v (1 + j 6cos 9) ( 2 + 5 Ky ) s s s s
+[Fb +ps(1+ Kys)] j 6sin9 }
(4.16 )
(4.17)
(4.18)
'"
_ t ~,.
--! , I
-1 1 .~.
't ;
, , .
51
ywMomentum:
w da (1+jbcos9)(p ... u v tL+p V ... v n..+p V ... u 0 4 ) ~ s s -"2 s 5 . ~ S S a x
= .l..{p v (u F1wv F3)+(D. - p)(2+j 6cos 9+Ky) y s s s S 'D S S
s
+[Pl:oub= +2Ps us2
(l+j 6 COS9)JKYs} (4.19)
Species continuity:
+ V 0 ) ~ Ps ... 3 dx
• -• W'b
+ (1 + Ky ) (1 + j bcos 9) W. I 5 IS
1 - - (C - c. ) (p u F - p v F ) y ib ISS S 1 5 S 3
s (4.20)
where F l' F 2' and F 3 are di mensi onl ess functi ons defi neel as
F 1 = (1 + Ky ) tan Il - (1 - Ky ) j 6sin e 5 5
(4.21 )
(4.22)
F = (2 + 3 Ky ) (1 + j 6cos 9) 3 s
(4.23)
The equations given above must be supplemented by the energy equation and the
equation of state in differential form.
""-----.~.-;;.. .," ..... , -
l
,-
52
Energy:
dlb c.!T + ~hib
d C ib + cpb
b 0 '1, "d.;( =
dx dx •
(4.24 ) ,
State: dl\ d'b dTb
dx - R~ T - - R~ Db dx b dx
(4.25) - p R* Tb~~ d C
ib = 0 b . M. dx , ,
Since the mass fraction~ C. ere linearly related to each other by the two mass , balance equations, only three of the five C. are independent. Therefore, , equations (4.17) through (4.25) constitute a system of eight :iimultaneous diffe
rential equations for the grddients of Pb
' ub
' ~, Tb
, (J, and three of the five
It may be seen from the conservation equations in the divergence form, and
from the discussion of the integral method, that the equations presented in this
section contain linear approximations for the following terms:
Continuity:
x-Momentum:
y-Momentum:
2 . (p+pu )r J,
P u v K r j - j p (1 - Ky) sin e
2 . (p+ pu ) K rJ + j P (1 + Ky) cos e
Species continuity: {p u C. ,
.'
, ,~
~ ,
~, ,
53
It is of interest to examine these approximations in some detail. First, it is
noted that it has no effect on the x-Momentum equation, Eq. (4.13), whether one
approximates the terms p u v K r j and j p (1 - Ky) sin e separately, or combined.
The situation is analogous for the y-Momentum equation. Second, the body sur-
face curvature K (x), and the surface inclination angle e (x) are constants as
far as the linear approximations in y-direction are concerned. It is thet. noted
that both of the momentum equations contain linear approximations for (p +p i~
and p u v rj. However, while the term p (1 - Ky) is linearized in tl,e x-Momer.-
tum equation, it is p (1 + Ky) in the y-Momentum equation. This is an obvious
discrepancy which requires particular attention.
Application of equations (4.8) through (4.10) to these terms leads to the
following relations for p (y) : ,.
Since two different values for the pressure at anyone point in the flow field are
unacceptable, P1 (y) and P2
(y) must be required to be the same. It can be seen
at once that this condition is satisfied for K = O. On the other hand, for K + 0,
it can be shown that this not only requires Fb and Ps to be the same, but also
that
,
l • - . '.
I .~ ~'.
, " :;
, ":',
•
;
, : !
:-
: 'i
--
54
p (y) = I\, = Ps = constant
Fortunately this discrepancy is the only one, since the application of
equations (4.8) through (4.10) to the remaining terms generally leads to only ane
2 nonlinear function of y, each, for pu, pu v, p + P u ,pu C., and W •• A I I
procedure to avoid the discrepancy altogether, and simultaneously reduce rhe
number of functions for which approximations have to bo introduced, is discussed
in the next section.
Apart from this it should be noted that certain terms disappear in the con-
servation equations when either j or K, or both, are zero. In this case, no
linearizations are introduced for the respective terms. Thus, for example, when
the equations are applied to wedge flow (K = 0, j = 0), the linearized terms
2 • which are actually used reduce to (p u), p, (p u ), (p u v), (p u C,) , and W ••
I I
4.2 .3 The Semi -exact Procedure
The general form of the conservation eqyattons for the one-strip approxi-
mation, Eq. (4.11), already indicC:1.ted -that the equations are formulated in terms
of the variables along the-shock wave and along the body surface, only. This
situation suggests the use of the flow tangency condition, Eq. (3.19), in order to
simplify some of the partial differential equations before they are integrated in
y-direction, and thereby converted to ordinary differential equations. As a
matter of fact, it is seen from equations (3.12) through (3.15) that the x-Momentum
.r~ -, ~-==-"'!"tllll_ .!"'--""---~'-'''''''-''.-='''',-''!'<:-;::: .. ~-iiiii!A~;!iaaE_, --- ~ ~",,;-~':...-". ~_ ...... t~
--
t ..
" .• ~ .. . ,
" " , f.
, 1
, , , ) > , ! 1 •
55
equation and the species continuity equation reduce to ordinary differential
equations in terms of the variables along the body surface when v !~ set to zero.
Therefore, only the overall continuity equation and the y-Momentum equation
need to be used in approximate form, which results in a much simpler set of
equati ons for the semi -exact procedure. Thi s set then consi sts of equati ons (4.17),
(4.19), and (4.21) through (4.25), while the x-Momentum and the species con-
tinuity equations are replaced by
dUb +
d"b = 0 ~ ub -dx dx
(4.26)
and
Pb ub
d Cib dx
- Wib
= 0 (4.27)
respectively. Thus, the introduction of the above exact equations eliminates the
need to approximate the terms pu C. r j , (1 + Ky) W. r j , and p (1- Ky). In I I
particulc!"" the d!s'crepancy discussed in the previous section is removed by not
using an approximation for p (1 - Ky).
A careful comparison of equations (4.26) and (4.27) with their approximate
counterparts reveals that they may be expressed in a combined form by introducing
certain indicators, say e and a.. Setting the indicators to zero results in the
exact equations, while assigning the value unity yields the approximate equations.
Hence,
•
\
, , , : ~ :1 I!.
I ,
r ~ ~ ,
56
x-Momentum:
d I\, 2 dPb d + eu -d + (1+ e) x b x
+ e (1 +j 5'cos 9) [I< Z (p T n 1 + P T n 2) 5 SCD CDS
(4.28)
= ~ {(p + P u 2) F - (Po +p u 2) F y 5 S s 1 'b b b 2 s
- P u v (1 +j &' cos 9) (2 + 5 Ky ) S 5 S S
+ [I\, +pS(l+ KyS)] j &sin 9 }
Species continuity:
dC ib - do d - a.(C·
b- c. )(l+j [) COS9)(PCD U D-+p VCD
(3) -d x I IS 5 -2 S x
(4.29)
- .l.. (C'b - C. ) (p u F 1 - P v F 3) ] y I IS S S S 5 S
Equations (4.17), (4.19), (4.24), (4.25), (4.28), and (4.29) now represent
a system of six simultaneous equations for the gradients of ub
' ~, 1\,' Tb, C ib
and o. Noticing that the y-Momentum equation and the species continuity
equations yield the derivatives of the shock wave angle 0 and the mass fractions
Cib
directly, the above set is easily solved for the remaining gradients. Using
..
~'" "
57
the energy equation, Eq. (4.24), and the equation of state, Eq. (4.25), to
eliminate the derivatives of ub and Pb' the following equations are obtained:
dP b .. a 22 A 1 - a 12 A 2 -dx all a 22 - a 12 0 21 (4.30)
d T b =
all A2-a
21 A1 dx all 0 22 - a 12 0 21
(4.31 )
do = A3
;.;' dx 033
(4.32)
,~"
dC ib = A4 (i =0, N, NO) dx a 44
(4.33)
The coefficients a.. and the terms A. are functions of the variabl es along the IJ I
body and the shock wave. They are explicitly given in Appendix C in their
most general form, valid for any combination of the appropriate values for j, K,
a., and e. For the present investigation which is primarily concerned with the
application of the semi-exact procedure (a.;:: 0, e;:: 0) to the flow obout circular
cones (j = 1, K = 0), this implies the use of nonlinear approximations for P u and
P u v ,
Pu = + (p u r - Q u
b r b )( y / y ) s S s "I:) s
(4.34) rb + y cos 9
pu v = P u v r s s s s (
rb + y cos e (4.35)
, ; ~ •
, I ; , . , " .'
r
"'~':r J ;
58
and a linear approximation for p,
y (4.36)
Since it can be shown that the right hand sides of equations (4.34) and
(4.35) are the sums of infinite alternating series in powers of y / y , where 5
o s: y / y s: 1 , the error of the method is determined by the linear approximation, s
Eq. (4.36), for the pressure p (y) across the shock layer.
Having determined the density Pb ' temperature T b' shock wave angle 0,
and the mass fractions Cib
(i = 0, N, NO) by numerical integration of equations
(4.30) through (4.33), the remaining Cib
(i = O2
, N2
) are calculated from the
mass balance equations. Knowing the temperature T b and all of the mass
fractions Cib
, the static enthalpy can be evaluated, whereupon the velocity ub
can be computed from the energy equation. Similarly, with density 1\,' tempera
ture T b' and the mass fractions Cib known, the pressure Pb is given by the
equati on of state.
As far as the variables along the shock wave are concerned, it suffices to
integrate equations (3.28), (3.32) an,:) (3.33) in addition to (4.32). Once the
shock wave angle 0, density P , and temperature T are known, all other s s
variables behind the shock can be computed algebraic!y from the expressions given
in Chapter III •
The advantage of the semi-exact procedure over the standard approximation
appears not only in its relative simplicity, but also in the fact t~at no
. .-Att'tt': . ( .~ '.' ._,~~, .. J
, I
, • ( .,
· '. ~ •
'.
,.
- &.
. .--
59
approximations are involved which contain the rate equations or the species r:!t!!~
fractions. This is important because, for chemically relaxing flow, the C. are I
key variabl es of parti cular interest.
4.3 Initial Solution
4.3.1 Frozen Flow
In order to start the numerical integration of the equations presented in
section 4.2, the values of all variables must be known at x := O. The calculation
of these values will be based on the assumptions that, first, the flow is chemically
frozen and, second, that, in case of a body with non-zero curvature (K + 0)
there is a small region including the tip where the body may be considered to be
a circular cone or a wedge (K ::: 0). Since the use of frozen shock conditions
implies that the flow is frozen at the tip, this assumption is not a new one.
With the moleculllr vibrations in equilibrium, and the gas composition
frozen, there exists no mechanism to destroy the self-similar character of the flow
field, and therefore the classical result of a straight shock wave and constant
properties along the body streamline may be applied. Mathematically expressed,
this means that in equations (4.17) through (4.19), the gradients of all flow
variables along the body and that of the shock wave angle cr can be set to zero.
This implies that also the right hand sides of these equations must vanish. Hence,
with 6, defined by Eq. (4.16), taking the form
11 .. ..$2 •. ) .. _.
,.
j
i v.
"
f , , t
" \
',I ' '
'. )'
'''; ,
6 = ys x sin e
and F1, F2, F3 reducing to
F 1 = (1 - j) tan ~
F2 = (l+j)tan ~
tan~ sin e
F 3 = 2 (1 + j cot 9 tan 13 )
60
(4.37)
(4.38)
(4.39)
(4.40)
the following algebraic equations, valid for chemically frozen flow about wedges
o = 0) or circular cones 0 = 1), are obtained from equations (4.17) through (4.19).
Continuity:
x-Momentum:
y-Momentum:
- 2 P v (1 + j cot 9 tan ~) = 0 s s
- 2 P u v (1 + j cot 9 tan ~ ) + j (0. + P ) tan ~ = 0 s s S 'D S
P u v (1 - j) tan ~ - 2 P v 2 (1 + j cot 9 tan ~ ) S S S 5 S
+ ~ - ps) (2 + j cot 9 tan~) = 0
(4.41 )
(4.42)
(4.43)
Equations (4.41) throlJgh (4.43), together with the energy equation and the
equati on of state, then represent fj ve equati ons for fj ve unknowns: u b' Pb
' T b ,~,
and a.
•. -", ..... ,I!
>
...
61
For a given flight configuration, that is given semi-vertex angle a and
free stream canditions, the calculation is started with an assumed value for the
shock wave angle (J, and calculating the temperature T , the density p, and s s
the pressure p from the frozen shock conditi ons, that is, from equati ons (3.29) s
through (3.31), respectively. With the velocity components u and v, which s s
are to be computed from equations (3.24) and (3.25), the velocity at the body
surface, ub
' is given by
v u b = u s + ...,...._,...,...;s....,.-~....,.._
2 cot ~ + j cot a (4.44)
This relation can be derived by first eliminating ("b - ps) from the two momentum
equations, and then replacing the term (Pb
ub
) from the continuity equation.
Knowing the velocity ub
' the continuity equation yields the density Pb •
With ub
and Pb
available, the surface pressure can be calculated from one of
the momentum equations. Finally, the equation of state serves to obtain the
temperature T b. The energy equati on, not needed so far, is now used to check
the value of ub
• If the velocity ub
so calculated does not agree with the value
obtained from Eq. (4.44), a new value for the shock wove angle (J must be chosen;
and the procedure is repeated.
4.3.2 Initial Derivatives
The necessity for determining the derivatives of all variables at x = 0 arises
from the fact that the equations for the initial values were abtained by equating
'<.$&2. &
."
.. i
,. ,
"
-,"
. •.. 1 I
, I
I
'.".:1
. ; , , .j .,
.--
62
the numerators of the right hand sides of equations (4.17) through (4.19) to zero.
Since, for an attached shock wave, also y (0) = 0, the right hand sides of s
these equations become indeterminate at the tip of the body. The condition of
frozen flow (C. = constant) lel]ds to the same difficulty in Eq. (4.20). I
A set of linear equations for the initial derivatives can be derived from the
governing equations by evaluating their limitir.g expressions as x'" O. For this
purpose let the right hand sides of equations (4.17), (4.28), (4.19), and (4.29)
be denoted by 8 l' 82
, 83
, and IS 4' ~fi:;pectively. Specializing them for the
case of zero curvature (K = 0, rb = x sin e), and introducing
Ys Ii :z:
x
they may be written as follows:
8 1 = + [ P sus (ton ~ - j Ii) - ~ u b (tan S + j Ii) s
- 2 P v (1 + j Ii cot 9)] s 5
82
= +[psu/(ton~-jli)- Pbub2(tonS+jli) 5
- 2 P v u (1 + j Ii cot 9) + (p - Dr ) ton ~ ] 5 5 5 $ 'I)
8 3 = -L [p u v (ton ~ - j Ii ) - 2 P v 2 (1 + j Ii cot e ) ys 5 S S S S
- (ps -'1,) (2 +j licot 9)]
(4.45)
(4.46)
(4.47)
(4.48)
,
.. _ ~.: ;:;;;;: . .!-:;~ ':a#;!LlU=-"-~"" .. ~,,-"'~""- ~
~ 'i
I ",
:~ . ,"
63
84
= Wib
+ a. {(I+ j bcot 9) Wis
- ~(C'b-C, )[p u (tanS-jb)-2p v (l+6jCot9)'}(4,49) y I IS 5 5 5 5 ~
5
Then applying L 'Hospital's rule in order to resolve the indeterminacy, the following
limiting expressions can be obtained:
lim 8 1 x'" 0
p v 5 5
, 2Q Sin tJ
- 2 (cotp + j cot 9) (poo v 5 O2 + Ps Va> (4)} ~ ~
- (1 + j) ..i... (p u ) d x b b
2 2
I ' . _s:!-...::...S --=.,5 ,.-.."...,,-:=--...;:.~:...... + 5 5 5 {(P+pu )-(Pb+Pbub) P uv
, 8 "E:-x!.."b 2 sin 2 S , 2 Q
Sin tJ
+(I-J')(p u 2 02
+2p V u (3)+ RZ (p Too 0 1 +Poo T ( 2 )
005 5 00 5 55 5
- 2 (cot p + j cot ~ (pooU 5 V 5 0 2 + Ps V 00 v 5 0 3 + P 5 V oou s(4)} ~ ~
(4,50)
d 2 d'b - E: ( 1 + j) dx (p bUb ) - € """"(j)( (4,51)
iLl&.._· '- 2 1 ;
l
.. ~., J
•
"
,
lim B = s s s {
p u v
x"O 3 51n 2 a + P +p v 2 _p.
5 5 5 • b
. 2 g Sin ...
- 2 (cot 13 + j cot e) (p v 02 + 2 P V"" 0 ~ ) v "" s s ... s
- R Zs (2 cot 13 + j cot 9) (ps T"" 0 1 + P"" T s 02)} ~ ~ dF\,
+ ( 2 cot 13 + j cot 9) d x
and, finally,
+ a [ (1 + j tan 13 cot 9) w. IS
d Cib ] -(1+j)~ub dx
(4.52)
(4.53)
where it is noted that the above expressions are to be evaluated at x = 0 only.
If the limiting process is now applied to equations (4.17), (4.28), (4.19),
and (4.29), and if the expressions given above are inserted on their right hand
sides, the following equations for the initial derivatives in terms of the initial
val ues result.
Centinuity:
x-Momentum:
dPb U -+ Q
b d x 'b
(1 +e) dd~
(4.54)
+ e(2 + j)
(4.55)
65
y-Momentum:
(4.56)
Speci es conti nuity:
• • :a W'b + a (1 + j tan 13 cot e) w.
I IS (4.57)
where 0S, ° 6' and 07 are dimensionless functions listed in Appendix 8.
Supplementing equations (4.54) through (4.57) by the energy equation and
the equation of state,
dUb d Tb 2
Pb Ub + Pb cpb +
Pb VOO Lb = 0 (4 58)
dx dX X t " 2 "
d~ d Pb
- Pb R Zb d Tb Pb T b VOO
S = 0 (4.59) dX - RZ b Tb dX dx X Too b
where X is a characteristic length, and Land S are dimensionless functions
defined in Appendix A, this set can be solved for the initial gradients. The
resulting expressions are:
where
"
", ,"'l':::
--
F = U b2 { [1+e(1 +j)] cpb - (1 +e)RZb } 07
- RZ b cpb Tb { [1+e(3 + 2 j)] ~~ rlS - e rl 6 + (1 + e)rl7 }
2 u b - eR Zb ub [(2+j) Vtt> Os -°6 ]
( dPb ) P", v! { [ ub
- = - 1+e (3 + 2j)]-d x x =0 [1 +e( 1+j)]u; V tt> °5
- e ° + (1 + e) eL} (~) 67 d x -0 x-
( d T b) =_ d x -0 x- Pb cpb { (:~) ~
e( 2+J') ~ ° -eO -Vtt> 5 6 + (1 + e) ° 7 +
( dCib) =
d x -0 x-
l+e(l+j)
• • W'b + a. (1 + j tan fl cot e) W •
I IS
Pb
u b [ 1 + a. (1 + j) ]
66
(4.61 )
(4.62)
(4.64)
It should be noted that, since no assumptions or approximations were used in
deriving the initial gradients from the governing equations, the compatibility of
equations (4.60) through (4.64) with equations (4.30) through (4.33) is assured.
Again, for a.;::; 0 and e= 0, the expressions given above reduce to the initial grad-
i ents to be used for the semi -exact procedure. Letti ng a. ::: 1 and e = 1 resul ts in
the initial gradients for the stnndard approximation.
--
67
4.4 Equilibrium Flow
As mentioned earlier, an equilibrium flow solution in connection with non-
equilibrium flow about wedges or circular cones is useful because it provides a
check, in an approximate way, for the values which should be approached
asymptotically by the nonequilibrium solution. There are two reasons for the
approximation. It will be shown in the discussion of the results that, for non-
equilibrium flow about cones, the shock wave is curved. By assumption, the
shock wave angle a at the apex is the same as that for frozen flow, while further
downstream, (] approaches the equilibrium shock wave angle which is always
found to be small er than the shock wave angl e cr for frozen flow. Consequentl y, .. ,
those streamlines which have passed through the stronger portion of the shock wave
1 • form a region close to the body surface which must have a higher entropy than the
. ~ stream lines farther away from the surface. This region is sometimes called an
entropy layer. Furthermore, it is known from theory that, in contrast to equi Ii-
brium, nonequilibrium dissociation and recombination are nonisentropic processes.
Thus, the chemical relaxation in the flow provides another mechanism by which
entropy is increased. It is therefore concl uded that the equi libri um state whi ch is
reached after complete relaxation has a higher entropy than an equilibrium flow,
originating from the same free stream conditions, would have. Since, by assumption,
the two flows have the same total enthalpy, but their entropy is different, they
cannot reach the same fi nal state •
. ,
b.
.--
68
Assuming that the air is in thermodynamic equilibrium everywhere in the
flow field, the same considerations as for frozen flow apply, namely, that the
flow field is similar. Again, this implies that all flow field parameters are
constant on planes, or coaxial cones intersecting at the tip of the body.
Newman (Ref. 36) has used equations (4.41) through (4.43), in connection
with thermodynamic data from tables, to calculate conical flow parameters for air
in equilibrium. He found that these algebraic equations yield results which are in
excellent agreement with the results from the numerical integration of the Taylor-
Maccoll differential equation by Romig (Ref. 21). Similar calculations by Thoenes
(Ref. 16, 17) for a simple air model, but using equations analogous to those pre-
sented in Chapter II for the thermodynami c properti es, confi rmed thi s agreement.
The more sophisticated air model employed in the present investig~tion requires a
method of calculation which differs slightly from that used previously, and is there-
fore outlined below.
For an assumed shock wave angle cr the equilibrium conditions behind the
shock wave can be calculated as described in Section 3.2.3, whereupon the body
surface velocity, ub
' is computed from Eq. (4.44). Substituting this expression
into the continuity equation, Eq. (4.41),yields
[ p u (1-j)-2p v (coHI+j cot9)](2cotl3+j cot 9) p = =-~s~s~ __ ~ __ ~s __ ~s ____________ ~~ __________ __
b (1 + j) [ Us ( 2 cot 13 + j cot 9) + v s ] (4.65)
» !
.'
J
, .'.
, . ,
, ., I
,~r.
i ) ,
iiiils :~ 7$--·-
69
Solving the y-Momentum equation, Eq. (4.43), for the body surface pressure
results in
2 P v 2 (1 + j cot e tan ~ ) - P u v (1 - j) tan ~ s s s s S
Po = Ps + -...::-~--=-:--:---,-~-::;,......;;......;~----• b 2 + j cot e tan ~
(4.66)
Starting from a first guess for the body surface temperature, Tb ' together with the
density at the body surfal:e, Pb
' from Eq. (4.65), the equilibrium species mass
fractions must now be calculated by an iteration, using th •• equations provided in
Section 2.6. This iteration can be terminated when the mass fractions C ib and
the temperature T b thus obtained, together with density Pb and pressure Pb
from equations (4.65) and (4.66), satisfy the equation of state.
The energy equation, not used so far, again provides the closing link in
the loop. If the value for the velocity at the body surface, ub
' obtained from
the energy equation does not agree with the one calculated from Eq. (4.44), the
procedure has to be repeated by selecting a new value for the shock wave angle a.
4.5 Numerical Techniques
For the numerical evaluation all equations were converted to dimensionless
form by introducing the following dimensionless variables:
-
70
u' = u ,- v • P' = ..L T'= T . V; v -Ya>
• Ta>
, Pa>
, , eo
c Teo ..L. . h
p'" h ' - . c'= p • (4.67) 2 ' V2 ,
y2 ,
p Peo Yeo a> a>
~, " = x , l
)"
Since infinite long cones have no typical dimension, the characteristic length )"
was defined as a dissociation relaxation length by
(4.68)
where Z is the compressibility factor defined by Eq. (2.17). The equations were
then programmed in FORTRAN Y, suitable for the UNIYAC 1108 located at the
University of Alabama Research Institute. For simplicity of program check-out
and debugging the problem was programmed in three separate parts for frozen,
equilibrium, and nonequilibrium flow.
The frozen flow program consists essentially of a double loop iteration,
following the process of solution as described in Section 4.3.1. The process is
started with an assumed value for the shock wave angle a in the main progrom,
which then calls a subroutine for the iterative calculation of the temperature
behind the shock, T , from Eq. (3.29). If the difference in the body surface s
velocity, ub
' which is computed from two independent equations, is larger
• -- £ = = , •
,.
."'-
71
than a specified minimum, the computation returns to the main program and resumes
with an adjusted value of the shock wave angle cr. The built-in limit of ten
iterations was in no case exceeded. Running times, including compilation of the
program, are in the order of 5 to 10 seconds, even with double precision.
The structure of the equilibrium flow program is analogous to the one for
frozen flow, however, with additional loops for the calculation of the equilibrium
compositions as function of density and temperature.
The nonequilibrium flow program is set up in terms of some eighteen sub
routi nes for the vari ous thermodynami c and other functi ons di scussed in Chapters II
through IV. Its core consists of the standard fourth order variable step Runga
Kutta integration routine RKVS, which is described in detail in Ref. 38. Using
the initial gradients and a simple Euler integration to calculate values for all
variables at an incremental distance away from x = 0, the computation switches
over to the Runga-Kutta integration of equations (4.30) through (4.33). Numeri
cal problems and running times depend on the case and will be discussed in
Chapter V.
.'
..
.,
! \
i l f ,
72
CHAPTER V
DISCUSSION OF RESULTS
5.1 Frozen Flow
The success of the one-strip integral method when applied ~o the com-
putation of supersonic flow of a perfect gas (y = 1.4) about circular cones was
previously demonstrated by South (Ref. 12). He showed that, particularly for
higher free stream Mach numbers (Mea> 3), the results from the integral method
are in excellent agreement with the charts in Ref. 39. In the present investiga-
ti on, chemi call y frozen f1 ow was therefore studi ed onl y to obtai n proper i ni ti al
conditions for the calculation of chemically relaxing flow, as pointed out in
section 4.3.
Figures 7 through 12 show the variation of some typical flow field variables
with cone semi-vertex angle and free stream Mach number, for undissociated air
in vibrational equilibrium. In Figures 7, 8, 11, and 12, the perfect gas results
are 01 so given for comparison. In view of previous findings (Ref. 12), the
difference between the perfect gas results and those of the present investigation
can be attributed to vibrational equilibrium. It can be observed that for those
combinations of free stream velocity and cone angle where the temperature in
the shock layer becomes high enough to cause significant molecular vibration,
the shock wave tends to be closer to the body than for the perfect gas case.
.'
Ii Lll .1
,..
" \ .,~
, t' , ~
i ., • , ,
73
Simultaneously, there is a slight increase in the surface velocity, while the
surface pressure is hardly affected at all •
5.2 Equilibrium Flow
Rather than to duplicate the work of Newman (Ref. 36), conical flow
parameters for air in thermodynamic equilibrium are presented here in order to
demonstrate the validity of the five component air mcx:lel for the range of cone
semi -vertex angl es and ',:ree stream conditi ons consi dered in this work. The use-
fulness of an equilibrium ::low solution in connection with calculations of non-
equilibrium flow was already discussed in section 4.4.
Figures 13 through 16 present a comparison of equilibrium flow lesults
obtained from the one-strip integral method, used in this investigation, with
results obtained by integrating the Taylor-Maccoll equation (Ref. 21). In
addition, these figures represent a comparison of the thermodynamic properties
of the air model used in the present investigation with the thermodynamic tables
of Blackwell et al., which were used in Romig's work (Ref. 21).
Thirteen cases for a variety of cone semi-vertex angles (30:S; e :S;45) and
free stream Mach numbers (10 :s; Moo S30) were selected to cover a wide range
of values of the hypersonic similarity parameter (5 s; Moo sin e:s; 18) at two
different .free stream pressures. In all cases, the results from the two calculations,
each using a different air model and a different method of solution, agree quite
well.
•
,
'~
" ·r " 1'"
.~
, .'
" ~
, •
l I I ,--
74
Additional c"ses for conical equilibrium flow were calculated for the
purpose of comparison with chemically relaxing flow. In particular, it will be
shown at the end of this chapter, how the state of equilibrium, which is approached
asymptotically by the nonequilibrium flow, compares with the state of equili-
brium found when thermodynamic equilibrium is assumed throughout the flow
field. In the next section, these equilibrium states are, for simplicity, labelled
"asymptotic equilibrium" and "conical equilibrium", respectively.
5.3 Nonequilibrium Flow
Nonequilibrium flow calculations have been carried out for various pur-
poses, and the free stream conditions were chosen accordingly. Table V gives a
detailed summary of all cases which were investigated, while Fig. 17 displays,
for each case, the three independent characteristic parameters (semi-vertex
angle a, free stream velocity, and al titude) in a velocity-al titude diagram.
Cases 1 and 2 were selected principally for the purpose of comparing the
results from the integral method used in the preser,t investigation, with those of
the method of characteri sti cs (Ref. 20). Case 3 was computed to check previ ous
calculations using a much simpler air model (Ref. 16). Cases 4 through 8 are for
similar cones at constant altitude but at different free stream velocities. Cases
9 and 10, together with case 5, were selected to study the effect of different
altitudes on all flow variables at the same cone angle and flight velocity. The
•
, ,. , ___ .. ~·zz
. "
"
1, '~ \'
~ 1 J ,t
, /~
. ~.
, " , "
i "
" , , :. "",
" f "
. ,,' .~
".1 ,f, « =, v
75
influence of the semi-vertex angle e on the flow parameters is studied by com-
paring cases 10 through 13, in which both altitude and free stream velocity are
kept constant.
Figures 18 and 19 show the species mass fractions on the cone surface for
cases 1 and 2, respectively. The slight difference i" the results near the tip of
the cone might be due to one or more of the differences in the calculation
procedures. A small difference in the initial values, for example, which is
undetectable from the graphs shown in Ref. 20, together with different expressions
of the equilibrium constants used for the reactions IV and VI (see section 2.5),
may be one reason for the discrepancy. Another cause may be the fact that, in
order to use the method of characteristics, a certain small but finite area of
frozen flow has to be assumed near the tip, while no such assumption is needed
for the integral method. In any case, the results for the gas composition from the
two methods are al most i denti cal for x > 4 mm. The temperature and densi ty
along the cone surface, plotted in figures 20and 21, show the same excellent
agreement.
The variation of the shock wave angle along the length of the cone is
shown in Fig. 22. The resul ts from the two methods agree insofar as both methods
predict that the shock wave angle C1 should decrease. While the characteristics
sol uti on from Ref. 20 seems to i ndi cate that the shock wave angl e C1 shoul d
approach its value for conical equilibrium flow monotomically from above, the
,
.,--- $ ~7._-:>.r=&, >-~t~·,'!.'~~ - :.~~~-" .~.:--~.:=;' ~.-=,~::::;J
..
. '.
;. j;). (-
"
'. "
, ,-
, " , ~:
,Ii i"t, , :,t·
" jJ
" I, t .~. ,
,
'i , i';J ,
.. -
76
integral method solution indicates a limiting value for cr which is slightly
higher than that. As a matter of fact, when the integration was carried out to a
distance of about 70 cm away from the tip for case 2, it was found that the shock
wave angle has a minimum value (cr= 44.39) between x = 25 mm and x = 75 mm,
then approaches its asymptotic equilibrium value (cr = 44 .41) from below. From
x = 23 cm onward none of the variables show any change in the first four sign-
ificant figures. Apart from this, it is noted that the difference in shock wave
angle between frozen flow and equilibrium flow is relatively small (only about
1 .5 degrees incase I, and even I ess than that incase 2).
Cone surface pressures for cases 1 and 2 may be compared in Fig. 23. In
both cases, the difference between the values for frozen flow and those for equili-
brium flow is less than 2%. The method of characteristics is seen to predict a
slightly lower surface pressure Oower by about 0.5%) than the integral method.
Case 1 also was the only case that presented numerical problems. The
der.sity, the shock wave angle, the pressure and the surface velocity exhibited
small damped oscillations, of which an example is given in Fig. 23a. It was
found that a systematic removal of all possible sources for numerical inconsistencies
in the program reduced the amplitude of these oscillations considerably. A
further reduction was achieved by finding an optimum size for the initial step
which was then taken to be AI;= 10-5 for all remaining cases. The fact that
case 1 has the lowest body surface Mach number, i" connection with other
, "
i , ", ~:
. \' • ~ ~'. ., ,. ~\ " i " 1 .' , ,
, . "
, .1
.,\ " ~
77
fir, lings, indicates that the probable cause for these oscillations lies in a viola-
tion of the stability criterion for hyperbolic equations. No definite conclusions
have been reached at the present time since it is felt that this problem requires
a more detailed analysis.
The body surface velocity and tile surface Mach number for cases 1 and 2
are shown in figures 24 and 25, respectivel y. No results from the method of
characteristics were available for these variables. However, the fact that the
authors of Ref. 20 present results for wedge flow where ub
was considered to be
constant, permit the conclusion that they also found thllt the change in surface
velocity along the body surface is negligible.
For both, case 1 and case 2, the flow field has been calculated up to a
dimensionless distance of s= 10.0 from the tip of the cone. This corresponds to
a physi cal distance of about 10 mm incase 1, and about 70 mm for case 2. Com-
putation times were 146 sec for case 1, and 93 sec for case 2. This contrasts with
computation times of about 1 hour per case on the BRLESC computer (Ref. 20) for
the method of characteristics calculations. For the integration of case 2 up
to S = 100.0, the running time was only 252 seconds in this work.
Figure 26 shows the influence of the free stream velocity on the surface
speci es mass fracti ons for constant al titude and cone semi -vertex angl e. Inc! udi ng
the nitri c oxyde mass fracti ons, whi ch show the lowest val ues for the highest
velocity, all variations are as expected. The large difference in the characteris·
tic relaxation lengths is particularly noteworthy. The surface temperatures,
'~'
,
,
I "
·f • .. , . ,f i
, . ,
78
shown in Fig. 27, again display the drastic drop near the cone tip, which is
caused by the strong dissociation gradients in this region. Surface density, shock
wave angl,., and surface pressure are shown in figures 28 through 30. Their
limiting values for large ~ show onl y a slight dependence on the free stream
velocity, in contrast to the gas composition and the surface temperatures.
Figures 31 through 33, representing a 300 cone at constant free stream
velocity but at three different altitudes, clearly demonstrate the validity of the
nonequilibrium scaling law
p • L = constant <XI
which is discussed in detail in Ref. 40. Here L is some char'lcteristic dimension
of tho f10w field or the body. According to the theory, nonequilibrium scaling is
a:f)plicable for that region of the relaxation zone where the reactions are predomi-
nontly dissociation reactions. As seen from the production rate equations given
in Chapter II, the dissociation rates are proportional to the density. Once the
state of equilibrium is approached, the rate of recombination, being proportional
to the square of the density, becomes of equal importance, and therefore the
nonequilibrium scaling law should cease to be valid. Figures 31 through 33 show
that thi s is preci sel y so. >
The length scole in these figures is somewhat arbitrary because the reference
density was chosen for convenience such that the characteristic length
•
• , . , ~,
' . .f .i-"
--
whi ch shoul d theoreti call y be the some constant for all cases, was equal to one
meter for case 10. This means that the; scale is practically identical with the
79
~ scale, which permits an easy conversion to the actual physical length in each
case •
The reason for the dose agreement in all variables shown, in spite of
differences in the free stream temperature, lies in the fact that, for hypersonic
conditions, the total enthalpy of the free stream is given mainly in terms of the
velocity, while the free stream temperature contributes only a negligible amount.
On the downstream side of the shock wave the situation is almost opposite, and
therefore the absolute temperature, which controls the reactions, is practically
the some in the three cases.
Figures 31 through 33 shaw that the results, plotted versus x, are identical,
or at least in very dose agreement, up to ~ = 10. For -; > 10, case 10 rapidly
approaches equilibrium, which is reached near -; = 80. In fact, for -; >80, no
changes in the first four significant figures for the density p and the tempel'ature
T at the body surface could be observed. The curves for case 9 show a tendency
towards equilibrium at -; = 100, but no such trend can be recognized for case 5.
It is also noted from figures 32 and 33 that the asymptotic equilibrium values of
..
80
the surface temperature and the surface density do not agree with the respective
values for conical equilibrium.
It is concluded that, under otherwise identical conditions, the altitude has
a decisive influence on the flow field variables. While, for practical purposes,
the state of equilibrium is reached at a distance of about 50 cm away from the tip
in case 10 (40 km altitude), this equilibrium state is not yet reached at 150 m
from the tip in case 5 (80 km altitude).
The influence of the cone semi-vertex angle e on the flow field is shown in
figures 34 through 37. It is foui"ld that, qualitatively, an increase in e has the
some effect as an increase in free stream velocity. Actually, the effect of a
change in e depends very much on the other parameters, namel y 01 titude and
free stream velocity. It is shown in Fig. 35, for example, that for a variation of
e from 20 0 to 35 0, the temperature at the cone tip changes by roughly 8000 OK •
The values for conical equilibrium indicate that, once equilibrium is reached,
the temperatures at the body surface sti II di ffer by 3000 oK, approxi motel y •
Fi gures 38 through 40 di spl ay the strong dependency of the characteri sti c
relaxation length A on free stream velocity, altitude and semi-vertex angle.
For the examples discussed herein, A varies approximately between 10-3 and
102 meters, that is, by five orders of magnitude.
Fi noll y, Fig. 41 shows a compari son of coni cal and asymptoti c equi I ibri um
in a Mollier chart for air in thermodynamic equilibrium. The three equilibrium
----'- -:
" ,
" .' '.'.
~ ::. ',"
" > , I
t , . ~
j
i
, , ,
,)
,--
81
states shown l Are fixed by their respective values for the enthalpy and the
density. First, the figure demonstrates that conical equilibrium flow is characte-
ri zed by an i sentropi c compressi on from the shock wave to the body, si nce the
two equilibrium states are located on the same vertical line. Second, it is
shown that the value of the entropy of the equilibrium state, which is asymptoti-
call y approached by the nonequil ibri um fI ow, di ffers from the entropy val ue for
conical equilibrium, as discussed in section 4.4. Whether the entropy increase
shown is mainly due to the difference in the shock wave angle for the two situa-
tions, or due to the relaxation process cannot be explained unless more detailed
calculations are carried out. Finally, the close agreement of all variables cal
culatedforconical equilibrium at the body (e.g. Tb ==55270
K, Zb =1.255)
with the corresponding values indicated in the Mollier chart (Tb"':l 5500oK,
Zb "':11.253) provides another demonstration for the accuracy of the air model and
its thermo<:lynamic properties as used in this investigation.
5.4 Conclusions
A semi-exact procedure, which uses exact forms of the x-Momentum and
species continuity equations along the body surface, was shown to yield results
which are in excellent agreement with those obtained by the method of characte-
ristics. Although this procedure was also successfully used in the investigation of
nonequilibrium blunt body flows (Ref. 3,4,5), it was reported to be always
= s
,.
i ,
. ' ~ '. ;.
'}
, ~
~ . ,. 1. ,.
a*
82
unsuccessful for the case of vibrational equilibrium flow past painted bodies
(Ref. 12,13).
The amplitude of some bounded oscillations, which were encountered in one
case with a relatively low surface Mach number near the cone tip (Mb
< 2.5),
could be reduced to a negligible amount by carefully removing all possible sources
for inconsistencies in the program. The rapid increase of the surface Mach number
near the cone tip apparently caused the oscillations to stabilize quickly, and no
further probl ems were encountered.
Although it is recognized that the method of characteristics furnishes infor-
mati on about the variables within the shock layer which cannot be obtained by
the one-strip integral method without additional calculation schemes, it appears
that running times for the semi-exact procedure used in this investigation are
drasti call y shorter than those reported for the method of characteristi cs .
The definition of a characteristic relaxation length for dissociation,
A'"' (d Z/ dxf 1, was found to be extremely useful. Its principal advantage is
that it transforms all problems to the same length scale, thereby avoiding the need
to specify integration ranges for each problem individually. The latter process
could be very time consuming because of the strong variation of A from case to
case (10-3 < HmJ < 102, in the examples discussed).
The application of the nonequilibrium scaling law, P.., • L = constant, to
cases coveri ng a wi de range in 01 ti tude cI earl y demonstrated its validity for the
type of fI ow consi dered in this i nvesti gati on •
1.
•
,.
- '-
- ,
f ! -' , r
, '~
) ,
,j
j
. ,I
-;1 't
--·1' . . . '"
----
83
It WtlS also demonstrated that, the three parameters, cone semi-vertex
angle, free stream velocity, and altitude, exert a strong influence on the
variation of most flow field variables. It is therefore concluded that, for cases
where the characteristic length is of the same order of magnitude as the length
of the conical body, large variations of the gas composition, the temperature and
the density along the streamlines have to be expected in the entire shock layer.
.... 5 -
,..
84
REFERENCES
1. Hayes, W. D., Probstein, R. F., Hypersonic Flow Theory, I, Inviscid Flows, Acodemic Press, New York and London, 1966.
2. Dorodnitsyn, A.A., "A Contribution to the Solution of Mixed Problems of Transonic Aerodynamics", Advances in Aeronautical Sciences, Vol. 2, Pergamon Press, New York, 1959.
3. Shih, W. C. lo, Baron, J. R., Krupp, R. S., and TONie, W. J., "Nonequilibrium Blunt Body Flow Using the Method of Integral Relations", MIT Aerophysics Laboratory, Technical Report 66, May 1963; also cataloged by DOC asAD No. 415934.
4. Hermann, R., Thoenes, J., "Hypersonic Flow of Air Past a Circular Cylinder With Non-equilibrium Oxygen Dissociation Including Dissociation of the Free Stream", paper presented at the Vlth European Aeronautical Congress, Munich, S9pt. 1-4, 1965; also UARI Research Report No. 28, University of Alabama Research Institute, Huntsville, Alabama, September 1965.
5. Thompson, K. 0., "Hypersonic Flow of Air Around Blunt Bodies With Finite Chemical Reaction Rate", Dissertation, University of Alabama, University, Alabama, 1967; also UARI Research Report No. 48, University of Alabama Research Institute, Huntsville, Alabama, September 1967.
6. Allen, H. J., "Hypersonic Aerodynamic Problems of the Future", The High Temperature Aspects af Hypersonic Flow, AGARDograph 68, edited by w. C:. Nelson, The Macmillan Company, New York, 1964.
7. Stephenson, J. D., "A Technique for Determining Relaxation Times by FreeFlight Test of Low-Fineness-Ratio Cones; With Experimental Results for Air at Equilibrium Temperatures up to 34400 K", NASA TN 0-327, September 1960.
8. Sedney, R., South, J. C., and Gerber, N., "Characteristic Calculation of Non-Equilibrium Flows", The High Tem~erature Aspects of Hypersonic Flow, AGARDogroph 68, edited by W. C. Ne son, The Macmillan Company, New York, 1964.
"
-~ - ;0* j
'" " ., ,. ,
, -, J ·f '" t
•
85
9. Sedney, R., Gerber, N., "Non -Equil ib ri um Flow Over a Cone", IA S Paper No. 63-71, also AIAA J., Vol. 1, No. 11, November 1963.
10. Capiaux, R., Washington, M., "Non-Equilibrium Flow Past a Wedge", AIAA J., Vol. 1, No.3, March 1963.
11. Wood, A. D., Springfield, J. F., and Pallone, A. J., "Chemical and Vibrational Relaxation of an Inviscid Hypersonic Flow", AIAA Paper 63-441, also AIAA J., Vol. 2, No. 10, October 1964.
12. South, J. C., "Appl i cati on of Dorodnitsyn's Integral Method to Nonequi I ibrium Flows Over Pointed Bodies", NASA TN 0-1942, August 1963.
13. South, J. C., "Application of the Method of Integral Relations to Supersonic Nonequilibrium Flow Past Wedges and Cones", NASA TR R-205, August 1964.
14. South, J. C., Newman, p. A., "Appl i cati on of "he Method of Integral Relations to Real-Gas Flows Past Pointed Bodies", AIAA Paper No. 65-27, January 1965.
15. Newman, P. A., "A Modified Method of Integral Relations for Supersonic Nonequilibrium Flow Over a Wedge", NASA TN 0-2654, February 1965.
16. Thoenes, J., "Nonequilibrium Flow Past a Circular Cone Including Freestream Dissociation", AIAA J., Vol. 4, No.8, August 1966; also UARI Research Report No. 25, University of Alabama Research Institute, Huntsville, Alabama, May 1965.
•
17. Thoenes, J., "Hypersonic Flow of Dissociating Air Past a Circular Cone", UARI Research Report No. 26, University of Alabama Research Institute, Huntsvi lie, AI abama, January 1966.
18. Lee, R. S., "A Unified Analysis of Supersonic Nonequilibrium Flow over a Wedge: J. VibraHonal Nonequilibrium", lAS Paper 63-40; also AIAA J., Vol. 2, No.4, April 1964.
19. Dejarnette, F. R., "Numerical Solution of Inviscid Hypersonic Flows for Nonequilibrium Vioration and Dissociation Using an "Artificial Viscosity" Method", Ph.D. Tl,esis, Virginia Polytechnic Institute, March 1965.
20. Spurk, J. H., Gerber, N., and Sedney, R., "Characteristic Calculation of Flowfields with Chemical Reactions", AIAA J., Vol. 4, No.1, January 1966.
:'
". :-
;,1
" . . i
, .\ i
•
--
86
21. Romig, Mary F., "Conical Flow Parameters For Air in Dissociation Equilibrium", Convair Scientific Research laboratory, Research Report 7, May 1960.
22. Herzfeld, K. F., Griffing, V., Hirschfelder, J. 0., Curtiss, C. F., Bird, R. B., and Sp.)tz, E. lo, Fundamental Physics of Gases, Princeton Aeronauti cal Paperbacks, Vol. 7, Pri nceton Uni versi ty Press, 1961.
23. Lee, J. F., $ears, F. W., Turcotte, D. lo, Statistical Thermodynamics, Addi son-Wesley Publ ishing Co., Inc., Reading, Mass., 1963 •.
24. Vincenti, W. G., Kruger, C. H., Introduction To Physical Gas Dynamics, John Wiley and Sons, Inc., New Yol'k, 1965.
25. Wray, K. L., "Chemical Kinetics of High Temperature Air", Hypersonic Flow Research, Progress in Astronautics and Rocketry - Vol. 7, edited by F. R. Riddell; Academic Press, New York, 1962.
26. Lin, S. C., Teare, J. D., "A Streamtube Approximation for Colculation of Reaction Rates in the Inviscid Flow Field of Hypersonic Objects", Proceedings of the Sixth S m osium on Ballistic Missile and Aeros ce Technol Reentry, Vo • 4, Academic Press, New Yo , 1961.
27. Treanor, C. E., Marrone, P. V., "Effect of Dissociation on the Rate of Vibrational Relaxat1on", Cornell Aeronautical Laboratory Report No. QM-1626-A-4, Feb. 1962.
28 •
29. Schwartz, M., Green, J., and Rutledge, N. A., ·'VectorAnalysis", Harper and Brothers, New York, 1960.
30. Wittliff, C. E., Curtis, J. T., "Normal Shock Wave Parameters in Equilibrium Airll, Cornell Aeronautical Laboratory Report No. CAL-111, November 1961.
31. Marrone, p. V., llNormal Shock Waves in Air: Equilibrium Composition and Flow Parameters for Velocities from 26,000 to 50,000 ft/sec, " Cornell Aeronautical Laboratory Report No. AG - 1729 - A - 2, August 1962.
1 !
-'
=!::::'--::===-" ___ """"'="""'''''''''''''=_:::_==.::-_'''._~.,_''''''=--1111'''''-=='''''_= .. -:::_-:;·~=·.~·~=, __ =.~:;O:.-""_,_7_~:'"'-=··~-;~""-~·-~~·#"'·· .. = __ '''_-'; ___ ",~~",,,,~_.- :-:::-=:-,.::;, ..... «zIt=.::;;;;:;.;;;:c:=iJ
, . ).
~ , l
, .
:'j' ,
. , / I, ...
" t.
32. Sokolnikoff, I. S., Redheffer, R. M., Mathematics of Physics and Modern Engineering, McGraw-Hili Book Company, Inc., New Yol'k, 1958.
33. Busemann, A., "DrUcke auf kegelformige Spitzen bei Bewegung mit Oberschallgeschwindigkeit", ZAMM, Vol. 9, pp. 496-498, 1929.
34. Hansen, C. F., "Approxi mati ons for the Thermodynami c and Transport Properti es of High Temperature Ai r", NA SA TR R-50, 1959.
87
35. U. S. Standard Atmosphere, 1962, Prepared under sponsorshi p of NA SA, U. S. Air Force, U. S. Weather Bureau, U. S. Government Printing Office, Washington, D. C., December 1962.
36. Newman, P. A., "Approximate Calculation of Hypersonic Conical Flow Parameters for Air in Thermodynamic Equilibrium", NASA TN 0-2058, January 1964.
37. Belotserkovskii, O. M., "Flow With a Detached Shock Wave About a Symmetrical Profile", PMM, Vol. 22, No.2, 1958, pp. 206-219.
38. UNIVAC 1107 BEEF MATH ROUTINES, Manual No. UP-3984, published by the UNIVAC Division of the SPERRY RAND Corporation.
39. Ames Research Staff, "Equations, Tables and Charts for Compressible Flow", NACA Repart 1135, 1953.
40. Hall, J. G., Eschenroeder, A. Q., and Marrone, P. V., "Inviscid Hypersonic Air-Flows With Coupled Nonequilibrium Processes".- lAS Paper, No. 62-67, 1962.
41. Feldman, S., "Hypersonic Gas Dynamic Charts for Equilibrium Air", Avco Research laboratory, Research Report 40, January 1957.
,
, .~
,.
., •
)
"
:.~
, , • \,
.. f j
.~
..
6'02)0
-'YN2)0
z = p /p RT o 0 0 0
Po
T 0
R*
N*
h
(C02)0 =';0
(CN2)0
M
R
p 0
88
TABLE I
GAS MODEL DATA
Adopted Primory Quantities
0.21
0.79
1.0
1.01325 . 105 N/m2
288. 15 oK
8314.32 J/kmol oK
6. 02257 . 1026 kmol -1
6.6237 . 10-34 J. sec
Derived Quantities
0.232 918
0.767002
28.850335 kg;kmol
288.187 988 J;kg oK
1.220 174 kg 1m3
r """"'~c:.==::::::o ____ """""""",======--.="""",,,,,,,,,,,=====,,, . ...,. ... ..,,,,=_ . .,,.. .. ,,,, .. ="....,..,L= ::..._" _~~ ... ,,«::- *
!' .
I . -.,;..;,...;~. . .. . .. .. -. : : ... - ..... '-""~'.":' ~: •• : •• ~, •• ~. '''''l<.~ .... <;. :.'I.'~_;_:.;': ....
TABLE II
ATOMIC AND MOLECUl.AR CONSTANTS
Molecular Rotationol Vibrational Char. Temp. Electronic Weight Temperature Temperature of Dissoc. Degeneracy
Species M. ]) x. a~ 2) a.v 3) * 3) 4) a. g. I I I I J
kg /kmol oK oK oK
O 2 I 31.9988 4.20 2256 59380 3
2 1
N2 28.0134 5.80 3374 113260 1 0 15.9994 5
3 1
N 14.0067 4 10
NO 30.0061 2.50 2719 75490 2 2
1) Ref. 35; 2) Ref. 24; 3) Ref. 20; 4) Ref. 34, Values for NO from Ref. 23.
>
II "
I
,I I ,
Electronic Temperature
a~ 4) J
oK
0 11390 ]8990
0 0
228 326
0 27700
0 174
.
CD -0
~- ~---
....
\
J
il . '" . il I
I: i
\1 ~ I
II i .I
I
, I
II
III
IV
V
VI
Reaction
.... °2+M -20+M
N +M .... 2N+M 2 -
NO+M" N+O+M -NO +0"" 0 +N
- 2
N2 .. 0 .:' NO + N
O 2 +N2 .:' 2 NO
'" "1111
T~BlE III
DISSOCIATION RATE CONSTANTS \
Third Body M 3 k d [m / kmol . sec ], Ref. 20
N2
, N, NO I _ 18 -1.5 *
kd (N2) - 1.2·10 T exp (- eol T)
O2 I _ '
kd (02) - 3.0 kd (N2)
0 I kd (0)
_ -3 I - 1.75·10 Tkd (N2)
O2,0, NO II 17 -1.5 *
kd (07) a 9.9·10 T exp (-eN2/ T)
N2 II _ II
kd (N2) - 3.V3 kd (02)
N II
kd (N) II
= 15. 15 kd (02)
°2,N2,0,N,NO kill 18 -1 5 * .. 5. 2 . lOT • exp (- eNO
I T) d
klV d
= 2.4 '108 TO. 5 exp (- 19230/T) .
kV _ 10 -5.0·10 exp(-38000/T) d
kVI d
if1 -2.5 / = 9.1·1 T exp (-65000 T)
8
J
ff y
"-'
11
!. I
! " "
r
I)
fi
~ ~
! ~
t, I , 1 '" ~ '1
;~: ~i
~i ~I I' ,
.... -"'.~ .... ,- --.~, ...
, I
II
III
IV
V
VI -
~ ".~_, ........... "r .••• .....:., :. ,-"J'- •• ' ... ;"
TABLE IV
EQUILIBRIUM AND RECOMBINATION RATE CONSTANTS \
Equilibrium Constant K M k = kd IK c r c
I 6 -0 5 * N2,N,NO kl = 1012 T -1 Kc • 1.2 ·10 T . exp (-902 /T) r (N2)
3 O2
I I Ckmol 1m] kr (02) = 3.0 kr (N2)
0 J
kr (0) 9 = 1.75·10
II 4 * °2,0,NO kll = 55·1013 T-1.5 K-:: =1.8'10 exp(-aN2/T) I
r (02) . I
N2 II
= 3.03 k~\02) [kmol 1m3] kr(N2) I
N kll = 15. 15 k~J(02) r (N)
111_ 3 * °2,N2,0, kill = 1.3.1015 T- 1.5 Kc - 4.0 ·10 exp (-a
NO IT) r
3 Ckmol 1m] N,NO
KJV =3.333 '10 -3 TO•5 exp (-16 110/T) kJV 10 = 7.2 ·10 exp (-3120/T) c r
KV = 4.5 exp (-37 no I T) kV 10 = 1. 111 . 10 exp ( - 230 /T) c r
KVI .1.35 103 T-0 .5 exp(-21660/T) kVI 18 = 6.741 10 exp(-43340/T) c r - -0
K~ through K~J from Ref. 25, K~V through K~J calculated. -
j
I~ .. ,-" ... !I
I~
t " Ii II "
II
I ~
•
, 1
2
3
4
5
6
7
8
9
10
11
12
13
e *)
45.43
41.07
25
30 II
II
"
" II
.. 35
25
20
." .-,6'" ",",.- "··'X· ~
TABLE V
TABULATION OF CASES FOR CHEMICALLY RElAXING FLOW
01 t .*) Va:> *) Pa:> 3 Ta:> Ma:> Comments
km m/sec kg 1m oK
\
31 6638.0 -2 1.287 ·10 273.16 20 For comparison with method II 5974.0 II II 18 of characteristics (Ref. 20)
40 6390.0 3.996 '10-3 250.35 20 For comparison with Ref. 16
80 11186.0 1.999.10-5 180.65 41.4
II 9350.0 II II 34.6 To study influence of free II 7909.6 II II 29.3
stream velocity on flow field parameters
II 6750.0 II II 25 II 5400.0 II II 20
60 9350.0 3.059 '10-4 255.77 29.1 To study intluence of
40 II 3.996 '10-3 250.35 29.4
01 titude Onc!. IJ 5)
II II II " .. To study i nfl uence of II II II II " semi vertex angle
One!. IJ 10) II II II II II
*) Parameters which are considered as independent.
;8
J
,I . -, ,,_ _ _. _.<
I~
·S. .,'.~ .. -_.." .... ~ ...... .-,~
Ii iii iii
~I II' "
i tl I' I
fi
~, ~I
r I: r , ~ Ii
~ 1>
~I
~; " ,J ~i
l00r--1 --,--------r--I--I---~l
1.01 1.10 1.30
80 1 #.1 \ \ \ . .. \ .
\ \
1.40 V
\ -=.. ./
\ \ \ \
6°1 f I \ V ' I I / \ I \ " ..... v V r ' :7'
l '--' .. ~ ::I -
"i
\ \ \
E 401 I I \ I \ I I' I 1\ / \ , <Cl I I. r \/ • 7"
\
201 1 I" f , V' / X \ r.. ,\y 7\.
\ = 2000 I 3000 4000 6000 7
\ \
01 A I I I I I I
O-V-- 2 3 4 5 6 7
Va> sin e • 10-3 [m/ sec]
Fig. 1 Temperature [oK] and compressibility f.'1ctor at the surface of 0 circular cone for air in thermodynamic equilibrium
(Data from Ref. 21).
...
\
~
j
~, "
" , , • "
( ~::
j,
" " • r·
,\ , , '; ; " 'I·
\
, 1
10~r-------T-------~------~-------r-------T------~ I I
1~~ ____ ~ ____ ~~ ____ ~ ____ ~~ ____ ~ ____ ~
d T -
1
1
cfI
0
l~ ____ ~i~' ______ ~----~----~~----~----~ o 2 6 8 10 12 T • 10 -3 [ oK)
Fig. 2 Ratio of chemical to vibrational relaxation time for molecular oxygen and nitrogen (CO = C N = C NO = 0).
94
=-----"""""""""""'===~==-:.=>======""'", ,,,..,,,, -='" "=,--=, ,=u¥2~,_=, ;~==, ~~-=.I::::::.dJ
,.
,,'
;,
. ,
. '
x
/ /
--
/ /
/
u
--- - -...-
-11-------
.-.-- .
R (x)
Fig. 3 Coordinate system on a pointed body with convex longitudinal curvature •
95
e (x)
•
,..
), , i
1 , ,
pI
/\ds2 = dy
dsj ..> /",.,<'" \
r
dS 1 \
\ \
\
~ \ \
\ \
\ \
\ \
\
d 51
d'P
d 51
={R+y)d'P
dx --R
=(l+Ky)dx
Fig. 4 Differential flow field geometry for the dfltermination of the metri c coeffi ci ents.
96
...
\ , , "'~ ..
!
, 1 , .,
. ;. , ,
, i I
a
..
shock wave
\ \ \
x
--}-__ - - e (x)
---- -- -
Fig. 5 Velocity diagram for locally oblique shock waves.
97
• •
,
,.
.~
) ,.
, .• ,
-.-.~-
./ /'
/'
./ ./
./
N
./ N-l ./
----Y (x)/N - 1 s _--
98
..... -._-
Fig.6 Strip arrangement in shock layer.
. 7'"ftF _. _. 15~ :; .. .. - :e--' .. J
,.
b .. II -r I 1
-,. 7.'
99
50~------------------------.--~--------~--------~
Integral method
--- Ref. 39 (Y = 1.405)
40~-------'--------~---------+--~~~-r--------~
30
20
10~------~~-------4--------~---------+--------~
O~ ______ ~~ ______ ~ ________ ~ ________ ~ ______ ~
o 10 20 30 40 50
Fig. 7
Semi -vertex angle, e Variation of shock wave angle a with cone semi-vertex ongle e (chem. frozen, vibr. equll., TID = 250 oK).
CD
I
t> II en .. .. .. !
..¥ u .-..I: ... .. u ~ -
..¥
~ .. a "'5 . ~ ~
"
" <
" i
, ,
.~ ~ " "
r ,
100
10 , , Integral method ,
\ -- - Ref. 39 (y = 1.405) \ M =S \ .
8 \
~ , I
I /
\ / \ /
6 /
\ \ \ \
.. \ \ \
4 \
~ 8 ... ~ ~
2 \
\ \ 20 , ....... .... -
o~ _______ ~~ ______ ~ ________ ~ ______ ~~ ______ ~ o 10 20 ::10 0 50
Fig. 8
Semi-vertex angl«;;, e Variatian of angular shock layer thickness a with e,:me semi-vertex angle e (chern. frozen, vibr. e4uil., Too = 250 OK).
. , ,
, ,. " . . ' ,.
50
~ . ........ 30 ~..a
..
i • ~ 20
10
o 0
Fig. 9
101
/ M =2 ..
J
, ,
~ ~
5 -L
10 .!O 10 ~ ~'O
Seml-vertex angle, e Variation of cone surface temperature with semi-vertex angle (chem. frozen, vlbr. equll., T GO = 250 OK) •
.. --
!IJ
. .
... '
\;;
1~' , \" ,
h. "', ~! . .. :'.
):' :1,. , . 'If' z
(.. 4;u ~-~~ r;
)1:-1, .••.
.. ~ ~f 1<.
1!: :~":~ ..... '~ -. &\'1 .~ .~~;;
'~~:-' .;e; ~.-,~>;
/.," ;)"
":~.' ".'.:,' .:) !.'-!
---. ,
1,0.' ,.,~ "~'''-'' • t.!'-; •. : .• ," ........ " • •• ,# ,.
• a,
" .Q a,
.. l:-'III
.j v
i
102
10
8
6
'"'
2
o~ ______ ~ ______ ~~ ______ ~ ________ ~ ______ ~ o 10 20 30 '"'0 SO
Fig. 10
Seml-vertex angle, 8
·v'arlatlon of cone surface density wfth seml-vertex angle (chem, frozen, vlbr. equll., T. = 250 OK).
I , :-$'
4 .~
l , ~ I
--j
, , ,
,
:::~ ;,.!!. ' '. ~>.~ ..
• :.,' .'l.
',' .
N
>8 8
Q.
"-
.. -
. ..... ,. ' .. . _ . : "', .'"
103
0.7r-------~--------_.--------._------_.--------,
0.6~------_+---------L--------L-------~-----,~~
Integral method
--- - Ref. 39 (y = 1.405)
0.5~------~--------~---------~--------_r~~----~
~0.4~------~--------r--------'-----~~+--------i .. i l II
~ ~ 0.3~------~--------+--------f-#------+--------i
0.2~------_+--------~-.~~--+-------~--------~
0.1~-------+ __ ~~~~ ________ +-______ ~ ________ ~
M =5 ..
O~ ______ ~~ ______ ~ ________ ~ ______ ~~ ______ ~ o 10 20 30 40 50
Fig. 11
;,;; "'
Semi-vertex ongle, e Variation of cone surface pressure with semi-vertex angle (chem. frozen, vibr. equil., T .. = 250 OK) •
I
i ,
<.~.' . . , :+,
" ~ . .. ::: \' ....
- 00;:' ,.": .... . ... ,,,._, .. , ~, "
104
1.0....--"...---.,.-------,----.-----r-----,
O.9~----~------~~r_---+~------+_------_i
0.8~-------4--------~--------~~~--~----__ --~
.. l:-1 'ii > G O.7~--------+_--------~----__ --~------~~~------~
i Integral method
---- Ref. 39 (y = 1.405)
0.6~------~--------._------_.-------+_--;~~
O~------~------~------~~----_.~----~ 20 30 40 50
Fig. 12
Seml-vertex angle, e
Variation of cone surface velocity with semi-vertex angle (chem. frozen, vlbr. equil., T CD = 250 oK).
, ! I '1 .j ;
" I,
" .... /. '",
". I.
'''''' . .
: . ;. , ... ;-,. I,
'. ;::..: .
l. , .. . '" '>J
: ,
? 75 .~'~, '~".;
~;:.' ~, , ... .')"
'~. ", : ':~" ~. .,:.
.,:'\' ,~l .;t.'.~~
;~ t,
.. ~ ....
~, l:I:
, 'tl" ?: .: . . :. '~ '.'.(
{ ~.
..0 N .. ~ .E >.. ... .--.-..0
'11 l G U ., u ~ ~
=
.--'
...... ,. ..,-.
105
1.5
0 Integral method
---- Romig (R.f. 21)
1.4
1 .3
1 .2
1 • 1
~ 1.0 o
Fig. 13
.'
I , I
// Pat -3' /
-=10 / Po I I
\~ / / ' I I
/' / 1 I
1/ ' / ,,\ /
all/ Pat -2
/' // -=10
Po
/'- / /
,,' l I / , ,
I ' , I
" /1 I I
1 I / / // .- /
/~. ~/ pe.
6 8 10 12 14 16
M .. sin e Surface comPressibility factor for equilibrium cane flow (T ::. 273.16 '1<, P = 1.0 atm).
.. 0
I 18
; , i
I , ~ , I
i , I
~ \
~:..-:,: ':, , '
, ' , .
" ,-, . ...,;. " "
.~ , " I~: ~f , ... ,
J ~ :,r, ' '11,1.
• I
~{;
t 1 .~ A~
~.~I #,:. ,~.
1 ~~ OJ,:'
" ' . '~
" .;
I :'b~ ': X, .. ~. i'l::
:1,/ ~,
" ,;,
- --------
, --
.--
,:~ ... ",~ .";, ...... ~ .• , .•.... , ... ,~ '.< .... - . ",-""
8 .... "-.... .0
.. ! .2 e &. ! III U
..E ~
~
2 5
0 Int'Jgral method
---- Romig (Ref. 21)
, i
2 0 , , " I /
I I I I
II i ,
II .../ I
1
1
5 .'J //
P.. -2 / , --10
Po \,/ ,p\ " ,l'
P... -3 -=10
I / Po
I '" / I / /~
0 ',' I
,I 1/'
... iIoo I
P
0 A 0 -, 6 8 10 12 14
M sin e ... Fig. 14 Surface telllfM!rature for equilibrium cone flow
(T ... = 273. 16~, P = 1.0 atm). o
/'
V/ /
/ / ./
I V/ /
p/
16
106
"
18
I
t
I I 1
I I 1 , , 1 l I j
.! ,
, ~.:
.£.
.. j:~' ~f
"
A-~ 't.: '\': ~
. -f ) .
;;t\' r,:. ·i~ ~::
"".: .,* . \rio~
l' ., }'<';
i . . .' ;,
I 't-',', ,
't; .:~~.
~~' ';~ ~. ~. "
~\
. -
"
..:. <.
-- "
'.~ > " ~:~. \ •. , ..... "', ,'" 'PC -,-,' .."',,.! ........... ...., ..•. , • I' • ~.'
16
1~
I Q.
12
"-~
Q. .. ~ .. ~ • u
~ 10
8
-c:
0
107
0 Integral method
---- Romig (It.,. 21)
I /'
p. -3 /
/
Po"=\ ,/ / /~
/ 1
/( 10
I' ,. \
" , P. _~ , -=10
'" / Po .... -.. 'r- ... ~, .. .,,' 1
,,' "",1 ~ .... """
,/ -.L / ,. ,
I 1 / 1
I 1 I I
I , I
~ I
/1 / / 1 /
/1 '1~/
>- r/ I , .A " o , 6 8 10 12 16 18
Fig. 15
M. sin e Surface density for equilibrium cone flow (1'. = 273.16 oK, Po = 1.0 atm) •
! I ,I
I
'I
,
. "-,,,~C&-ib.·.,_:'.dif_ .. " '; "7'# ·'1. ... '!,'!1tii . ..,:-oi· . '''."",'~- p. ~:t"d\ •• J
.,
':.,"
~,J;' •. #.,
, ' , ,
,.,.:". "~. ,- .
" , ; .. ..... , .. , ' , ': f:·~. "
' ..... ~ , "
.~. "
'., .. ~ ",\', .... ~ .. . ... " ....
20
• Integral Method Po. _
~ ~~ Romig (Ref. 21) -=10 f- ---- Po ~
A' .I.: (/
//
# 16
~; {'-.
r- // ;//
,;(/ ,~
~ {/
- P .f
-' ,Ii
12
8
" /
~' ~
... ~ 0 ~
0 '(
Fig. 16
-6 8 10 12 1" M. sin 9
Shock wave ooole a for equilibrium cone flow cr. = 273.16 OK, P = 1.0 atm). o
P. _" ~=10 Po
16 18
108
1 ..... l< ....
~I;
1,':'
J \;
ti
, <~
,....., E ~
t.......J .. " "0 ::I ----~
100
so
60
40
20
o
J ' ' "IlIii'i~~",,."',""k~-""~~' '.' "'"''
A - v
'" .. ' '.' " i "~-;--"~~'-;:'ii'.l:~.~-:t--:j . .. ~.!il.::-.r .... ~~~~~,.?~~ ~¥;.~~-~~-?\ .. t; .. '.' '\.~: . -: r:\ ' .. : ~ 9~<. -.: :i~~'\ -~~-_ . . i .. -.~~~ .. ~·:~~~~-:;, ~ .... ~:t-~~;~~!'~
30° ....
' .... ..&. -1 ..&. ~ ITS "1 6 ITS IT.
.J.. -y
19,30°
-
12,41.07
•
13, 2So .J.. T
o~ .... t 1,4S.43°
" 6 V..,[lcm/ sec]
.-
...&.
I 10, Ji00 I 11, 3So I 12, 2So I 13, 20°
Fig. 17 Survey of investigated nonequilibriurn flow cases.
-'"-.,-" "-', .• ~., ....... "A" .. ", .. ,\ ....-"_",",." _, •. '
.... o 2
-~
... ---~ '-...-.......~-_ ..• _'. » •
....
• • . -; ,
• \
j
',~ .
:p
:;}\ ." ,I! . ... ,\:-: ~
., . ,~ .
. ~" '. "t: 'I "". ;. " '.
, , ,
.< . ;,:. ~::
-:' .' "
.;e U
- .. -._--- . __ .. _._----------------.. ..,
.-..... , ..
110
O.4r-------~~------~--------~--------_r--------~
1 T C N2 -.-"
O.3~------_+--------~--------+_------~--------~
--0--
_._.-
Integral method
Method of character! stl cs (Ref. 20)
Ir:tegral method (conical equlllbrlull1)
o
.-
0.1~------~~------_,--------~--------_+--------~
-- N
o NO -!-k'-'-'-
o~~~~====~==~~==~~----~ o 2 " 6 8 10
Fig. 18
x [m mJ
Cone surface species mass fractions for case 1 (9 = 45.43 0,
V .. = 6638.0 m/sec, T .. = 273.16 OK, p. = 0.01 atm).
i ; •
."
. . ,
',~'-
. '
, . .... ....... ,'
. ' :
••
. ~,... ~,' ..... . .
111
O.4r--------.---------r--------~------_,--------~
_.-
O.3~------~r--------L--------~--------_r--------~
Integral method ~ .-
U .. .. fj
':I: u
.! :I ~ .. • . -j
--0--_._.-
0.2
~
Method of characteristics (Ref. 20)
Integral method (conical equilibrium)
o
0.1r----,~~--------4---------+_------_4----.----~
----
--- .-.-N-t-__
o~ ..... _~==~~==~==~~ bm!d o 2 .. 6 8 10
Fig. 19
• i;,
x Lm mJ Cone surface speci es mass fracti ons for case 2 (9 = 41 .07 0
,
V." = 5974.0 mlsec, T ",= 273. 16 OK, P", = 0.01 atm) •
~"
•
."
'. ','
" ..
'"
',-, '.!
'.,.-",
,.'~ ,
"f ::.:
-.'
8 ~
35
"..030 ~ ..
j J JI 25
20
.. o
- Int.gral method
--0-- Method of characteristic. (Ref.20)
, 1: 9 = 45.43 0, V. = 6638.0 m/sec
'2:9 =41.07°, V,.. = 5974.0m/sec
\
" ~ \ \ \
\ \ \\ ~,
\\ ~ ~ \
~ , I / , ! ' 1 r'-~ -- coni col
equillb rI um \
, ~ <- Il._
• ~
x [mm]
Fig. 20 Cone surface temperatures for cases 1 and 2 (T ... = 273. 16 oK, p .. = 0.01 atm).
112
111
} 1
•
r
: ' 0'
I. .~',
~ ," r " , "
I • ,i.
I . '
..:~. :',' , ....
,'.J.. ,. : ;.>' '
'" " .:
..
-' ',,\..f " I..
113
l1r---------,----------r----------r--------~--------_,
, , 1 ~ 'l'.
.---
10~------~~~~~~--------_+---------~--~----~
, I
8 I ,~ I
..0 9 t-+i-'----+-~,----'------~-----'-----~ Q. .. ~ ';I
-3 ., ~ Jl
I I I
Integral method
- -0 - - Method of characteri sti cs (Ref. 20)
, 1: a = 45.43 0, V." = 6638.0 m/sec
, 2: a = 41.07 0, V." = 5974.0 m/sec
_._.- Integral IT'ethod (conical equilibrium)
8 __ ~-----+-------~--------,_--------._--------~
OL-______ ~ ________ ~ ________ ~ __ ~ __ ~ ________ ~ o 2 4 6 8 10
x [mm]
Fig. 21 Cone surface densities for cases 1 and 2 (T ... = 273.16 OK, POD = 0.01 CJtoo"~.
.,
==,.....,,.....-:-... '\i!i j 53 "'"
r
I .• " ~ , , .
, .'.
, . ."
I . ,.:,,"
: .....
·':t "\ '. ,.
1.;_, ,
i " ,"?, :. ):.
, , .,'
·r,.~.' .,:',: .. ~: ','. , ,
.. ~:,
:$.t, ~ :.
":;f'l . .-!~i ,<' '''':' ~:~" ;. , ~
,·;t .. ,,-, '~,
.-~:.~ '~ft
if .)1: ~f'"
.. ..:.. t' ' •
.. .~~:",
'., ;,,,,,"" .' ., ,'.
t) .. • -r i
.iii
8 ~
t) .. • 1 • > ~
.iii
8 ~
.. -
, ... " ",. . - , I' ,- , .•
114
51
Integrol method _._-- Integral method (conIcal equlllbrfum) - - • - - Method of charocterf atl CI (Ref. 20)
50
49 ', ....
-...... \"'- ---- ..... _---- - ----~ ---_._--'III >
o 0 2 4 6 B 1 o
xCm,n]
0) case 1: 9 = 45.43 0, V", = 6638.0 m/sec. .~ ,
Integral method _. - . - Integral ...,ethod (coni col e~uil ibri um) --O--Method of characteristics e'.20)
45 , r"- __ .. , .-- ---- --..4 It--- ___ ,...--
_._.-
.... > o
o 2 4 6 8 10 x Cmm]
b) case 2:9 =41.07°, V", = 5974.0m/sec.
Fig.22 Shock wove angles for cases 1 and 2 (r", = 273.16 oK, p", = 0.01 atm).
.: ......
'.' . . ' .; ~ .' . "'. '. -',. ~ .... ,
. '~~''''
I,'
~, , .,'
" ...
N >1
I a.
-'
0.535
~ 0.530
( • 0.525
i ." o
0."52
0."50
0.-448
0.446
. -'
Integral method
--0-- Method of character! stl CI (Ref. 20)
......... o scill atl onl (encountered In' 1 only)
.~ . , ..•. . :.:' .' . ~,. .: . . ". ", . . : • .... '0 0 '. • . ... . ~ .. . ' . . . . . . . . : . . . . • '.' '.' '. . . '. . .
........... - '.' . . . . > p.-- - - --p...----- -----....j --o 2 .. 6
x [mm] 8
a) case 1:9 "' .. 5.43°, VCD =6638.0m/sec.
Jlltegral method
--0-- Method of characteri sti cs (Ref. 20) ~ 1\ \~
" ''-I .... ---- ------11'------ ---
o oe; >-0 2 4 6 8
x [mm]
b) case 2: 9 = "1.07 0, V CD = 597".0 mlsec.
Fig. 23 Cone surface pressures for cases 1 and 2 (r ... = 273. III oK, p ... = 0.01 atm).
115
. ......
10
.-
10
I ~ I
1 i ! t
.,. ; .. ' r- ,.
, .~ ... ·i·. ~, : ., ..
I ,. ~ .... . '
I I' I
~:.
.~~. "
.....
' .... '" .
0.65 2
0.650
0.648
r 0.646
oC .. 0 o
0.716
0.71 4
~
~ 0.71 2 .-u
° ~ 3 ~ 0.71 0 V J!
oC > 0 o
Fig. 24
,.' : :
coni cal equlllbrf um:
ub/v. =0.665
2 4 6 8 x [m m]
a) case 1: 9 = 45 • .043°, V"" = 6638.0 m/sec •
coni cal equilibrium:
~ Iv"" = 0.723
2 4 x [m m] 6 8
b) case 2: e = 41.W ° , V"" = 5974.0 m/sec.
Cone surface velocities for cases 1 and 2 CT ... = 273.16 oK, P .. = 0.01 atm).
116
10
10
.D
~ ~ ., 1 ..c u a ~ ., u
i
. .
.----~ ---- .. --.-.--.-.----------- ~-.
. ~, .. ,
3.0
2. 8
2. 6
2. 4
2. 2
'C
0
. .'
," ·,0
'2 ---.
~
I
'1-...
-..... , 1: e = 45.43 0, V'" = 6638.0 m/sec
, 2: e '" 41.07 a, V'" = 5974.0 m/sec
~
o 2 4 6 e x [m m]
Fig. 25 Cone surface Mach numbers for cases 1 and 2 (T.., = 273.16 oK, P. = 0.01 atm) •
117
10
iiiiIiiII=: ..... _ .... s=;;_;~~~Vt::ll*iIii. ;;;;:;:·~.;-~liiiit,"; !~i~.!li.ilfiiiiiEE:\liiiii".iiiiii.i!. ·iiiJii!_~· .~:eiii;;::;;=.:;:. =:.,~";:;:.-;;:;._ z::r!iiiiliiwir..::;.=_!!!:."iiAE_e_;; ... i<==;;::;; .. ;;-;: .• !:::;"'l'-~~'';.i\.£.~ -;:t=- L .c_
';...:
~l' • '
" ,
:,..,:; .. ::,f.;'," .,' "
. , .... , ;'~ ..
I ,:
i I' I'
'~~~:.' '. .' ;;y>;
)l ''r
.' ..
'-
--' --.' .
118
0.4r------r-------,r-------,-------,
~.~~.~--~.==~.==~.==~.=-~~--t_----------~r_~;::-:~I--C-N~ ....... ... _--- -_ ... _----------- -
0.3 case , V GO [miseC] ~[m]
5 ---- 9350.0 1.451 6 7909.6 7.042 7 _._.- 6750.0 61.59
--------.... ---O.2r/~~-~"--~==F===::::::::::=:=r~=, -.... -
/' "'--0 / I I I
......... -' ....... --.- --.-- . .-- . . -'
_._0 . -'
. / O.IHl~,~---~,----~r_----~-------~
NO
0_._._._.-. ._._._ . . ------.-.-. -- ...... --::. ..... --- --
N --
o~~~ __ ~_~ o 2 <4 6 8
Fig. 26
~ =x /~
Influence of free stream velocity on cone surface species moss fractions (9 = 30°, altitude 80 km).
, , ,j ~
, :' . ''''', '.
:>~> :~,~~::' I:~;'i i, ";'
• ,,.: ,'f;f
" ;:,;..' ~ . .
...
60
50
• 40 ... ........
..Q ... ..
t ! II 30
i
20
. ---'"
r
I' I , \ \
f-\ \ ' ..... ..... ...
f-\
"-. \
" ...... '-.-
.. =-o
o 2
.'-
119
CaM , V.[m/sec] ~ [m]
5 9350.0 1.451 6 7909.6 7.042 7 6750.0 I 61.59
-- "-~5 ___ --------. t----__
r'6 ~-.--.- '- --_.-.
r:,~-'-
6 8 10 ~ =x/~
Fig. 27 Influence of free stream velocity on cone surfoce temperature ( 9 = 30°, altitude 80 km) •
. . Ii " - • •
,
I:,..'.
;., . .. ".
"
... ,.' , ..
. ; ... ~ , ..
i !
.~ .
:U" '" ,"
.... _---.- --.~ .. -~-----------
'1
-
"" ....
120
11
---_ .... -------- .~ , 5 --'6~"'" ........ .
..-: ........ -..-. ,," ~ 0 . ..,..' ,// ~f'-
/ l • .; ",', 7
1
/ •
/ I
I i . •
Case , V. [m/see] ~ Em]
5 9350.0 1."51 6 7909.6 7.042 7 6750.0 61.59
8
r
--
c :..
o o 2 4 6 8 10
~ =x /~
Fig. 28 ... fluenee of free stream velocity on eone surfaee density (9 = 30 0,
altitude 80 km).
I ..
. ,.
,.".j , .. . : . .L'
,,' . . ~:. , .~
~; ~~
;';"
"', >8 0:
......... .tI
0..
b .. • "i3
0
~ J
.JI. g '6i
0.262
0.26 0
--.. ----.-- ------ - -- -.-.- --"--'-'-'~---"--------------""
--.
-.
o 1 2
Case I
5 6 7
3
9350.0 7909.6 6750.0
1 •• 51 7.042 61.59
5
Fig.29 Influence of free stream velocity on shock wove angle (9 = 300,
altitude 80 km) •
Case , V", [m/sec) ). [ m) r
5 9350.0 1 •• 51 6 7909.6 7.042 7 6750.0 61.59
~: 16~ -.-.~ .!.7.::'!. ., ----~.-.-.-r---- _. -.... _-- "Is-::;;r-- ------ ------ ------
121
.. :. 0 o
Fig. 30
1 2 '!: =x /A 3
Infl uence of free stream velocity on cone surface pressure (9 = 30 0, altitude 80 km).
5
.,
..0 .-u ~ .. g -... u
~ l:I 0 E .. II -u
1
. .. " ' ~.
,~ ,"
.. ,,',
'." .' . . , ,.,-, ;' .
'-~ ...
.... -' . '. .
0.4r---,--------------r-------------r-------------,
1 T C N2
.-.
0.3~--4-------------~-------------+------------~
o --0.2
Case I nit. [km] ).[m] ). imJ t: _._.- 80 1.023 1.451 9 ----- 60 0.998 0.093
10 40 1.000 0.007
). = ( D .. I D f)). re
122
0.1r---~r_----------~--------.------._------------~
0.1
Fig. 31
NO
-..
Species moss fractions an cone surfaces for different altitudes (9 = 30 0
, V .. = 9350.0 m/sec).
j i 1 ! l
1 ,
't..J." . ,,~ : ; ;'. ~.f/. . i·.r~" ... \ . ,
~".,y.. '
,-.. :;
... !
I-
"-1--'1
II
11--'1
• ! l
I fj ~ it 1 .x
.. -
--' ...... -,.. ......
35
30
25
------ --
Case I
5 _._--9 ----
10
Tb
alt. (km)
=
80 60 "0
( ~ :f) (
Hm)
1.023 0.998 1.000
Tb
TCD ) T = 250.35 0 K ref
r • (p",iP f) >. re
conical equilibrium
Fig. 32 Scaled temperoture on cone surfaces for different altitudes (9 = 30°, V CD = 9350.0 m/sec).
). (m)
1."51 0.093 0.007
123
I i I
! , t ! ! ! ; I
,~
,~
i ,. ,
! , , .,
,', /"
"
.;. ...
"'" ',I." ,
:>, l" .. J~ .• ' I~
... -: I . ,. I" ' , .'
" ....... ,' . to', .. .... . ....
a.' "'
..Q 0.
12r---~-------------,--------------.-------------~
conical equlllbri um
/'
"r---t------------~------------r_~~~----~
10~--~------------~----~~------~------------~
9
C ... se I alt. Ckm) ).[m) ).[mJ
5 --_.- 80 1.023 1.451 9 ---- 60 0.998 0.093
10 40 1.000 0.0071
X = (p ... /p fl). re 8
;=~l[mJ
124
Flg.33 Density ~1 cone surfaces far different.altitudes (8 = 30°, V ... = 9350.0m/sec:).
•
j
i i \
i ,
-
r
;~,. , ......... .
I',.
r "
','
I '
~, ".
,','.: , .,1.
,,~,
. : ,
..0 .-U .. .. ~ .-.. ~
," .. u 0 .:: .. .. a E .. " u
",." oJ
, -
---' . , '.
125
O •• ~-------,--------,---------,-------~--------,
" ....
0.3
0.2
I I
Fig. 34
.... - 1 T CN2 ---- --- -- ----
Case , 9 Hm)
11 --- 35 0.0024 13 20 1.1216
---------------------
-....
-----
o
N
---- ..... -- ------- NO
Influence of (.;one semi-vertex angle 011 surface species moss fractions (V"" = 9350.0 m/sec, altitude 40 km).
,I ,
, 1
:.':-.,'
,,"
- -------------------------
.--"
60
5
..... " " .... ..0
0
0
~
\ , \ \ \
f- \
\ \ \
\ \ , ,
0 \ , . '"
2 0
~ .. .. 0 o
"'-"'-
.......
.. '
126
Case' e ). em]
10-·_·- 30 0.0071 11 ---- 35 0.00237 13 20 1.1216
... -
"" ..... ........... ....... -.. -------- .. - -- -----.
""'- ....... -'- --..--.-.-
'r;~-_._._.-
conical .. -. equi I ibri um
1 2 ~=x/). 3 5
Fig. 35 Infl uence of cone semi -vertex angl e on surface temperature N. = 9350.0 m/sec, altitude 40 km}.
1 '. , , A I !
'!
1 t
" .. ,:
" "
'.
:'."" " .' :
~,~' . I ,:
I· •
, ~'.~ , .- .
/~. r-r , '
.. ~ . "
, .1
. +,:~" ; ';~'':'''-
8 Q.
........ ..a
Q. .. >.. -'II! C .,
"tI ., u
~ ::I
.."
-, .. . ... "',, .. , '.
127
12r-------~r_------_,--------~--------_r--------~
7'-llr---------+---------+--------~~~------r_------__4
coni ca; _-+--</ equilibrium
1
------0
9
r·/ liJ L
8
Case'
10-----11 ----13---
e 30 35 20
-' .... -\ ,
0.0071 0.00237 1.1216
;i o 1 2
~ =x/). 3
Fig. 36 Influence af cone semi-vertex angle on surface density (V ... '" 9350.0 m/sec, altitude 40 km) •
4
---
5
.,
".,.u .'"
"
.'
, ' 'i ' ~ . ," ,
"'.'
" .. ....--'. ~ ....
.,;. ...... . ,. '. .' , .
128
8
, 7
6
Case , e ~[m]
10 _._.- 30 0.0071 11 ---- 35 0.00237 13 20 1.1216
5
1--.- .-.- I- .. -.-.-._._._ ..
-.- ,....---.--._. /
/' Ii ..
------ ------r----- ------ ... ------"...-~.
""Ii ~
0 o 1 2 3 5
Fig. 37 Influence of cone semi-vertex angle on surface Mach number (V .. = 9350.0 m/sec, altitude -40 km).
- ;" ~
-- -.-~---.-------------.,
"-
.,' "
. . '.~~~" ""~ " , ,,',.'"Y", ';"~""'~''='''' ..,...... .... "' ...... . '_ .. ,. -',,- .;... .. . .
10
r= I~ ~
~ I-~
"" l-
F I~ f:
~ I- altitude l-i- --- 80km
-~
~ ~ ~
~ :: --, 60 km
= 19u
= ----
~ 40km = 0 = , 10 ----
10 .~
A 0 Y 7 8 9
v •• 10~3[ m/sec] 10 II
Fig. 38 Influence of free stream velocity (and altitude) on characteristic ~elaxatian length (9 = 30°).
129
I 2
, 1 , j ,
I
1
#' •
"
.:.;,':'
:~\. $;:,
"
tr. 'r '.
.\ ,,;. ,...., 1:(1 E
~ ......
,.' :".: ..< .. .J: I . . . ..
,. 21 ,
I." ., -I:: 0 .-*;, D
',~, ~ "" J.; 1 ~. "
-":"'t,: u
'1ft;: ti ·c
~"<' U 1*.' , ... ~ ~~, U
~. f 0
.J:
II u .. " . -t". ~"~
'i~ ~ ," ~ ......
~:' '!\' )o? ,. ;~ . .\.
"'-
--"
10
1
... ~ ... ' ... -,..,.;. ...
~ I-r r l-
I-
I-
~ l-I-l-
I-
r
l-
~ l-I-I-
I-
l-
I-
2 -----I-
-
-
,~
o 20
, . ' .. ' -...
" 13
, 12
, ' " '
Seml-vertex angle 9
I 20 0
19
250
I
l
"0) / 300~
'11 I 35 0
40altltude [km~O
' St
-'-
80
Fig. 39 Influence of altitude (and e ) on characteristic relaxation length (V ... = 9350.0 m/sec).
~ ,'-'" "'0/'" ",., _""'-'"~~' "'~\d··"·
130
100
\. ' . . ,~ '" , .
I
; ..... ,.'." "; "
,
-
: "'I~"'" ........ 'lwtllio' ..... _ ........ " ..
10
1
....., E
'-' .< .. ..c ... r .. fi 10-1 .-1; ~ ! u .-1; ·c II .. u
G '0
10-2
'.
'.-'. " ." .. -
altitude
'5 80km , 13
9 60 km
20 30 e
Flg.40 Influence of cone semi-vertex angle (and altitude) on characteristic relaxation length (V .. = 9350.0 m/sec).
131
40
£ LE " =.=~ .~. oO--"U & ._. __ ..... 2£ __ ... n-:., ,..~."=e;:;tJ!iJ:::.;;'ia:i· ."C:~ 2
. '!" ~" . ;~:t ~i' .,
~. :' .
',.;::, " ,,- .
.-,~:- ", ;: " ., "
~:,~:' ~;-r .\
I:i!.'-:',
.,)~ I~
,,j , i . ,- . I" ):; I
.,,'
0 .... aI!
"'-..c: .. t J .£
'-~ .. ,
, '
, "
170r-----------~--------------~~--------~--~ asymptotic
" , "--- ....... -
equlllbri um .,ody)
"-160r-----~~~-+--~~------_A~~----_n~----~
150 ./ . conical
equi IIbri u", ........ ~ock)
........
/' ./
',// ./ ..... ·~cS
./ '
I~'" conical equilibrium
f:,ody)
, " /'~-t
140~~~------_.~--------+_--~--~_+----------'~--~
'" ..... I
'0 ..... , .... . , I
14-+-1 6 (siR)
I 40 41 42
Entropy, $,/R
Fig. 41
132
Comparison of conical and asymptotic equilibrium for case 10 (9 = 300
, V ... = 9350.0 m/sec, altitude 40 km) in a Meiller 3 diagram (Ref. 41; To = 273.16 OK, P = 1.0atm, p = 1.288kg/m,
R = 283.188 J / kgOK). 0 0
.,
! ,
r=r _ j - r-.-----~·"""""";!:===I:::Ii==:;:;;;;s;;===========_======= !::" =5!!:i;;?===;;::-~-~-C-=:=C-==""'-'-""I,!!:' =;=_~
,.,'
. ,
0, ,
... ,": .' .'
lo:~ "z~oo ..... ~. o o. . ;;<.
, .:~" , .1 ....
. ;;~'" o ~"
t-0<
.--.----.. -_ .. _--- .. --.. _._-----_.---.---._.-~------------""
"-0<
" , '
'~";-;" 0 • 0 0 ,
......... 'W' .. "t~.~._ w.. .. '", .. ' .......... ,' .•• _ ............ ~, ";... .. . . ..... . .. -...
APPENDIX A
SOME EXPRESSIONS ~ELATED TO THERMODYNAMICS
Frozen speed of sound, a f :
Dimensionless functions L, S:
c RZT P
c - R Z P
d C.
•
L ..
R* T CIQ
S = A--:-y2
CIQ
I
dx
L:*-• I I
d C. I
dx
o •
'. " ". ... ..... ,... ~~ •• ' ,>~.' '""
133
(A-1)
(A-2)
(A-3)
(A-4)
-al e. v ... "p 0:":;:1 S
. ~.;:.', .. ,"'.
, ,: I: !.ti.
'~9JI),;;" . , ..•.. (,":' .. ; .~'
..... '.
'.
.: ", "
! • :~
. -.--- . ,'~.
'.' .~~,."".,. \I:-""".~'~ o,j ................ , • .:. ...... ,ok • • ", -", •• ,~._ .... "-"'" ....... ", '." . '- ..... , ~., .... ,,' ... .,
where
APPENDIX B
DIMENSIONLESS FUNCTIONS 01 FOR DIFFERENTIAL SHOCK RELATIONS AND INITIAL DERIVATIVES
2 V",cos a [V", sin a (V",sin a - B) - (2 - b: (hs - hJ]
T",[(RZ", -2c ) B + (V",~in a)bc ] ps ps
(L c ° -V 2 sin a cos a) ] - ps 1 '"
RZ", T", b = 1+ ---,...
(V",sin a)2
134
(b-1)
(b-2)
(b-3)
(b-4)
( P", ) (p", )2
03 = P -1 (sin a c\,;'s ~ + cos asin ~ ) - P sin a sin ~ 02 s s
(b-5)
2
04 = (-:-~- -1) (sin asin ~- cos a cos ~) - (:: ) sin a cos ~ 02
(b-6)
The following expressions are valid only at x = 0 for K = 0:
J
1 i , ., I 1 #
i 1 I 1 i '.
q
~ ,;",
;:;:;:;=_~ -.,- ~~!:$'2'lIiI5F=' ===IOOlI==;:::;&I;;;;iIii;========;:;;a=====..::= tl!:::_;:1!o:;;""e~·===~_ ... a_.~--_ -
t
r,' .'
• t\
~, . . .~.,
,r :';!
:'
•
'-~., ..
, '
,t~ ''''', ~ .... ~. ,". ._ .... ~ ...... ' .. ' .. - , .. -., .' ...
" . 5
.... e=-, .... 3 la· _J. 11
135
PCD V~ (2 +j) { +
+ 2 ( cot j3 + j cot e) (p ... v 5 "2 + Ps v ... "4 ) $-7)
+ j (1 + ton j3 cot e) (p u "2 + P V "3)} ... 5 5 CD
+ j (1 + tan j3 cot e) (p CD u 2 "2 + 2 P V CD U " 3 ) 5 5 5
$-8)
+2(cotj3+jc:ote)(p ... u v "2 +p V ... v "3+ P v ... u 0 .. ) 55 5 5 S 50t
+ j tan j3 cot e (p 5 Teo ° 1 + PCD T 5 02 ) R Z 5 }
1 { [
P U V 5 5 5
p ... V; sin 2 j3 +
P +p V2
-D. 5 5 5 'I)
, 2 Q
sin '"
- J' (1 + tan j3 cot e) (p u v 02 + p V v rL + p Veo U 0 .. ) "'ss seos3s Sot
- 2 (cotj3 + j cot e) (p ... v~ 02 + 2Ps Veo vs(
4)]<2 cotS + j cot ef1
- R Zs (ps Teo "1 + Peo T 5 ( 2 ) } $..(9)
q
" :"0.
..--- .,.,.
. .... . "
APPENDIX C
FUNCTIONS a •• AND A IN EQUATIONS (4.30) THROUGH (4·.33) IJ I
= 2 a 11 Ii,
a 12 = - Pb C pb
a 22 ; Pb [ R Zb - (1 + e) ":pb ]
136
(C-l)
(C-2)
(C-3)
(C-4)
a 33 = (1 + j 6 cos 9) (p U v "2 + P Va:> v 03 + n Va:> U ( 4 ) (C-5) a:> s s s s s s
A 1 = ~ { P U [(1 + K y ) tan f:l - (1 - K y ) j 6sin 9] y 5 5 5 5 S
PbUb[(1+KYs)tanl:3+j 6 sin9 ]
- P v (1 + j bcos 9) ( 2 + 3 K y) } s S 5
- u. (p U 02 + P V (3
) (1 + j '6 cos 9 ) b a:>s sa:>
(C-6)
(C-7)
-_ .. _._,'-
. , " ~ • j ,,~
, .
~ . . ;.~.: .'
,. " ,.' .
/:-;:r .
..... . ':
.. '
• '"
, ~',
.' '.',
.--.- ,.
. f,'" . ~: .... ~ • ., •
+ ~{(p +p u 2 ) [{1+ Ky )toni!-{l-Ky )j5sina] y 5 S S S S
S
-(~ +Pb u;) [{1+Kys)ton~+ j6 sin a ]
-P u " (l+j 5cosa){2 +5Ky) s s s s
+ j "6 sin a [~ + ps (1 + Ky s)] }
- € (1 + j "6 cos a) [R Z (p 1;" ° 1 + q,., T ( 2 ) S s s ,M
2 ] A3 + P u 02 + 2 P Va> u 03 --<X> S S S 0 33
A3 = ...!.. {p u v [( 1 + Ky ) toni! - (1 - Ky ) j"6 sin a ] y S S S S S
S
- 2-+ (n - p ) (2 + j beos a) - P v ( 1 + j 0 cos a) (2 + 3 Ky ) "D s s s s
+ [ ~ - Ps + Db u; + 2 Os us2
(1 + j 6 cos a)] Kys }
(1 (C.b - C. ) { [ - _I IS P u {1 +Ky )ton~ - (1- Ky ) j
y s s s S S
o sin 6 ]
-;:J v (1+j 5 cos6){2+3KY) } S S S
A3 +(1 {C.
b - C. )( 1 + j 0 cos e)(Q" u 02 + p Va:> (3)
liS 5 5 033
137
(C-S)
(C-9)
(C-10)
.'
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