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On the generation of pressure waves at a plane flame front
GENERATION OF PRESSURE WAVES AT PLANE FRAME FRONT
603
77
ON T H E G E N E R A T I O N OF P R E S S U R E W A V E S AT A P L A N E F L A M E F R O N T B y BOA-TEH CHU SYMBOLS
c Cp G m M M1
sound velocity specific heat at constant pressure universal gas constant mean molecular weight of the medium Mach number ( = u / c ) Mach number of the flame front ( = u l / c l = &/c3
p Q R S~ ~S~
pressure heat release per unit mass of the medium gas constant flame speed apparent flame speed (relative to our coordinate system) S~, S~ entropy ahead of and behind the flame T temperature T,t stagnation temperature u velocity of flow, (coordinate system being chosen such that ua -- St) 3' ratio of Cp to the specific heat at constant volume an operator signifying "an increment o f " X ratio of the stagnation temperature p density
A flame front considered as a discontinuity is characterized by two quantities; the flame speed St and the amount of heat release per unit mass of the mixture Q. Once a combustible mixture is specified it will be assumed that both St and Q are given (generally, as some functions of temperature and pressure). I n addition, we shall assume that the combustible mixture and the product of combustion are perfect gases with mean specific heats at constant pressure Cv~, Cp 2 and mean specific heats ratios 3'1, 3''-' respectively. Under normal conditions, the flame speed of a mixture is usually a few fps; the Mach number of the flame front St/c~ is therefore of the order of 11)-3.
Flame
1L u~.= k S t
uI = St
o
I* It
FIG. 1. Position of the flame before change Subscripts
1, 2 indicate conditions of flow ahead of and behind the flame resp. Incd., refl., tran. designate the incident, reflected, and transmitted wave. INTRODUCTION
I t is well known that pressure waves are generated during the build-up and decay of a flame. Pressure waves are also generated when a flame front propagates into a medium of different physical and chemical properties. These pressure waves are known to be the cause of m a n y phenomena associated with the propagation of flames such as the pressure build-up in a detonation wave, oscillatory flow in tube induced by the flame, etc. [e.g. Ellis and Kirkby (1) and Lewis and yon Elbe (2)]. We shall collect for convenience of reference in the following some of the well-known equations and formulas [e.g. see Tsien (3) and Emmons (4)] to be used in subsequent consideration.
The flow conditions at two sides of a flame front are related by the continuity, momentum and energy equations. If we choose our coordinate system (fig. 1) such that the flow ahead of the flame is seen to be approaching it with the velocity ul ( = the flame speed St) these equations assume the form: plUl
=
022t~2
Pl + plul 2 = P2 + P2U22 Q = Cv~T~t~ - C p i T a 1
(l)
(2) (3)
where T . is the stagnation temperaturc defined bv C,,~T.~t2 = C.~T~ + 1/~u;2
(4a)
CplT,~ q = C,,tTI + 1/,~u12
(4b)
and the temperature T~, T2 are related to the pressure and density by the gas law: p~ = p~R2T2
(5a)
Pl = p l R I T 1
(51))
604
CELLULAR FLAMES AND OSCILLATORY COMBUSTION
Of course, if m~, m.~ are respectively the mean molecular weights of the combustible mixture and the product of combustion, a n d G is the universal gas constant, then R i m , = R2m2 = G. From equations ( l ) to (5), one can solve for P~, p2, T2 and u~ in terms of p~, pl , TI a n d MI ( = u~/cl = S,/Cl), the M a c h n u m b e r of the flame front. Since M~ is usually very small, we can expand the solutions in power series of M~ a n d drop out all terms of order M~ ~ and higher. T h e result is simply:
P~
+ 0(M~)
(6a)
Pl p2
-
pl ~2
T1 u2 Ul
M~
31~
-
-
-
R1 1 R2 k
(66)
+ 0(M~)
(6(',)
x + 0(M~) R2 R1
~ -F 0(M~)
3"lR~
"r2R~
k,R,
6S, = -~u~ + ~St
X -b 0(M~)
This will be referred to as the "flame condition". I n addition to this equation, we h a v e the continuity, m o m e n t u m a n d energy equations at the flame. These equations can be simply written down b y replacing S t , Q, p l , P 2 , 0 1 , p2, T1 , T2
Florae'
(6e) o
-
1/+
(8)
(6d)
where k = T,t~/T~tl -~ [1 -F Q/C,IT~,,] Cpl/C~,~ and is therefore known. Similar equations t h a t are valid to the order of M) 3 can be derived. I n subsequent discussion, we shall h a v e occasion to use only one of these more exact expressions. This is the formula for pressure drop across the flame p,
Supposing t h a t due to some causes, S t , p~, P2, 9 9 9 are changed to S, § 6S~, pl + ~p~, P2 + 6 p 2 , . . . respectively. Before this change, the flame is seen to be s t a t i o n a r y in our reference system (mentioned in the Introduction). After the change the flame is disturbed a n d it will be seen to p r o p a g a t e with an a p p a r e n t flame speed of 6S, in the negative x-direction (fig. 2). Then, it follows from the definition of flame speed t h a t
0(M~)
(7)
BOUNDARY CONDITIONS AT THE FLAME FRONT
Let us suppose t h a t due to some cause or other there is a change of some variables a t the flame front. For example, a flame propagates into a medium of different chemical a n d / o r physical properties with a resultant change of either St or Q or 3' or a combination of them. Or, we can imagine t h a t a pressure wave or a weak contact surface hits the flame front causing a change in flow variables a t the flame front. Since the continuity, m o m e n t u m and energy equations m u s t remain valid a t all instants, pressure waves m u s t be generated a t the flame f r o n t ? I t is our purpose to discuss q u a n t i t a t i v e l y the nature of these waves. l In actuality, when a flame of finite thickness is considered, we have to take into account not only addi-
g
FIG. 2. Position of the flame after change with St + 6St, Q + 6Q, . . . etc. in equations (1) to (5) and replacing u l , u2 by Ul + ~ul + 8S, and u2 + ~u2 + 6Sa respectively. T h a t we have to insert a n additional q u a n t i t y ~.7~ in the increment of velocities is a consequence of the fact t h a t equations (1) to (4) are only valid with respect to a coordinate system fixed with the flame front. If we assume t h a t 6Q/Cp~T1, b S t / o , 6 u l / o , 6u2/c2,6pl/pz, 9 9 9 are small compared with u n i t y so t h a t their products can be neglected, the above system of equations can be simplified (with the help of equations (1) to (5)) to: tional equations such as equations of diffusion and the rate of chemical reaction, but also the temporal changes. The transient behavior of the interaction and the resultant change produced by the detailed interaction is "essentially" established in a finite time--the "relaxation time" whose value depends, among other things, on the thickness of the flame. When the "thickness" of the flame approaches zero, the relaxation time must go to zero so that the interaction between a flame considered as a discontinuity and external changes takes place instantaneously. Consequently, by considering a flame front as a single discontinuity in fact means that we have ignored the detailed process of interaction which now takes place instantaneously and we have chosen to study the flow field upstream and downstream of the flame before and after the interaction takes place, but not during the interaction.
GENERATION OF PRESSURE WAVES AT PLANE FRAME FRONT Continuity Equation
--
+ (,+-+ 6Pl Pl
6Sa~
\ N1
6p -- = p
6~,~a"~
6Ul It1
112 t
t12
M o m e n t u m Equation
6p2 _ 6pl Pe Pl
6u c
( 13b)
We would therefore like to reduce the above set of boundary conditions to be satisfied at the flame front by the pressure waves generated there to another system in which the only unknow~)s involved are 6pt , 6p.,. , 6ul and 6u~ . Carrying out the elimination, we obtain
'~,t1 / (9)
\ lie
-v--
605
(10)
6p~ _ 6pl P2 Pl
(14a)
6u~ - - 6ut
u,
- ~ X -- 1
-t- - ~
1-}-
~
U,l
3'2 -- 1
1 +
~'2
Cp~ X (6Cp~
u,/\Cm
~. -- 1
+ E/
6T? 6R1 R1
601 pl
_ 6p2 p2
6 ~'1 T1 6R~ R2
602 p2
67'2 = 0 T2
(12)
where 5u~/ul, 6Sa/ut could be small but need not
be. 2 Now, according to the characteristic theory, the pressure disturbance generated at the flame front is propagated outward with the velocity of sound. Furthermore, for pressure waves propagating in the positive x-direction, we have Sp
~u -
p
~
c
+
(ll)
Gas Law 6pl pl
C ,, ~ T ,
u, \
1
1) '~
E X/\~
( ~ 67t 1)el -
-- 1
-
--
p,
(14t))
where the change of entropy 6S, ahead of the flame is introduced to convert 6pl/p~ into 6pt/p~ in accordance with
6T~'~
_~
-- E
R~ X ! J L ~ x ( w -
u--T
Energy Equation
6Q
u-~
R'-22~)--(1-'t'- Ul/',,U:,,,X-- l)] &S' Cp21-
+~LU--T
C,,, T, - C,,
u,/
(13a)
and for pressure waves propagating in the negative x-direction, we have 2 The original calculation was made with the assumption that ~ut/u, , ~St/St << 1. The author is indebted to Dr. F. H. Clauser for pointing out that the main results should still be valid even when ~u~/ul, ~St/St a r e not small but ~m/c~, ~St/ct are much less than unity.
aO2 = 1 6 p ! _ Pl "Y1 pt
~82s
(15)
CYPl
Equations (14a) and (141)) form the two houmlarv conditions to be satisfied at a flame front. It is clear, if any one of the terms at the right hand side of equation (14b) is different from zero, pressure waves will be generated at the flame front. Consequently, pressure waves will 1)e generated at a flame front whenever there is: l) a change in flame speed St ; 2) a change in heat release Q; 3) a change in entropy of the oncoming medium St;
4) a change in "y; 5) a change in pressure (or velocity) ahead or behind the flame front as those caused by interaction of pressure waves and the flame front a phenomenon more customarily known as the transmission and reflection of pressure waves at the flame front? It is observed that if 6St/ul << 1, the abm-e set of boundary conditions becomes simply 3 Transmission and reflection of pressure waves at a flame front has been studied previously by Manson (81
606
CELLULAR FLAMES AND OSCILLATORY COMBUSTION
8u2 -- 6u~
6p2
3pi
P2
P~
R2 X _
1
(16a)
6S__,q_
ul
ul
R~ Cvz Tt
-R-Z\Glx-1 ~ +
572
IR2
9 ~X(7~_
I) ~
72-- I ( C w 7,
72
571 I (7~ -- i) ~
\CruX-
)@1
1
~
1. Effect of change of flame speed Let us imagine a flame propagating in a medium (in one-dimension) uniform up to a certain point at which there is an abrupt change of property of the medium in such a manner that all the physical constants except the flame speed remain unchanged and the flame speed increases from St to St + 6St where 6St need not be small compared with St but is small compared with the velocity of sound in the unburned medium. Then at the flame front, we must have: 8p2
(16b)
in which the various effects enter into the expression only linearly. In other words, the effect of a change of heat released can be investigated quite independently of a change of flame speed. We also observe that when 6St/ul is not very much less than unity, then the various effects can still be analysed independently with the understanding that the resultant effect is not merely a simple superposition of the two individual effects. In addition, we must add a small correction due to the presence of terms 8St/u~ in the coefficients of the various terms in equation (14b). A final remark concerns with the order of magnitude of the quantities neglected in equations (10) co (12) and hence in the Boundary Conditions (14) and (16). Two types of quantities are neglected: 1) terms involving M12 or higher powers of M1 ; 2) terms involving the product of 6p/p, 8p/p, 6T/T, 6u/c and 8St/cx (not 6u/u and 6St/ux). It will be noticed from equation (13) that indeed 6u/c and 6p/p are of same order of magnitude. That 6p/p and 6T/T are of the same order as 6p/p follows from the assumptions 6S1/RI and 6Q/CpIT1 << 1 (cf. equation 23). GENERATION OF PRESSURE WAVES AT THE FLAME FRONT
The application of the boundary conditions [equation (14)] is best illustrated by examples. To facilitate the discussion of the nature of pressure waves generated, we shall consider the various effects one at a time, although, as will be seen, many of the remarks, which are made in connection with one special case, can be traced back to the form of the boundary condition [equation (14)] and are therefore valid for all cases.
p~
-
6pl
(17a)
pz
6p\c,l(c"X --
9p
)
1
(17b)
These boundary conditions can be interpreted as follows. To maintain the conservation laws at the flame front, a change of flame speed St necessitates a change of pressure and velocity ahead of the flame (@1 , 6ul) as well as a change of pressure and velocity behind the flame (@2,6u2). These changes Forward-propagating wove
Rearward -propogating wove
Flame
lk C2- U2
Fm. 3. Waves and contact surface generated by the flame. are brought about by pressure waves propagating away from the flame front (Fig. 3). The forward propagating pressure wave causes the changes 8pl and ($Ul which, according to the characteristic theory, are related by @_2 + . y ~6 _u 1 =Q 0 pl cl
(18a)
The rearward propagating pressure wave causes the changes @2 and 6u2 which are related by
@2 ---p2
8u2 7~-- = 0 c~
(lSb)
GENERATION OF PRESSURE WAVES AT PI,ANE FRAME FRONT We thus obtain four equations with four urlknowns ~Pl, 5P2, ($ul and 5u2. Solving for the pressure rise, we obtain:
~p,
~p.,
pi
p.2
The fact that pressure waves of equal strength are generated at the flame front is a d{rect consequence of the form of the houndarv condition
~" \ R ~ "n
)
R~
% ")'2 - - --
I
9[(' + as,~(c,,, -E-,,,~ x The coefficient of the expression is exact up to the order of M1. As M, is small, we may write (to zeroth order approximation) simply:
6P' -- ~P' -- Yi ( ~ X - - 1 ) pl
A/'giR2
p2
&~t
(19b)
ct
,]/ y T ~ t ) ' n u l We thus see that whenever there is a change of flame speed, pressure waves of equal strength are generated at the flame front. Moreover, since hR,~R1 is in general greater than unity, compression waves are generated when the flame speed increases; and expansion waves of equal strength are generated when the flame decelerates. If we neglect the differences in the gas constants R~, R2 and in the 3"s, these waves are of the magnitude 3,(~,/X
+
cl St
--
pl
P2
--
"Y1
M~ (//,2 X -- 1"~ q- O(M~) \Rt
-
1 - 51 -
M,
(19a)
_ [equation (14a)] and is therefore true for all cases, whatever is the cause of the generation. More precisely, pressure waves of equal strength will be generated by a change 6Q, 6St, or &y. (Generation of pressure waves by interaction of flame front and pressure waves are not included here. See the last section of this paper.) It shouhl be remembered, however, that equation (14a) are obtained from the Momentum Equation by neglecting terms of order M~"- or with higher power. Therefore, the conclusion is, strictly speaking, not exactly" true, especially when MI is large) On the other hand, if we take the M~2 terms into account and neglecting all higher order terms, equation (14a) should be replaced by
ap2 -- 6pl = - p 1 % ] i ~
(R2 ) ~X
- 1
1) SA aS_j
Hence, when St and ~St are of the same order of magnitude, the pressure waves generated are of the order of S,/A. On the other hand, according to equation (7), the pressure drop at the flame front is given by Pl
607
/
and hence is of the order (S,/c,)2; it is thus clear that the pressure waves generated by a flame may be of (o/SJ (or roughly 103) times larger than the pressure drop across the flame front. This may account for the role played by a seemingly innocent accelerating flame (across which there is only a slight change of pressure) in the formation of the detonation wave during which the flame accelerates from a very low speed to a very high speed, contributing, at least in part, to the tremendous pressure build-up at the detonation wave? 4 This is first pointed out to me by Dr. Clauser in a discussion of this paper. See footnotes on page 604.
> (a~t
9 u~- \~-~-t -k- 2
) -k 0(~/1) ]4
(20)
which shows that actually the forward propagaling compression wave is always slightly, greater than the rearward propagating compression wave. This accounts for the fact that a small pressure drop of the order of M12 [equation (7)] is developed across the flame front when a steady state of affairs has been established, in fact if we consider a hypothetical case and put, ul - O, signifying that we start with a "flame" which does not propagate or propagating very sh)wly, and put 8S~ = St, i.e. observing a change of flame speed from 0 to ,S't , such as in the initial build-up of a flame front, then the above equalion reduces lo ~Similarly all other formulas obtained here are exact up to the order of M,. In case M, is not very small, but all the changes aSt/q, aQ/Cpfl'i, etc. are small, there is no difficulty, except for the more complicated algebra involved, to obtain the exact fornmla valid for all M1.
608
CELLULAR FLAMES AND OSCILLATORY COMBUSTIO~
~P2 -- 3pl = - p 1 3 " 1 \ c ~ !
~ ~-
1 + 0(M~)
Thus the resultant strength of the forward propagating wave is greater than that of the rearward propagating wave by an amount exactly equal to the pressure drop at the steady flame given by equation (7) so that, when the pressure waves have eventually disappeared from our sight, a steady state of affairs is established with the proper drop of pressure at the flame front. The drift velocity generated by the pressure waves can be calculated from equations (18a) and (18b). For convenience of discussion, we shall consider the case R1 = R2 and 3"1 = y~. The corresponding formulas for the more general case can be derived in like manner without difficulties The drift velocity generated by the forwardpropagating pressure wave is given by [see equation (18a) and fig. 3] 6ux = - ( V ~
- 1)~St
The general expression for the entropy change across the contact surface produced downstream of the flame front can be readily derived from the definition of entropy and equations (8) to (12). For the case, R1 = R2 and 3q = 3'2, this is given by 6S2 R
1 8S~ XR
3" 3"--1
8S2_
-R
~-
g
(21b)
(22)
The fact that 6u~ = - Vr~6ul is again a direct consequence of the boundary condition equation (14a) and the characteristic relations, equations (18a) and (18b), and therefore is always true whatever is the cause of generation of pressure wave (excluding again the case of transmission and reflection of pressure waves at the flame front). Together with the generation of pressure waves at the flame front, there will be a corresponding change in entropy downstream of the flame as a result of change of flame speed. If there is neither dissipation nor heat transfer, the entropy of a flow particle will remain constant for all instants (except, of course, when the particle crosses a discontinuity). As a result, a contact surface separating a region of low entropy from a region of higher entropy will be produced at the flame front as soon as the pressure waves are generated there, and will be carried down with the flow.
(23)
1
3"(V~--
~St
1)-cl
(24)
Hence, as the flame speed increases, a contact surface is generated along with the pressure waves and is carried downstream with the flow (fig. 3) c
6Sa = v ~ 6 S t
Q
For the case in consideration 6St = 6Q = 0. Therefore
(21a)
which is greater than the drift velocity produced ahead of the flame by the factor %/h. As a result of the changes taking place, the flame itself is seen (in our reference system) to be propagating with an apparent flame speed of equation (8)
X
l ~Q
}, - - l ~ p , >, pl
that generated behind the flame is [equation (18b)] 6u~ = %/~(~v/~ - 1)8S,
~ -
p,
F
!J
\,
-----_+__
FIG. 4. Generation of pressure waves at flame front The medium between the contact surface and the flame front has a slightly lower entropy, and hence a slightly lower temperature and higher density, than that between the contact surface and the pressure wave propagating downstream of it (fig. 3). [The change of entropy in equation (24) can be shown to be due to the change of X resulting from an increase of Tt produced by the pressure wave propagating ahead of the flame.] The various changes resulting from an increase of flame speed are illustrated in figure 4. In the figure P t , P2 are the compression waves, F is the flame front and C is the contact surface. The particle line is shown dotted in the figure. For clarity, the graph has not been drawn to scale. The graph is constructed for a reference system fixed with respect to the undisturbed medium ahead of the compression wave P1.
GENERATION OF PRESSURE WAVES AT PLANE FRAME FRONT
2. Effect of change of heat addition We next investigate the case 8Q ~ 0. The flame is assumed to be propagating at a constant speed ,St until a certain point is reached at which there is an abrupt change of heat release per unit mass from O to O + 8Q. This may be caused by various factors such as change of composition of the mixture or heat loss near a wall etc. The boundary conditions to be satisfied at the flame are
~p2 _ ~P~ p2 (~U2 -- ~Ul
ul
-
(25a)
p~
R2 ~Q R1 Cp~ T1 - / = - - 1 (~p, FC_~p2 ~2 p, L C p , )' -
] 1
(25b)
Proceeding as before, we can solve for 5p~ and 5p: with the help of equations (18a) and (18b). We find ~
St in the same l)ercentage. Since X is a simple function of Q, equation (27) can ill principlc be integrated, (if we assume that S remains constant) and an equation giving the strength of compression wave for large increment of O can be obtained. Unfortunately, the result cannot be written as a simple quadrature when the variation of n in equation (27) is taken into account. Generally speaking, a change of Q is usually accompanied by a change of the flame speed, e.g. in the decay of a flame. For such cases, Iht. resultant effect, as mentioned already, is not merely a superposition of lhe two effects acting individually unless 6St << St. It can be easily verified that the pressure wave generated by Ihe combined effect of increase of the flame speed and Q is p 1
p~
3"(x - 1) ~s__,+ 3" S~ . . . . ~Q cl
5P~ _ 5P~
Pt
609
cl Cv ~1'1
p2 R~ aQ ~,~ ~ M~ C~, T-----~
= ~/X-t-I+
( y - - 1)(X-- 1)
(2S)
(26)
=
x +1 + ,,-,(c,,
9
)
~p~ p~
3' ( V ~
St ~Q -- 1) - c~ Q-
l-
c~--
where the differences in the R's and the "/'s haw'
which shows that an increase in Q will also generate compression waves of equal strength (or, more exactly, of practically equal strength, see p. 607) propagating away from the flame front. If we ignore the differences in the R's and the 5"s, and omit the small term with the factor M~ in the denominator, we can write the above equation as ~p~ p,
l+&]+==
(27)
where h = 1 + Q/CpT~q has been used in deriving the above equation, Thus, it turns out that the pressure wave generated by an increase in Q is the same as that generated by an increase of As we have neglected in all our analysis terms of order M~, it might be argued that the term M~(= u~/ct) in the denominator should be dropped out. This term is indeed very small. However, it is not incorrect to retain it here, for if we had included all the M~ terms, they will turn out to be of the order of M~ in this formula. Of course, the term M~ can be shifted to the numerator,
been ignored for the sake of simplicity. If we ignore the real mechanism of the initial build-up of a flame front and replace il with a simplified hypothetical model which may be cquit :~ lent to the real one as far as lhe g{t.lleralir r pressure waves is concerned, we w(~ut, l he [n('lim',l to think of the initial build-ul) ()f a flame fr~,nl as equivalent to a gradual increase ()f l]le slw~'d (d propagation from 0 to 5;, of a hyl),lb(.tical disc<,, tinuity which we shall call "tlame,'" t~)gc)hcr with a~ lent to a gradual increase of the speed ()f [)r(q)aga lion from () to St of a hypolhclical di~c(mlinuily which we shall call " f a m e , " logether with a gradual i)~crease of 1he heat release acr()ss ~}~(" discontinuity from 0 to O. During this buihl-up, compression waves of nearly equal strength are generated at the flame front. If the mixture is ignited from the open end of a tube, the rearward propagating compression waves are almost immediately reflected from the open end as expansion waves of equal strength. These expansion waves catch up with the forward propagating comprcs sion wave and practically annihilate it completely.
610
CELLULAR FLAMES AND OSCILLATORY COMBUSTION
In this case the second order difference in the st~'ength of the forward and rearward propagating waves [equation (20)] becomes important. As we have seen, the forward propagating wave is slightly stronger than the rearward propagating wave so that the resultant effect of reflection at the open end of a tube and interaction of the expansion and compression waves is that there will be a very weak compression wave propagating ahead of the flame. On the other hand, if the mixture has been ignited at the closed end of a tube, the rearward propagating compression waves will be reflected at the closed end as a second family of forward propagating compression waves of practically the same strength as the original forward propagating compression waves. The second family soon overtakes the first and reinforces it. These compression waves create a considerable drift velocity and temperature rise, so that the flame is seen to travel with a much greater speed than if the mixture has been ignited at the open end of a tube. A rough estimation of the order of magnitude of the apparent flame speed for a mixture ignited at the closed end of a tube shows that it may be ten times larger than the speed of propagation of a flame of the same mixture ignited at the opened end of a tube. (In this estimation X is taken as 9.) The compression waves generated during the initial build-up of a flame soon steepen up into a shock wave and are responsible for the eventual development of a detonation wave. The compression waves propagating ahead of the flame compress the unburned mixture ahead of the latter, shortening the induction period of the mixture and thereby causing an increase of flame speed. But as we have seen already, an increase in flame speed produces additional compression waves which reinforce the original compression waves and cause a further increase of flame speed. The effect is therefore cumulative, leading to a considerable build-up of pressure across the shock wave and a considerable increase of flame speed9 In the meantime, additional compression waves are generated as a result of the self-heating of the mixture which may or may not be of significant amount, depending on whether the nature of the chemical reaction involved is more of the type of thermal explosion or of chain branching. However, before long a tremendous pressure will build-up at the shock wave propagating ahead of the flame. The high temperature behind this very strong shock shortens considerably the in-
duction period of the unburned mixture behind it so that either the flame speed increases so much as to catch up with the shock wave, forming a detonation front, or the mixture auto-ignites, 7 sending out new flame fronts which coalesce with the shock to form the detonation wave.
3. Effect of non-uniformity in entropy in the unburned medium When a flame front propagates in a medium with non-uniform entropy distribution, pressure waves are generated at the flame front. Such a circumstance may occur when the flame front is preceded by a shock, whose strength varies with time. Such shock wave may either be produced artificially or is formed by the steepening up of the compression waves generated at the flame front. Proceeding as before, it is found that the pressure waves generated due to the propagation of the flame from a region of entropy $1 to one of entropy $1 + 6S1 is given by
(29) That is, expansion waves are generated when a flame front propagates into a region of higher entropy. As an example, when a flame traverses a contact surface and propagates into a region with slightly higher temperature, rarefaction waves are generated. The drift velocities behind the waves, the entropy change generated downstream of the flame, and the apparent flame speed can be found from equations (18), (23) and (8) respectively. Likewise, when there is a change-of mean "1'~in the unburned mixture, pressure waves are generated. As this case is essentially the same as the previous ones and is not of too much practical interest, we shall omit the details.
4. Transmission and reflection of pressure waves at a flame front Equations (6c) and (6d) show that across a flame front there are discontinuous changes in temperature and velocity: T2 -X T1
u2 ul
R2 X RI
That auto-ignition plays an important part in the formation of the detonation wave has ample experimental support, and has been stressed very strongly by Zddovich (5) and Kistiakowsky (6).
GENERATION OF PRESSURE WAVES AT PLANE FRAME FRONT Hence, if Rt = R2, the temperature and velocity are increased by the same factor. Intuitively, however, it is felt that the nature of transmission and reflection of pressure waves at the flame front should be essentially similar to that at a temperature discontinuity rather than a velocity discontinuity. In fact, it is precisely under such assumption that Rayleigh discussed this problem [Rayleigh (7)]. We shall demonstrate here that this is indeed the case if the flame speed is small [Also, see Manson (8)]. Let us consider a compression wave overtaking a flame front. To satisfy the boundary conditions [equation (14)] at the flame front, the wave is partly transmitted and partly reflected. Thus, downstream of the flame the pressure rise is made up of two parts: that due to the incident wave and that due to the reflected wave, i.e.
ap~ = (ap)i.r
011
2C2 (~p)t ..... --
72
cA+c2 "IT
9
T2
3'2 - \ C m X -- 1 ct q_ __
Co
"Yt " (ap)i~ed. C2
el
e~ + ~ 3'1 "/2 . I 1 _ T~ -- 1 (C,= 72 \Cv~
)
2 c2 <
Ahead of the flame, the pressure rise ~pt is due to the transmitted wa~e only. Hence,
6ul
.....
(~p)t
= (au)t .....
Substitute these into the boundary conditions [equation (14)] (i~p)incd.._~_ (SP)refl. _ ((~p)t .....
P~
P2
p~
((~U) incd....~ ({~U)refl; ({~U)t ..... ~/4 7Zl ZQ
")'~
\ C v ~ X -- 1
P,
Now the reflected wave propagates in the positive x-direction, while the transmitted wave propagates .in the negative x-direction, therefore, by equation (I3) (~P) ~m. -
-
p~
--
(6u) ~ l . c2
T 2 ~
(6p), ..... pl
]
+ (~p)~o.. 9(ap)i.~<
~p~ =
(3Oh)
(~u)~ ..... "
g
l
-
-
Cz
With four equations and four unknowns it is therefore a simple matter to solve for the transmitted and reflected waves
(30b)
Thus, the nature of transmission aim rettecti(m of pressure waves at a f a m e front is essentially the same as that occur at a contact surface separating two regions of temperature T1 and T2 ( = XT1). In fact, it becomes identical to Poisson's result [see, e.g., Rayleigh (7)] if we neglect the correction term in the bracket. Thus, a compression (or expansion) wave approaching the flame from the rear is transmitted as another compression (expansion) wave and is amplified. The amplification factor can not be greater than 2. The reflected wave also has the same sign as the incident wave and is always weaker in strength. The effect of discontinuity in velocity across the flame is to cause a slight decrease in the amplitude of both the transmitted and reflected waves as calculated by treating the flame front as a contact surface separating a region of lower temperature from one of higher temperature. A similar analysis for a compression (expansion) wave in a head-on collision with a flame front shows that it is transmitted as a weaker compression (exl)ansion) wave and reflected as a weak expansion (compression) wave. The effect ~)f discontinuity in velocity across lhe flame front is to cause a sligh~ decrease in lhe amplitude of the transmitted wave together with a slight increase in the amplitude of the reflected wave as calculated by treating the flame fronl as a contacl surface.
612
CELLULAR FLAMES AND OSCILLATORY COMBUSTION
The x - t diagrams for these cases are shown in figures 5 and 6 in which P stands for the incident wave and all other notations have the same meaning as in figure 4.
/s/ F
I
I
.---------'7
t
F. H. Clauser for his very suggestive comments (see footnotes, pp. 605 and 607). The author also has the pleasure to acknowledge his indebtedness to Mr. Bertrand des Clers who first introduced him to a few of the interesting combustion phenomena associated with fluid mechanics which leads to the present study. To Miss Vivian O'Brien and Miss M. Ann Emmart, the author is indebted for their assistance in the preparation of the manuscript. To the reviewers o f this paper, the author is indebted for their comments and suggestions, in response to which the footnotes on pp. 604 and 605 have been added. REFERENCES
FIG. 5. Compression wave overtaking a flame front
,\,,, \
., 1
,x
Fie. 6. Head-on collision of a flame and a compression wave. ACKNOWLEDGMENT
The author wishes to thank the faculty of the Department of Aeronautics, The Johns Hopkins University, for their participation in the discussion of the paper, and is particularly grateful to Dr.
1. ELLIS, O. C. DE C., Am) K~KBY, W. A.: Flame, especially Chap. II. Methuen's Monographs on Chemical Subjects (1936). 2. LEwis, B., AND VON ELBE, G. : Combustion, Flames and Explosions of Gases, Chaps. VII and XI. New York, Academic Press Inc. (1951). 3. TsmN, H. S.: J. Applied Mech., 18, 2, 188-194 (1951). 4. Em~oNs, H. W.: Flow Discontinuities Associated with Combustion, Vol. III, Section E, of M. Summerfield's High Speed Aerodynamics and Jet Propulsion. Princeton University Press (unpublished). 5. ZELDOVmH,K. B.: Translated as NACA T.M. 1261. 6. KIS~AKOWS~CY,G. B.: Tech. Rept., Gibbs Chemical Lab., Harvard University, December 7, 1950 (Office of Naval Research, Contract N5orl-76, T.O XIX, NR-053-094). 7. RAYLEmH, LORD (J. W. Stutt): The Theory of Sound, Vol. II, Section 270, pp. 78--86, New York, Dover Press. (1945). 8. MANSON, N.: Proc. Seventh Int. Cong. Applied Mech., 2, 187-199 (1948).
603
77
ON T H E G E N E R A T I O N OF P R E S S U R E W A V E S AT A P L A N E F L A M E F R O N T B y BOA-TEH CHU SYMBOLS
c Cp G m M M1
sound velocity specific heat at constant pressure universal gas constant mean molecular weight of the medium Mach number ( = u / c ) Mach number of the flame front ( = u l / c l = &/c3
p Q R S~ ~S~
pressure heat release per unit mass of the medium gas constant flame speed apparent flame speed (relative to our coordinate system) S~, S~ entropy ahead of and behind the flame T temperature T,t stagnation temperature u velocity of flow, (coordinate system being chosen such that ua -- St) 3' ratio of Cp to the specific heat at constant volume an operator signifying "an increment o f " X ratio of the stagnation temperature p density
A flame front considered as a discontinuity is characterized by two quantities; the flame speed St and the amount of heat release per unit mass of the mixture Q. Once a combustible mixture is specified it will be assumed that both St and Q are given (generally, as some functions of temperature and pressure). I n addition, we shall assume that the combustible mixture and the product of combustion are perfect gases with mean specific heats at constant pressure Cv~, Cp 2 and mean specific heats ratios 3'1, 3''-' respectively. Under normal conditions, the flame speed of a mixture is usually a few fps; the Mach number of the flame front St/c~ is therefore of the order of 11)-3.
Flame
1L u~.= k S t
uI = St
o
I* It
FIG. 1. Position of the flame before change Subscripts
1, 2 indicate conditions of flow ahead of and behind the flame resp. Incd., refl., tran. designate the incident, reflected, and transmitted wave. INTRODUCTION
I t is well known that pressure waves are generated during the build-up and decay of a flame. Pressure waves are also generated when a flame front propagates into a medium of different physical and chemical properties. These pressure waves are known to be the cause of m a n y phenomena associated with the propagation of flames such as the pressure build-up in a detonation wave, oscillatory flow in tube induced by the flame, etc. [e.g. Ellis and Kirkby (1) and Lewis and yon Elbe (2)]. We shall collect for convenience of reference in the following some of the well-known equations and formulas [e.g. see Tsien (3) and Emmons (4)] to be used in subsequent consideration.
The flow conditions at two sides of a flame front are related by the continuity, momentum and energy equations. If we choose our coordinate system (fig. 1) such that the flow ahead of the flame is seen to be approaching it with the velocity ul ( = the flame speed St) these equations assume the form: plUl
=
022t~2
Pl + plul 2 = P2 + P2U22 Q = Cv~T~t~ - C p i T a 1
(l)
(2) (3)
where T . is the stagnation temperaturc defined bv C,,~T.~t2 = C.~T~ + 1/~u;2
(4a)
CplT,~ q = C,,tTI + 1/,~u12
(4b)
and the temperature T~, T2 are related to the pressure and density by the gas law: p~ = p~R2T2
(5a)
Pl = p l R I T 1
(51))
604
CELLULAR FLAMES AND OSCILLATORY COMBUSTION
Of course, if m~, m.~ are respectively the mean molecular weights of the combustible mixture and the product of combustion, a n d G is the universal gas constant, then R i m , = R2m2 = G. From equations ( l ) to (5), one can solve for P~, p2, T2 and u~ in terms of p~, pl , TI a n d MI ( = u~/cl = S,/Cl), the M a c h n u m b e r of the flame front. Since M~ is usually very small, we can expand the solutions in power series of M~ a n d drop out all terms of order M~ ~ and higher. T h e result is simply:
P~
+ 0(M~)
(6a)
Pl p2
-
pl ~2
T1 u2 Ul
M~
31~
-
-
-
R1 1 R2 k
(66)
+ 0(M~)
(6(',)
x + 0(M~) R2 R1
~ -F 0(M~)
3"lR~
"r2R~
k,R,
6S, = -~u~ + ~St
X -b 0(M~)
This will be referred to as the "flame condition". I n addition to this equation, we h a v e the continuity, m o m e n t u m a n d energy equations at the flame. These equations can be simply written down b y replacing S t , Q, p l , P 2 , 0 1 , p2, T1 , T2
Florae'
(6e) o
-
1/+
(8)
(6d)
where k = T,t~/T~tl -~ [1 -F Q/C,IT~,,] Cpl/C~,~ and is therefore known. Similar equations t h a t are valid to the order of M) 3 can be derived. I n subsequent discussion, we shall h a v e occasion to use only one of these more exact expressions. This is the formula for pressure drop across the flame p,
Supposing t h a t due to some causes, S t , p~, P2, 9 9 9 are changed to S, § 6S~, pl + ~p~, P2 + 6 p 2 , . . . respectively. Before this change, the flame is seen to be s t a t i o n a r y in our reference system (mentioned in the Introduction). After the change the flame is disturbed a n d it will be seen to p r o p a g a t e with an a p p a r e n t flame speed of 6S, in the negative x-direction (fig. 2). Then, it follows from the definition of flame speed t h a t
0(M~)
(7)
BOUNDARY CONDITIONS AT THE FLAME FRONT
Let us suppose t h a t due to some cause or other there is a change of some variables a t the flame front. For example, a flame propagates into a medium of different chemical a n d / o r physical properties with a resultant change of either St or Q or 3' or a combination of them. Or, we can imagine t h a t a pressure wave or a weak contact surface hits the flame front causing a change in flow variables a t the flame front. Since the continuity, m o m e n t u m and energy equations m u s t remain valid a t all instants, pressure waves m u s t be generated a t the flame f r o n t ? I t is our purpose to discuss q u a n t i t a t i v e l y the nature of these waves. l In actuality, when a flame of finite thickness is considered, we have to take into account not only addi-
g
FIG. 2. Position of the flame after change with St + 6St, Q + 6Q, . . . etc. in equations (1) to (5) and replacing u l , u2 by Ul + ~ul + 8S, and u2 + ~u2 + 6Sa respectively. T h a t we have to insert a n additional q u a n t i t y ~.7~ in the increment of velocities is a consequence of the fact t h a t equations (1) to (4) are only valid with respect to a coordinate system fixed with the flame front. If we assume t h a t 6Q/Cp~T1, b S t / o , 6 u l / o , 6u2/c2,6pl/pz, 9 9 9 are small compared with u n i t y so t h a t their products can be neglected, the above system of equations can be simplified (with the help of equations (1) to (5)) to: tional equations such as equations of diffusion and the rate of chemical reaction, but also the temporal changes. The transient behavior of the interaction and the resultant change produced by the detailed interaction is "essentially" established in a finite time--the "relaxation time" whose value depends, among other things, on the thickness of the flame. When the "thickness" of the flame approaches zero, the relaxation time must go to zero so that the interaction between a flame considered as a discontinuity and external changes takes place instantaneously. Consequently, by considering a flame front as a single discontinuity in fact means that we have ignored the detailed process of interaction which now takes place instantaneously and we have chosen to study the flow field upstream and downstream of the flame before and after the interaction takes place, but not during the interaction.
GENERATION OF PRESSURE WAVES AT PLANE FRAME FRONT Continuity Equation
--
+ (,+-+ 6Pl Pl
6Sa~
\ N1
6p -- = p
6~,~a"~
6Ul It1
112 t
t12
M o m e n t u m Equation
6p2 _ 6pl Pe Pl
6u c
( 13b)
We would therefore like to reduce the above set of boundary conditions to be satisfied at the flame front by the pressure waves generated there to another system in which the only unknow~)s involved are 6pt , 6p.,. , 6ul and 6u~ . Carrying out the elimination, we obtain
'~,t1 / (9)
\ lie
-v--
605
(10)
6p~ _ 6pl P2 Pl
(14a)
6u~ - - 6ut
u,
- ~ X -- 1
-t- - ~
1-}-
~
U,l
3'2 -- 1
1 +
~'2
Cp~ X (6Cp~
u,/\Cm
~. -- 1
+ E/
6T? 6R1 R1
601 pl
_ 6p2 p2
6 ~'1 T1 6R~ R2
602 p2
67'2 = 0 T2
(12)
where 5u~/ul, 6Sa/ut could be small but need not
be. 2 Now, according to the characteristic theory, the pressure disturbance generated at the flame front is propagated outward with the velocity of sound. Furthermore, for pressure waves propagating in the positive x-direction, we have Sp
~u -
p
~
c
+
(ll)
Gas Law 6pl pl
C ,, ~ T ,
u, \
1
1) '~
E X/\~
( ~ 67t 1)el -
-- 1
-
--
p,
(14t))
where the change of entropy 6S, ahead of the flame is introduced to convert 6pl/p~ into 6pt/p~ in accordance with
6T~'~
_~
-- E
R~ X ! J L ~ x ( w -
u--T
Energy Equation
6Q
u-~
R'-22~)--(1-'t'- Ul/',,U:,,,X-- l)] &S' Cp21-
+~LU--T
C,,, T, - C,,
u,/
(13a)
and for pressure waves propagating in the negative x-direction, we have 2 The original calculation was made with the assumption that ~ut/u, , ~St/St << 1. The author is indebted to Dr. F. H. Clauser for pointing out that the main results should still be valid even when ~u~/ul, ~St/St a r e not small but ~m/c~, ~St/ct are much less than unity.
aO2 = 1 6 p ! _ Pl "Y1 pt
~82s
(15)
CYPl
Equations (14a) and (141)) form the two houmlarv conditions to be satisfied at a flame front. It is clear, if any one of the terms at the right hand side of equation (14b) is different from zero, pressure waves will be generated at the flame front. Consequently, pressure waves will 1)e generated at a flame front whenever there is: l) a change in flame speed St ; 2) a change in heat release Q; 3) a change in entropy of the oncoming medium St;
4) a change in "y; 5) a change in pressure (or velocity) ahead or behind the flame front as those caused by interaction of pressure waves and the flame front a phenomenon more customarily known as the transmission and reflection of pressure waves at the flame front? It is observed that if 6St/ul << 1, the abm-e set of boundary conditions becomes simply 3 Transmission and reflection of pressure waves at a flame front has been studied previously by Manson (81
606
CELLULAR FLAMES AND OSCILLATORY COMBUSTION
8u2 -- 6u~
6p2
3pi
P2
P~
R2 X _
1
(16a)
6S__,q_
ul
ul
R~ Cvz Tt
-R-Z\Glx-1 ~ +
572
IR2
9 ~X(7~_
I) ~
72-- I ( C w 7,
72
571 I (7~ -- i) ~
\CruX-
)@1
1
~
1. Effect of change of flame speed Let us imagine a flame propagating in a medium (in one-dimension) uniform up to a certain point at which there is an abrupt change of property of the medium in such a manner that all the physical constants except the flame speed remain unchanged and the flame speed increases from St to St + 6St where 6St need not be small compared with St but is small compared with the velocity of sound in the unburned medium. Then at the flame front, we must have: 8p2
(16b)
in which the various effects enter into the expression only linearly. In other words, the effect of a change of heat released can be investigated quite independently of a change of flame speed. We also observe that when 6St/ul is not very much less than unity, then the various effects can still be analysed independently with the understanding that the resultant effect is not merely a simple superposition of the two individual effects. In addition, we must add a small correction due to the presence of terms 8St/u~ in the coefficients of the various terms in equation (14b). A final remark concerns with the order of magnitude of the quantities neglected in equations (10) co (12) and hence in the Boundary Conditions (14) and (16). Two types of quantities are neglected: 1) terms involving M12 or higher powers of M1 ; 2) terms involving the product of 6p/p, 8p/p, 6T/T, 6u/c and 8St/cx (not 6u/u and 6St/ux). It will be noticed from equation (13) that indeed 6u/c and 6p/p are of same order of magnitude. That 6p/p and 6T/T are of the same order as 6p/p follows from the assumptions 6S1/RI and 6Q/CpIT1 << 1 (cf. equation 23). GENERATION OF PRESSURE WAVES AT THE FLAME FRONT
The application of the boundary conditions [equation (14)] is best illustrated by examples. To facilitate the discussion of the nature of pressure waves generated, we shall consider the various effects one at a time, although, as will be seen, many of the remarks, which are made in connection with one special case, can be traced back to the form of the boundary condition [equation (14)] and are therefore valid for all cases.
p~
-
6pl
(17a)
pz
6p\c,l(c"X --
9p
)
1
(17b)
These boundary conditions can be interpreted as follows. To maintain the conservation laws at the flame front, a change of flame speed St necessitates a change of pressure and velocity ahead of the flame (@1 , 6ul) as well as a change of pressure and velocity behind the flame (@2,6u2). These changes Forward-propagating wove
Rearward -propogating wove
Flame
lk C2- U2
Fm. 3. Waves and contact surface generated by the flame. are brought about by pressure waves propagating away from the flame front (Fig. 3). The forward propagating pressure wave causes the changes 8pl and ($Ul which, according to the characteristic theory, are related by @_2 + . y ~6 _u 1 =Q 0 pl cl
(18a)
The rearward propagating pressure wave causes the changes @2 and 6u2 which are related by
@2 ---p2
8u2 7~-- = 0 c~
(lSb)
GENERATION OF PRESSURE WAVES AT PI,ANE FRAME FRONT We thus obtain four equations with four urlknowns ~Pl, 5P2, ($ul and 5u2. Solving for the pressure rise, we obtain:
~p,
~p.,
pi
p.2
The fact that pressure waves of equal strength are generated at the flame front is a d{rect consequence of the form of the houndarv condition
~" \ R ~ "n
)
R~
% ")'2 - - --
I
9[(' + as,~(c,,, -E-,,,~ x The coefficient of the expression is exact up to the order of M1. As M, is small, we may write (to zeroth order approximation) simply:
6P' -- ~P' -- Yi ( ~ X - - 1 ) pl
A/'giR2
p2
&~t
(19b)
ct
,]/ y T ~ t ) ' n u l We thus see that whenever there is a change of flame speed, pressure waves of equal strength are generated at the flame front. Moreover, since hR,~R1 is in general greater than unity, compression waves are generated when the flame speed increases; and expansion waves of equal strength are generated when the flame decelerates. If we neglect the differences in the gas constants R~, R2 and in the 3"s, these waves are of the magnitude 3,(~,/X
+
cl St
--
pl
P2
--
"Y1
M~ (//,2 X -- 1"~ q- O(M~) \Rt
-
1 - 51 -
M,
(19a)
_ [equation (14a)] and is therefore true for all cases, whatever is the cause of the generation. More precisely, pressure waves of equal strength will be generated by a change 6Q, 6St, or &y. (Generation of pressure waves by interaction of flame front and pressure waves are not included here. See the last section of this paper.) It shouhl be remembered, however, that equation (14a) are obtained from the Momentum Equation by neglecting terms of order M~"- or with higher power. Therefore, the conclusion is, strictly speaking, not exactly" true, especially when MI is large) On the other hand, if we take the M~2 terms into account and neglecting all higher order terms, equation (14a) should be replaced by
ap2 -- 6pl = - p 1 % ] i ~
(R2 ) ~X
- 1
1) SA aS_j
Hence, when St and ~St are of the same order of magnitude, the pressure waves generated are of the order of S,/A. On the other hand, according to equation (7), the pressure drop at the flame front is given by Pl
607
/
and hence is of the order (S,/c,)2; it is thus clear that the pressure waves generated by a flame may be of (o/SJ (or roughly 103) times larger than the pressure drop across the flame front. This may account for the role played by a seemingly innocent accelerating flame (across which there is only a slight change of pressure) in the formation of the detonation wave during which the flame accelerates from a very low speed to a very high speed, contributing, at least in part, to the tremendous pressure build-up at the detonation wave? 4 This is first pointed out to me by Dr. Clauser in a discussion of this paper. See footnotes on page 604.
> (a~t
9 u~- \~-~-t -k- 2
) -k 0(~/1) ]4
(20)
which shows that actually the forward propagaling compression wave is always slightly, greater than the rearward propagating compression wave. This accounts for the fact that a small pressure drop of the order of M12 [equation (7)] is developed across the flame front when a steady state of affairs has been established, in fact if we consider a hypothetical case and put, ul - O, signifying that we start with a "flame" which does not propagate or propagating very sh)wly, and put 8S~ = St, i.e. observing a change of flame speed from 0 to ,S't , such as in the initial build-up of a flame front, then the above equalion reduces lo ~Similarly all other formulas obtained here are exact up to the order of M,. In case M, is not very small, but all the changes aSt/q, aQ/Cpfl'i, etc. are small, there is no difficulty, except for the more complicated algebra involved, to obtain the exact fornmla valid for all M1.
608
CELLULAR FLAMES AND OSCILLATORY COMBUSTIO~
~P2 -- 3pl = - p 1 3 " 1 \ c ~ !
~ ~-
1 + 0(M~)
Thus the resultant strength of the forward propagating wave is greater than that of the rearward propagating wave by an amount exactly equal to the pressure drop at the steady flame given by equation (7) so that, when the pressure waves have eventually disappeared from our sight, a steady state of affairs is established with the proper drop of pressure at the flame front. The drift velocity generated by the pressure waves can be calculated from equations (18a) and (18b). For convenience of discussion, we shall consider the case R1 = R2 and 3"1 = y~. The corresponding formulas for the more general case can be derived in like manner without difficulties The drift velocity generated by the forwardpropagating pressure wave is given by [see equation (18a) and fig. 3] 6ux = - ( V ~
- 1)~St
The general expression for the entropy change across the contact surface produced downstream of the flame front can be readily derived from the definition of entropy and equations (8) to (12). For the case, R1 = R2 and 3q = 3'2, this is given by 6S2 R
1 8S~ XR
3" 3"--1
8S2_
-R
~-
g
(21b)
(22)
The fact that 6u~ = - Vr~6ul is again a direct consequence of the boundary condition equation (14a) and the characteristic relations, equations (18a) and (18b), and therefore is always true whatever is the cause of generation of pressure wave (excluding again the case of transmission and reflection of pressure waves at the flame front). Together with the generation of pressure waves at the flame front, there will be a corresponding change in entropy downstream of the flame as a result of change of flame speed. If there is neither dissipation nor heat transfer, the entropy of a flow particle will remain constant for all instants (except, of course, when the particle crosses a discontinuity). As a result, a contact surface separating a region of low entropy from a region of higher entropy will be produced at the flame front as soon as the pressure waves are generated there, and will be carried down with the flow.
(23)
1
3"(V~--
~St
1)-cl
(24)
Hence, as the flame speed increases, a contact surface is generated along with the pressure waves and is carried downstream with the flow (fig. 3) c
6Sa = v ~ 6 S t
Q
For the case in consideration 6St = 6Q = 0. Therefore
(21a)
which is greater than the drift velocity produced ahead of the flame by the factor %/h. As a result of the changes taking place, the flame itself is seen (in our reference system) to be propagating with an apparent flame speed of equation (8)
X
l ~Q
}, - - l ~ p , >, pl
that generated behind the flame is [equation (18b)] 6u~ = %/~(~v/~ - 1)8S,
~ -
p,
F
!J
\,
-----_+__
FIG. 4. Generation of pressure waves at flame front The medium between the contact surface and the flame front has a slightly lower entropy, and hence a slightly lower temperature and higher density, than that between the contact surface and the pressure wave propagating downstream of it (fig. 3). [The change of entropy in equation (24) can be shown to be due to the change of X resulting from an increase of Tt produced by the pressure wave propagating ahead of the flame.] The various changes resulting from an increase of flame speed are illustrated in figure 4. In the figure P t , P2 are the compression waves, F is the flame front and C is the contact surface. The particle line is shown dotted in the figure. For clarity, the graph has not been drawn to scale. The graph is constructed for a reference system fixed with respect to the undisturbed medium ahead of the compression wave P1.
GENERATION OF PRESSURE WAVES AT PLANE FRAME FRONT
2. Effect of change of heat addition We next investigate the case 8Q ~ 0. The flame is assumed to be propagating at a constant speed ,St until a certain point is reached at which there is an abrupt change of heat release per unit mass from O to O + 8Q. This may be caused by various factors such as change of composition of the mixture or heat loss near a wall etc. The boundary conditions to be satisfied at the flame are
~p2 _ ~P~ p2 (~U2 -- ~Ul
ul
-
(25a)
p~
R2 ~Q R1 Cp~ T1 - / = - - 1 (~p, FC_~p2 ~2 p, L C p , )' -
] 1
(25b)
Proceeding as before, we can solve for 5p~ and 5p: with the help of equations (18a) and (18b). We find ~
St in the same l)ercentage. Since X is a simple function of Q, equation (27) can ill principlc be integrated, (if we assume that S remains constant) and an equation giving the strength of compression wave for large increment of O can be obtained. Unfortunately, the result cannot be written as a simple quadrature when the variation of n in equation (27) is taken into account. Generally speaking, a change of Q is usually accompanied by a change of the flame speed, e.g. in the decay of a flame. For such cases, Iht. resultant effect, as mentioned already, is not merely a superposition of lhe two effects acting individually unless 6St << St. It can be easily verified that the pressure wave generated by Ihe combined effect of increase of the flame speed and Q is p 1
p~
3"(x - 1) ~s__,+ 3" S~ . . . . ~Q cl
5P~ _ 5P~
Pt
609
cl Cv ~1'1
p2 R~ aQ ~,~ ~ M~ C~, T-----~
= ~/X-t-I+
( y - - 1)(X-- 1)
(2S)
(26)
=
x +1 + ,,-,(c,,
9
)
~p~ p~
3' ( V ~
St ~Q -- 1) - c~ Q-
l-
c~--
where the differences in the R's and the "/'s haw'
which shows that an increase in Q will also generate compression waves of equal strength (or, more exactly, of practically equal strength, see p. 607) propagating away from the flame front. If we ignore the differences in the R's and the 5"s, and omit the small term with the factor M~ in the denominator, we can write the above equation as ~p~ p,
l+&]+==
(27)
where h = 1 + Q/CpT~q has been used in deriving the above equation, Thus, it turns out that the pressure wave generated by an increase in Q is the same as that generated by an increase of As we have neglected in all our analysis terms of order M~, it might be argued that the term M~(= u~/ct) in the denominator should be dropped out. This term is indeed very small. However, it is not incorrect to retain it here, for if we had included all the M~ terms, they will turn out to be of the order of M~ in this formula. Of course, the term M~ can be shifted to the numerator,
been ignored for the sake of simplicity. If we ignore the real mechanism of the initial build-up of a flame front and replace il with a simplified hypothetical model which may be cquit :~ lent to the real one as far as lhe g{t.lleralir r pressure waves is concerned, we w(~ut, l he [n('lim',l to think of the initial build-ul) ()f a flame fr~,nl as equivalent to a gradual increase ()f l]le slw~'d (d propagation from 0 to 5;, of a hyl),lb(.tical disc<,, tinuity which we shall call "tlame,'" t~)gc)hcr with a~ lent to a gradual increase of the speed ()f [)r(q)aga lion from () to St of a hypolhclical di~c(mlinuily which we shall call " f a m e , " logether with a gradual i)~crease of 1he heat release acr()ss ~}~(" discontinuity from 0 to O. During this buihl-up, compression waves of nearly equal strength are generated at the flame front. If the mixture is ignited from the open end of a tube, the rearward propagating compression waves are almost immediately reflected from the open end as expansion waves of equal strength. These expansion waves catch up with the forward propagating comprcs sion wave and practically annihilate it completely.
610
CELLULAR FLAMES AND OSCILLATORY COMBUSTION
In this case the second order difference in the st~'ength of the forward and rearward propagating waves [equation (20)] becomes important. As we have seen, the forward propagating wave is slightly stronger than the rearward propagating wave so that the resultant effect of reflection at the open end of a tube and interaction of the expansion and compression waves is that there will be a very weak compression wave propagating ahead of the flame. On the other hand, if the mixture has been ignited at the closed end of a tube, the rearward propagating compression waves will be reflected at the closed end as a second family of forward propagating compression waves of practically the same strength as the original forward propagating compression waves. The second family soon overtakes the first and reinforces it. These compression waves create a considerable drift velocity and temperature rise, so that the flame is seen to travel with a much greater speed than if the mixture has been ignited at the open end of a tube. A rough estimation of the order of magnitude of the apparent flame speed for a mixture ignited at the closed end of a tube shows that it may be ten times larger than the speed of propagation of a flame of the same mixture ignited at the opened end of a tube. (In this estimation X is taken as 9.) The compression waves generated during the initial build-up of a flame soon steepen up into a shock wave and are responsible for the eventual development of a detonation wave. The compression waves propagating ahead of the flame compress the unburned mixture ahead of the latter, shortening the induction period of the mixture and thereby causing an increase of flame speed. But as we have seen already, an increase in flame speed produces additional compression waves which reinforce the original compression waves and cause a further increase of flame speed. The effect is therefore cumulative, leading to a considerable build-up of pressure across the shock wave and a considerable increase of flame speed9 In the meantime, additional compression waves are generated as a result of the self-heating of the mixture which may or may not be of significant amount, depending on whether the nature of the chemical reaction involved is more of the type of thermal explosion or of chain branching. However, before long a tremendous pressure will build-up at the shock wave propagating ahead of the flame. The high temperature behind this very strong shock shortens considerably the in-
duction period of the unburned mixture behind it so that either the flame speed increases so much as to catch up with the shock wave, forming a detonation front, or the mixture auto-ignites, 7 sending out new flame fronts which coalesce with the shock to form the detonation wave.
3. Effect of non-uniformity in entropy in the unburned medium When a flame front propagates in a medium with non-uniform entropy distribution, pressure waves are generated at the flame front. Such a circumstance may occur when the flame front is preceded by a shock, whose strength varies with time. Such shock wave may either be produced artificially or is formed by the steepening up of the compression waves generated at the flame front. Proceeding as before, it is found that the pressure waves generated due to the propagation of the flame from a region of entropy $1 to one of entropy $1 + 6S1 is given by
(29) That is, expansion waves are generated when a flame front propagates into a region of higher entropy. As an example, when a flame traverses a contact surface and propagates into a region with slightly higher temperature, rarefaction waves are generated. The drift velocities behind the waves, the entropy change generated downstream of the flame, and the apparent flame speed can be found from equations (18), (23) and (8) respectively. Likewise, when there is a change-of mean "1'~in the unburned mixture, pressure waves are generated. As this case is essentially the same as the previous ones and is not of too much practical interest, we shall omit the details.
4. Transmission and reflection of pressure waves at a flame front Equations (6c) and (6d) show that across a flame front there are discontinuous changes in temperature and velocity: T2 -X T1
u2 ul
R2 X RI
That auto-ignition plays an important part in the formation of the detonation wave has ample experimental support, and has been stressed very strongly by Zddovich (5) and Kistiakowsky (6).
GENERATION OF PRESSURE WAVES AT PLANE FRAME FRONT Hence, if Rt = R2, the temperature and velocity are increased by the same factor. Intuitively, however, it is felt that the nature of transmission and reflection of pressure waves at the flame front should be essentially similar to that at a temperature discontinuity rather than a velocity discontinuity. In fact, it is precisely under such assumption that Rayleigh discussed this problem [Rayleigh (7)]. We shall demonstrate here that this is indeed the case if the flame speed is small [Also, see Manson (8)]. Let us consider a compression wave overtaking a flame front. To satisfy the boundary conditions [equation (14)] at the flame front, the wave is partly transmitted and partly reflected. Thus, downstream of the flame the pressure rise is made up of two parts: that due to the incident wave and that due to the reflected wave, i.e.
ap~ = (ap)i.r
011
2C2 (~p)t ..... --
72
cA+c2 "IT
9
T2
3'2 - \ C m X -- 1 ct q_ __
Co
"Yt " (ap)i~ed. C2
el
e~ + ~ 3'1 "/2 . I 1 _ T~ -- 1 (C,= 72 \Cv~
)
2 c2 <
Ahead of the flame, the pressure rise ~pt is due to the transmitted wa~e only. Hence,
6ul
.....
(~p)t
= (au)t .....
Substitute these into the boundary conditions [equation (14)] (i~p)incd.._~_ (SP)refl. _ ((~p)t .....
P~
P2
p~
((~U) incd....~ ({~U)refl; ({~U)t ..... ~/4 7Zl ZQ
")'~
\ C v ~ X -- 1
P,
Now the reflected wave propagates in the positive x-direction, while the transmitted wave propagates .in the negative x-direction, therefore, by equation (I3) (~P) ~m. -
-
p~
--
(6u) ~ l . c2
T 2 ~
(6p), ..... pl
]
+ (~p)~o.. 9(ap)i.~<
~p~ =
(3Oh)
(~u)~ ..... "
g
l
-
-
Cz
With four equations and four unknowns it is therefore a simple matter to solve for the transmitted and reflected waves
(30b)
Thus, the nature of transmission aim rettecti(m of pressure waves at a f a m e front is essentially the same as that occur at a contact surface separating two regions of temperature T1 and T2 ( = XT1). In fact, it becomes identical to Poisson's result [see, e.g., Rayleigh (7)] if we neglect the correction term in the bracket. Thus, a compression (or expansion) wave approaching the flame from the rear is transmitted as another compression (expansion) wave and is amplified. The amplification factor can not be greater than 2. The reflected wave also has the same sign as the incident wave and is always weaker in strength. The effect of discontinuity in velocity across the flame is to cause a slight decrease in the amplitude of both the transmitted and reflected waves as calculated by treating the flame front as a contact surface separating a region of lower temperature from one of higher temperature. A similar analysis for a compression (expansion) wave in a head-on collision with a flame front shows that it is transmitted as a weaker compression (exl)ansion) wave and reflected as a weak expansion (compression) wave. The effect ~)f discontinuity in velocity across lhe flame front is to cause a sligh~ decrease in lhe amplitude of the transmitted wave together with a slight increase in the amplitude of the reflected wave as calculated by treating the flame fronl as a contacl surface.
612
CELLULAR FLAMES AND OSCILLATORY COMBUSTION
The x - t diagrams for these cases are shown in figures 5 and 6 in which P stands for the incident wave and all other notations have the same meaning as in figure 4.
/s/ F
I
I
.---------'7
t
F. H. Clauser for his very suggestive comments (see footnotes, pp. 605 and 607). The author also has the pleasure to acknowledge his indebtedness to Mr. Bertrand des Clers who first introduced him to a few of the interesting combustion phenomena associated with fluid mechanics which leads to the present study. To Miss Vivian O'Brien and Miss M. Ann Emmart, the author is indebted for their assistance in the preparation of the manuscript. To the reviewers o f this paper, the author is indebted for their comments and suggestions, in response to which the footnotes on pp. 604 and 605 have been added. REFERENCES
FIG. 5. Compression wave overtaking a flame front
,\,,, \
., 1
,x
Fie. 6. Head-on collision of a flame and a compression wave. ACKNOWLEDGMENT
The author wishes to thank the faculty of the Department of Aeronautics, The Johns Hopkins University, for their participation in the discussion of the paper, and is particularly grateful to Dr.
1. ELLIS, O. C. DE C., Am) K~KBY, W. A.: Flame, especially Chap. II. Methuen's Monographs on Chemical Subjects (1936). 2. LEwis, B., AND VON ELBE, G. : Combustion, Flames and Explosions of Gases, Chaps. VII and XI. New York, Academic Press Inc. (1951). 3. TsmN, H. S.: J. Applied Mech., 18, 2, 188-194 (1951). 4. Em~oNs, H. W.: Flow Discontinuities Associated with Combustion, Vol. III, Section E, of M. Summerfield's High Speed Aerodynamics and Jet Propulsion. Princeton University Press (unpublished). 5. ZELDOVmH,K. B.: Translated as NACA T.M. 1261. 6. KIS~AKOWS~CY,G. B.: Tech. Rept., Gibbs Chemical Lab., Harvard University, December 7, 1950 (Office of Naval Research, Contract N5orl-76, T.O XIX, NR-053-094). 7. RAYLEmH, LORD (J. W. Stutt): The Theory of Sound, Vol. II, Section 270, pp. 78--86, New York, Dover Press. (1945). 8. MANSON, N.: Proc. Seventh Int. Cong. Applied Mech., 2, 187-199 (1948).