Reaction-resisted shock fronts

Mi(

_{CoA R E P O R T No. 150}

THE COLLEGE OF A E R O N A U T I C S

CRANFIELD

REACTION-RESISTED SHOCK FRONTS

by

REPORT NO. 150 May, 1961.

T H E C O L L E G E OF A E R O N A U T I C S

C R A N F I E L D

It is shown that shock waves whose structure is determined solely by the effects of chemiical reactions (reaction-resisted shock fronts) a r e possible and completely analogous to relaxation - resisted waves. A single dissociation reaction is considered and numerical results Indicate that such waves could be observed experimentally. Bulk viscosities equivalent to reaction effects a r e possibly 10* or more times shear viscosity values. (Examples a r e based on Lighthill's ideal dissociating gas).

CONTENTS

Page Summary

List of Symbols

1. Introduction 1 2. The Basic Equations 2

3 . The Fully Dispersed, Reaction-Resisted, Wave 5

4. Numerical Example 10 5. Weak Reaction-Resisted Shocks 11

6. Conclusions 12 7. References 12

LIST OF SYMBOLS a , a. Equilibrium and frozen speeds of sound

e I

c Atom m a s s fraction

c (p, s) Equilibrium atom m a s s fraction evaluated at local p and s c ' See equation 19

D Dissociation energy per unit m a s s h Enthalpy per unit m a s s

k-, k Specific reaction rate constants

M , M. Equilibrium and frozen Mach numbers p P r e s s u r e

R Universal gas constant 8 Entropy per unit m a s s T Absolute temperature u Gas velocity

W Molecular weight of molecules X Streamwise co-ordinate

•y , 7 , Equilibrium and frozen "polytropic exponents" , see equations 30, 31

K Equivalent bulk viscosity

X See equation 22

H Shear viscosity

ju , ju Chemical potentials, atoms and molecules respectively p Density

a See equation 12

T , T ' . T * See equations 16, 18 and 28

1

-1. Introduction

Inviscid flow theory shows that a plane compression wave in a gas will always steepen until, finally, the physically impossible condition is reached wherein three different values of p r e s s u r e , gas velocity, etc. are predicted to occur at one point in the flow field at one and the same t i m e . Before this state is reached of course, inviscid flow theory begins to lose validity and it becomes n e c e s s a r y to include the effects of viscosity and heat conduction. The diffusive effects of these phenomena act in such a way as to halt the convective steepening of the wave and eventually a shock front will be formed (so-called diffusion-resisted waves). The width of such a shock front is measured in units of /Vpu (^ p , u a r e shear viscosity, density and velocity, respectively).

At least, this is true if viscosity and heat conduction a r e the only dissipative effects present in the gas. If the gas is polyatomic, relaxation of the internal degrees of

freedom provides a further mechanism which will exert its influence on shock s t r u c t u r e . Ordinarily, this influence is confined to regions behind the thin, viscosity-resisted wave (the relaxation zone), except for rapidly relaxing modes like rotation. In the latter c a s e s , where the relaxation time may be l e s s than ten molecular collision time i n t e r v a l s , the effective "viscosity" of the mode remains comparable with /u , and can frequently be included as a bulk viscosity factor, m e r e l y serving to widen the

diffusion-resisted wave. F o r the modes with much longer relaxation times (many hundreds or thousands of collisions), the effective viscosity is v e r y much g r e a t e r and, a s a direct result of the dispersive c h a r a c t e r of a relaxing g a s , the possibility a r i s e s that a compression wave can be entirely resisted by relaxation effects alone. F o r this to happen, the equilibrium Mach number of the shock front must be l e s s than the

appropriate frozen to equilibrium sound speeds ratio. These facts, and the analysis to support them, have been given by Lighthill (1956), who called the resulting waves

"fully-dispersed" shock waves.

In broad t e r m s , the dispersive power of a relaxing gas is measured by the sound speeds ratio mentioned above, whilst its absorptive power depends both on this and on the relaxation t i m e . When a gas is reacting chemically, dispersion and absorption are still present and behave in a v e r y s i m i l a r way to that encountered in relaxing g a s e s . However, the sounds speed ratio has the possibility of being much l a r g e r in a reacting than in a relaxing gas and, furthermore, the appropriate "chemical t i m e s " a r e l a r g e r , or certainly no s m a l l e r , than many practically encountered vibrational relaxation t i m e s . F o r these reasons we might speculate that " r e a c t i o n - r e s i s t e d " shock fronts can occur in chemically reacting gases and it is the purpose of the present paper to prove that this is Indeed so. It is also shown that r e a c t i o n - r e s i s t e d waves may be two or even m o r e , o r d e r s of magnitude thicker than diffusion-resisted waves, i . e . of thickness 10* nlpn or m o r e .

We consider only plane waves in one dimension and, treating the flow a s steady, find the conditions under which a purely r e a c t i o n - r e s i s t e d shock is possible. In this

we follow Lighthill, but the method used here differs slightly from his in that here we s t a r t from the inviscid, non-heat conducting differential equations governing the gas

2

-flow, w h e r e a s L i g h t h i l l ' s a n a l y s i s s t a r t e d f r o m the i n t e g r a t e d f o r m s of t h e s e e q u a t i o n s . B r i e f l y , the r e a s o n i s t h a t , in the r e a c t i n g c a s e , the b e h a v i o u r of the c o n c e n t r a t i o n s i s a l i t t l e m o r e difficult to handle t h a n i s that of the i n t e r n a l m o d e t e m p e r a t u r e in the r e l a x a t i o n p r o b l e m .

In t h e a n a l y s i s to follow, r e l a x a t i o n effects a r e i g n o r e d (but c o m m e n t e d upon in Section 5) and t h e r e a c t i o n i s a s s u m e d t o b e the d i s s o c i a t i o n and r e c o m b i n a t i o n of a s y m m e t r i c a l d i a t o m i c m o l e c u l e . That i s t o s a y , ^f A + A — ^ 2A + A . (1) m o <T— a o k r

A and A a r e c h e m i c a l s y m b o l s for m o l e c u l e s and a t o m s , r e s p e c t i v e l y , and (in the p u r e g a s m i x t u r e ) t h e " c a t a l y s t " in t h i s r e a c t i o n , A , m a y be e i t h e r A o r A . E i t h e r i s a s s u m e d to be e q u a l l y e f f e c t i v e , k , and k a r e t h e a p p r o p r i a t e r a t e c o n s t a n t s . In e x a m p l e s below n u m b e r s a r e u s e d which a r e a p p r o p r i a t e to an o x y g e n - l i k e i d e a l d i s s o c i a t i n g g a s ( L i g h t h i l l , 1957). 2 . T h e B a s i c E q u a t i o n s T h e m a s s c o n s e r v a t i o n e q u a t i o n in s t e a d y o n e - d i m e n s i o n a l flow i s s i m p l y p u = Q (2) w h e r e Q i s a c o n s t a n t , o r , in d i f f e r e n t i a l f o r m , u Ë £ . + p Ë l i = 0 . (3) dx '^ dx

Since p i s a function of p r e s s u r e , e n t r o p y and n a a s s f r a c t i o n of a t o m s (p, s and c , r e s p e c t i v e l y ) we c a n w r i t e u Ë £ = ( | £ \ . „ d p V 8 £ \ . ^ d s / a A ^ d c dx V Spy dx \ds J dx Vac/ dx ^ ' ^ s . c ^ ' p , c ^ ' p , s (4) The e n e r g y e q u a t i o n i s dh dp „ ,^.

ilj and M are the chemical potentials, per unit m a s s , for atoms and molecules,

a m

respectively). Making use of the r e s u l t s in equations 4 and 6, and manipulating some of the thermodynamic derivatives which appear in the course of the analysis it can be shown that equation 3 leads to

dp 2 du 2 dc _ .„.

" d l ^ ^ ^ f d ^ ^ P^f^^ö^ =«• <^>

(Details of the derivation of this equation can be found in Clarke, 1958). The quantity a? is identifiable with the frozen speed of sound and

^i<^l

s, c

The momentum equation in a one-dimensional steady flow is

which can be integrated at once to give

p + pu* = F = p + Qu (10) where F is a constant. Alternatively equation 9 can be used a s it stands to eliminate

u (dp/dx) from equation 7, resulting in

/ 2 2\ d u 2 dC _ #1 1 \

(a. - u )-J— + a a u T - = 0 . (11) _{f dx f dx} The quantity a which appears here and in equation 7 above, is a purely t h e r m o

-dynamic function. In general, for the dissociating diatomic gas,

I

(^).

^ 7

(i^L

•

p.

-pf ^ " ^ ' p , T '' ^ " " ^ p , T

^f % 9 T ) : ^pf ^ ( ? T )

p , c ' ^^ '' p , c

If we t r e a t the gas as an ideal dissociating gas (Lighthill, 1957), a can be evaluated analytically. We find that

a = rDWjj/RT+ l l [4 + c T ' - J j + c l " ' , (13)

where D is the dissociation energy p e r unit m a s s , W^ is the molecular weight of the molecules and R is the universal gas constant. We note for future reference that the combination a' a h a s the following value for an ideal dissociating g a s ,

a j j = [ D ( 1 + c ) / 3 ] - RT/Wj , (14) (since a^ = [ ( 4 + c ) / 3 ^ (p/p) and p = p ( l + c)(R/W^)T) .

The atom continuity equation, with atoms being produced according to the reaction described in the Introduction, can be written in the form

T u g ^ - K ( l - c) - c* (15) where T is the "chemical tinae"

T = w j / 4 k p ' ( l + c) (16)

and

K - (W^kj/4 p k^) . (17) k, and k a r e the forward (dissociation) and r e v e r s e (recombination) specific reaction

rate constants measured in t e r m s of moles per unit volume (raised to appropriate powers) per unit t i m e , respectively. We shall shortly demonstrate that a reaction-r e s i s t e d shock wave Is necessareaction-rily a weak wave. In that event, the m a s s freaction-raction c will not change greatly from the free s t r e a m to downstream regions, and it s e e m s legitimate to linearise the reaction rate t e r m on the right-hand side of equation 15. Indeed we shall go a little further and assume a constant value for the chemical time (and write it a s T ' ), so that equation 15 now appears in the form

T' u — ^ c (p, s) - c . (18)

c i s an a r b i t r a r i l y chosen local equilibrium value and c ( p , s ) above indicates that we have chosen local p r e s s u r e and entropy for the evaluation of c . We shall subsequently be able to evaluate both the difference c (p,s) -c and c itself, so that T' can be found from the mean value of l<c(l - c) - c*l / ( c ( p , s ) -c J a c r o s s the shock wave. e

_{Tc .}

e

It i s r a t h e r m o r e convenient to work in t e r m s of the quantity c ' , defined a s

c ' = c - c^(p,s) . (19) Then equation 18 can be written a s

d c ' d^P

T ' u ^ ^ . + T ' u - r ^ + c ' = 0 . (20) dx dx

Now we may write

\ 8 p / " dx \ 8 s i " dx dc„ /8c \ J /8c

e " d T

5

-/ oi;

V i p "

= ^ " ^ + i ^ ) • (P - ^ ^ ) T " \ ( C ' / T ' ) . (21)

dx \ 8 s / a m by using equations 6 and 18. We have written

ac

3p^; <^^> for brevity- It has been shown elsewhere (Clarke, 1958, 1960) that in - u )T is of

a m o r d e r (R/W2)c' and,that the derivative (8c /9s) is l e s s than order (W^/R).

Consequently it is legitimate to use equation 21, together with these r e s u l t s , to write equation 20 in the form

.r' u 1^' + T' X u ^ + c' - 0 (23)

dx dx

The approximation involved in neglecting the last t e r m in equation 21 when this result is used in equation 20 is no worse than that leading to the linear rate law involved in equation 20.

It is important to note that in adopting equation 23 as the atom continuity equation we have not in any sense "linearised" the problem. We have m e r e l y made as

convenient an approximation to the dissipation effect as possible. The convective, steepening effect, remains with us in the presence of the ud/dx operators etc, 3. The Fully Dispersed, Reaction-Resisted, Wave

The chemical reaction rate equation (equation 15 or its "linear" version equation 18) shows that u dc/dx •• 0 as equilibrium is approached. The shock transition takes place between two states of equilibrium and we must have du/dx •• 0 on either side of the wave. Also, u dc/dx must remain finite everwhere. In order for the solution of the equations in the preceding section to be meaningful in the present context, clearly du/dx must remain finite during the transition from one equilibrium state to the other. It follows at once from equation 11 that u must be l e s s than a, everwhere. Since u d e c r e a s e s and a . i n c r e a s e s in passing; with the flow, through the shock front, it is sufficient to specify that u < a. (suffix i signifying the " u p s t r e a m " equilibrium state). . . T

-.•

Using equations 18 and 19 with equation 11 we have .

<-|-"^i^ - - f - 7 = «• <24)

Equations 10 and 23 show that

- 6

and elimination of c' between these two equations leads to an equation in derivatives of u only.

/ d r / 2 v-'/ 2 2\ du j

, u - [(a^) ( a f - u ) - J

+ (a*ff)''|^a* - u*(l +Pa^aX) J

du dx

(26)

Equation 26 is still somewhat awkward to handle a s it stands and we intend to make two more approximations, which are not, however of a very drastic nature. In the first place we recall the result in equation 14. The dominant t e r m there is D / 3 , since DWj/R is " 6 x 10* K for a typical gas (oxygen, for example) of the type that we are considering. Typically, T may be of order 4 x l O ' K in the "interesting" dissociation range and we are concerned with relatively small variations of T about the mean value through the wave (of order 10* K say, but see the example below). Variations of c will also be small and we may fairly take the t e r m (a*a)~' as constant in equation 26.

It has been shown (Clarke, 1958) that, if all of P, a , o and X are evaluated in an equilibrium s t a t e , then 1 + pa* a X is equal to the square of the ratio of frozen to equilibrium sound speeds in that state. In equation 26, p , a. and a a r e evaluated at actual values of the thermodynamic v a r i a b l e s , which a r e not equilibrium values in general. X is evaluated in t e r m s of c ( p , s ) . We r e m a r k that the maximum value of the sound speeds ratio squared s e e m s to be about 1.4, roughly speaking, whence it follows that any variations in the product p a* a X a r e relatively (about 0.4 : 1) l e s s important h e r e . In fact it s e e m s reasonable, firstly to treat 1 + p a* aX as constant everywhere and secondly, to write it a s ( a , / a )*, where a i s the equilibrium sound

I e e

speed. This latter approximation becomes exact far in front of and far behind the wave, Accepting these approximations, the gas velocity u satisfies the differential equation

We note that although ( a . / a )* is treated as a constant h e r e , so that we shall write

\ a . = T* = constant (28)

7

-It can be shown that

a .2 3 ^ + ^ E _{f 3 • p} (29)

for the ideal dissociating gas, but it is quite consistent with the previous approximations

to write a* 3 7 . £ (30) f 'f p and also a" » 7 . ^ , (31) e e p

and to t r e a t 7 . and 7 hereafter as constants, (or better still, a s constant suitable mean values),

Proceeding on this b a s i s we find from equation 10 that

a* = 7f ( | - u)u . (32) with a s i m i l a r result for a* . Then equation 27 can be integrated once to give

, , a 2» d u . F ._ , . ,v u 2

T ' ( a ' - u ) ^ + 7 i: u - ( 7 + 1) ^ = Constant (33) I dx e Q ' e 2

8 2

Since a, - u ^ 0 and we expect to find du/dx •• 0 upstream and downstream of the wave, the integration constant can be found when u = u,, the upstream value (which we can take a s given at x = - 00 ). Then clearly the t e r m

2 2

- ^ - ^ e - ^ ^ ^ l ^ ^ e i " - ^ e | ^ ^^ye^'^1 <34) must be zero again (at x =« + 00), where u = u , say. It follows that we can now write

equation 33 as

^'[-^f l^-^-^f^^^^'jar =- - ^ ^ (u-u,)(u-u.). (35)

Making use of the fact that

a 2

a, - u

r "^f F -] J^ " . , ,

n ( _ 7 f + l Q n j 7 ^ + 1 equation 35 can be integrated to give the result

2 2 2 2

, a. - u , va, - Uj \

8

-T h e c o n s t a n t h e r e i s a r b i t r a r y and m e r e l y s e r v e s to l o c a t e t h e o r i g i n of x . -T h i s r e s u l t m a y be c o m p a r e d with L i g h t h i l l ' s (1956) r e s u l t for the r e l a x a t i o n r e s i s t e d s h o c k . Not u n n a t u r a l l y , in view of the c o n s t a n t 7 and 7 a s s u m p t i o n s i m p l i e d in

f e

e q u a t i o n s 30 and 3 1 , the r e s u l t s a r e s i m i l a r : in fact the t e r m ( 7 . + l ) u d o e s not a p p e a r in L i g h t h i l l ' s e q u a t i o n , but s i n c e u i s r e s t r i c t e d t o an e v e n s m a l l e r r a n g e of v a r i a t i o n , in h i s c a s e no s e r i o u s e r r o r would be i n c u r r e d b y w r i t i n g u « u t h e r e and a b s o r b i n g ( 7 . + l ) u . in the a r b i t r a r y c o n s t a n t .

We r e i t e r a t e L i g h t h i l l ' s r e m a r k s h e r e , t h a t t h e coefficients (a* - u* )(u - u )

i n n 1 2

( w h e r e n = 1 o r 2) a r e t h e s c a l e f a c t o r s for t h e r a t e of change of u n e a r the head and t a i l of t h e s h o c k f r o n t . In p a r t i c u l a r we s e e t h a t if u, •• a . , the c h a n g e s in u with x in t h e h e a d of the wave b e c o m e e x t r e m e l y r a p i d . Since u , will a p p r o a c h a , b e f o r e u •• a , it follows t h a t t h e f u l l y - d i s p e r s e d w a v e b e g i n s to b r e a k down n e a r the head

• 12

f i r s t , t h i s p a r t of the wave b e i n g g o v e r n e d b y r e l a x a t i o n * o r v i s c o u s e f f e c t s . Some e x a m p l e s of the v e l o c i t y p r o f i l e t h r o u g h a fully d i s p e r s e d wave a r e given below ( s e e a l s o Griffith and K e n n y , 1957). O n c e t h e v e l o c i t y p r o f i l e h a s b e e n found, the d e n s i t y p r o f i l e follows at once f r o m e q u a t i o n 2.

T h e v a r i a t i o n of t h e c o n c e n t r a t i o n d i f f e r e n c e c ' follows f r o m e q u a t i o n s 24, 26 and 3 3 , which give

( 7 „ + l )

c ' = - — S (u, - u) (u - u ^ . (37) 2 a* CT

e

With t h e a s s u m p t i o n s of c o n s t a n t 7 and 7 . it follows t h a t we m a y , a s with a* a, t r e a t a'a a s constant .Whence the m a x i m u m v a l u e of c o c c u r s when 2u = u + u^ and h a s the v a l u e

( T _ + 1) c ' = (u, - u ) (38) m a x „ 2 ' * 8 a a, e i < We c a n find t h e v e l o c i t y i n c r e m e n t u , - Uj b y s o l v i n g t h e q u a d r a t i c e q u a t i o n obtained on s e t t i n g e x p r e s s i o n 34 e q u a l t o z e r o . T h u s

2(uJ - a* ) 2(a* - u*) (Tp + i ) u , " (-y^ + I K

(39)

9

-and it follows from equations 38 -and 39 that

- 1 2

M - M c = - V ei ei

^ . (40) 2a/7g + 1)

If we take M ^ 1 . 2 (about the maximum possible) and o " 1 (which is conservatively s m a l l for a l a r g e p a r t of the dissociation range), then c ' ~ 0.04, which i s s m a l l enough for a linear r a t e law approximation to appear reasonable.

It i s interesting to observe that c' (at a given u) is independent of the reaction rate (as s u m m a r i s e d in the chemical t i m e r' ) . Since the significant thickness of the shock is c l e a r l y m e a s u r e d in some multiple of T ' U , (see equation 36), the entropy and reaction equations show that the entropy r i s e is independent of T ' . In other w o r d s , the connection between initial and final equilibrium s t a t e s is independent of the shock s t r u c t u r e and m e r e l y r e q u i r e s that some dissipative mechanism shall be present to r e s i s t the convective steepening effects. That mechanism is provided by the chemical reactions in this c a s e .

We may use the r e s u l t in equation 21 (neglecting the last t e r m t h er e ) in o r d e r to e s t i m a t e the changes in c ( p , s ) . Assuming a suitable mean value for X (X will do a s a first approximation) we can readily show that

c ( p , s ) - c^ « X^(p - p^) = \p (u^ - u), (42) using the momentum equation (equation 10). The maximum value of c - c occurs

when u = Uj (since u d e c r e a s e s monotonically). Whence equations 39 and 42 show that 7^ P

(c ( p , s ) - c ) = c - c. «2X . -—-. (M' - 1) . (43)

e 1 max * « i 7 + 1 ei e

The value of p^X, can be found in Clarke (1960) and i s v e r y roughly of o r d e r (RT^/W^D), o r about 10 , to give it some n u m e r i c a l magnitude. Thus with M = 1.2 it follows that c , - c , ~0.04.* Changes in atom m a s s fraction a r e therefore e x t r e m e l y small throughout the wave and the assumptions involving constant 7,, 7 , a ^ e t c . a r e

apparently quite justifiable. (Note that p^p^/p^p^ ^ 1 +(2/7 + 1)(M - M *)(7 M -M ) =• T / T (1 + c / I + c ) so that T v a r i e s only a little a c r o s s the wave too),

*

It should be emphasised that these e s t i m a t e s a r e quite crude, so that not too much significance should be attached to the s i m i l a r i t y between c - c and c ' _{m a x}

10

-4. Numerical Example

To illustrate the r e s u l t s just obtained we consider the following situation. The upstream (suffixi) flow is assumed to be at a p r e s s u r e of one atmosphere and a t e m p e r a t u r e of 4,250 K. With an oxygen-like Lighthill g a s , it follows that c, = 0.78, Also ( a , , / a ) = 1.35. We may therefore take 7 . = 1.59 and 7 = 1.18. The value of

ft e i I e

p X for these conditions i s 0.086.

1 1

By a suitable choice of the constant in equation 36 the velocity profile equation can be written in non-dimensional form as

(Ml* - Dlogd - u/u ) - (M."' - l)log(u/u, - u / u , ) - (7, + l)(u/u )

f, 1 f2 , 2 1 f ^^^j

= (7g + 1)(1 - U ^ U , ) ( 7 J / 7 ^ ) ( X / 2 T ' U , ) .

The ratio (u/u^) calculated from this expression i s plotted against ( X / T ' U^) in Fig. 1, for two values of M , namely 1.1 and 1.2. The sharpening up of velocity variations in the head of the wave as M "• (a, / a f is apparent, a s is the d e c r e a s e in wave

ei fi ei

width in t e r m s of (X/T' U ) units under the same conditions. Under the chosen

conditions u = 1.6 x 10* c m / s e c when M* = 1.1 and 1.67 x l O ' c m / s e c . when M* = 1.2. 1 ei ' ei The wave thickness is roughly given by ( X / T ' U ) ~ 1 0 , so that practical wave thickness

will be ~ 1.6 T' cm. if T ' is measured in microseconds. In fact it would seem that T' is in the region of 1/u sec for oxygen dissociation in the range of variables considered in the present example (see below). It s e e m s that measurable profiles could

therefore occur with a judicious choice of initial conditions.

(If we choose to define wave thickness as the change in X / T ' U^ in going from velocity u, - 0.05 (u, - u^) to velocity u^ + 0.05 (u, - u,) we find that A ( X / T ' U^) = 15.23 at M* = 1.1 and =• 7,69 at M* = 1,2).

ei ei

Some values of m a s s fraction c and the equilibrium m a s s fraction c ( p , s ) have also been calculated for the one case of M* = 1,1, using equations 37 and 42. They a r e plotted in Fig. 1 a s a difference from c^ (which equals 0.78).

The temperature change through the wave whose equilibrium Mach number is VT7Ï can be found from the expression at the end of the previous section, and turns out to be given by (T^/T^) - 1 = 0.0195. The small variations of both c and T through this wave a r e ample confirmation that some of the averaging approximations made in the analysis a r e justifiable.

11

-The value of T' in equation 18 can be obtained via the method proposed in the paragraph following that equation. Carrying out an o r d e r of magnitude, slide r u l e , calculation for the M* =1.1 case using three points at u / u . = 0.97, 0.95 and 0.93

e i ' _ _

it is found that T' ~ 1 0 ' / k , where k is measured in ( m o l e / c . c ) sec , An r r

acceptable value for k s e e m s to be about 10 in these units, confirming that T ' is Indeed of the o r d e r of 1 fjsec in the present problem,

5, Weak Reaction-Resisted Shocks

Following Lighthill (1956), we may ignore the difference between (a* - u*) and (a* - u*) when the shock is v e r y weak ( i . e . M « 1 ) . We may also write

f» « e i

(7, + l)u « ( 7 . + l)u on the right-hand side of equation 36, whence the velocity profile is approximately given by , (7 + 1) (u - u )x , U . " U e 1 2 / y , , . 1 « exp j , (45)

I 2

- a M J

having put u « a in the denominator of the exponential t e r m . The profile of a shock wave of s i m i l a r velocity amplitude which is resisted by a bulk viscosity K is given by

u - u ( 7 + l ) ( u - u ) x

u

(46)

assuming the same 7 in both c a s e s . It follows that the action of the chemical reactions here is s i m i l a r to that of a bulk viscosity K^ given by

K " p T ' (a* - a ' ) . (47)

1 1 f i e i

With the values given in the numerical example above, this implies that K^ ~ 0 . 3 T ' g m / c m . s e c . if T ' is measured in m i c r o s e c o n d s . The ordinary shear viscosity probably has a value of about 10 gm. / c m . s e c . , so that if T ' is about 1 iusec. , we see at once how very much m o r e powerful the reactions a r e in r e s i s t i n g the

convective steepening than is shear viscosity (and indeed also thermal conduction, which is comparable in effectiveness with the l a t t e r ) .

The strong dependence of K^ on the difference between a* and a* is clear from 1 11 e 1

equation 47. An exactly s i m i l a r result to this holds for relaxation effects in like circumstances (Lighthill, 1956) and we note that, if vibrational relaxation in a

diatomic molecule is considered, so that 7 =» 9/7 and 7 , = 7 / 5 , the maximum value

2

of (a. / a ) then is only 1.09, compared with 1,35 in our chemical example. It s e e m s

II e i

12

-chemically-resisted wave front (under the conditions of our numerical example,

anyway) since we do not expect the vibrational relaixation time to be much g r e a t e r than the chemical time T' . Rather do we expect it to be somewhat smaller than V ; and it must be remembered too that only 22% of the mixture consists of molecules in this c a s e . When the "chemical (a, / a )* " begins to fall off and the atom m a s s fraction

ii e i

d e c r e a s e s too it is possible that vibrational relaxations play a l a r g e r role in d e t e r -mining u v s . X profiles. P r e s u m a b l y one must always consider the "secondary" effect which vibrational relaxation may have on the chemical time T'. Such detailed considerations a r e most interesting, but outside the scope of the present, heuristic, discussion however.

6. Conclusions

The foregoing analysis demonstrates that r e a c t i o n - r e s i s t e d shock fronts a r e possible and that their general character is precisely the same as that of Lighthill's fully dispersed, relaxation r e s i s t e d , shock. This in itself is not perhaps a v e r y startling conclusion; one would suspect a s much without the need for analysis to prove the point. However, the numerical o r d e r s of magnitude in the present case a r e interesting since they suggest that a r e a c t i o n - r e s i s t e d shock should be observable experimentally without undue difficulty.

7. References

1. Clarke, J . F . (1958) College of Aeronautics Report No. 117. (1960) Jnl. Fluid Mech. , 7 pp 577-595.

2. Griffith, W. C. (1957) Kenny, A.

3. Lighthill, M . J . (1956)

Jnl. Fluid Mech. 3, p 268.

"Survey in Mechanics", Ed. by G.K. Batchelor and R. M. Davies. Cambridge University P r e s s . (1957) Jnl. Fluid Mech. 2, p p l - 3 2 .