Sonic line in nonequilibrium flows

APRIL 1966

SONIC LINE IN NONEQUILIBRIUM FLOWS by

Duvvuri Tirumalesa

SONIC LINE IN NONEQUILIBRIUM FLOWS

by

Duvvuri Tirumalesa*

,..

ACKNOWLEDGEMENTS

The author wishes to thank Dr. G. N. Patterson for his en-couragement and interest in the present work and Dr.!.!. Glass for rnany helpful discussions and critical comments.

I arn indebted to the Faculty of Applied Science and Engineering for a Senior Research Fellowship.

This work was supported by NASA under Contract NsG633, and the Canadian National Research Conncil and Defence Research Board.

·

.

•

'j

•

SUMMARY

An approximate equation valid in the region near the partially frozen sonic line is derived. Solutions of this equation are obtained and it is shown that the curves of constant velocity, the partially frozen sonic line, the line of horizontal velocity and the limiting. characteristic are all parabolic. In sorne cases the sonic line and line of horizontal velocity intersect on either side of the nozzle centerline. It is shown that the line of horizontal velocity lies upstream of the sonic line .

TABLE OF CONTENTS Page NOTATION v 1. INTROD DC TION 1 2 . THEORETICAL CONSIDERATIONS 1

..

a) Assumptions 1 b) Rate Equation 1

c) Basic Flow Equations 2

d) Relation Between Sound Speed and Flow Speed 4

3. TRANSONIC APPROXIMA TION 7

3.1 Equilibrium Flow 9

3.2 Solutions for the Nozzle Flows 10

3.2.1 Limiting Case of Equilibrium Flow 12

3.2.2 General Nonequilibrium Case 12

4 CHARACTERISTICS 15

4. 1 Characteristics for Near- Equilibrium Flow 15

4.1.1 Approximation in the Sonic Region 17

4.1.2 Limiting Characteristic s 17 5. SPECIFIC CALCULATIONS 18 6. DISCUSSION 20 " 7. CONCLUSIONS 20 REFERENCES 22 F IGDRES APPENDIX A APPENDIX B APPENDIX C

..

iv

a AI A Bi f

Bf

C g ~Ol) ~Oi!, h k k à , k r Kc

K*

L m ma M Mf n

..

N P p NOTATION sound speed

expression defined in Eq. (23) expression defined in Eq. (26) expression defined in Eq. (32) expression defined in Eq. (34)

slope of the velocity along the nozzle axis in the sonic region

expression defined in Eq. (65')

(

statistical weight of ground energy l.ev..el for at om and molecule respectively

specific enthalpy or Pla.nc;k constant or nozzle height at throat Boltzmann constant

dissociation and recornbination rate constants respectively equilibrium constant,

~!

expression defined in Eq. (43) expression defined in Eq. (A.5) mass of an atom'

mass oÎ a.toms per unit mole

Mach number; expression defined in Eq. (48) frozen Mach number

coordinate normal to streamlines expression defined in Eq. (48) pressure

q R s t T x,y

x*

j

m

velocity vector

x and y components of velocity speed

gas constant per unit mass referred to diatomic gas

,

cdordinate along strecirnlines time

temperature

Cartesian coordinates

coordinates of the point where sonic line and line of horizontal velocity intersect

point where line of horizontal velocity meets nozzle centerline throat location with respect to part~ally fI?Oz:en sonic point deg,ree of dissociation (mass concentration of atom)

d~fined

in Eq. (48),

~i

fictitious specific heat ratios for froz.en and equilibrium flow

respectively (B.13), (B.14)

true specific heat ratios for frozen and equilibrium flows respectively (A.33), (A. 34)

defined in Eq. (23)

constant defined in Eq. (78)

J •

transformed y coordmate streamline angle

characteristic temperatures for rotation, vibration and dis-sociation respectively

Mach angle

transformed x coordinate

ex,pression defined in Eq. (66') vi

.

'

Suoocripts: Superscripts:

*

j density

characteristic density for dissociation (A. 8) perturbation parameter

characteristic chemical and flow times perturbation velocity potential

rate parameter (A.4)

equilibrium

partially frozen (vibratiofi in equilibrium with> active modes; f is used for conveniei1ce)

reference state

..

'0.

1. INTRODUCTION

It is known that in the case of supersonic reacting gas flows in nonequilibrium, the flow field can be calculated by means of the characteristics method witb the frozen Mach number playing a role similar to the usual Mach number in non-reacting. flows (Refs. 1 to 7). In order to calculate such flows through a nozzle, Der (Ref. 3) has studied the various aspects of the charac-teristics method for the supersonic part of the nozzle. Similar studies were done by others, for example Ref. 4. However, in all the studies of the nozzle flow, the flow was computed by quasi-one-dimensional methods up to a point where the frozen ,Maçh number is slightly.g.reater than one. The flow prop-erties so obtained are then assumed to be constant along a line perpendicular to the nozzle axis throughthis point. Using this as the initial data, line, the two-dimensional supersonic flow is computed. There appears to be no justi-fication for this assumption oLa . .straight line with constant flow properties as the initial line for the supersoriic flow calculations . It therefore appear,ed worthwhile to investigate the nature of the sonic re.gion (i. e., the region where the frozen Mach number is near unity) so as to establish a correct initial line for the supersonic flow calculations.

In this note it is proposed to study the nature of the sonic

re-.gion by applying the small perturbation technique. This study mayalso be useful, at least in a qualitàtive way, in the transonic region of a reacting flow over a blunt body or in rocket exhaust plumes (Ref. ,8).

2. THEQRETICAL CONSlPERATIONS a) ASsumptions

(1) The analysis is r_estricted to the case of a pure diatomic gas like 02, giving a binary mixture of atoms and molecules.

(II) It is assumed that while the vibrational and translational degrees are in equilibrium, dissociation is in nonequilibriumtpartially frozen). (IIl) It is further assumed, that the dissociation is only slightly out of equilibrium so that the dissociation rate equation may be linearized. (IV) Only steady flows are considered.

b) Rate equation

The rate equation for the atomic mass fraction

oe

may be written as (Ref. 6, see also the Appendix)

1

where ~

=

19rad and \.V is the rate parameter and L -+ 0 for equilibrium

flows.

Consider the flow to be a perturbation from a reference state, which may or may not be in equilibrium, such tha t

p

=

p.4-

pi

}: t

+P'

0{ ::: 0( • .jo

rl.

1

(2 )

(where stars denote the reference state and primes the perturbations) and expand L (P ,Plo() in a Taylor series about this reference state as

(keeping only the first order quantities). Define a local equilibrium value O(~ = o(e(~,f)=ct"

+rxé

such that L

<P,P.'i)

= O.

Then expanding L (P JP.~

=

0, one has

-Thus one can write

(3 )

(5 )

(6 )

Now, putting this in the rate equation and writing the rate parameter also as a perturbation from its value for the reference state

"t

= -.t+~

and keeping only the first order quantities, one obtains

c) Basic flowequations mass . momentum ener.gy enthalpy state Expanding (7 ) (8 ) (9) (10) (11) (12) (13)

\

one obtains

-(~-

f)p Cf .

~f

-

f~cUvf

-+

Jt..,.~i

==

0

where, subscripts denote partial differentiation with respect to that

variabie. From the definition of frozen speed of sound (Ref. 6)

Eq. (14) may be written as

From the linearized rate equation

(7)

where the equilibrium speed of sound CZ~ is given by (Ref.6).

Eliminating ~ on the LHS in Eq. (17) by the use of Eq. (16), one

finally obtains:

(14)

(15)

(16)

(18)

This equation can also be obtained without the assumption of the flow being only slightly out of equilibrium (see Appendix A).

It was shown by Vincenti (Ref. 6) that for flows which are slightly out of equilibrium, one rnay introduce a velocity potential (see Apperidix C) .

Now writing the velocity as a perturbation from the reference state

ve-locity ~

*

\

Cj~ =-

1(1/

f

x )

~,=-

'l!ff

₁

(20)

where t.p is the perturbation velocity potential,and substituting,.e above in

Eq. (19), one obtains ~z. . ~ .

[(

J

--+

\f~)-h

+

<p~~J{

ti; [

<f~~

-I-t,OY1-

f((r+~K)-h

-+-

c,PV-%J[Cf""+

~j~<lkJ]]

-1f~

Lol'

p

a",

l~t(eJ»{ l.f,,~

-t

LPy

~

-

-k-

[(t

~

+LI)()-};~ ~~~l:"~~+lPYJI=O(21)

cr

d) Relation between sound speed and flow speed:

In order to sirnplify.Eq. (21) for the transonic case, a relation between 'the sound speeds (frozen and equilibrium) and flow speed is needed.

Since the vibrational and translational de..grees are assumed to be in

equilibrium, the frozen sound speed occurring in the above equation is

actually the partially frozen sound speed (in a partially exitèd

dissocia-ting gas) as derived by Glass and Takano (Ref. 5),

A I -1-2

a;=

A' (/4-o()RT=f!(/+Dt)RT (22)

where

(23)

and Ris the gasconstant. This expression can be also derived starting

from Eq. (15) as shown in the Appendix~-\

The .specific enthalpy may be written as

(25)

Using Eq. (24),

where

(26)

From Eqs. (9) and (10) for momentum and energy,

or

-Il.

+.

'1/2,

=

constant

=

ho (27 )

(Since the flow is from a reservoir, ho is the same on all streamlines and hence throughout the flow). Substituting for h from Eq. (26)" one obtains

t

~

+

A

af

= constant

In terms of the critical speed

1

*

=

0./

or

1

~ -#-

A

a.i

=

(A

/.0 -+ I )

a..;

2.

at

== ((

Al-+- /)

at~- f~J/A

~

((At-r /)

4.tt~

-

tj~]/A"

Under the assumption of small perturbations A is replaced by A*.

(28 )

(29)

(30)

From Ref. 5, the equilibrium speed of sound (see also the Appendix ikfor derivation from 'Eq. (18), is given by

where

(32)

Using Eq. (25) for h,

2,

4t

= Bta:

(33 )

where

Denoting by

(J.eltlthe

value of ~ when ~

::a;,

~ qs. (27) and (33) may be

combined to give an equation similar to Eq. (29), namely,

(35)

(36 )

Under the assumption of small perturbations, this may be approximated

by

(37 ) It may be sh0wn that Bf ~A

;

a~~ where the equality occurs in the limits

0/.. - 0", 0/.. ~ I •

It is sn0wn in the..ÀppendixA.thattlle.err-OOil introduced in replacing A by A* and Bt by Bt* in deriving Eqs. (30) and (37) from Eqs. (29) and (36) are not large.

.

.

.. " ,

3. TRANSONIC APPROXIMATION

As noted previously the purpose of this note is to study the flow field in the sonie region in a nozzle for flows whieh are slightly out of equi-librium. For sueh flows it ean be shown from a quasi-one-dimensional ana-lysis that the sonic point oeeurs a short distanee downstream of the

geomet-rieal throat. Clarke (Ref. 9) has also shown that, in nonequilibrium flows, the flow speed at the geometrieal throat is always equal to the equilibrium

sound speed at that point. It ean also be shown that in the case of frozen or equilibrium flows, the sonie point is again at the geometrieal throat.

It is also known that the frozen and equilibrium sound speeds at any point in a nonequilibrium flow may differ up to 15%. Noting these eomments, one may derive the transonic approximation of Eq. (21) as follows.

GeometrIe Throat SKETCH 1 I Real Throat

~---

Centerline I I I . 140 Region of lnvestigation

Te determine the order of the various terms in Eq. (21), eonsider new variables ~

''1 '

cp

(~,1) givenby.

and

_{q .. ::.}

0..-. f

(38)

where ho is the senli-height of the nozzle at the throat and

cp,

0/ a~

,

,

'0/

all'}

are of order unity (see Appendix C for details about this transformation) Also from Appendix C

7

"'~ . "

(38 I)

and

where RI l and _{R' 2 are given in Eqs.} (C19) and (C. 21) of Appendix C.

Transforming Eq. (21) to the new variables by the

transforma-tion of Eq. (38), substituting for

Qh./

h ..

and

~ (h~

+

hrJ.(j.e~)1

h..

from Eq. (38 I) and dividing throughout by the quantity ~

(_p.

be" ) , one obtains

h~

\

ha.~

rr

l

_+1'cjl.)

~

₊

_1'2.

~ :~(I

_+1'

R.'.) {

_l'

CPuL' -

(~R' +I'CP\!)~

+'I1.

_{ce (.}

\

_-r"!o

d!1.

cp,1.)_l-r~

.

Q.~)l.

ce

~

(\

+"'(~

)} _

ho"P ..

L\IL.(ht.+hQl1't~e~

..

)<

'"'Ij

Q.t

i

\

0.+

1~,

\

e

o.'t

.

h~...

(31)

~ +'I'R~)\

l'

'fe,~

-~)(

'+'l''f"tl+

'l'''ëë,,(I-

I"[€

cp~

)-2.

1"(~r

ce,

êfe,tl

+

'l'Cfe)} "

0

One ean further show by the use of Eqs. (30) and (37) that

and

(4.1)

80 that Eq. (39) finally reduees to

~

+

'l'

Cf'~HI,;

+

i'

~

CP,

:'11(

I

+

l'

R:) { -

1"1"

CPa

<p~~

+

1'~

CP"

+

0

CI")

}

-

ho~

.

(\

+

'\

R~){"'M cp~~ -î'~N cp~cp~~

+

,1.CP'1

+

oCï?)} :.

0

~

..

where

M-N

(43)

If one keeps only terms of the lowest order in Eq. (42), one has the trivial case <.p~" = 0 or cp~ = f(i). Thus to the next higher order (i. e.

"r

2), one has

Transforming back to the original variables

cp,

x, y,

(45)

The parameter {3

=

K* / af * in this equation tends to infinity to the limit of equilibrium flow. ~ Thus small deviations fr om equilibrium for which this equation is derived, imply very large bl,lt finite values of {3. Hence the limit of frozen flow given by {3 --.0 cannot be logically derived from this equation. However, it wiU be seen that, by putting {3

= 0 in Eq. (45), one obtains a

transonic equation valid for frozen flows. This re sult! can be considered only fortuitous. Hereafter the limit of frozen flow will not be considered.

3. 1 Equilibrium

Flow:-In this limit {3 -+ -a and hence Eq. (45) simplifies to

(46)

Referring t{ie perturbation to the equilibrium critical speed of sound

tt:

(i. e.

whe~

de

=-1

==-12:) and writing

3:

.

~;{JI't~,

ofot in Eq. (46) is to be replaced by

t{e ' tht:ls,

Bl

-+

1

l.f..

1

c.;

1 "

-

2

el

.

~

Jtl(

-r

Llv~:='

0

(47)

where

8:

is now t!1e value fOF -

~:;:.

b

~

of: .

3. 2 Solutions for t.te nozzle

flows:-To study the f1.ow in the sonie region, one has to solve the

follow-ing equa:ion, where

p:=.

Z

~:~

..

L

f~

=

~+

~

~

I -

t2//tl;d

..,~ III-i! (45)

N

=-

e

~

( ( -+-

~&)

In the case ü<! a nozzle sy metriGT Wlth respect to the centerline(ó.. e., Y-,- axis).

the ~ -cornpor 8::t oÎ the velocity is antisymmetric in

y

while the Z -component

is syr.lm~t!'i8. Also for supersonic flow, the perturbation component ~

changes frem !-~gaüve to positive values as one passes through the sonic llne.

Th\ls the perturbation velbcity potential ma.y be written as

. ."

which gives for

'P,

t~

-ftP, :::

(-(3N

-+

pt)Cf()><Lf()()()-r~MIf())("

e

-~x

lp.

~

f

e-p>t ( -

f

N

-+-

Fix)

(lp())C

'fOA-K)

d

X

+

r

p

M

e

-(l)CI.f~)(,Ix

+

constà~i

'

~

(>()::.

e

px {

J

e-p)t[t_

(!> N

-+

P~)(

'fox

tf().()()

~

f

tv1

~",)cJd><

-+

A}

(49) where A is an integration constant.

In the region ne ar the throat, one may write, (as in perfect gas flows, see Ref. 10),

Lfox

==

C)( where c is a positive constant

found .. _~ by taking' the origin of the axes X,

'1 :.

0 at the sonic point. Sub -stituting for ft\(, ~O)()( in Eq. (49), one obtains, ,

(50)

The equationfor

l..fLx)

is

dd~

-f

'P~:.

(-f3N+ P

}x){

lI()(tf,l(~

_+'f,>t'foxx

)-+

(3

tv1'-1,KJr

~~::: e~)({

f

e-P[C-pN-t

r~)~()vf,)f)f t"'x"o)r~)4(3MtPLtJdx4oB}(51)

where B is an integration constant.

Substituting for

lf

_h .lfolt>t .J

LP

1)( "

'I,

)Ot ' one has

'f~(X).: N~c.l ~ A~~~ r:($X[~:)(

(7'-N)

+ ('

(~P-NJ-+

,B'\It]

~ Be~)(

(52)

Thus the perturbation velocity potential

'f(

x~

1)

and the ;t, ~ components of the perturbation velocity

tP)(

.J

'f>

~ are: .

~(X.~ )=fc~~

4-

f(t

N~VC'"

-M

c

-+

IJ

Ö

4-Ae~./-I;rN"è·+tI~~;«(l'-")-I-.

c

(G P - N)

+

r&

M]

+

B

e~)(

}

.

(53)

Cf

x=

ex

+

-f

(Ne

~-f-ltfSe~)()

-+

~

{A(J~C( P-N):te~~

A(l-:r

ef,Xr!~X(P_N)

(54)

+

CC 2 p-

N)

-t-

(!>fV1]

+

Sf

e,BX}

Cf#;:

~

{

(Ni!)

C 2.. - fv1 c

+tJc~~

f

A

efo

x

+

~2.

C

N?:.~Ap.:l-'ltePxL8Sz'"

(P-N

)+-c

(~f-N)

-+ffv1].}

8eP

1}

(55)

3. 2. 1 Limiting case of equilibrium flow:

,. In this limit (3-7-t><J,and replacing ~~by tfe:fl- as in Section 3.1,

M

~

0& N

='~l~/!~where

5;

is the value at the point where equilibrium critical

speed is obfained. Thus the components of the velocity are:

10 -==- /' 'V

+

Be

~

.f I h J.

"'~

'1'x L I \ , B"" ,'c ' - ( (56)

e ,:

Equation (56) shows that the constant perturbation velocity curves

'-fl<'

are parabolas.

The sonic line and the line of horizontal velocities are 9btaineçi by 'put1ing

tfJ( ;..

'flV ::. 0

,

respectively,

Sonic line is (58)

Line of horizontal velocity is (59)

Both lines , are parabolas and have the common point

'

Y::'I

:::::Stas in perfect gas flows. (See Appendix B)

3.2.2 General nonequilibrium

case:-...

The(-x., ~)-components of the perturbation velocity for the general nonequilibrium case are,

'Ix -

ex

-+

fc

NC"'...,..

A~

ePX)-I-

-1~

{A

~2:reP~[f3C(P-N)(

1+

f

)+('('l.P-N)~

+

B~e~)(}

t.f;-=

~{CN1)C:a._

M CA

!1L.

~

=t+-,4.efJ~-f{N-z.ci-+A(3·::re(ltfl'.;

(P-N)+C(-z-P-tJ)+

f>

M)

of

B

e

p. )(}

(54) (55)

As noted previously,

f>

is negative and very large for smal! de-viations frorn equilibrium (see also .Appendix). So, all the exponential and

other terrns in ~ in Eqs. (54) and (55) IIlay be neglected giving the approxirnate

results . . ,~

'f>(

= (

(x-+

NC;!~)

(54)

(55')

These results could have been obtained as wel! by neglecting the contribution of the first term in Eq. (45).

Eq. (54') shows that th~ curves of constant velocity

3

~ ~. -I- tiK

are parabolic . Thus the initial data curve for supersonic flow calculations by the method of characteristics can be taken as a parabolic arc with consta.nt flow properties on this curve.

The frozen sonic line and the line of horizontal velocity are given by

I.f

x ...

"-ti

=0 respectively.

Sonic

Line:-Line of Horizontal Veloci ty

(56)

(57 )

.E;-qs·. (56), (57) show that these two curves do not meet on the

axis as in per:t'ecLgas flows (SeeAppendix B). The point where the line of horizontal veloc ity m eets the À:: axis i. e.

i -:: ()

is obtained by putting ~::. 0 in

Eq. (57),

(58)

,.;-;.,

iince

a.t>

_Cl.e", M< 0.. Also N ~O. Thus.(;"" is ups~ream of the sonic point. In other words, the line of horizontal velocity intersects the x axis upstream of the sohic line. This is physically sound since,one should have parallel flow in the vicinity of the geometric __ throat

Geometric Throat Sonic Line Line of Horizontal Velocity

I

tt ---

SKETCH 2

o

,; , This displacement of the sonic 'point from the geometric throat ,,-' -, can be obtair:i~.d by usio.g the boundary condition Olf .the nozzle wall that Lf:;f == 0

when the wal! is parallel to the centerline or x-axis. If XTis the abscissa of the throat and h the throat height, then

0 -

'V -~ -t

NC{4-_ - A.T

Ne.

- b -v _"'-'ï;

...M

Ne

N

e

"'-l.

~ (59)

If the sonic line and line of horizontal velocity cross, this crossover point is given by the solution of Eqs. (56), (57) for x, ~ This point is

X ' 3 M

C.-~Nc

!Jc

==

{

i/cM

This crossover occurs only ti Yc= Yw (x=x_c

>'

wall equation.

where Yw

=

yw(x) is the (60)

Other limiting -cases: - Two other lirniting cases can be considered where the amount of dissociation 0( tends to 0 or 1 and correspondingly I~O ~~. In

these cases

t2/--a/ ..

-a..*and

B/

~ A

lt-,0 Ar.,. / :l- ~

"f,(:: L)(-t

A* -

.

C

Y

(61)

and (62)

Line of horizontal velocity is ....

(64)

both of which are parabolas and have the common point", Co

1

=0 .

4. CHARACTERISTIOS

The flow through a nbzzle can be divided into two distinct parts: (i) the sub.sonicre~ion and that part of the supersonic re.gion which influences

'the subsonic part, (ii) the remainder of the supersonic field. Consider a point

Q on the nozzle wal!. such th at thefrozen Mach line emanating from it meets the sonic line on the nozz1e centerline. Mach lines coming from all points upstream ofQ.· wil! r~flect on the sonic line and thus influence the subsonic and transonic part of the flow.

Tluis-

the knowledge of the location of the limiting

/

.Mach lipes and the point ~ wil! be of interest in the nozzle flows, particularly, for the inverse nozzle problem.

SKETCH 3

Limiting Characteristic

Line of Horizontal '~. _Centreline

Sonic Point Velocity

__________ ~~ ____ ~ __ ~L-________________ _ L _ _ _ _ _ _ _ _ ~

In the. perfect.gas case, it is known that the point Q .uès upstream of the throat (Ref. 10). In this seGtion, the situation in the reacting gas flows wil! be considered.

4. 1 Characteristics for ~ar-Equilibrium Flow

In section 2. c., the basic system of Eqs. (7) to (12) was reduced

to /

(16)

In terms of streamline coordinates SJ 0, these may be written as

( M/ -

I )

~

-

~ ~

-+?

~

0

q~=r::u

V

~s

where

Mt:$

flat

the frozen Mach number,

?::

~~

(iJ =-

tf .. ~

(0(

-deJ

(7 ) (65) (66) (651 ) (66' )

For small deviations from equilibriumJ Vincenti (Ref . . 6) has

shown th at the flow may be considered irrotationalJ thus

(67 )

E;q. (66) is alreadyin character.istic formJ the characteristics

.being streamlines: Forthe syst~m of Eqs. (~·5)J (67), the characteris.tic directions

.1.,

I i~ can be shown.to be given by

(68)

and the compatibility relations along these characteristies are

(69)

4.1.1 Approximation in the Sonic Region

In Eq. (40), the approximate value of ~i~-

,

is derived for the . . and' . b

sonlC reglon • ... .... ls"glven y

(70)

neglecting higher order terms. If

p

is the frozen Mach angle i. e. ~)A:;:-L

then "1f)

Cdt)'-

=J

Mf a.._1

~

[ptt]

V2. (71)

where

P

is given in Eq. (48).

The approximate form of the compatibility relation valid in the sonic region is obtained by replacing

1

Dy

a./""+1

1.-v 4f{l4tPx)as

(72)

given as

In Cartesian coordinates '):, ~ the characteristic directions are

(73)

In the sonic region)which is in the vicinity of the geometric throat. if the nozzle contour is sufficiently-smooth and slowly varying", () will be small compared to

ft

which is nearly ïT /~ and h~nce one may approxirnate for the characteristic directions in Cartesian coordinates

(74)

4. 1. 2 Limiting-··Cha-racteristics

By use of the solution

"f)(

given in Eq. (54) or (54'). one can

find the (;haracteristic curves in the sonic region by integrating Eq. (70). As

thj.s'integration is a little complicated, restricting our attention to the limiting

chRrade.d stics i. e. those passing through the origin

x.

=~ ~ 0, consider if any

par-abolas

(75)

caD. cobcid,= with the characteristics. The slope of the parabola is

or

dAd.~

_q :=.

~

.

~

U

_q

₍₇₆₎ Substituting in Eq. (70) and using Eq. (54') for

lf

N , one finds

~

4-

,~

=

(C

~

+

Ni)

P

(77 )

Pc

±

J

p~t::L-+ g"PNè

8

(78)

Si;;~.~,~ E f\!lCl Pare positive, ~ is real and is positive or negative according to

tht; sig:~. ~}dcre the root and hence the limiting characteristics are parabolic.

emanate:;)

The point Q on the wall (·where the limiting',charaderistic

ean be obtained by solving the wall equation

(79) anC! t1.!.c 1f::;;; !·uT.I.;:üng characteristic equé'.tion

(80)

5. SPECIFIC r~ALCULA TrONS

•

.

.

a hyperbolie nozzle for reservoir conditions 10 =.!5 qOO°.f(and

Po::

~

e

~.

is calctilated for which quasi-one dimensional results were available (although this example is drawn from a completely nonequilibrium case. it serves the purpose. since in the throat region

Tv

and Ttdiffer only slightly and i~··equi­

valent to the partially excited model). The values of

r,

0(* were taken·from the quasi-one dimensional results •. from which the parameters in

LPl(.

tf'j were

ca1culated and found to be

A'* = 3.905.

al

= 1. 494. Bf>!c = 31. 218. a~ = 1. 340. A* = 9.487.

B* = 11. 783. P = 1. ·4797. M = -0.242. N = 1. 8378 .

f · .

The equation of the hyperbolie nozzle was

;

where the. origin of the coordinate axes are now taken at the geometrie throat

for convenience. Tb.e variation of the perturbation velocity

cgJt.

~ t.p>t~ ~ -d.f *""

along the ~xis is shown înEig. 1 ior various cases. from which the constant C isdetEl-rmined. Thus the x and y components of the perturbation velocity were found to be

lf-,: .:

3.z95 (X-r

b.037

yïz)

Lf~ ~

,

9.632

eX

-t 0.04- -4-

b.

037

va.)

'f'Je

= 'f

1 =-0 giving the sonie line and line of horizontal velocity respectively. The li~iting characteristics were found to be

!

x/~,.

= - /

.

3D8

X/.y"-

= 2.5.;:>3

The displac e.rn ent "of ;the frozen sonic point from .the .geometrie

throat and the point where the line of horizoJ;ltal velocity meets the centerline

were found to be respectively. is found to be -:t-/~

= -

(J.

8

5

..

Xj.Jc

=

-0.8

The point where the sonie line and line of horizontal velocity cross

'Xc./-l

=

-1.2-VC/--A-

~

±

Z.

9

The limiting characteristics, frozen sonic line and the line of horizontal velocity' are shown in Fig. 2.

6. DISCUSSION

One important objection that may be raised about the analysis of Section 3 conçerns~ the linearizatüm of the rate equation while keeping terms of order 'L~ in the approximation for the potential equation. If the variation of 0< in the sonic region is m uch smaller than the variation of fj

one may feel justified in the linearization of rate equation. The results of quasi-one dimensional calculation for two cases, where the flow is slightly out of equilibrium (~ ~O(e ~f%), showed that, in the sonic r~ion, ~ varied by 0.3% while

1-

varied by 3% from the critical state values. Thus the linear-ization of the rate equation appears justified.

This analysis will. give at least a qualitativ.e description of the "flow, .if not a quantitative one, and is carried out in the same spirit as that of

.. Vincenti (Ref. 11). It may be· noted that it gives the correct trends in the pre-diction of the sonic line downstream of the gometric throat and the line of horizontal velocity in the vicinity of the geometrie throat. Since the curves of constant velocity are shown to be parabolic, the calculation of the supersonic flow field by the 'method of characteristics can be started from· an initial data curve which is the arc of a parabola or a circle. The values of the various flow variables, obtained by quasi-one dimensional analysis, may be taken con-stant along. this curve.

A similar analysis can be done if one considers nonequilibriuI!l in a single mode (e.

g.

vibrational nonequilibrium with no dissociation). In that case the rate equation has to be replaced by the Landau-Teller equation for vibrational nonequilibrium .

It was found from some preliminary analysis that the case of simultaneous vibrational and dissociational nonequilibrium cannot be reduced to a single equation as in the present case regardless of whether one considers coupled or uncoupled models for the vibrational and dissociational rate pro-cesses.

7. CONCL USIONS

The qualitative picture .. _ one obtains for the nozzle flow in nonequilibrium is:

of the geometric throat.

iii} The line of horizontal velocity is parabolic and meets the nozzie centerline upstream of the frozen sonic line.

iv} The frozen sonic line and the line of horizontal velocity meet on either side of the nozzle centerline, in case M ~o

(i. e.

a../

~

a;- )

or do not m eet at all as in the exam pie gi ven, in contrast to the perfect _gas flows where they meet on the nozzie centerline. In the earlier case portions of the hori-zontal velocity curve near the nozzle walls will be supersonic while those near the centerline will be subsonic or in the latter case the whoie curve is subsonic in contrast with the

-perfect gas-flows-where the whoie curve is supersonic.

v} The limiting characteristic which divides the nozzie flow into two distinct regions, (nameIy, 1. the subsonic flow and that part of the supersonic flow which influences the subsonic flow, and Ilo the fully supersonic flow) is parabolic and

emanates from a point on the nozzie wall which is downstream of the.geometric throat in contrast to perfect gas flows where it is upstream of the geometric throat.

vi} The initial data curve for the computation of the supersonic

flow by the method of characteristics may be taken as a'parabolic or circular arc with constant flow properties on it which may be obtained from quasi-one-dimensional caiculations.

vii} It appears from a rough analysis given in Appendix A that the qualitative picture for simultaneous nonequilibrium in vibration and dissociation may be similar to the present case. However, the partially frozen speed of sound used in the present analysis is to be replaced by the fully frozen

speed of sound.

1. 2. 3. 4. ' 5. 6. 7. 8. 9. J 10. 11. Broer, L. J. F. Wood, W. W. Kirkwood, J. G. Der, J. J. Brainerd, J. J. Levinsky, E. S. Glass, I. I. Takano, A. Vincenti, W. G. Vincenti, W. G. Bowyer, J. et al, Clarke, J. F. Guderley, K. G. Vincenti, W. G. REFERENCES

Characteristic s of the Equations of Motion

of a Reacting Gas, J. F. M., Vol. 4, Part 3 pp. 276 (July 1958)

Hydrodynamic s of a Reacting and Relaxing Fluid, J. Chem. Physics, Vol. 28, No. 4.

(April 1957)

Theoretical Studies of Supersonic Two-Dimensional and Axisymmetric

Non-equilibrium Flow. NASA TR R-164, (1963)

Viscous and Non- Viscous-Nonequilibrium Nozzle Flows, AIAA Journal Vol. 1, No.11 (November 1963)

Nonequilibrium Expansion Flows of Dis-sociated Oxygen Around a Corner, UTIA Report No. 91, (June, 1963)

Nonequilibrium Flow Over a Wavy Wall,

Stanford Univer sity Report, SUDAER No. 85, (March 1959)

Calculations of the One- Dimensional Nonequilibrium Flow of Air Through a Hypersonic Nozzle, AEDC-TN-61-65. (May 1961)

Transonic Aspects of Hypervelocity Rocket Plumes, Artic1e in Supersonic Flow,

Chemical Processes and Radiative Transfer,

Pergamon Press, Oxford, (1964) The Flow of Chemically Reacting Gas Mixtures, COA Rep. No. 117, (Nov. 1958) Theory of Transonic Flow, Pergamon Pre ss, Oxford (1962)

Linearized Flow Over a Wedge in a Non-equilibrium Oncoming Stream, Stanford University Report, SUDAER No. 123, (March 1962)

Figure 1. Variation of q Along the Nozzle Axis. (quasi-one dimensional calculation)

Hyperbolic nozzle. qN1f'T o 1.2 -0.4 -U.2

o

02 I point where q = af

A vibrational and dissociational nonequilibrium - - -preferential and nonpreferential.

T o=5900oK, po=82 atm.

B, C Vibrational and dissociational nonequilibrium ---preferential and nonpreferential respectively.

T o=5900 oK. po=9. 4 atm.

.04 .06

UB

Subsonie region Flow direetioll

•

o

Flow direetion 0.8

•

Subsonic

0.4

Line of horizontal veloeity

0'

-0.4

o

eometrie throat

Charaeteristies

Supersonie region

Line of horizontal velocity

Center line

x/h

PE R FECT GAS FLOW

Geometrie throat Y/h = 1

+O.Ol73~~t

Char aeteristie s Sonie line region

0.4

Supersonic region Center line 1.6 x/h

REACTING GAS FLOW

APPENDIX A

DERIV ATIONS

Rate Equation

It was shown in Ref. 5 that for dissociation and recombination

of a pure diatomic gas described by the process

(A. 1)

where .A2 and.A are a diatomic molecule .and an .. atom, respectively,: and X is

a third body, -kcJ. , ,-k.~ :;l.re the dissociation and recombination rate constants respeètively, the rate equation for the net production of atoms in terms of the

atomic mass fraction C( , may be written as

(A.2)

where

ma..

is the mass ·of atoms per unit mole,

p

is the density and Kc is the

equilibrium constant defined by

In this derivation the atoms and molecules are considered to

have the same efficiency in causing dissociation. If they are considered to

have different efficiencies, then the factor ( 1-4-0( ) in Eq. (A. 2) has to be

re-placed by (1- eX.

+

z>.D() where À is the relative efficiency of atoms and

mole-cules. Comparing Eq. (A. 2) with Eq. (1) in the text, one finds ;

~ ~

,kAl'-( 1--1

~) o(~/-111~

_{(A. 4)}

and

'- ( b!

rl ,"'\

rJ):!-

~

mIJ

K 0-0() _ /

c o<r (A.5)

It may be noted that L is dimensionless and 1/

1f

has the dimensions

of time and is taken as the characteristic chemical time ~ . If

?j.

is the· .

characteristic flow time, then for ?-y~ ~ 0 , one obtains the limit of equili-.

brium flowand for

n

f-'l1°C, the limit of

fr~özen

flow. Also in the limit of

equilibrium flow

L

~ 0 , giving the equat~oIl. for the equilibrium mass fraction

of atoms as

(A.6)

In Ref. 5, an expression for Kc is obtained from thermodynamics

as (A.7)

and (A.8)

where ().J).., ()-v-~ () 1> are characteristic temperatures for rotation, vibration

and dissociation respectively, m is the mass of an atom, k, hare Boltzmann

and Planck constants, ~Ol J ~o~ are statistical weights of the ground

energy level for atoms and nVolecules respectively.

Expressions for

K!ti

J

a.

e ~

a.l

Now expressions for these parameters explicitly in terms of the

state variables

p

~

p

~ 0<. I

-r

wil! be derived. In this Appendix, the function L

is obtained in terms of

..p,

ï ,

ex..

whereas in the main text, it was in terms of

P,P,D(·

Using the equation of state

(A. 9)

one can write

(A. 10)

dL:c:

U-~)f,-r

r1

rX -t

(Lp)-r,oc

dJ

+(LT)PI~

ct

T

=

Lr>(

d

c<+

Lp

df

+

L,{

CTp~.~

J

p+tJ;)p~d.!

+(T.)p,!

doéJ

=((LJ

_f

,ï1- (

LT)fIOtiToJp,Mc!~

t-[é

~\~+Q..ïJ"JTI')f~df+(L",J>"tr,;~ctp

=(loJ,pJpdoL+CL'\t<dt

+U-t»/,o(dp

(A.11)

LLolJp"

=

lL-l)t,,+lL

T)p,oiC-rcJp;f

(A. 12)

From Eq. (A. 9)

•

•

Thus

(A. 17)

The enthalpy h is

-{Cl"

o<}==-

7+-i

cX

'RT

-+ (

J-od

R

~

-I- 0(

R

f}J> (A. 18)

.

~

lT,

tX)

==

,{[T(f:~,J)o(

),0( ]

=

~

Lp,

f,

o()

(A. 19)

d~= (~)()(dl

-+

Ciq(),d~

==

(.{-r)t:([(~)flo(dp

+tr;)"o(df

+

(t;\.fd~] -+l.-A~)Td.O(

==

(·4vr

~(Tp),

,o(dp

+

(iT)J-Tf)p

,o<df

+ [(

{~O(tT;)p" +~h]do(

=-l~p)/o(

d

p

-t.

C-1t/)P.r(

d!

-+~)p,ydo(

•

(A.20)

. ~ .. :

Similarly the local equilibrium ma.ss fraction of atoms ()te is

(Á. 21)

A.3

/

•

where

rXe (

p,

T )

is given by Eq. (A. 6) and

drX~

=(O(et).,.

d.;+{deT)t

dT

=

(rxe,)-r

lf ...

(o(eT}((Tp)p,GletÁp-+(TP)~;c(e4.'p;-(l;,e)PlpdlX'eJ

drie

[1-

(rXe-r

)ptte)p,~

.= (( r{o/)-r

+(

rXe-r

~t1f)~, ~Jdf+

((Áe, }

(T

_p

},ol41

ti.

P

cl

~

=

[(o(ef)ï+(c(eT~(te)p,o(J

dp+(o(er)p(7jJ

_e,

_oIp

clP

[ I -

(o(e-r)j>

(Tale)p ,p

J

(A. 22)

Thus from Eqs. (A. 20) and(A. 22)

i,

+

-A.ot

eXee _

(i.,.

)JV)p,J

I-LcieT)p(,Dt4!)p,p]

+(O.T~lTot),.

.

p

+

~)-d (ldet'),+(~h)t'("f)p,~J

--A.I)

-

I I

J

~T)O(

(if)p#.

[1-(rXer)p {tote\"f]

=

C--h-r

)Q([tTp)f'~

+

er

I'(

)p,p(

~el').rJ

of

~Ol ~ot"/~

+

(ci~t"f)~,O(J

(iTl

(ïp)p,O(

(1-

C~e-r~

('O(e)p

JfJ

(A.23)

From

Eqs.

(A. 6), (A.7), (A. 9), (A. 18)

/0(, ) -

rXe (,

-o(e)

(.1.

+

BI>

-~)

t.: e-r:'f - (z-o(~)

-r

2-

-r

(~

) = _

o(e (/-o(Q)

ep.,.

P

(:l -Ot'e)

(~)~II

== -

T

II-to<

o-p)p,oe=

- I / . f

tAt."')0{

=

R [

'7'

-t (/-(JO

~

J

tt~)T

:=

RT

ti-

+

~~

s: )

(A. 24) (A. 25) (A.26) (A. 27) (A.28) (A. 29)

Substituting these in Eq. (A. 23) and simplifyipg,

,

The frozen sound speed is

where

_{A' =}

I~o(

_((5"+ot)

₊

;e-tl-"')~]

n

_f -

Pr' ....

z

A'

(A. 31) \ (15) . (A. 32) (A.33)

The equilibrium sound speed is

.

()i!_~ ~'p)",o(-t(~o<.)t.?(c(ef\'

"1.t>- l ( 18 )

( fLp).f, 0<+( -lo()p.j> (~ef)j-

.P

[

11 ) (0 \ 1

(o(ef\·d~eJ{~p(

___ (.J,..,)o«(Te\,rx+

t"tTltrroi

p,y+\.-11.olHj 1-(!Xer)JI(t;.t)

.•

~

j;

-

(~T)iTp1Io(-t[(ld/--CJf,,+~o(),](o<eT)ptTj.~"e

__

I

• .I-(CXer)J...ToIe)t>,p ?

(~-r)o(

[èrY)fJpt-t"

l--C

)~.p(o(e,).] +~),.[(o(e,)1" -+(o(e~etTt)t,J

:b -

fl

T

1.

tr;.~,o(- ~er

}[f

(ï;J

_p1

+l.(o(),.(1p )fJo(]

Substitutirlg for the various quantities and simplifying (A. 34) ,.:. where (A.35) (A.36)

Estimation of the errors in the approximate Eqs. (31) and (37).

In deriving the relations between sound speeds and flow speeds in section 2. d,

(31 )

(37)

A and Bf in the denorninators were replaced by A* and B:r* . The errors involved in these approximations are evaluated here.

A

(0( I -r)

==

-he

15

+

ol

~

:2 (

I-"d

~

]

[7 of 3oC'

+

G ( 1-0()

~

-+

~4(~

J/[7+

34(+

zu...(~]

where

and

Expanding A (eX,T)in Taylor's series about ,O(~T~ , and keeping only first 'order terrns:

-(A.38)

Frorn Eq. (A. 37)

l~

A=-0

A'

-t-

l

1

~

-

~CA~Z)

-17(,+(xJ

t~ =f~~ +t~

-

A'~~~~

-ïb

~

= -

I)~

-

~~

+

G

~~

[ -

I~c(

-

Q;j;~J

-

_ 3 - (/fdf)

(j+

rXj

TT

~.,.

=

~ ~ ~re ~/T

/) A)* _

A*[-

-1-* -

~

(lof

~r~)

-

+

11j

\.~ lof I)(

/1'*(

A'''.fz)( ~~*)z.

(A.39)

-Ld.A-

=

-L

)t\'

-+

I

M _

.l_-L

''tJA' " d

T

A')

T

~ ~T

-r

A'+~

dT

2A!.. _

Z (1-t)<J ~

àï -

I +~ d.T~ ()A

*

~l

\

4

(1-0<.*)

Ei

-{.

~},)

=

A

--r

+U-+~*)A/f(A/;+~)

TT

+

7J

(A. 40) A.7

where

(A.41)

B lo(T)= (2 -ex:') (5-t0('

+~U-o<')~]

+0((1-0('

)C±+

(~~~f 7~3e(

T+(t-od

.fJ~l(A.

42)

f

7+3o(-tZU-ri)f.~-fO«(I-Il('~)(t.+t)J)':;'f.).

z 'î

~

==

~

.

~

11

Sf :::

0

C

-1.a;

D

+

lCtJ

~/R

J-

à

Bt

=.1

~

.-L

à.D.

+

J,.. è»).. Bf)o(

c..

dcX. -

D ~o{

--K.

~

.dd~

=

-3-2O<+2f'.,.(-3

-I-Zot)+{/-'201.)(i

+tl~Sf'

~

== :5 -ZE..,.

+ (,-

3o(~)(-t

+

&b

~f)'

(A. 43)

~~-J-k

_1-~

+ '

dAt

B;

~-r

-

c.

Toe::

D"aT

T}T

~

=

Z ( /-ol)(

Z-ot) FT'"

-+

Zo( (,-ol)

(+

-+

~I>

-z

)(t-~

-

h)

;)T -r

-r

-r

(A. 44)

Expanding Bt ( r:;/, T ) in Taylor's series and keeping up to first order te rrns,

(A. 45)

~+-It c an be shown, that f or 00 ~ f}z, / , ~ 0

o~éj-r~1

O~

é

_r

~

I

and

T~TT~O

(A. 47)

, , Considering only cases where

e.,,/I

is equal to or less than one,

E/T

and

Er

may be replaced by unity for the error estimation.

Thus it can be shown that

A'::.

5+t>IC..+2. (.I- oe)

Er

(!

+0<. )

7-f><.

'V __ _ (A. 48)

I

+C1<...

is always greater than 3 which value it attains when a

= 1, and always

less than 7 which value it attains wh en a

= O.

From this

(A. 49)

Thus the 2nd term in the coefficient a' /l+a* in Eq. (A. 41) is always less than O. 54. The 3rd term in the coefficient of T' /T* in Eq. (A. 41) is very nearly zero.

Therefore

(~ft(.1

k) :::

3/~

T

+

Bo -

€.

7-ri-tX..

i

+

(1-0< )

E.

+-

Di.

GD

or

(1-

'

hot/h. ')

~

4+-0,50<.-

0-0<.)

BJ)/ï

4

-

5+0·50(.+

ex

Bl>/r

Also,

o·s

+-

Bb/r

4'5

+01...(0.5+

fk>/T)

(A. 50)

<I

-/+

T-h.-r

=

-[~ -tÓ-o()EIr+Q(94iJT[~+<1-"')sa

O<9,,1r

(A. 51)

~

7+30<.

-~

"

<I

~

+

(t-c<)

'i:/

_r

-+-rx..(}~/7 "·S+·5o<.ttl.&,Jr since the denominator is always larger than the numerator. From Eqs.

(A. 49) to (A.51), 'one can show that the coefficients of a'/l+a* and T'/T* in Eq. (A. 41) for A, are always less than 1. i. e.

In a similar way, it ean be shown that the eoeffieients of 0( I and TI

in Eq. (A. 46) for ~-t are also less than unity. Thus replaeing A by A* and

B:fby ~* in deriving Eqs. (31) and (37) in the test wiU not lead to large errors.

Alternative Derivation of Eq.

(19):-In deriving Eq. (19), it was assumed that the dissociation is

only slightly out of equilibrium. This restrietion ean be removed as

fo~lows:-Consider the Taylor ' s series of the quantity 1p. L in Eq. (1)

about the referenee state values. Then

(A. 52)

analogous to Eq. (3). Then

(A. 53)

From the energy equation, Eq. (10) and Eq. (13),

D<:>t -=.

~

_

-(hp -

\/~)

rLE -

hEI

~

Dt Dt - -

_hal..

D t -

_h",

_Ct

(A. 54)

whieh upon substitution in Eq. (A. 53) gives

(A.55)

Now define _{a e ,} the loeal equilibrium value of a, as that given by

as long as

\\J

is finite. The total differential of Eq. (A. 57) gives

From Eq. (A. 58), one obtains

d.Cl'p

= _

tw

L)~

l'1l

L ) at.

CCL)e

î.V L)~

which upon substitution inEq. (A. 55) gives

1L

(Dd..)

:.l~L)

{[-()ler-

_lh,.-

\/~)J ~

+[-tlL

ap -

hh

tl]

DP

~

Di:

\Dt~"

h~

Dt:

at

Dt)

=-l'\.VL)

(h~+h~ClU){_1 Q~9.

Q! -

Ad'

-+9}

0.4

hd.

Cl~

Dt.

\' '\I.

_ 11J'"

L

L

.

~rb~

L

\Ç>(h~+ hGlClt~

Îd'\I.q -

.L

~.

nq

J

- '1" t:J.'iJ

~ +

'L\J-

L~.)

h(1.

[

0..\

Dt

(A. 58) (A. 59) (A. 60) (A. 61)

by the use of Eq. (8), (9) and the def:i.nition of a 2 given in Eq. (18). It m~.ebe

noted that Eq. (A. 61) differs from Eq. (17) in tfie text by the factor(, + L \

'tV'"

LC1-")

Finally eliminating Da/Dt on the LHS of Eq. (A. 61) by the use

ofEq. (16), oneobtains,

in the place of Eq. (19) in the text.

Vibrational and Dissociational

Nonequilibrium:-hl any real floVl, the vibration as weU as dissociation wiU be out of equilibrium and one would have to deal with fuUy frozen characteristic s

for the calculation of the supersonic flow region by the method of characteristics.

The analysis of the flow in the fully frozen sonic region is much more complex

than the one considered in the text which is for the partially frozen sonic region. However a gener al idea of what one might expect from the results of the partially frozen flow may be obtained as follows:

In this case either the vibrational energy

E

v or vibrational temperature Tv is to be taken as an additional variable. Thus

which on substitution in the energy equation, Eq. (10), gives

where, hp' h~ , ha in Eq. (A. 65) are different from those in Eq. text which contain also the contributions from

h

_{t "} D

tv

where, function of T alone, i. e. T:: TI . D

~

v

(A. 65)

(13) in the

ê,'1 is a

For uncoupled vibrational and dissociational nonequilibrium, Eq. (A. 53) for D jDt (DajDt) is still valid and é.'1 satisfies the London-Teller equation, namely,

(A. 66)

,

.

where,

1:,

is the vibrational relaxation time and é.~ is

é.",

evaluated by replacing Tv by T. Let Tv differ from T by a small amount, then

. ~.

1:, ::. \

oT-

'T'.

(A. 67)

E.."

=

é.CID

+

E.'

Also

_h

t." -

h

f... =. (\ -

~)

Thus

h

_D

é,,,

=

h

Dé.CIO

+

h

Dê.'

E~

Ot

Eo.

Dt.

E ... t) t.

where the sub scripts denote differentiation. Substituting Eq. (A. 68) in Eq.

(A. 65) and rearranging one may write

(h ...

+

hE..f-.

T

-r:.)

g~

:::

-(h ..

+

hE.E:. ..

.,.T~

-

~)-(h~+~~

..

.;r~)%t -h~_DD~t

(A. 69)

Substituting Eq. (A. 69) for Dal/DT

=

DalDt on the RHS in Eq. (A. 53) and

making use of relations (A. 59), (A. 60), one has

(A. 70)

Since the coefficient of Df.1 IDt _{can be shown to be smaller than unity and a e}

is the equilibrium speed of sound given by

a.~

=. _

h~

+

h~~e~

+

h€.1>0

t.ClDT

(T"

+

~

C)le.~)

\

ht-

+

hal

~e.\?

+

he.co é.ooï

l

T~

+

Tty,

~e.",)- /~

Using Eq. (A. 65), the RHS of Eq. (A. 70) may be written as

D

{h.

f \

D

P

DP]

l

Dt

hu.

L

Oot

Dt. -

Dt.

S

where the fully frozen speed of sound af is given by

(A. 71)

(A. 72)

(A. 73)

and the

Df...,/Dt

term was neglected since it can be shown that its coefficient

he..., /

h~ i:s smaller than unity. Equations (A. "70), (A. 72) together give~ by use

of Eqs. (8) and (9)

(A. 74)

One may note that the expression for

a~

given in Eq. (A. 73) explicitly exclu~s all vibrational energy contributions and thus it is the fully frozen speed of sound while the equilibrium speed of sound given by Eq~ (A. 71) is the same as before since it includes the vibrational energy contributions.

Since Eq. (A. 74) differs little from Eq. (A. 62) except in the definition of

ar,

one can accept the vibrational-dissociational nonequili-brium results to be very similar to that of vibrational equilinonequili-brium-

and the line of horizor.:tal velocity or the locus 'of points where the velocity vector is parallel to the nozzle axis is given by putting l{J~::: () as

(B.7)

From the approximation

'1-

~

a.

*"+

LP\(,

it wiU be seen fromEqs. (B. 4), {B. 6), and (B. 7) that the constant velocity curves, sonic line and line of horizontal • velocity are all parabolas .

It is a1so shown that the characteristic directions are approx

i-mately given by

(B. 8)

along which

l'

and () are related as

constant (B.9)

where ~' is the devi8.tion of the velocity from the critical speed.

. ti fI: 111d1::

1. e. 0 I

=

~

-

0.. '$'(AJ tp~

,

From Eqs. (B. 4) and B. 8), two special curves which are characteristicl? are showl'l to exist and are given py

.z-

_~

_..

:::..

-c

-1+/

_z

_{(B. 10)}

/G

-

C

y+

1

~'

4

(B. 11)

The first curve (see line OB in Sk. 4) is. known as the lirniting éharacteristic since H divi des the flow into two distinct regions:

I) thai in the subsonic part of the nozzle and that in the super-sonic part of the nozzle (AB) which influences the subsonic flow

and II) the remaining supersonic flow beyond, B.

It is also shown that the sonic line and the limiting characteristic meet the nozzle wall upst'!t!::am of the geometrie throat C and that th~ s-onic line and line of horizontal veloéity meet on the axis as shown in Sk. 4;Any changes in the nozzle c,ontour downstream of point B will not influence the flow in the region upstream of thecharacteristic passing

--

APPENDIX B

TRANSONIC FLOW IN A NOZZLE FOR PERFECT GASES

The flow field in the sonic region of a de Laval nozzle for per-fect gas flows iS'clescribed in this Appendix as a ready reference for comparison with the reacting gas flows described in the text. All of these results are

taken from Ref. 10.

Let ~ ,

a.. ,

-A. , ()

be the flow speed, sound speed, specific enthalpy and f:?treamline angle respectively. It has .Deen shown in Ref. 10 that a perturbation vebcity potential

Lf

can be introduced such that

~)(

..: a.

~6

+

L/

x )

O~: afLf~

(B.1)

where a* is the critical speed (i. e. where

I::

a,) and

1><

I

I.fx

and Cf ~ I

t.e'l

.

are

the x and y components of the velocity and perturbation respectively and l.p

satisfies' the equation

(B.2)

The solution of this equation valid in the sonic region of a de Laval nozzle is shown to be

(B.3)

where C is a positive constant and Y is the ratio of specific heats. The x and y components of the perturbation velocity are then,

lof" :::

c.:t

-I-

Cl

-+

I )

C~V~

(B. 4) .

(B.5)

,~,

~: ..

.

"

,

.

~:

Perfect Gas Flow

, Sonic Line

~

\

Sub ~onic Region \

,. --Line of Horizontal Velpcity

Super sonic Region Flow Direction - - -... .-.

Centerline

o

SKETCH 4

through B, whereas changes in the nozzle contour between Band A wiU

Ll1-fluence the sonie line and thus affect the entire flow inclyding the subsonic region. To.give an insight into the chara.cteristic network in the ragion·

be-tween the son~c line and the lirniting characteristic, an exaggerated sketch of this re.gil)n is shown below. The charactedstics are inclined to the strearn-lin€s at the Mach angle

Y

}L wherefo

p-

=i-. For M= I , P ==

ïT/z.

and as

M increases

IJ..

decreases from /TIl-

Th~p

characteristics

A

P

/

---Lirniting

Scnic Line Q-. \

r

Characteristie _ _ ----A\

_1----

_ - -

-\

-,

J.:i'low Direction \ \

...

,\

---}

Streamlines Char acteristic s (two.families) Subsonic '\

Supers.!!~-

-Centreline

o

SKETCH 5

at a point ~ on the sonk line are perpendieular to the strearnline passing through it whereas" at a point p on the nozzle wal!, which is also a" streamline, they are inclined at an apgle slightly less than _{1r/2 '} and along each one of these characteristics thechanges in

l'

and Gare related by Eq. (B. 9)

Comparison of Reacting a~d Perfect Gas Flows

For a perfect gas, the specific enthalpy -IL and the sound speed

are related as

(B. 12)

Comparing Eq. (B. 12) with Eqs. (26) and (33) in the text, one

may ::lote that

,)

-I=-È:-'r

A (B. 13)

(B. 14)

Eqs. (E. 13) and (B. 14) may be taken as the definition of

'I;

1d

Yt:.

They may be consj.dered as fictitious ise.ntropic indices for partially

frozen and equilibrium cases in reacting gas flows. The true. expressions

for these quantities in the present casee are given by Eqs. (A. 32) and (A. 34).

From Eqs. (B. 13) and (B. 1,~), the parameters Pand N given by Eq. (4:8) in

the text wiU be

p=

(B. 15)

(B. 16)

In the limit of equilibrium flow, (see Sec. 3.1), N reduces to

(B. 17)

tj:~J1.\

,- ,.$t1bstituting Eqs. (B. 15), (B. 16), (B. 17) in the variolls equations giving the

, V~locity cornponents, sonie Ene, line cf horizontal velocity and the limitip..g

APPENDIX C Existence of a Velocity Potentia+

For small deviations fr om equilibrium of a reactinggas flow, the flow may beassumed to be nearly isentropïc" giving rise to the ·existence of a velocity potential. This may be shown as follows:

The entropy equation for reacting gas flows is given by (Refs. 6, 9)

T

grad

S ::

grad h

-~

grad p

-+ Q

grad

~ (Cl)

where T, S, h,

e '

pare temperature, entropy, enthalpy, density and pressure re sp ectiv ely, 0( is the atomie mass fraction and Q is the difference between

the specific chemical potentialof atoms and molecules given by (Ref. 5)

(C2)

.;j ' ..

where '~~,\ _{is the characteristic dissociation density, 9v , 9d are characteristic} temperatures for vibration and dissociation, respectively, and R is the gas constant per unit mass referred to the diatomic gas. In term's of thelocal equilibrium v~lue of (Xe , Q takes the form

Q

=

R T

Log [

<X!

'

.

,-

ot ]

. 1- C(e d.' (C3)

....

By Scalar multiplication of Eq. (C 1) by

Cf '

one obtains the variation of entropy along a streamline as

T9

"grad

S

=.

T DS -::.

Q~.8radO(

=

Q

D~

Dt

ot

(C4)

~ince

from the energy equation Eq. (9),

~~

-

k

~~

=. 0 , where,

~t

=.

q.

grad ,

. rl;~'"- ~. Eq. (C4) may be written using streamwise coordinate s as .

:-".- .

(C5)

': R Log

r

(j.~

.

L

I - ot.

\

~~

J

~~

..

-

(C6)