Determination of the detonation wave structure

DETERMINATION OF THE DETONATION WAVE STRUCTURE A. K. OPPENHEIM AND J. ROSCISZEWSKI

In order to investigate the influence of transport properties on the coupling between the shock and deflagration that may occur in a steady, plane detonation wave, the structure of the wave has been determined for some specific cases by numerical analysis performed on an IBM 704 computer. After a preliminary check with some representative cases of Hirschfelder and Curtiss,1,2 the procedure was applied to detonation in dissociating ozone. Although the agreement with the solutions of Hirschfelder and Curtis8 is not perfect, their contention concerning the strong coupling between the shock and deflagration has been essentially confirmed. The ozone computations included the consideration of the variation of transport properties with temperature and the change in the number of moles due to chemical reaction, while the rate of the reaction was described in terms of the best available data on kinetics. For this purpose the reaction rate has been expressed in terms of the quasi-steady-state approximation for the oxygen atom concentration which, with exception to the immediate vicinity of the hot boundary, represents quite accurately the exact chain reaction mechanism of the thermal decomposition of ozone. On this basis the structure of the "laminar" detonation wave in ozone has been determined for three fundamental models: the coupled-wave, the yon Neumann-I)6ring-Zeldovich model, and the decoupled deflagration wave. The results have been compared from the point of view of the approximation proposed by Spalding.~ The validity of the continuum treatment has been checked by evaluating the variation throughout the wave of the average number of intermolecular collisions per molecule of product. It appears from this that, although the theory may not be realistic close to the hot boundary, it is certainly quite reasonable in the regime of coupling between the shock and deflagration--the primary objective of our inquiry. In conclusion it is contended that only a thorough understanding of the so-called "laminar" wave structure can provide proper basis for the assessment of the effects of turbulence and other time dependent and multidimensional phenomena that may accompany the detonation process. being the nondimensional space coordinate, with m representing the mass flow rate per unit area and

The Problem Under the usual assumption of steady, onedimensional flow and the essentially perfect gas behavior of reactants and products, the structure of the detonation wave is governed by the following equations3: d~/dx = 1du/dx = ~-

( I ~ / m ~ f l ) ~ ( u , X)

(1)

Xa(u)~/7~3u

(2)

where the symbols have the following meaning:

F = Dp = X/cp = ~1~

expressing the "transport" or "exchange coefficient"--a single quantity that, as a consequence of the assumption that Prandtl number is ~ and Lewis number is 1, represents at the same time mass diffusion, thermal conductivity, and viscosity, while u is the nondimensional velocity or specific volume. Finally ~2 -- q/c~,r = Xh - - Xe

c~T + ( ~ / 2 ) X~

(3)

(c~,T)e

is the nondimensional stagnation enthalpy, B = dx/dl~

is the nondimensional stagnation gradient, ~ - - ~ 0 n / r ) dl = ( r e / r ) d~

(4) enthalpy (5)

(6)

(7)

is the nondimensional heat release, the equation above representing in fact the over-all energy balance for the wave, while: ~(u, x) is the volumetric reaction rate expression and Xa(u) is determined by the Rayleigh Line. For our purpose both these functions have to be given a pr/or/. The knowledge about the process is completed by the information on the composition of the

424

DETONATION

WAVE

reacting mixture. This is obtained from the condition of constant total (i.e., thermal, dynamic, and chemical) enthalpy that results from our assumption concerning the Prandtl and the Lewis numbers. ]f the instantaneous composition can I~e expressed by a single parameter, for example, Y, the mass fraction of the product which changes from 0 to 1 in the course of the reaction, the constant total enthalphy condition, taking into account Eq. (7), yields simply: y = X -- Xr Xh -- Xc

(8)

With the use of the equation of state, the above permits to determine c0(u, X) from the conventional form of co = co(Y, T, p). The problem lies in the solution of (1) and (2), subject to boundary conditions: at

X =

Xr

u=

u~

/3=

0

at

X =

Xh

U =

Uh

f~ =

0,

..

(9)

where u~, the Mach number of the detonation wave, and uh, the volumetric contraction produccd by the wave, are specified by the Rayleigh Line relations.

425

STRUCTURE

Computations For numerical computations, it is convenient to regard x as the independent variable, this having been indeed already suggested in the form of our Eqs. (1) and (2). The major advantage of x in this respect is that it varies monotonically in the domain of the solution. The computational procedure is then, in principle, quite straightforward: one evaluates the change in/~ from (1) and the change in u from (2). Themain difficulty lies in the fact that both boundaries are nodal singularities where, for the physically correct solution, du/dx is infinite. In order to start from any boundary, one must therefore know the initial slope fl(X). Since at the boundaries both the numerator and denominator of the expression for 1 " (dfl/dx) in Eq. (1) approach zero, one can apply for this purpose the l'Hospital rule, according to which: Liana(

dfl) Fft(d~/dX)h 1 -- ~ = m2 (d43//dX)h

(16)

As indicated by Spalding 3 dfl/dx cannot be zero at the boundaries, while physical considerations imply that at the cold boundary d~/dx must be essentially zero. Consequently

T h e R a y l e i g h Line

(d43/dx)c = 1

For a perfect gas the Rayleigh line equation is, in our coordinates,

while for the hot boundary Eq. (16) yields directly:

XR(U) = --[-(~f + 1)/2"]U(U -- UO)

(10)

an inverted parabola passing through u = 0 and u = u0 on the X = 0 axis. In accordance with Eq. (7) of ref. 3: 40 =

2 Xr ~ , + lur - -

+

uo =

2 -y+ 1 -

-

E'Y"o +

(11) the second equality above being the consequence of Eq. (3) from which it follows directly that Xo = 1 + [-(~' -- 1)/2-]u~ The Chapman-Jouguet

condition,

(12) 2~lh2 =

Vh2/(~ -- 1)%Th = 1, leads from Eq. (3) to: Xh = E(T Jr- 1)/2-]Uh 2

(13)

which is satisfied at the peak of the Xa(U) parabola, where according to (11),

Uh ----89

['),Ur-~- u~--lff(T "Jr-1)

(14)

Finally, for the Chapman-Jouguet detonation, ur is determined from the overall energy balance , (7), which with (12), (13), and (14) yields ur

ur

[2(v + 1)~zJJ

(15)

(d43/dX)h = I

{ [ 1 --

1 -t- 4 ~

(10a)

x-xb (165)

With the known finite value of dfl/dx at the start, one has to guess only the value of u corresponding to the specified first step in X, so that the solution will pass through the other boundary. It is also important to test the sensitivity of the solution to the value of/iX, the step in the independent variable used for the computation, in our analyses, with ~2 = Xh -- Xo of an order of 10, it was demonstrated that/iX = 10-3 produced essentially the same results as /iX = 10-4 (.see Figs 5 and 6). The computations were made starting from the cold, as well as from the hot end. The approach to hot boundary from below was, however, quite unstable, making the computation much more difficult than that corresponding to the reversed procedure when the cold boundary was approached from above. As a matter of practical convenience we recommend therefore the start from the hot boundary, in spite of having to solve in addition Eq. (16b) actually a trivial task in comparison to that of having to hit the hot boundary from below. Our solution is given by the functional relation

426

DETONATIONS

B = B(X). According to (4) the nondimensional space coordinate is then: (17)

e -- Go = f E e ( x ) ] -' dx

while the physical coordinate can be obtained from (5) by the quadrature: X - z0 =

f

while X is related with u by the Rayleigh Line equation X ---- - - [ ( 7 + 1)/2]u(u -- 2Uh)

dx -- (7 + 1)(Uh -- u) du

(19)

2 -- 3" + 1 (u -- u~)(u -- us)

(20)

where us = 2Uh -- u~ = u 0 - Uois the coordinate of the yon Neumann spike. Equation (19) upon integration gives therefore: 3"

at

X = Xh

u=

Uh

~=

0

at

X = X~

u=

us

~#

0

2Uh -

S p e c i f i c a t i o n of the P r o b l e m

Computations have been made for two representative cases of irreversible, unimolecular reactions which have been originally analyzed by Hirsehfelder and Curtiss I (they will be referred to here as the H&C reactions) and for thermal decomposition of ozone. For the H&C reactions the expression for the volumetric reaction rate is mk

= - - (1 -- Y) exp ( - - E * / R T ) ua~

23" [ u~ 3"+lLu,--uN

(27)

AXh-- X

in ( u -- tu'~]

u'----~

\u

--

u~,/j

X exp ( _

(26)

The numerical solution is obtained in the ~ m c manner as for the coupled wave, but this time it must be started from the hot boundary, since the value of ~8 at the yon Neumann spike is not known a w/or/.

d~ u.

(2s)

so that, with (8), Eq. (1) becomes

[In (u -- u~)(u -- UN)

- G0= 3" +-----~

+

Xo = 3'"[u -- uh du ~ l m jf ~

Actually, as pointed out already by Hirschfelder and Curtiss, I the influence of transport properties on the deflagration can be evaluated quite simply; the structure is governed by Eqs. (1)-(8), as before, with different boundary conditions, namely:

With Eqs: (10), (12), and (14) it can be shown that the denominator is simply: 2 X9 - - )(.R(u) = 3" _ . ~ [ ( U - - Uh) 2 - - (~/~ - - Uh) 2-]

(24)

The structure of the wave is determined therefore by the quadrature: x--

d~ = ( ~ d u ) l [ x . - ~ ( u ) - ]

(23)

according to which

Model

A simple illustration of the solution is provided by the yon Neumann-D6ring-Zeldovich model which is based on the postulate that the shock is completely decoupled from the deflagration. Since such a solution serves as a convenient point of departure for our coupled wave solutions, it is here derived from our basic Eqs. (1) and (2) which now become deeoupled by virtue of the model. Under such circumstances we obtain directly the classical shock solution, since now, as a consequence of the assumption that Prandtl number is I, it follows that x -- eonst -- Xc. Equation (2) becomes then:

(22)

d x l ~ = (~lm),o(u, x)

(18)

since 1~, given usually as a function of temperature and composition, can now be expressed, with the use of the solution in the X-U domain and Eq. (17), in terms of ~ alone. Finally the dimensional coordinate is simply l = ( F c / m ) x . The N-D-Z

In the classical N - D - Z model the deflagration is devoid of transport phenomena and proceeds along the Rayleigh Line. Consequently Eq. (1) reduces to:

ln(u~--u)

UN In (U - - UN)] u, -- UN

X-

[ ( 7 ~- 1)/2-] u2)

(28)

where for a Chapman-Jouguet detonation: (21)

Fk

3" + 1 ro 3' l"~'u~(Tv~ " -{-- 1)

(29)

DETONATION

WAVE

and

427

STRUCTURE

follows: = Et/RTr

= r0--'

(30)

w = Ms \

+"

(1 - - Y) exp ( - - E t / R T )

while the Mach number of the wave is determined from the equation: uc

_

1 =

uc

2(~ 2

1)

(31)

The right-hand expressions in Eqs. (29), (30), and (31) are given in terms of the notation used by IIirschfelder and Curtiss2 The two H&C reactions considered here have been specified in the above reference as follows:

(33) This time the variation of transport properties with temperature is taken into account by assuming F = F r t. Eq. (1) becomes then: d/~

A (Xh -- X) ( X -4- X) Ot exp (--e/O) uS

dxx-- 1

fl

(34) where

1 0.0028 0.02 1.12

ro

T Tc

1 0.0045 0.02 1.12

x + x Ix 2~

89 -

since 2~ cp/(%)r =. 1 -4- ~ Y =-- X "4- X

They will be denoted here by H&C 1 and H&C 2, respectively. For the purpose of our computations they are given as a consequence of (29), (30), and (31), in terms of the following parameters:

An u,

H&C I

H&C 2

159.043698 50 7.23766

61.57543665 50 7.23766

In the above, with Fc = 2 X 10-4 gm/cm see., k = 4.61 X 10 ~5 emS/gm mole sec, E$ = 24 Kcal/gm mole (the latter two based on d a t a of Benson and Axworthy4) M3 = 48 gm/gm mole, "r = 1.255, for a Chapman-Jouguet detonation propagating into a mixture where the initial velocity of sound ar -- 255.4 m/see ( Tc = 300~ at a Mach number ur -- 7.20, so that ~2 = [-u~ -- u~-V]2/2(-r -{- 1) -- 11.0552, we have A~

I'r - 2.3618 Maac2G

X ~ 2Xh -- 3Xr = 14.5014 both representing a Chapman-Jouguet detonation. For the start of the computation from the hot end, the initial slope of 13 is evaluated from (16b) with 'm'2

~x h

Uh

X exp { - - ~ / [ X h -- 89

and = Et/RTc

For the determination of the slope in/~ at the hot end Eq. (34) yields for the constant in Eq. (16b) : {_d2

-- 1)Uh2-]}

(32)

yielding for H&C 1 : (dfl/dx) h = --0.07974745 and for H&C 2: (d43/dx) h = -- 0.5002388

For the thermal decomposition of ozone we considered only the case of 100% initial 03 concentration yielding undissociated 03 at the Chapman-Jouguet state. Under the assumption of the quasi-equilibrium oxygen atom concentration, the reaction rate can be expressed then as

= 40.25

m2 \dxJx-xb = A

X + Uh2

exp

(35)

leading to (d~/dX)h = --0.6876897

Results and Conclusions The results of our computations are presented in Tables 1-5 and Figs. 1-9. Figure 1 shows the comparison between our results and those of Hirschfelder and Curtiss t in the x - u plane. The discrepancy here is larger than in other coordinates, as indeed-demonstrated in Fig. 3 where the corresponding differ-

428

DETONATIONS

19_

I

16~

14_

Z

12 1 1:~ N _[ 0

1

[

I

I

2

3

4

]-----'l-----T-5

6

0

7

FIo. 1. Coupled wave for the H&C reactions in the x - u plane.

1

1

I

1

10

12

14

16

J'~ 18

FIG. 2. Coupled wave for the H&C reactions in the /~-x plane.

TABLE 1 Coupled wave for the H&C 1 reaction X 18.7486875 18.600000 18.399905 18.199811 17.999717 17.599528 17.199339 16.799150 16.198866 15.598583 14.998300 14.598111 13.997827 13.397544 12.797261 12.196977 11.796788 11.396599 10.996410 10.596221 10.196033 9.795844 9.393655 8.995466 8.595277 8.195088 7.994990 7.794976 7.594963 7.394949 7.547969

~ 0.000000 0.173270 0.437926 0.696549 0.952000 1.447040 1.912991 2.342946 2.909932 3.373409 3.726208 3.897974 4.059434 4.106185 4.041119 3.868223 3.696078 3.481134 3.226250 2.934872 2.611144 2.260019 1.887287 1.499399 1.102864 0.702936 0.502666 0.302571 0.102472 --0.097628 0.000000

U 4.082340 3.185000 3.037427 2.952060 2.893826 2.822355 2.788271 2.779717 2.802787 2.860630 2.947813 3.020615 3.150378 3.303949 3.481247 3.683221 3.832593 3.994877 4.171414 4.364153 4.575655 4.809392 5.070169 5.364864 5.703996 6.106017 6.341939 6.614880 6.959233 7.280266 7.237662

0 16.665500 17.331972 17.246760 17.110579 16.953038 16.603917 16.227632 15.833397 15.217015 14.575783 13.912199 13.457696 12.757317 12.033134 11.282475 10.501312 9.960792 9.401819 8.821424 8.215592 7.579055 6.904662 6.182428 5.397845 4.528431 3.534757 2.967566 2.325497 ..1.541198 0.769764 1.000000

t~ --~ 0.000000 0.696328 1.055581 1.300422 1.638620 1.878185 2.066752 2.295624 2.486601 2.655478 2.760405 2.911000 3.057764 3.204857 3.356403 3.462178 3.573676 3.693012 3.822970 3.967411 4.131932 4.325381 4.562747 4.872831 5.323833 5.659189 6.166819 7.249812 --

DETONATION WAVE STRUCTURE

0u

K 1016.

' ~ o

~s,~ H&C RESULTS

Y

6016 1012

1: p

5014

1016

12 40 lOS

429

:m:

~

~

0

1012 "501

10 30 8

40 IOe

,

104

104 - lO ~ 2

10 4

1

0 2 ~22

"~[ ~ ~~' 1 2

4

~

I-

6

FI(]. 3. Structure of coupled wave for the H&C 1 reaction.

0

~ I

I 2

I

I 4

I

I 6

i

I 0

I~h~:::1

FIG. 4. Structure of coupled wave for H&C 2 reaction.

TABLE 2 Coupled wave for the H&C 2 reaction x

fl

u

18.7486875 18.600000 18.399014 18.198029 17.997044 17.595073 17.193102 16.590146 15. 786204 15.384233 14.781277 14.148320 13.776350 13.173393 12.771422 11.766495 10.962554 10.359597 9.756641 9.354670 8.952699 8.550728 8.148757 7.746771 7.545774 7.547969

0.000000 0.074380 0.223274 0.361348 0.500504 0.780033 1.057263 1.461133 1.963119 2.193039 2.504745 2.768482 2.912400 3.070934 3.131996 3.095026 2.843072 2.5221 l0 2.098969 1.768308 1.406320 I. 021322 0.623370 0.221605 O. 020605 O. 000000

4.082340 3.328600 3.117939 2.978558 2.867148 2.688617 2.544404 2.366093 2. 174462 2.093876 1.990600 1.910293 1.872202 1.845551 1.854411 2.012428 2.331079 2.709083 3.219947 3.642226 4.140256 4.731429 5.450472 6.390552 7.098907 7.2376623

o 16.665,500 17.415033 17.184821 17.090053 16.970476 16.692490 16.384853 15.891346 15.196168 14.827194 14.286966 13.723168 13.339207 12.740636 12.342567 11.261262 10.284312 9.443206 8.46 1633 7.697444 6.810984 5.753426 4.436302 2.642877 1.247465 1.000(0)0

-- | 0.00000 1.43437 2.13454 2.60480 3.24256 3.68318 4.16550 4.63725 4.83074 5.08739 5.31586 5.45728 5.65044 5.78791 6.10817 6.37757 6.60186 6.86270 7.07082 7.32487 7.67858 8.15723 9.19335 1.59091 ~,

430

DETONATIONS __

~

_ B ~ - 10.4 o ~ = 10""

18

I- 3.4~,~ / / A 161~ ~ ~'/ \ 3 . ~

\

A COLD EFIr

START

\ (~y- lO-~

3.470

12

1

~

0

.o

1

2

~6.0

3

4

5

6

7

U

FIo. 5. Coupled wave for ozone in the x - u plane. (Numbers denote initial values of u(x = 18.6) and u(x = 8) for curves starting from the hot and cold boundary, respectively).

ences in wave structure are less noticeable. Figure 2 gives the corresponding f~-X plots. T h e y exhibit indeed the form of the approximation introduced by Spalding in rcf. 3; the extent with which t h e y adhere specifically to this approximation is discussed later in connection with Fig. 9. Figure 4 represents the wave structure in the case of the H&C 2 reaction demonstrating the effect t h a t can be produced by a decrease of 61.3% in the transport coefficicnt-steric factor product, Fk. Figure 5 is the X-u diagram of solutions obtained for ozone. There has been a n u m b e r of curves plotted here, depending on the initial value of u(;~ -- 18.6) for start at hot boundary, and u ( x -- 8) for start at cold boundary, t h a t have been used for the computation. T h e y serve to demonstrate specifically the character of singularities in this domain as well as the sensitivity of the solution to the value of the assumed initial step in u. The final solution has

TABLE 3 Coupled wave for the ozone reaction x 18.664974 18.600000 18.399905 18.199811 17.999717 17.599528 17.199339 16.799150 16.398961 15.798677 15.198394 14.798205 14. 398016 13. 797733 13.197449 12. 597166 11. 996883 11. 396599 10. 796316 10. 396127 9.995938 9.795844 9. 395655 8. 995466 8.595277 8.195088 7.994990 7.794976

7. 594963 7.547969

~

u

0.000000 0.044631 0.143765 0.271141 0.408830 0.706112 1.023433 1.352206 1.684659 2.173084 2.624647 2.891228 3.120773 3. 380095 3. 521282 3. 534739 3. 420446 3.186248 2. 845389 2. 566762 2.252995 2.082611 1. 728367 1. 351276 0.960006 O. 56 1444 0.361245 0.161154 --0. 038946 O. 000200

4.0686868 3.475000 3.204380 3.037404 2.910828 2.718514 2.573543 2.459674 2.369967 2.274078 2.221431 2.210332 2. 218962 2. 270895 2. 372400 2. 526549 2. 726323 3. 005056 3. 337367 3. 597725 3.892952 4.055120 4. 412826 4. 823603 5.303790 5. 885303 6.236327 6.662164 7. 319855 7.200000

0 11.035699 11.395632 11.485377 11.510212 11.510256 11.473182 11.407123 11.321897 11.221702 11.047415 10.846279 10.697161 10. 535386 10. 266629 9. 961985 9. 614317 9.213783 8. 746909 8.194942 7. 766831 7.278030 7.006336 6. 396072 5. 673230 4.794872 3.681396 2.984278 2.118235 O. 764059 1.000000

~ -- co O. 000000 2.586516 3.589104 4.187715 4.926779 5.395898 5.735384 6.000142 6.312859 6.563469 6.708569 6. 841629 7. 025893 7.199353 7. 368989 7. 541100 7. 722383 7.921124 8. 068949 8.234994 8.327253 8. 537537 8. 798529 9.148053 9. 686783 10.127875 10.935158 11. 960647 oo

~ -- o~ 0 . 000000 2.584411 3.584606 4.181856 4.919284 5.387339 5.726014 5.990093 6.301920 6.551701 6.696252 6. 828752 7. 012135 7.184639 7. 353196 7. 524087 7. 703917 7. 900838 8. 047174 8.211367 8.401308 8. 630093 8. 765921 9.106781 9. 624691 10.038589 10.757406 13.113231 ~

x -- ~o O. 000000 8.741526 12.133369 14.159827 16.660239 18.243619 19.385479 20.272170 21.312901 22.139453 22.613869 23. 045575 23. 637031 24.185665 24. 713024 25. 237358 25. 776248 26. 349352 26. 762713 27.213012 27.715337 28. 293199 28. 622093 29.401440 30. 465649 31.222143 32.371185 35. 447094 co

DETONATION WAVE STRUCTURE

431

TABLE 4 Deflagration in t h e N - D - Z model for ozone x

fl

u

18.664874 18.464014 18.263029 18.062043 17.861058 17.459087 17.057116 16.052189 15.047262 14.042335 13.037407 12.032480 11.027553 10.022626 9.017699 8.414742 8.012772 7.811775 7.610777 7.409780

0.00000 0.036819 0.082037 0.132985 0.188752 0.312742 0.451195 0.849337 1.304862 1.796805 2.300426 2.784342 3.208322 3.521835 3.664567 3.641041 3.571511 3.519306 3.455023 3.378414

4.0686868 3.647665 3.472434 3.338088 3.224867 3.034981 2.874927 2.546731 2.277699 2.044106 1.834808 1.643506 1.466228 1.300280 1.143731 1.053701 0.995146 0.966281 0.937682 0.909342

0

z

11.035699 11.246590 11.287297 11.299706 11.297294 11.266143 11.212514 11.018258 10.764860 10.465138 10.123307 9.740026 9.313969 8.842400 8.321366 7.982908 7.745707 7.623468 7.498731 7.371435

-~ 0.000000 3.587711 5.498923 6.763028 8.467749 9.484514 11.084910 12.035362 12.689471 13.172190 13.577920 13.912853 14.210447 14.489623 14.653201 14.764469 14.821100 14.878679 14.937447

TABLE5 Co~t~t~tMenth~pydefla~a~onbehindthe

~o~inozone

X

~

U

0

}

X

18.664874 18.660000 18.399014 18.198029 17.997044 17.595073 16.992116 16.389160 15.585218 14.982262 13.977335 12.972408 11.967481 10.962554 9.957626 9.555655 8.952699 8.550728 8.148757 7.947769 7.746771 7.545774

0.000000 0.044631 0.144979 0.274381 0.415146 0.722837 1.235016 1.793916 2.593629 3.223579 4.308964 5.408729 6.484543 7.489787 8.363673 8.658741 9.022767 9.201825 9.320226 9.354072 9.369751 9.366503

4.0686868 3.471660 3.194392 3.020883 2.887071 2.677334 2.436602 2.243372 2.028939 1.889865 1.686905 1.510686 1.355224 1.217975 1.099810 1.059036 1.007133 0.980549 0.962528 0.957498 0.955620 0.957318

11.035699 11.395632 11.490857 11.518554 11.521626 11.491520 11.398418 11.271861 11.068092 10.894387 10.570480 10.206209 9.801717 9.355075 8.862595 8.651560 8.318465 8.08444 7.839897 7.713346 7.583739 7.450895

--~ 0.000000 2.578225 3.572867 4.165016 4.892458 5.525745 5.929291 6.300583 6.508787 6.777623 6.985286 7.154600 7.298477 7.425131 7.472336 7.540466 7.584548 7.627918 7.649435 7.670895 7.692342

--~ O.000000 8.721776 12.095678 14.105623 16.573681 18.716811 20.075810 21.317201 22.007267 22.888412 23.558022 24.093761 24.539210 24.921598 25.061285 25.259747 25.385982 25.508353 25.568350 25.627693 25.686485

432

DETONATIONS

3X = I0"4 o 8

3X =

IO -3

z~ COLD END START ( S X = 10 .3 )

DEFLAGRATION

COUPLED WAVENDZ

7

8

9

10

11

12

X

Fro. 6. Solutions for the ozone reaction in the fl-X plane.

Figure 9 presents the interl)retation of our solutions in'the light of the approximation introduced by Spalding in ref. 3. According to it the relationship between fl and X should resemble a function = (:~ -- ~c) -- (:~ -- :~c)"

where ~ = ~3/~t and :~ = X/~. Consequently a logarithmic plot of [-1 -- (d~/dx)J versus :~ -- :~ should be a straight line whose slope is (n -- 1). Figure 9 is then such a plot based on data of Figs. 2 and 6. Indeed in the vicinity of the cold boundary the plots are quite liuear, the value of the exponent in (36) being about 3.5 for all three coupled-wave solutions. However, starting from about 35% of the interval in x between the cold and hot boundary, they deviate quite rapidly from the straight line dependcnce. Lines of constant n have been added as an aid in the interpretation of the variation of this parameter throughout the wave. Of particular significance is the plot of K in Figs. 3, 4, 7, and 8. I t represents the variation that occurs throughout the wave process in the ratio of the reaction time tr to the mean molecular collision time, tr or in the average number of inMrmolecular collisions per one molecule of product formed by reaction. Thc collision time has b u n expressed for this purpose, from the classical relation between the mean free path and viscosity yielding ~

been obtained with a 10-4 step in X starting from the hot end; marked also on the diagram are some representative points corresponding to a 10-~ step in X, and for the start from the cold end, in order to illustrate the agreement which has been attained in all these cases. The same is shown in the fl-X plot of Fig. 6 which describes dranmtieally the difference between the coupled detonation wave and the decoupled deflagrations~ with and without the effect of transport properties taken into account. Finally, Figs. 7 and 8 show the structure of the detonation wave in ozone for the coupled wave and the N - D - Z model respectively. In contrast to Figs. 3 and 4 the linear scale'of the horizontal axis in Figs. 7 and 8 is that of the physical coordinate x = (m/F~)l which is obMined by iz~tegration of Eq. (18) on the basis of the solution O = 0(~). The values correspond to p~ = 1 arm; for other initial pressures the physical dimension is inversely proportional to Pc. With Fc = 2 X 10-4 g m / c m see and m = 7p~(u~/a~) = 358 ( p , / a t m ) gm/cm2sec the proportionality factor between l and x is: Fr = 5.6 X 10~ (atm/p~) mm.

(36)

lr ~ = ~8 pp

1.13

(37)

It should be noted that our value of this parameter is about half that of Wood 5 who based it on the rigid sphcre model for thermal conduc~ rp ~ L1016-I

~i

~6016}

"1

4I

7

' 50 i 1012" ~ 12l- 6

0I

i

"'4010 i

~8

1

P

I~ I

104- .20 .I0

)r"

*l" . 0 L 0

1 ,1 4

2 , [ 8

3 4 5 . I . I. 12 16

6 L, 20

7 8 9 ,I . I . 24 28

_ 32

X

Fie. 7. Structure of coupled wave in ozone.

433

DETONATION WAVE STRUCTURE

lO ~

1012./-50

10~

3o

10 4.

20

t0

1o I

o

8

12 0

.

8

X

16

20

X

0

4

8

16

X

FIG. 8. Structure of uncoupled wave in ozone.

tivity; our results in this respect are therefore twice as conservative as his. For the II&C reactions, with the hell) of Eqs. (1) and (28), we obtain for the evaluation of (37): 9~ exp (~/O) K = 1.13,),AU(Xh - - X )

n

5

4

3.5

(37a)

3.0 2.75

i

2.5

while for ozone, in accordance with (34) : g

o.1

0.2

o.3

0.4

o.6

o.8 Lo

FIG. 9. Interpretation of solutions in the light of Spalding's approximar

(37b)

It is of interest to note that in all cases analyzed by us, x, starting from a high value at the cold boundary, approaches quickly a value of an order of 10 which it maintains for some time until, only in the immediate vicinity of the hot boundary equilibrium, it rises to infinity. For the II&C 1 reaction its minimum value in Fig. 3 is about twice as small as for H&C 2 in Fig. 4, essentially in accordance with the difference in the Fk product. For the ozone reaction, under the quasi-steady-state assumption of our theory which does not take into account the attenuating effect of oxygen recombin'~tion, the minimum value of about 10 collisions per molecule of oxygen formed as shown in Fig. 7, should be considercd quite reasonable. Incidentally according to recent shock tube results of Jones and Davidson ~ the value of the steric factor appears to be an order of magnitude smaller than that used here. Consequently the minimum value of should be about 100, which is well within reasonable expectations. In closing we would like to make the following comment. The advantage of our specific analysis is the fact that it takes into account the aerothermochemical effects with sufficient accuracy to render the results compatible with experimental evidence. Any deviations from the observed structure could be, therefore, considered in principle as a measure of some significant parameter governing the physics of the wave process. The most remarkable in this respect

ill

!

~Ot exp (~/0) l'I33'A(xh" -- X) ( X q- X)

434

DETONATIONS

turns out to be, not without surprise, the effect of the "transport coefficient," F. Moreover, its influence on the solution in the X-u and ~-X planes is much smaller than in the straightforward transformation between $ and the corresponding physical dimension, l. The experimentally observed spatial structure of the wave m a y be reflected, therefore, much more in the value of r than in other parameters, as for instance those involved in the description of kinetics. In particular F may be regarded as a convenient, although admittedly somewhat oversimplified, measure of the effect of turbulence or any other time dependent and multidimensional phenomena, since the over-all "stretching" of the wave thickness due to their effects can be certainly taken into account by proper adjustment in its value. Turbulence in detonation waves has been studied b y White ~ who demonstrated the influence it can have on the properties of the Chapman-Jouguet state. However, in contrast to the essential, but relatively small modification in this respect, the effect of turbulence on "wave stretching" should be orders of magnitude larger. This throws then a special light on our study. Although t o d a y the investigation of the "turbulent" wave structure may very well be of the greatest significance to the understanding of detonation phenomena, it is only the intimate knowledge of the so-called "laminar" w a v e t h a t can provide the correct point of departure for this purpose.

Nomenclature % E$ k l m M n p q R T u V x Y

t~

~, F e 0

Specific heat at constant pressure Energy of activation Steric factor Dimensional distance Mass flow rate per unit area Molar mass Exponent in Spalding's approximation (36) * Pressure Heat release per unit mass Gas constant Absolute temperature Nondimensional velocity or specific volume Velocity Nondimensional distance (ml/Fc) Mass fraction of products Nondimensional stagnation enthalpy gradient (4) *

= t~/u

Specific heat ratio Transport or "exchange" coefficient (6)* Nondimensional energy of activation Nondimensional temperature ( T / T c )

9 Numbers in parentheses denote equations where the symbol is defined specifically.

K

p X

Ratio of the reaction time to the mean intermoleeular collision time or the average number of intermolecular collisions for one molecule of product formed b y reaction :(37) * Conductivity Nondimensional transport coefficient (rk/.~) Viscosity coefficient Nondimensional space variable (5) * Density Nondimensional stagnation enthalpy (3) *

2

~-~ x / ~

X

~

w 12

Volumetric reaction rate Nondimensional heat release (7) *

k A tL

2Xh - - 3Xc

Subscripts c h 0 R

denotes cold boundary denotes hot boundary Denotes the nonzero root of the Rayleigh parabola on the X = 0 axis Denotes the Rayleigh line ACKNOWLEDGMENTS

The authors wish to express their appreciation to Professor D. B. Spalding for directing their attention to the advantages of the novel formulation of the problem that has been used in this work, and for the valuable incentive he provided in its execution. They would also like to acknowledge their gratitude to Mr. J. R. Bowen for helping them in establishing the most reliable description of reaction kinetics for the thermal decomposition of ozone. This research was supported by the National Aeronautics and Space Administration under Grant No. NSG-10-59 and by the United States Air Force Office of Scientific Research of the Air Research and Development Command under Contract AF 49(638)-166. REFERENCES 1. HIRSCHFELDER, J. 2.

3. 4. 5. 6.

7.

O. and

CuRrms,

C. F.:

J. Chem. Phys. 28, 1130 (1958). HIRSCHFELDER, J. O. and CURTISS, C. F.: Propagation of Flames and Detonations, in Advances in Chemical Physics, Vol. III, pp. 59129. Interscience Publishers, 1961. SPALDING, D. B.: Ninth Symposium (International) on Combustion, this volume. Academic Press Inc., 1962. BENSON, S. W. and AXWORTHY, A. E., JR.: J. Chem. Phys. 26, 1718 (1957). WOOD, W. W.: Phys. Fluids, 4, 46 (1961). JONES, W. M. and DAVIDSON, N.: The Thermal Decomposition of Ozone in a Shock Tube. Paper presented at the 135th National Meeting of the American Chemical Society, Boston, Mass., April, 1959. WHXT~, D. R.: Phys. Fluids, 4, 465 (1961).

DETONATION

WAVE

STRUCTURE

435

Discussion DR. W. W. WOOD (Los A l a m o s Scientific Laboratory): The analysis b y Hirschfelder and Curtiss (see references 1 and 2) of idealized (A -~ B, first order Arrhenius kinetics, no back reaction, constant heat capacities, P r = 88 Le = 1, etc.) steady s t a t e Navier-Stokes detonations, modified b y me 3 for the case of small reaction rates, was left incomplete as regards the case of very rapid reactions. Figure 1, a modification of Fig. 3 of reference 2, represents the existing interpretation of our analysis. Steady state strong (hot boundary M a c h number Ks < 1) detonations were shown to exist in the shaded region of the figure, while weak (~s > 1) detonations were shown to exist for values of the dimensionless rate parameter R lying on the curve R=. For values of R above this curve, i.e., for very fast reactions, no steady state solutions were found. As Hirschfelder and Curtiss indicated, and as Professor Spalding has mentioned, for such fast reactions the quasi-unimolecular reaction mechanism would be expected to bec0mebimolecular, so t h a t the physical significance of solutions for very fast first order kinetics is somewhat unclear. However, as Hirschfclder and Curtiss indicated, the changeover to a bimolecular mechanism has perhaps no marked effect as regards the coupling of the s h o c k and reaction zones. Furthermore, even if nature abandons the unimolecular mechanism in such cases (it is doubtful if it really provides it for any realistic detonation in the first place), one can envisage performing a numerical integration of the time-dependent, one-dimensional hydrodynamic equations, for the flow produced by a piston which at time t = 0 moves a t the constant speed U p (relative to the reactants at rest) into a quiescent medium for which we prescribe such a fast first

I0

order reaction. I t is clear t h a t for suitable piston speeds the initial shock produced b y the piston motion will initiate the reaction, a n d intuitively one might expect t h a t the resulting long-time behavior would approach some sort of steady s t a t e flow. Here I would like to indicate t h a t such a behavior can be predicted from the existing analysis if the steady state assumption is slightly relaxed to admit, in the region above the curve R= in Fig. 1, two-wave solutions which are similar to the doublewave shock structures observed b y Minshall 4 in iron, and which in the present case consist of a weak detonation followed by a slower moving shock. The dimensionless reaction-rate parameter R of Fig. 1 is given, in the n o t a t i o n of Hirschfelder and Curtiss, by R = (ko'Xo/poD2C~) e x p ( - E ~ / k T , ~ ) .

I t will clarify our thinking if we consider an initial state with completely specific kinetic and thermodynamic properties (i.e., fixed values of the initial steric factor ko', initial thermal conductivity X0, initial density p0, activation energy E * , etc.), and try to determine the nature of the flow as a function of the piston speed Up. W h e n the flow is truly steady (e.g. no rarefaction or Taylor wave is present) this piston speed coincides with the h o t boundary mass velocity U of the steady state solution. This'velocity is given in terms of the Hirschfelder-Curtiss variables b y U2/co ~ = (uo -- 1)2(Ks/S0).

(2)

For a fixed initial state, u0 and 0'0in Eq. (2) depend on the hot b o u n d a r y M a c h n u m b e r Ks, SO t h a t US/co s (co is the initial sound speed) is a function of Ks. I t increases monotonically as one moves up the detonation Hugoniot, beginning at the value 0 for the constant-volume detonation ( ~ = r a t the C - J point, Ks = 1, we find US~cos = Ucj~/co ~ = [2(e -- 1)]/(~, -}- 1);

5

(1)

(3)

as one moves indefinitely up the strong detonation branch U2/co2 - ~ r a~ K~ --~ ('y - 1)/2T. I n the subsequent discussion we will take U as our independent variable instead of KL In Eq. (1), R depends for fixed initial state on the Mach n u m b e r Ks, and therefore on U, through the detonation velocity D a n d the h o t boundary temperature T=. I t is accordingly convenient to separate it into two factors, one of which depends only on the initial state:

2 I 0.5 0.2 i

0.1

i

0.5 K2 Fro. 1.

i

1.0

R = Aoi~,

Ao = ko%/TpoCp, = (coVD') e x p ( - E : / k T ~ ) .

(4)

436

DETONATIONS

Similarly we define the eigenvalue A, corresponding to the eigenvalue R~ of Fig. 1:

(5)

t~ = R,,/R.

The eigenvalue R,, and consequently Au, depends on ~ (i.e., on U/co), and the medium parameters % ~ = Q/E ~, a n d ~ = 1 -I- Q/CpTo. In Fig. 2 is shown the dependence of A~ on U/co for the parameters of reference 1 (~ = 1.25, f~ = 1.12, ~ = 12.2), as determined by an iterative R u n g e - K u t t a integration of Eqs. (1.10-1.12) of reference 3. T h e important fe'~ture is t h a t A~ is found to increase monotonicly as one moves away from t h e C - J point in either direction along the detonation Hugoniot. The nature of the flow produced by a given piston speed Up is thus seen to depend upon the value of the parameter A0 for the initial state, relative to the eigenvalue h , (c-z) for the C - J detonation. If A0 < A,(c-J) then one has the classical situation, in which for Up > Uc-J the steady strong detonation with hot b o u n d a r y mass velocity U = Up is produced; while if Up < U c - j then the usual plausible extension 5 of the steady state analysis predicts a C - J detonation wave followed by a rarefaction wave, the latter reducing the mass velocity from the value Uc-~ to the value Up. If A0 :> A~(c-J) then the C - J detonation does not exist. Instead, if the value of A0 corresponds to the value of A~(A) = A~(A'), A and A ~ being the points indicated in Fig. 2, then the situation is as follows. If Up :> U(A') then the steady strong detonation wave in which the hot b o u n d a r y mass velocity U = Up is produced. If Up < U (A), then b y reasoning analogous to t h a t referred to above

I

I

I I~llll

\ ~

I

,

O.I

, ~ ill,

I

I

I

I i ii

~, f.12 9 12.2

\ )

\

/

I i IzKII

STRON

G

II I I Ucj/C O

CO DETONATION FOLLOWED BY RAREFACTION

,d

J

~

RAREFAGTIO

I

followed by a slower moving shock wave which increases the mass velocity from U (A) to Up. The present discussion of course leaves unanswered the question of the stability of these laminar flows to small perturbations. If the reasonable values k0' = 1013 sec -1, ;k0 = 5.10 - s c a l cm -I sec-* deg-*, Cp = 89cal g-1 deg-1 ~, = 5/4, p0 = 1 a t m are substituted in Eq. (4), one obtains A0 ~ 1200; the value of A~(c-J) from Fig. 2 is ~ 4 6 0 0 . Thus for the values quoted, the classical C - J strong detonation continuum mentioned above would be expected. However, a fourfold reduction in initial pressure would suffice to p u t such a system in the weak detonation, two-wave, strong detonation regime. I do not wish to overemphasize the physical significance of this particular calculation, in view of the highly idealized assumptions involved. However, the possibility of pseudosteady-state solutions of the types described should perhaps be kept in mind in connection with more realistic calculations. A more detailed account of the present analysis will probably be published elsewhere. I would like to mention m y indebtedness to m y colleague: Dr. J. J. Erpenbeck, whose suggestion of a double-wave solution for a different problem led me to the present

WEAK DETONATION //Aq~u FOLLOWED BY S H O C K /

~o,towEo BY

I

[-U = U ( A ) J ,

~'-k25

WEAK DETONATION

A o ~o4

for underdriven C - J detonations, one predicts a weak detonation wave with hot boundary mass velocity U = U (A), followed by a slowe." moving rarefaction wave. In the intermediate range U (A) < Up < U(A') the same argument predicts a solution consisting of the same weak detonation

i

I I HI J ( lilil

I

I IO

U/C O , Up/C O

FIG. 2.

r

I

r r Ir

lOG

437

DETONATION WAVE STRUCTURE results. This work was performed under the auspices of the U. S. Atomic Energy Commission. PROF. C. ADAMSON (University of Michigan) : I t is not clear t h a t the results of this calculation prove t h a t strong coupling does indeed exist between the shock and deflagration. Instead, as with the solutions presented by Hirschfelder and Curtiss, it is shown t h a t for given values of the physical parameters, the solution of the equations shows a coupling effect. The point, of course, is how closely the physical parameters agree with a real physical case. In the discussion presented for the ozone detonation, it is mentioned t h a t the steric factor found in most recent experiments appears to be an order of magnio rude lower t h a n t h a t used in the calculations. Yet the amount of coupling between the shock and defiagration is very sensitive to the steric factor. In fact, a consideration of Figs. l, or 3 and 4 of the paper, indicates t h a t a decrease of 63% in the steric factor decreases the coupling significantly. A decrease by an order of magnitude could well make the coupling become negligible. Finally, it is quite possible t h a t later experiments might result in different values for the steric factor as well as the other physical parameters. Quite apart from the above considerations is a question which should be discussed before further calculations of this type are attempted. In all theoretical work in which the structure of the detonation wave has been studied, the Navier-Stokes equations have been used both in shock and in the reaction zone. For most detonation waves, the Mach number at which the wave propagates is 5 or above. Yet in the study of shock wave structure, it is well known t h a t the Navier-Stokes equations are accurate only for relatively weak shocks with propagation Mach n u m b e r no greater than 2. For strong shocks, near equilibrium theory simply does not describe the molecular transport of m o m e n t u m and energy adequately. Hence, it should he born in mind t h a t Navier-Stokes solutions to the coupling problem can give only qualitative results, at best, for the strong waves considered. PROF. A. K. OPPENHEIM ~University of California): I am in full agreement with Professor Adamson t h a t indeed a strong coupling between the shock and deflagration is doubtful and the use of N a v i e r Stokes equations m a y not he entirely satisfactory. However I t h i n k t h a t an a t t e m p t to evaluate exactly the extent of coupling t h a t can be provided by transport properties is worthwhile. After all there is a difference between religion, the outcome of beliefs, and science, the code of knowledge. I am grateful to Professor Adamson for pointing out the marked difference of the steric factor on coupling. We are aware of this and we are continuing our work to investigate this very matter systematically.

As to the Navier-Stokes equations, until we get a better formulation---perhaps from Professor Adamson h i m s e l f - w e are, albeit aware of their fundamental limitations, stuck with them. Actually they have been demonstrated to produce surprisingly accurate results for shocks propagating at even higher Mach numbers t h a n 2. Da. W. W. WooD: Steady detonation solutions of the Navier-Stokes equations have been investigated in the case of arbitrary Prandtl n u m b e r P r and zero Lewis n u m b e r Le by Friedrichs,6 and for Pr = ~, Le = 1 by Hirschfelder, Curtiss, and their co-workers, 7-1~ Adamson, 11 Spalding, TM and Wood. 13 The conclusion of these investigations was t h a t t h e classical continuum of overdriven ("strong") detonation solutions, including the C h a p m a n Jouguet (C-J) solution, exists if the reaction rate is not too large. For any particular "weak" detonation a solution was found, but only for a corresponding eigenvalue of the reaction rate (the value being quite large, corresponding to a rather fast reaction). For still larger reaction rates no steady solutions were found; the author TM has recently suggested t h a t in these cases pseudo-steady, two-wave solutions exist. The object of the present note is to indicate that, contrary to a recent suggestion, no departure from the above-described behavior is likely to appear in the inviscid limit P r ~ 0. As our starting point we will take Eqs. (11)-(14) of Hirschfclder and Curtiss, 7 written in a slightly different notation 13 with the Prandtl and Lewis numbers inserted: (4/3) P r ~K~uRl(du/dt) = 0 - 8R(U) R~(dO/dt) = aG ~

0--1

-~

-r--1

")' --

~[(-~

(u -

T --

1)r2](u

R I L e (dx/dt) = x -- G

1) 1) ~

(2)

(3) (4)

dG/dt = -xr(O) O~(u) = ~ u ( 2 u ~

--

(1)

- u)

(5)

um ~ (~K2 -~ 1)/23~K2

(6)

r(O) = exp[-(0 -- 1)/T~O].

(7)

In order to simplify the discussion we will set Le = 0, remarking t h a t on physical grounds one expects t h a t in the limit Le--~ 0 the composition gradient d x / d t will remain finite, in which case E q (3)shows t h a t x -~ G. I n the subsequent discussion, then, we set o = ~.

(s)

In the limit P r 2~ 0, however, we may expect a more complicated behavior to appear. Namely, shocks are likely to form; t h a t is, a t certain points

438

DETONATIONS

in the flow one expects du/dt ---, = a s P r ---) 0, the dimensionless mass velocity u, as well as the density and pressure, developing jump discontinuities at these points. T h e dimensionless temperature 0, being governed by Eq. (2) for all Pr, is of course not expected to develop such discontinuities. For a discussion of the closely related problems of nonreactive shock structure in the inviscid limit, see Grad. 1~ A t points where du/dt remains finite, Eq. (I) shows t h a t 0 --* #n(u) as P r --* 0. R a t h e r than giving a detailed discussion of the limiting process Pr--*0, which entails a discussion of a threedimensional vector field, we set 0 = 0R(u),

- u,~),

dx)dt = ~ x r [ O e ( u ) ],

(10) (11)

where h(x,u)

= ] ( 7 + 1)~2(u - 1)(u - - f ) 2(1 -- K') f-~ 1 + (7+1)

z /o

(9)

and admit solutions in which # and x are continuous while u m a y h a v e a jump discontinuity. I t is easy to show t h a t Eq. (9) is equivalent to the familiar Rayleigh line relation of the standard p-v diagram. Equation (9) can of course be solved for u ( 8 ) , and the four-dimensional (u, 8, x, G) system ( 1 ) - ( 4 ) reduced with the aid of Eq. (8) to a two-dimensional (0, x) system. However, u(8) is double-valued, and it is somewhat more convenient to substitute 0e (u) for 0 in Eq. (2), obtaining the two-dimensional (u, x) system 27x2R~(du/dt) = h(x, u ) / ( u

o~

- ax,

(12) (13)

In Fig. 3 we have sketched the phase portrait of Eqs. (10)-(11) for the C - J ease x = 1; for the sake of clarity the separation between u ~ and u = 1 has been exaggerated. The principal features are:

(1) the quasl-saddle point C a t the intersection of u = u,, and h(x, u) = 0; (2) termination of integral curves on u = u= at points above C, origination at the points below; (3) the integral curves in the triangular sector CR1 belong to a nodal fan tending to the C - J point 1; (4) the presence of the separatrix integral curve ~q in sector QNC, and its continuation through C into point 1. Let us consider the case of slow reaction rate (small R~), and choose an ignition temperature TI= slightly above To. Then, since diffusion is absent, the state point begins to react a t a point I near 0 on x = I, as indicated in Fig. 3. I t w~I1 evidently remain near x = 1 and terminate on u = u m near


*u um

I

FIo. 3. point Q, unless we introduce a j u m p discontinuity. The figure shows t h a t a solution is obtained if a jump is introduced from the point A', at which the integral curve from point I intersects the reflection S' of separatrix S in the line u = u=, to its source point A on S ; note t h a t according to Eqs. (5) and (9), O(A) = O(A'). The state point then moves along ,.g until it reaches point C from which two integral curves depart; we choose the one in sector CRI, so t h a t we reach the C - J point 1. We believe t h a t a detailed examination of the singular perturbation problem P r -* 0 indeed shows t h a t Friedrichs' solution continues to exist in the limit and approaches the above solution I A ' A C I . The latter exists in the limit of vanishing rate, at which it approaches the cla~ical yon N e u m a n n solution. For an intermediate range of large reaction rates, the integral curve from point I intersects h (x, u) = 0 below point C, and to the left of point 1, entering the sector CR1 and eventually reaching point 1. In this case no jump is required and point C is not encountered. For sufficiently large rates the C - J solution does not exist. In the interval 7 -1 < ~ffi< 1, a strong detonation exists (unless the reaction rate is too large), having essentially the behavior just described for the C - J ease.

For xI < -r-~ point C lies in the physically meaningless region of negative x, so t h a t the complication connected with the choice of its exit curve disappears. However, a jump A'A is now always necessary, until the reaction rate becomes so large t h a t the solution no longer exists. Finally, for any value (7 - 1 ) / 2 7 < P < 1 there is an eigenvalue R ~ / ~ ) of the reduced reaction rate

DETONATION WAVE STRUCTURE R~ for which the integral curve from point I reaches the weak detonation h o t boundary point u = f , x~0. Thus, we see t h a t detonation wave structure with Pr--* 0 and Le--* 0 is not qualitatively different from t h a t found previously e-14 for P r ~ 0, so far as the existence of the C - J detonation and the continuum of overdriven detonations is concerned. This work was performed under the auspices of the United States Atomic Energy Commission. DR. W. I. AXFORD (Defense Research Board, Ottawa) : I n principle we could solve all problems involving combustion waves and the like b y using the appropriate exact equations, given a sufficiently large computer and enough time. I n practice, however, we must proceed in two stages using suitable approximate equations as follows: (I) Solve the " e x t e r n a l " flow problem fully, treating the reaction zone (or its equivalent) as a discontinuity, b u t satisfying all external b o u n d a r y conditions a n d allowing for the possible occurrence of shock waves (also treated as discontinuities). (2) Having determined conditions on b o t h sides of the reaction zone, solve for its structure. This entails transforming to a coordinate system moving with the discontinuity so t h a t conditions can be assumed to be quasi-steady. The solution as far as step (1) is certain to be indeterminate, since all possibilities of weak and strong detonations and deflagrations mns~ be allowed for. A unique solution can only be found b y completing step ( 2 ) - - t h a t is b y sorting out which of the m a n y possible solutions of the external flow problems involve reaction zones which have a real structure. In combustion theory the situation is often confusing because the distinction between (1) and (2) is not always made clear, and also because t h e reac~ tions t h a t take place tend t o be rather complicated. However a case has recently been discovered for which the complete solutions (1) and (2) can be carried out, hence providing an example of the propagation of a type of gas dynamic discontinuity which is a t least instructive if not completely analogous to ordinary combustion. The discontinuity in this case is an "ionization front" (I.F.) which is produced when ionizing radiation is incident upon a mass of unionized gas. Ionization fronts occur when a newborn h o t star irradiates the surrounding interstellar hydrogen-photons emitted a t frequencies above the L y m a n limit ionize a n d heat the neutral hydrogen and a n expanding spherical nebula of hot ionized gas is thus formed around the star. The ionization front (corresponding to the reaction zone in ordinary combustion) separates regions of unionized a n d fully ionized gas (corresponding to unburned and b u r n e d gas, respectively), and the

439

front itself must propagate according to Jouguet's rule. Goldsworthy 1~ has worked out p a r t (1) of t h e solution for a n u m b e r of different cases of I.F. propagation in which the flow pattern is "similar" or "self-preserving." P a r t (2) of the solution has been worked out b y Axford ~7 for two cases in which the radiation is assumed to come from a line source and the flow p a t t e r n has cylindrical symmetry. I n the first case no recombination or cooling effects are considered so t h a t b o t h the ionized and unionized regions behave a a perfect gas with ~ = 5/3 a n d each proton ionizes a fresh atom a t the I.F. I n the second case (which is more realistic) recombination and cooling effects are included and it is found t h a t the ionized region must be treated as isothermal and only a small fraction of the ionizing photons ever reach the I.F. due to absorption b y recombined atoms in the ionized region. I t is intuitively obvious t h a t if a strong source radiates into low density surrounding, then t h e resulting I.F. will move out very rapidly, corresponding to a detonation. On the other hand a weak source radiating into high density surroundings will result in an I.F. which eats its way only slowly into the unionized gas, corresponding to a deflagration. If R is used to denote a suitable nondimensional form of the ratio (source strength to density of the surrounding unionized medium) then Goldsworthy finds in fact t h a t for large values of R the I.F. can propagate as a weak detonation, while for small R the I.F. can propagate as a weak deflagration; for one particular intermediate value of R a strong detonation is possible. T h e flow p a t t e r n in the case of the weak detonation contains a shock in the ionized region, and in the case of the weak deflagration a shock moves ahead in the unionized region. The shock and the I.F. m a y be considered as moving together in the special case of the strong detonation. I t is also possible however to have, for every value of R, an infinity of flow patterns in which the I.F. is a strong deflagration and shocks occur in b o t h the ionized and unionized regions. The presence of these additional solutions produces an indeterminacy which is not resolved until part (2) is completed. I n the case in which recombination and cooling are neglected one does not have to embark on detailed calculations to see t h a t the correct set of unique solutions for all values of R is obtained b y ignoring the strong deflagrations. Since the reaction is entirely exothermic a strong deflagration would require a rarefaction shock to bring the solution from the subsonic to the supersonic branch of the Rayleigh line (see Fig. 4), and this is impossible. (The width of a n I.F. is much greater t h a n a m e a n free p a t h and therefore i t is permissible to consider I n the second case with recombination and cooling t a k e n into account the results are not as obvious. For large values of R the I.F. is a weak detonation

440

DETONATIONS

M=I

M=I

T$

TS M
T T (a) (b) FIG. 4. Diagrams showing a form of the Rayleigh line, where Ts is the nondimensional stagnation temperature, T is the nondimensional temperature, and M the Mach number. The reaction processes are assumed to be such that TB has a maximum at some point within the reaction zone. If this maximum value is equal to the maximum on the Rayleigh line then it is possible to change smoothly from M < 1 to M > 1. The solutions must follow the Rayleigh line except at shock waves which are transitions at constant Ta in the direction of increasing T. A weak detonation containing a shock is shown in (a), where the initial and final conditions are indicated by 1 and 5, respectively, the shock wave is 2 --* 3 and 4 is a sonic point. A strong deflagration is shown in (b) where 1 and 3 are the initial and final conditions and 2 is a sonic point. and for small values of R the I.F. is a weak deflagration as above. However in the intermediate range there are differences due entirely to the presence of the cooling process which can cause the reaction to be endothermic after being initially exothermic-t h a t is, the stagnation temperature has a maximum somewhere in the interior of the reaction zone. In this case a smooth transition from subsonic to supersonic flow can take place provided the maximum stagnation temperature is equal to the value at the maximum of the Rayleigh line. I t is easy to see that strong deflagrations can now occur and t h a t the special condition on the stagnation temperature constitutes an over-determinacy within the reaction zone which compensates for the externally underdetermined flow pattern associated with strong deflagrations. Furthermore a weak detonation can now contain a shock since the flow can return smoothly to supersonic conditions-- the position of the shock in the reaction zone is then determined by the requirement on the maximum stagnation temperature. The over-all picture then, is t h a t as R decreases a shock appears in the weak detonation (internally) and gradually moves forward--eventually it becomes detached and moves ahead, causing the I.F. to become a strong deflagration; with further decrease of R the shock in the ionized region catches

up and becomes absorbed in the I.F. which then becomes a weak deflagration--strong detonations do not occur in general. Clearly the presence of a cooling process has a great influence on the character of I.F.'s and this is presumably true of gas dynamic discontinuities in general--this has also been remarked upon by Zeldovich and Kompaneets.~ Furthermore the strong deflagration, if it can occur at all, is not indeterminate as the solution carried as far as step (1) would suggest--hence Hayes 19 proposal t h a t the strong deflagration is "internally unstable" is incorrect. White ~~ has suggested that the turbulence commonly observed to be associated with detonating combustion waves is a feature of fundamental significance, and he has shown that weak detonations can be produced if turbulence is present. Since a detonating combustion wave must begin with a shock in general, the turbulence must somehow allow a smooth transition to occur from subsonic to supersonic flow in much the same manner as a cooling mechanism. The process is a little more complicated than described above, but its nature can be understood from the following argument. In a turbulent medium a larger fraction of the total energy is associated with translational motions of the mole-

DETONATION WAVE STRUCTURE cules than is normal. If the turbulence decays, then in suitable circumstances its energy is adiabatically transferred to molecular motions (and in particular to nontranslational degrees of freedom) so that the effective sound speed is reduced. By this means it is possible for a weak detonation to occur provided the turbulence induced in the reaction zone is sufficiently intense for the effective Mach number of the flow to become unity at some point. The detonation wave (considered as the ensemble of the initial shock, the reaction zone, and the zone of decaying turbulence) then has an "internal" ChapmanJouguet point beyond which the flow can become supersonic as the turbulence decays. Presumably instabilities lead to exactly the right degree of turbulence to make the internal C--J point possible. The position of the shock is not disposable in this type of weak detonation wave, but this is compensated for by the fact that the Rayleigh line itself is variable, depending on the degree of turbulence at each point.

REFERENCES

1. I~I1RSCHFELDER,J. 0. and CURTISS, C. F.: J. Chem. Phys. 28, 1130 (1958). 2. CURTISS, C. F., HIRSCHFELDER, J. 0., and BAI~NETT, M. P.: J. Chem. Phys. 30, 470 (1959). 3. WOOD, W. W.: Phys. Fluids ~,, 46 (1961). 4. MINS~ALL, F. S.: Metallurgical Society Con-

441

ferences, Vol. 9, p. 249. Interscience Publishers, New York, 1961. 5. WOOD, W. W. and SALSBURG, Z. W.: Phys. Fluids 3, 549 (1960). 6. FRIEDRICHS, K. O.: On the Mathematical Theory of Deflagrations and Detonations, NAVORD Report 79-46, 1946. 7. HIRSCUFELDER, J. O. and CURTISS, C. F.: J. Chem. Phys. 28, 1130 (1958). 8. LINDER, S. CURTISS, C. F., and HIRSCHFELDER, J. 0.: J. Chem. Phys. 28, 1147 (1958). 9. CURTtSS, C. F., HtRSCHFELDER, J. O., and BARNETT, M. P.: J. Chem. Phys. 80, 470 (1959). 10. HIRSCItFELDER, J: 0., CURTISS, C. F., and BARNETT, M. P.: Phys. Fluids 4:, 262 (1961). 11. ADAMSON~JR., T. C.: Phys. Fluids 3, 706 (1960). 12. SPALDING,D. B.: These Proceedings. 13. WOOD, W. W.: Phys. Fluids 4i, 46 (1961). 14. WOOD, W. W:: See Discussion of ref. 12. 15. GRAD, H.: Communs. Pure and Appl. Math. 5, 257 (1952). 16. GOLDSWORTHY, F. A.: Phil. Trans. Roy. Soc. (London) A253, 277 (1961). 17. AXFORD,W. I.: Phil. Trans. Roy. Soc. (London) A$53, 301 (1961). 18. ZELDOVICH,YA. B. and KOMPANEETS, A. S.: Theory of Detonation. Academic Press, 1960. 19. HAYES, W. D.: Fundamental of Gas Dynamics. Princeton University Press, 1958. 20. WHITE,D. E.: Phys. Fluids ~, 465 (1961).