FlameletVortex Interaction and The Gibson Scale J. C. VASSILICOS* Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK
and N. NIKIFORAKIS Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, UK Under certain circumstances a flamelet interacting with a vortex can undergo a topological transformation whereby a connected double spiral flame produces a detached pocket of unburned fuel. The dependencies of the detachment time on the parameters of the vortex flow and of the flamelet are obtained by numerical integration of the eikonal equation. These dependencies are then used in a simple phenomenological model of smallscale turbulence leading to the definition of a topologically critical length scalethe detachment scale 1o. The detachment scale is found to be nearly proportional to the Gibson scale. Vortices of size larger than l o can generate shortlived double spirals on the flamelet that give rise to detached pockets of unburned fuel. Vortices of size smaller than ID simply wrinkle the flamelet. These pockets and double spirals and these wrinkles can all have fractal properties with significant effects on the turbulent flame speed. © 1997 by The Combustion Institute
I.
INTRODUCTION
In premixed combustion, flames move relative to the fluid because of a reactiondiffusion mechanism [5]. When the chemical reaction is so fast that the flow has no time to influence the reactiondiffusion mechanism, the thermochemistry can be decoupled from the dynamics of the flow and the reaction zone between burned and unburned fuel is so thin that it can be successfully modeled by an interface moving locally normal to itself with a laminar flame speed u L relative to the fluid [5]. Furthermore, the flamelet does not alter the fluid flow, but is advected and deformed by the fluid flow. In this limit, the premixed flame is called a flamelet, and all the thermochemical information is stored in the sole parameter u L. The fluid flow can influence u L because of flame stretch. In this paper we only consider the limit where the effects of flame stretch on u L are negligible (hence u L is a constant independent of flame curvature and flow divergence). In vortical flows, the magnitude of these effects is proportional to the ratio of the Markstein length _~ to the length scale l of the vortex. Experimental values of the Markstein length range between two to six times the flame thickness [14], which we assume to be much smaller than l, in accordance with the flamelet limit. Therefore, .~,~ I and the effects of flame
stretch on u L can be neglected in a good first approximation. We compute the evolution of flamelet interfaces for given constant flame speeds u L and given twodimensional vortical flows u = (ur(r, t), u,~(r, t)) (in cylindrical coordinates ur, u6 for velocities, r, ~b for positions). These computations are carried out numerically by integrating the eikonal equation (also called Gequation in combustion theory). A detailed derivation of the eikonal equation in the context of flamelet propagation is given in the paper by Vassilicos and Hunt ([19]; see also references therein) with a discussion of the assumptions leading to this equation. The flamelet interface is represented by an arbitrary isopleth of a tracing field F(r, ok, t), which evolves according to the eikonal equation. 0   F + u . VF = ULWFI. (1.1) ot
Of particular interest are the geometry and topology of the flamelet. The burning speed Ur(t) is directly proportional to the surface area A ( t ) of the flamelet and is therefore a function of the flamelet's geometry ( u r ( t ) is called the turbulent flame speed when the velocity field u is turbulent). Specifically, A(t)
u r ( t ) = UL
Lo
2 '
(1.2)
COMBUSTION AND FLAME 109:293302 (1997) © 1997 by The Combustion Institute Published by Elsevier Science Inc.
00102180/97/$17.00 PII S00102180(96)00165 4
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where L02 = A(t
critical length scale is defined, called the Gibson scale l o, which is such that I c / u L = l~/u(lc), that is, u L = u ( l ~ ) ~ (~lG)1/3, and therefore
= 0) (see Vassilicos and Hunt [19] and references therein). Gouldin [7] and Peters [12] hypothesised that flamelet interfaces in equilibrium turbulent flows are fractal and that their surface area is constant in time. Hence, they parameterised the turbulent flamelet area with a single timeindependent dimensionless n u m b e r   t h e fractal dimension of the interface (which may depend on properties of the flow such as the ratio of the root mean square turbulent velocity to uL). However, the fractal dimension is not enough to calculate the flamelet area; it is also necessary to specify the range between the largest length scale L 0 and the smallest length scale /mi, where the fractal dimension D of the flamelet is well defined (2 < D _< 3). According to fractal theory, the surface area of a fractal interface is proportional to lminZ(lmin/Lo) a, and Eq. 1.2 becomes
[/min 12D uv ~ uct~o )
(1.3)
(see [7] and [12]). Some experimental support for the hypothesis that turbulent flamelets are fractal in a certain range of length scales does exist as well as some attempts to measure D (e.g., Yoshida et al. [21], Smallwood et al. [15], and references therein). Experimental measurements suggest the existence of a length scale below which the fractal properties of the flamelet change, perhaps to the point of not being fractal at all, in which case this length scale should be identified with lmin" However, experimental estimations of this length scale are difficult and do not provide strong support for Peters' [12] theoretical estimation of lmin (see [21] and references therein). Peters' [12] estimation of lmin is as follows: the eddy turnover time of a turbulent eddy of length scale 1 is l/u(l), where, by virtue of the Kolmogorov phenomenology, u(l) ~ (el) 1/3 and E is the average rate of kinetic energy dissipation per unit mass. The time that a flamelet takes to propagate across an eddy of length scale l is l / u L. Where l / u L > l/u(l) the flamelet should have a wrinkle of length scale l and where l / u L < l/u(l) the fiamelet should have no significant contortions. Thus, a
l~ ~ uL3/e.
(1.4)
Peters [12] argues that the length scales of all significant contortions on the interface are larger than or equal to Ic and that the interface is not fractal below l c. Hence, setting lmin = IG in Eq. 1.3 and using Eq. 1.4, Peters [12] concludes that UT ~ lgL 7 3D(ELo ) D  2
(1.5)
The only currently identified fluid mechanical structures in smallscale turbulence are vortex tubes. Vortex tubes are a predominant form of organised vorticity at the small scales of the turbulence [3]. In this paper we study the interaction of one flamelet with one vortex tube and investigate the topology and geometry imposed on the flamelet by the vortex. Can a flamelet acquire fractal properties by the action of one vortex alone and if so under what conditions? Can lmin be estimated by Peters' argument and how do the fractal properties of a flamelet in a single vortex compare qualitatively with the fractal properties of a flamelet in smallscale turbulence? 2. A CHANGE IN FLAMELET TOPOLOGY Recent works [4, 1, 22] have shown that a pocket of unburned fuel in surrounding burned fuel may be formed when a flamelet crosses a vortex core. Once formed, this pocket of unburned fuel burns independently from the remaining fresh mixture. In the present paper we start by confirming the existence of such a pocket on a single and realistic vortex structure. The vortex structure that we chose is derived from the spreading line vortex (Batchelor [2], p 204), which is an incompressible solution of the NavierStokes equation where u r = 0 and
F ( u + = 27rr 1  e x p   ~ 
r2) (2.1)
F L A M E L E T  V O R T E X INTERACTION F being the total circulation of the vortex, l = 2 f ~  , and v being the kinematic viscosity of the fluid mixture. In the limit of large vortex Reynolds numbers where F / v ~ 1, advection around the vortex is so fast that a fluid element may complete many turns around the vortex before I can change appreciably. In this paper we consider the case where F / v ~ 1 and can therefore take l accordingly to be a
295 timeindependent characteristic length scale of the vortex flow (see Ting and Klein [16]). We simulate the flamelet propagation across this vortex flow by solving the eikonal equation 1.1 numerically (a brief description of the numerical method and initial condition used to solve Eq. 1.1 is given in the Appendix). In Fig. 1 we present a selection of numerical results for the evolution of the flamelet inter
Fig. 1. The evolution of the tracing field F for three of the numerical experiments, showing (a) the formation of a double spiral with two welldefined turns leading to a pocket of u n b u r n e d fuel (u L = 0.025, x/l = 2), (b) the formation of a double spiral with only one turn also leading to a detached pocket of u n b u r n e d fuel (u L = 0.125, x/! = 2), and (c) the formation of a cusp and failure to f o r m a detached pocket in the case of a high laminar flame speed (u L = 0.3, x/! = 0.5). The light coloured side is b u r n e d and the dark coloured side is unburned. The flamelet moves from the burned to the u n b u r n e d side.
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(c)
(b) Fig. 1. (Continued).
face as it approaches and interacts with the vortex. It is clear from these pictures that a pocket of unburned fuel is indeed created around the vortex core when u L is small enough, but that such a pocket is not created when u L is too large. For small enough values of u L, a critical (or detachment) time t o exists when the topology of the flamelet interface changes from a singly connected double spiral interface to two disconnected flamelets: one mostly weakly curved and moving away from the vortex; the other closed and burning inward in the vicinity of the vortex core. We find
that when u L is too large, the flamelet does not undergo a change in topology because it crosses the vortex very rapidly and has no time to become significantly spiral. The detachment time t o is therefore defined only for small enough values of uL. The only parameters in this problem are F, l, uL, and x, the initial distance of the flamelet (initially plane) from the centre of the vortex. Hence,
to = t ° ( F ' l ' u L ' x )
t(xtuL)
= ~Lf
~'F
'
(2.2)
FLAMELETVORTEX
INTERACTION
297
given by t h e f u n c t i o n a l f o r m
w h e r e f is a f u n c t i o n o f t h e m a x i m u m n u m b e r of independent dimensionless variables (here two) a n d w e chose ( w i t h o u t loss o f g e n e r a l i t y ) t h e s e two d i m e n s i o n l e s s v a r i a b l e s to b e x / l a n d luL/F. I n t h e limit w h e r e x / l ~ 1, we expect t o = x / u L ; h e n c e f ( x / l , lUL/F) = x / l as x / l ~ oo. The d e p e n d e n c e o f t o o n t h e p a r a m e t e r s F, l, uL, a n d x is less trivial w h e n x / l = O(1). By studying the d e p e n d e n c e o f t o o n u L for c o n s t a n t l, x, a n d F a n d c h e c k i n g t h a t this d e p e n d e n c e d o e s n o t c h a n g e significantly as w e vary x a r o u n d l, w e can d e t e r m i n e the d e p e n d e n c e o f f on l u L / F that is valid for x / l = O(1), In Fig. 2, w e p l o t t o versus uL for v a r i o u s n o n d i m e n s i o n a l i s e d initial d i s t a n c e s x / l = O(1) (specifically for values o f x / l bet w e e n 1.5 a n d 3.5) a n d find t h a t t o ~ u L  1/2 independently of x/l. Hence,
[3/2 to
=
l 3/2 (x) t o = C~ u c l / 2 F t / 2 e x p C2~ ,
w h e r e C~ a n d C 2 a r e positive c o n s t a n t s b o t h o f o r d e r 1 (in p a r t i c u l a r , C 2 = 1 / 3 ) . T h e scaling t o ~ 13/2/uL1/2F ]/2 can b e exp l a i n e d if it is r e c o g n i s e d t h a t the d e t a c h m e n t m u s t h a p p e n at a r a d i a l d i s t a n c e r ~ 1. I n s i d e the v o r t e x c o r e w h e r e r ,~ l, the f l a m e l e t interface t u r n s rigidly by solid b o d y r o t a t i o n , w h e r e a s o u t s i d e the v o r t e x c o r e w h e r e r >_ I the f l a m e l e t i n t e r f a c e is s u b m i t t e d to a differential r o t a t i o n which is such that the d i s t a n c e A r b e t w e e n successive coils o f t h e spiral f l a m e l e t i n t e r f a c e is smallest at r ~ l. A s n o t e d by P e t e r s a n d W i l l i a m s [13], at a given t i m e t, Ar = 27r2r3/Ft o u t s i d e the v o r t e x core. T h e first successive coils o f t h e spiral f l a m e l e t i n t e r face that b u r n into e a c h o t h e r a r e t h e r e f o r e t h o s e at r ~ l, a n d this d e t a c h m e n t h a p p e n s at a t i m e t o given by 2~r213/Fto = Ar = 2uLt o. H e n c e , t h e c r e a t i o n o f a p o c k e t of u n b u r n e d fuel h a p p e n s at t D ~ 13/2/uLWZF t/2. ( P e t e r s
[X
u il rll gt7 ),
(2.3)
w h e r e g is a f u n c t i o n o f x / l only. In Fig. 3 we p l o t (UL1/ZF~/Z/IS/2)to against x / l for cases w h e r e x/1 = O(1). T h e b e s t fit t h a t w e find is
2.0~>
.~
x a~
1.0
+
[]
O
4"
x
~
E)
A
x
0
[]
0
+
x
0
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~
x
[]
,,
0.8
0.6
~ . . . . . . . . .
0.08
I
0.1
'
'
'
,
,
,
,
,
(2.4)
,
[
i
0.2 lamin~ flame sp¢~
,
i
i
,
i
,
i
i
[
0.4
Fig. 2. Loglog plot of tD versus u L for various initial distances x in the case where l = 0.1 and F = 2.0. The dependence of to on uL is well fitted by a straight line in the loglog plot. By least squares regression, the slopes of these lines are 0.49 for x = 0.1S, 0.5 for x = 0.17, 0.45 for x = 0.2, 0.53 for x = 0.22, 0.53 for x = 0.25, 0.59 for x = 0.3, and 0.63 for x = 0.35. similar slopes were obtained for different values of F and L Note how these slopes are all close to  1 / 2 when x = 0(1) and how the slope decreases, presumably toward  1, as x/l increases. Indeed, tD = x/ul. for x/! ~ 1.
298
J. C. VASSILICOS A N D N. N I K I F O R A K I S
3025
i
!
20
152
10
1.2
m
$

I
I
1.4
1.6
I
I
1.8 2.0 n~distancc
I
[
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2.2
2.4
2.6
and Williams [13] considered the case where the vortex has no viscous core, i.e., l = 0, in which case their relation Ar = 2uLt gives the timedependent radius r . ( t ) of a growing burned flame core and no pocket of unburned fuel.) 3. THE GIBSON AND T H E D E T A C H M E N T SCALES
Fig. 3. LoglinearplotofuL1/ZF1/213/2to versus x/I. There are 30 points on this plot and they all collapse in five groups of six points. These points are well fitted by a straight line in this loglinear plot, and the slope of the line is about 1/3.
eddy turnover time of one such vortex tube scales as 12/F(l). The time that a flamelet of laminar flame speed uL takes to propagate across a vortex tube of length scale l scales as I / u L. In our collection of turbulent vortex tubes the Gibson scale is therefore defined by 16/u L ~ lc2/F(la). Hence,
l~ ~ F ( l c ) / u L
The experimental observations of Cadot et al. [3] indicate that the turbulence contains vortex tubes of many different sizes. Following these authors, we consider a simple phenomenological model of turbulence where the turbulence is a collection of parallel vortex tubes (all normal to the (r, ~b) plane) with different scales l and circulations F. In order to be in agreement with the Kolmogorov phenomenology, t we assume that the smallest possible vortex size l is r/, the Kolmogorov viscous microscale, and that F depends only on l and e. By dimensional analysis we obtain F(1) El~314/3. From Eq. 2.1, the characteristic time scale or ~
1The turbulence considered here is not dynamicallytwodimensional even though the velocityfield induced by the vortex tubes effectively has two components. Models where the turbulence is a collection of vortex tubes and has a Kolmogorov k 5/3 energy spectrum have been discussed most recently by Lundgren and Mansour [10].
(3.1a)
and therefore
I a ~ UL3/E,
(3.1b)
which is, perhaps, not surprisingly, identical to Peter's result Eq. 1.4. However, a different critical length scale that is directly relevant to the topology of the flamelet can be defined by comparing the detachment time tD(1) to the eddy turnover time 12/F(l) of a vortex of size 1. The topological transformation of the flamelet leading to a detached pocket of unburned fuel occurs if to(l) >_ 12/F(1) and does not occur if to(l) < 12/F(l). This defines a critical or detachment length scale l D by tD(l D) = lo2/F(lD). Because t o ( 1 ) F ( l ) / l z is an increasing function of l, pockets can only form in those vortex tubes of size larger than l o and cannot form where l < l o. F r o m Eq. 2.4 with C a = 1 / 3 and using
FLAMELETVORTEX
INTERACTION
299
Eq. 3.1b, x
The way in which the Gibson scale appears naturally in Eq. 3.2 is remarkable and is not a mere consequence of dimensional requirements. In Fig. 4 we plot the function l o / x = F ( l a / x ) derived from Eq. 3.2. 2 Note that (l D + 2 x ) ~ lc
(3.3)
for l o / x , > 1. Where I a < x, l D is not even defined, and we explain why this is so in the next paragraph. A flamelet at a distance x from a vortex tube of size l ~ x takes a time x / u L to reach the vortex tube. The time needed for such a vortex to coil the flamelet around it once is of order x Z / F ( l ) . If x / u L < x 2 / F ( l ) , that is, if F ( 1 ) / u L < x, no double spiral is formed and therefore no detached pocket either. Hence, vortex tubes that are too far away from the flamelet do not influence the formation of double spirals and pockets of unburned fuel. If the size l of a vortex tube is smaller than l o,
2 In principle the function F is doublevalued, but we only keep the values for which lo/x_> 2 so that 1o is an increasing function of la. Physically,we expect l o to be an increasing function of lc, because l n should be an increasing function of u t. Less double spirals and detached pockets should form at higher values of uL.
the flamelet overtakes the vortex before the vortex can significantly affect the flamelet. Only vortices of size l larger than 1~ and at a distance x from the flamelet smaller than F ( 1 ) / u L can roll up the flamelet and produce spirals which then give rise to pockets of unburned fuel. Because F(I) is an increasing function of l and because F ( l o ) / u L ~ l o, only those vortex tubes at a distance x < I c can potentially wind the flamelet and produce spirals, which then lead to detached pockets. Indeed, it is only when x _< l G that the detachment scale l o is defined. The average detachment scale i o can be obtained from Eq. 3.3 by averaging only over those vortex tubes that are at distances x from the flamelet significantly smaller than l c. All the other vortices do not contribute to the detachment scale. If 2 is the average distance from those vortex tubes that are close enough to the flamelet ( 2 .~ l a ) , then
],9 + 2Y: ~ l 6 .
(3.4)
The average detachment length scale ]o should be interpreted as a critical vortex size. Vortex tubes of size l larger than iz) and close enough to the flamelet may be expected to generate double spiral patterns on the flamelet which then suddenly give rise to pockets of unburned fuel that burn away from the main flame front.
30
25~2015i 105\. ............................................. I
5
I
~......... I
? ...... I
10 15 20 n~mnlisedGibsonl~tle
I
25
!
30
Fig. 4. Linear plot of l n / x versus Ic/J The solid curve represents the dependence of lD/x on lu/x, the dashed curve also follows from Eq. 3.2, but is unphysical, and the straight line intersecting the origin is line where l o ~ l o.
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J. C. VASSILICOS AND N. NIKIFORAKIS
Vortex tubes of size l smaller than ID only weakly wrinkle the flamelet.
Power law defu~g the golmogorovespaeity
2.2
4.
FRACTALPROPERTIES
In the framework of our simple phenomenological model we may therefore expect double burning spirals and pockets of unburned fuel to coexist over a range of length scales above 1D and the flamelet to be weakly wrinkled over a range of length scales below i D. T h e fractal properties of such a flame are qualitatively different above and below lb. Below I o, the fractal properties of the flamelet are an imprint of the distribution of vortex sizes below lo, if that distribution is fractal. Above ]o, the fractal properties of the flamelet are those of the double spirals and of the distribution of pocket sizes, if that distribution is also fractal. Isolated spirals can have welldefined fractal properties (Vassilicos and Hunt [18]). A passive interface coiling around a vortex tube such as Eq. 2.1 with u, = 0 adopts the spiral form 3 r ~ 4)1/2. The evolution of such a passive interface corresponds to the case where UL = 0 in Eq. 1.1. Flamelets are scalar interfaces or isopleths with uL 4:0 in Eq. 1.1. Typically, spiral flamelets do not have the time to develop more than one or two spiral turns before running into themselves and producing a pocket of unburned fuel, unless uL is extraordinarily small. The fractal dimension (Kolmogorov capacity) of the passive (u L = 0) spiral interface r ~ ~b1/2 is D r = 1 + 4 / 3 or D r = 4 / 3 if, like in Fig. 1, the flamelet is idealised as a moving line interface (see [18]). Spiral flamelets (u L ~ 0) usually have a fractal dimension equal to 1 because they do not have the time to develop more than one turn. However, we find that when u L is nonzero but so small that the double spiral has at least two turns, the fractal dimension of the spiral is well approximated by 4 / 3 (see Fig. 5). Even when they do have more than two turns, spiral flamelets do not have these turns for long. They are soon replaced by pockets of unburned fuel. However, in a quasiequilibrium
3From Eq. 2.1, r(dch/dt)= uo, which implies ~b ~ r z, where r ~ l.
& a' & a ,
~ 2.0
~ 1.8~
1.6
9. ~1.4
log( N( 6)(f(Dd)
~ 1,2 ~ 1.0
+ +++++++++++++ + +
~ +
3.0
 5
.+ , +
.,
+1+'r'r~+I~~t++'~'~'brq.r,
2.0
Zog(~) Fig. 5. The Kolmogorov capacity (fractal dimension) is measured by means of the boxcounting algorithm. The spiral curve is repeatedly covered with square boxes of size 8 for many different values of 8 and the number N(`5) of such boxes is counted. The Kolmogorov capacity D K of the spiral is defined if N(`5) ~ `5rig in a range of scales bounded from below by the distance `stain between the innermost consecutive coils of the spiral and from above by the distance `smax between the outermost consecutive coils of the spiral. This plot of log[N(`5)`5 °k ] versus log(`5) is a typical result obtained from a double spiral with two turns formed on a flamelet with u L = 0.01 and x / l = 2. It is shown that 1 is not the right value for D K and that D r seems to be well approximated by 4 / 3 in a range of scales ,5 between `stain = 0.04 and `smax= 0.2.
situation where spirals are continuously generated and continuously break into detached pockets, we may expect a coexistence of both these features in the reacting flow. Whilst the fractal dimension of the double spirals may be expected to be 1 + 4 / 3 = 2.33, the study of the fractal dimension of the distribution of pocket sizes is beyond the scope of the present paper. This latter fractal dimension can be expected to depend on the fractal properties of the intermittency of the turbulence, and it may be that future models of flamelet combustion will require a bifractal (if not multifractal) description of the flamelet so as to take into account both the spiral and the pocket features in the range of scales above i D. In the range of scales below ]o, the wrinkled flamelet may also have nontrivial fractal properties. In contrast with Peters' [12] interpretation of IG, the detachment scale i o may be such that the flamelet is significantly wrinkled below iz~ even though
FLAMELETVORTEX INTERACTION there are no spirals and detached pockets at these very small scales. If, for the sake of argument, we assume that there exists one unique welldefined fractal dimension Dse of spirals and pockets above i D, and one welldefined fractal dimension D w of the wrinkled flamelet below iD, then the turbulent flame speed u T should be given by (see Eq. 1.3) 
cal length s c a l e   t h e average detachment scale i o  can be defined. Those vortex tubes of size I larger than i o and at a distance smaller than one Gibson length scale l c from the flamelet generate double spirals on the flamelet which subsequently collapse to form detached pockets of unburned fuel. Those vortex tubes of size smaller than i o simply wrinkle the flamelet. It is found that
7~ t 2Dw '
(4.1)
where rt is Kolmogorov's viscous microscale. 5.
301
CONCLUSION
A flamelet interacting with a vortex tube can either coil around the vortex tube or be wrinkled and cusped and overtake the vortex tube. In the case where the flamelet coils around the vortex tube and adopts a spiral shape, the flamelet undergoes a sudden topological transformation whereby the spiral breaks and a detached pocket of unburned fuel is left behind the main flame front. The detachment time t D when this topological transformation occurs has been calculated by numerical integration of the eikonal equation 1.1 with a prescribed vertical velocity field. When the initial distance x of the flamelet to the vortex tube is comparable to the size l of the vortex it is found that
]~ + 22 ~ l a, where $ is an average distance of the vortex tubes from the flamelet. However, in contrast with Peters' [12] interpretation of l o, the flamelet may have fractal properties both above and below i o, albeit of a different nature. The fractal properties of the flamelet above i o are determined by the double spirals which have a fractal dimension equal to 2.33 and by the distribution of pocket sizes. The fractal properties of the flamelet below lD are those of a singly connected wrinkled flamelet and depend on the distribution of vortex sizes below iz~. An important question that remains unanswered and needs to be addressed is whether pockets of unburned fuel still form with volume expansion a n d / o r flame stretch. We are thankful to Norbert Peters .for motivation and to Rupert Klein for a helpful discussion. JCV acknowledges support from the Royal Society and N N acknowledges support from the DoE.
13/2 lD
e x/31
ULX/ZFW2 where u L is the laminar flame speed of the flamelet and F is the circulation of the vortex tube. This information on the detachment time t o can be used in a phenomenological model of turbulent flamelet combustion where the turbulence consists of a collection of vortex tubes of different sizes l and circulations F ( 1 ) ~ e~/3l 4/3, in accordance with Kolmogorov's dimensional arguments (e is the average rate of kinetic energy dissipation per unit mass). In the context of this model, a topologically criti
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Cambridge UniversityPress, Cambridge, U.K., 1967. 3. Cadot, O., Douady, S., and Couder, Y. Phys. Fluids 7:630 (1995). 4. Chat6, H., and Cant, S. R. Combust. Flame 74:1 (1988). 5. Clavin, P., in Combustion and Nonlinear Phenomena (P. Clavin, B. Larrouturou, and P. Pelce, Eds.) Les Editions de Physiques, Paris, 1985. 6. Godunov,S. K. Math. Sbornik 47:271306 (1959) (in Russian). Translated US Joint PuN. Res. Serv. JPRS 7226 (1969). 7. Gouldin, F. V. Combust. Flame 68:249 (1987).
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8. Hirsch, C. Numerical Computation of Internal and ExternalFlows, vols. 1 and 2. Wiley, New York, 1988. 9. LeVeque, R. L. Numerical Methods for Conservation Laws. Birkhauser, Boston, 1992. 10. Lundgren, T. S., and Mansour, N. N. J. Fluid Mech. 307:43 (1996). 11. Nikiforakis, N., and Toro, E. F., in Numerical Methods for Fluid Dynamics (K. W. Morton and M. J. Baines, Eds). Oxford University Press, London, 1995. 12. Peters, N. TwentyFirst Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, 1986, p. 1231. 13. Peters, N., and Williams, F. A. TwentySecond Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, 1988, p. 495. 14. Searby, G., and Quinard, J. Combust. Flame 82:298 (1990). 15. Smallwood, G. J. Giilder, I~. L., Snelling, D. R., Deschamps, B.M., and G6kalp, I. Combust. Flame 101:461 (1995). 16. Ting, L., and Klein, R. Viscous Vortical Flows. Lecture Notes in Physics Vol. 374. SpringerVerlag, Berlin, 1991. 17. Toro, E. F. Phil. Trans. Roy. Soc. London Ser. A 341:499 (1992). 18. Vassilicos, J. C., and Hunt, J. C. R. Proc. Roy. Soc. London Ser. A 435:505 (1991). 19. Vassilicos, J. C., and Hunt, J. C. R. Combust. Sci. Technol 87:291 (1992). 20. Yanenko, N. N. The Method of Fractional Steps. Springer Verlag, New York, 1979. 21. Yoshida, A., Kasahara, M., Tsuji, H., and Yanagisawa, Y. Combust. Sci. Technol. 103:207 (1994). 22. Zhu, J., and Ronney, P. D. Combust. Sci. Technol. 100:183 (1994).
can be written as a hyperbolic conservation law
Received 30 August 1995; revised 15 May 1996
APPENDIX NUMERICAL
A. THE METHOD
Equation 1.1 was integrated numerically using the fractional steps approach (Yanenko [20]), where the time and space operators are treated separately. The lefthandside of Eq. 1.1, which expresses the dynamics of the physical system,
OF 
Ot
O(uF) +  
Ox
O(vF) +
Oy
0
(A1)
for the divergencefree velocity field (u, v) in Cartesian coordinates. Equation A1 was integrated using the weighted average flux (WAF) method, which is a generalization of Godunov's [6] method (see the paper by Toro [17] for the formulation of WAF and Nikiforakis and Toro [11] for assessment of the method for forced advection simulations). WAF belongs in the highresolution class of methods; desirable features of these schemes include at least secondorder accuracy in space and time, conservative discretisations, resolution of discontinuities without excessive smearing, and absence of spurious oscillations in the vicinity of sharp gradients (Le Veque [9]). The partially updated answer given by Eq. A1 is the initial condition for the ordinary differential equation
dF dt = uLIVFf'
(A2)
which expresses the effects of combustion on the flow, integrated over the same time step as Eq. A1 to yield the final answer. A finite difference approximation (five point symmetric formula [8]) was used to solve Eq. A2. The initial condition for Eq. 1.1 is a smoothly varying function along the longitudinal direction (max(F)  1, rain(F) = 0), with no initial lateral variation (the longitudinal (lateral) direction is the x(y) direction). Every isopleth along this profile describes a possible flame front. For the simulations shown here we invariably follow the evolution of the 0.5 isopleth.