Interaction of a flame front with its self-generated flow in an enclosure: The “tulip flame” phenomenon

C O M B U S T I O N A N D F L A M E 88:201-220 (1992)

201

Interaction of a Flame Front with Its Self-Generated Flow in an Enclosure: The "Tulip Flame" P h e n o m e n o n M. GONZALEZ, R. BORGHI, and A. SAOUAB Universit~ de Rouen, Mont-Saint-Aignan, France The propagation of a flame front under nonturbulent condition in a closed tube ignited at one end is numerically investigated using a computing procedure based on finite volumes technique and devoted to two-dimensional, compressible, reacting flows. A global one-step reaction for the chemical process and an Arrhenius law for fuel consumption are assumed. The detailed analysis of the results of computations in which wall friction, tube aspect ratio and initial flame configuration are varied allows to highlight the influence of different parameters and to get more insight into the tulip-shaped flame phenomenon. In particular, Darrieus-Landau instability is examined by comparing the shape variations of an initially perturbed fiat front in a tube closed at both ends to those in a tube in which the ignition end is open while the opposite one is closed. Attention is also given to the computed flame generated flowfield; the flame front-confined flow interaction is specially scrutinized. Furthermore, the oscillatory acoustic regime occurring during tulip flame appearance, as well as the collapse of the tulip shape in tubes of large aspect ratio, already experimentally put into evidence but never numerically addressed, have also been simulated and discussed.

INTRODUCTION Propagation of flames in tubes and enclosures has been extensively investigated for a long time (see, for example, Ref. 1). Ellis [2], studying experimentally the propagation of flames in closed cylinders, put into evidence that for an aspect ratio greater than 2 the flame front may undergo a radical change in its curvature, that is, the appearance, during the last stage of combustion, of a backward directed cusp that was called "tulip flame" by Salamandra et al. [3] (Fig. la). The works reported by Gu6noche [ 1] show that the changes occurring in the shape of a flame front propagation in a tube depend on many parameters: initial pressure, aspect ratio, equivalence ratio of the mixture, and also whether the tube has open or closed ends. As a matter of fact, the problem is complex and the tulip flame phenomenon, in particular, has given rise to experimental [4-6] and numerical studies [4, 7-9] that aimed at explaining its appearance. Various possible mechanisms have been put forward: quenching and viscosity effects [2, 10], burned gases vortex motion effect [11, 12], the DarrieusLandau flame instability [4, 13] and a flamepressure wave interaction [1]. The problem of tulip-flame formation has been numerically tackled by means of various methCopyright © 1992 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc.

ods. Dunn-Rankin et al. [4] and Rotman and Oppenheim [7] have used a zero Mach number model in which the flame front is an infinitely thin interface dividing the computation domain into two regions (burned and unburned), each of which has a spatially uniform density; they traced the motion of the flame interface by the use of the Simple Line Interface Calculation (SLIC) technique [14]. Fernandez [8] has also computed the appearance of a tulip-shaped flame by solving the set of Euler and diffusion-reaction equations on an adaptive mesh [15] and using an hybrid finite elements/finite volumes method; he assumed a global one-step chemical reaction and, contrary to Dunn-Rankin et al. [4] and Rotman and Oppenheim [7], his predicted flame front is thick. Lastly, Hwang et al. [9] have addressed the problem with the aid of the CONCHAS-SPRAY code [16] devoted to the simulation of viscous reacting flows. In their calculation, the problem of the numerical singularity of the laminar flame is overcome by introducing an artificial flame thickening. Similarly to Fernandez, they modeled the chemical process assuming a global one-step reaction. The above-mentioned studies have partially shed light on the tulip flame phenomenon. At least, the mechanisms that are not necessary to the tulip appearance have been identified. First, it is now clear that neither the calculation of the 0010-2180/92/$5.00

202

Fig. la. Stroboscopic flame records in cylinders of various aspect ratio (Ellis [2]).

detailed flame front structure nor the use of a complex chemical scheme is required to simulate a tulip-shaped flame. Second, the previous calculations have been performed with slip and adiabatic conditions at the walls, and, as a consequence, the effects of viscosity and heat losses at the walls cannot be put forward. On another hand, the hypothesis of a flame-pressure wave interaction is likely to be eliminated too; as a matter of fact, Fernandez [8] and Hwang et al. [9] have obtained the appearance of a tulip flame using implicit time integrations that preclude the simulation of pressure waves propagation. Moreover, Dunn-Rankin et al. [4] and Rotman and Oppenheim [7], despite the fact that the zero Mach number approximation they used does not include the effect of pressure waves on flame propagation, showed the formation of a tulipshaped flame. They also assumed the existence of a velocity potential for their flow calculations and could not simulate the rotational nature of the burned gases flowfield; the effect of burned gases rotational motion has thereby been ruled out. Although the previous numerical studies agree on some points, the authors have proposed dif-

M. GONZALEZ ET AL. ferent interpretations of the tulip flame formation. According to Rotman and Oppenheim [7], the tulip-shaped flame would result from a flame stretch effect due to unburned gas recirculating flow at the downstream corners of the vessel while Jeung et al. [6] and Hwang et al. [9] suggest that a radial velocity gradient along the front, on the fresh side, resulting from confinement, could be the main cause of the inversion of the front curvature. On the other hand, DunnRankin et al. [4] explain the appearance of the backward-directed cusp of the flame by setting forth the growth of the Darrieus-Landau instability after the flame front has been perturbed at a wavelength equal to the width of the box by the transversal velocity gradient that Hwang et al. have also put forward. Indeed, after the calculations of Fernandez [8], who obtained a folded front from an initially flat one, it seems that the Darrieus-Landau instability is really important. It is worth noting that although all these authors stress the fact that hydrodynamics is essential to understand tulip shape appearance, they disagree on the mechanism to be put forward as the leading cause. Anyhow, each of their studies is focused on some particular aspect of the phenomenon and does not confront the whole problem, which in fact is very complicated; most likely, the tulip-shaped flame is generated under the simultaneous actions of combined mechanisms. The study we have undertaken aims first of all at obtaining, as far as possible, a complete and detailed description of tulip flame formation and hence at addressing the aspects of the problem that were missing in previous works. We have also tried to derive a shortened description of the process, emphasizing the main mechanisms. However, distinguishing a single mechanism from which the tulip shape would exclusively originate is probably not suitable. We have simulated the propagation of a flame front in a closed vessel under nonturbulent condition using a numerical procedure for two-dimensional, compressible, reacting flows. Our reference test case consists of a rectangular closed box, the aspect ratio of which is equal to 3, filled with a combustible mixture at rest. Adiabatic and slip conditions at the walls are prescribed. Ignition is simulated in a restricted zone at one end of

FLAME FRONT/FLOWFIELD INTERACTION IN A CLOSED VESSEL the vessel. The simulation of the propagating flame front clearly shows the appearance of a tulip-shaped flame in the last stage of propagation. The changes occurring in the flame front shape as well as its generated flow confined in the box are scrutinized and discussed; a detailed analysis of their interaction, which is a crucial point of the problem, is reported. The study of three other cases makes it possible to highlight the alterations of the tulip flame development subsequent to various "experimental" conditions. The first of them is concerned with the effect of wall friction, the second with the effect of a large aspect ratio of the enclosure, and in the last, by examining the evolution of the shape of an initially perturbed fiat front, we try to point out what makes the tulip flame case different from the simple case of a plane flame front undergoing Darrieus-Landau instability during its propagation in a tube. It is also found, in agreement with the experiment of Starke and Roth [5], that in tubes of large aspect ratio the tulip shape does not persist up to the end of propagation: after it has taken a very slender shape, the flame front undergoes a collapse that causes the disappearance of the backward-directed cusp. Moreover, as reported by Starke and Roth [5], it is found that an oscillatory acoustic regime occurs during the tulip flame formation process and that the amplitude of the oscillations increases with the aspect ratio of the tube.

SHORT SUMMARY OF EQUATIONS AND NUMERICS Equations to Be Solved The simulation of the flow under study requires the solving of the conservation equations for mass (Eq. 1), momentum (Eq. 2), and energy (Eq. 3):

ap --

at a

a +

-~pu~

--pu~

=

ax~ o

+ ~puc,

ax~

(1)

O,

u~ +

3p axa

a

+ - -

ax~

( Ou~ au~ 2 au~ - -6~B--It ~ + -cgX~, 3 a x , ]I = 0 ,

(2)

-~ph s + ~pu~h~ ax~

203

+

~x~

Pr ax~

ap -

(3)

oI;VFAQ,

at

where

1

h s = CpT + -~u~u~.

(4)

An equation of state (Eq. 5) and a balance equation for the fuel mass fraction (Eq. 6) are added:

(5)

p = prT, OX a

Sc

= pl;Vr.

(6)

Hypothesis and Values of Constants The dynamic viscosity is stated to a constant value: /.t = 3 × 10 -s kg m - t s - t ; with the grid that has been chosen, it is the lowest value for which numerical stability is ensured. Concerning chemistry, a global one-step reaction is assumed and the consumption rate of fuel is expressed owing to an Arrhenius law:

Wr = - K Y F Y o e-rA/T

(7)

with K = 3.3 × 107 s - l and TA = 10000 K. Although those values do not correspond to any defined fuel, they are realistic for hydrocarbon combustion; the activation temperature TA is rather small, but a higher value would require a finer gridding. At the moment, calculations have been performed with Prandtl and Schmidt numbers equal to unity: the Lewis number (Le = Sc/Pr) is thereby equal to unity and thermodiffusive instabilities cannot occur. The specific heat Cp and the constant r are averaged values; their dependence with respect to composition and temperature is not accounted for. We state

Co

3,=--= Co

1.4,

r = 2 8 7 J k g -I K - I ,

204

M. GONZALEZ ET AL.

and for the molecular weight of the mixture, M = 29

x

10 - 3

kg,

which gives

Cp =

1004 J kg - I K - I

With AQ = 4 x 10 7 J kg -~ and an initial temperature equal to 300 K, the computed temperature of the burned gases is found to be around 2300 K. Following the analysis of Clavin [17] we can derive the characteristic asymptotic values of the flame: Laminar flame speed: lm s i, Flame thickness: 3 x 10 ~ m. Those values are substantially greater than the usual ones. However, the exact value of the laminar flame speed is immaterial because all the results may be nondimensionalized by a timescale that is equal to the ratio of box width to flame speed. Moreover, length may be nondimensionalized by the box width. On the other hand, the flame thickness is five or ten times larger than normal; this is necessary to avoid the use of an overly refined grid and has often been adopted (see for instance Ref. 18) for this purpose. It may play on the flame stability, but at the smallest scales only, which are much finer than the tulip itself. The influence of the flame thickness-box width ratio probably deserves to be investigated. However, in our case, that study cannot be conclusive because the flame thickness is too small compared with the box width, and thus, it has not been undertaken. The low Mach number approximation is not used and the equation of state takes the classical form; as a matter of fact, the numerical method we have chosen is implicit and does not require the low Mach number approximation. Furthermore, in order to account for all the physical aspects of the problem, we aim at computing the acoustic fluctuations; with the method used, that may be achieved simply by lowering the timestep.

Spatial discretization is performed with a finite volumes technique, the interpolation scheme is centered and is then of second order. It is well known that the centered scheme leads to numerical instability when the grid Reynolds number (or P6clet number) exceeds a critical value. In the present numerical method, that could be overcome by adding to the physical diffusion term, a false diffusion, linked to the cell Reynolds number, that ensures numerical stability. Nevertheless, the amount of such an artificial diffusion depends on the computed local velocity, and the calculation would thereby be performed with an effective viscosity that would be higher than the physical one that, furthermore, cannot be estimated beforehand. Instead of that, we have chosen to use a sufficiently high fluid dynamic viscosity, ensuring that the physical diffusion term is large enough to preclude numerical instability without the requirement of a local false diffusion; with the chosen grid, that is achieved with a dynamic viscosity # = 3 x 10 - 3 kg m-~ s -~. Furthermore, it has been shown [17] that the Darrieus-Landau instability does not depend, at the first order, on the fluid viscosity. It will be seen below that the tulip flame phenomenon is not deeply modified by the high value of the dynamic viscosity (except if it plays at the walls). The time integration technique consists of two phases: 1. An explicit phase providing a first-order approximation of the temporal derivatives. 2. An implicit phase correcting the explicit one that aims at solving the linearized equations in which the unknown terms are the time increments. Time discretization is centered (Crank-Nicholson scheme) and is thereby of second order. The implicit phase is based on a factored scheme and includes two steps:

The Numerical Method

a. Solving tridiagonal systems resulting of the spatial discretization with respect to the longitudinal coordinate. b. Solving tridiagonal systems resulting of the spatial discretization with respect to the transversal coordinate.

The set of Eqs. 1-6 is solved using a numerical method devised by Dutoya and Michard [19] and further improved by Dupoirieux and Dutoya [20].

That method is very similar to that of Beam and Warming [21]. Note that pressure is accounted for in an implicit manner, replacing p in

FLAME F R O N T / F L O W F I E L D INTERACTION IN A CLOSED VESSEL the Navier-Stokes equation (Eq. 2) owing to the following relation:

P:

--'O "y

hs-

1]

~u,u~

.

(8)

205

Case I: Reference Test Case Ignition in a 6 x 6 cells square zone at one end wall (Fig. la). Slip condition at the walls.

Case II: Effect of Wall Friction

That results in a coupling among the Navier-Stokes, continuity, and energy equations that we treat as follows: during step a, only density, energy, and the longitudinal component of momentum are coupled while the transversal component of momentum is computed separately. During step b, density, energy, and the transversal component of momentum are coupled while uncoupling is now applied to the longitudinal component. The balance equation for the fuel mass fraction is solved separately. The implicit treatment allows timesteps larger than the timestep given by the CFL criterion, A tcF L. For noncombusting flows, timesteps as large as 50 times the CFL timestep can be used while for flows with combustion that ratio must be smaller. The boundary conditions at the walls are

Ignition all along the left end boundary; the ignition zone thickness departs slightly (sinusoidally), from a constant value and the wavelength of the perturbation is equal to the width of the box (Fig. 4a).

zero normal momentum component,

Open or closed ignition end.

zero heat and mass fractions fluxes,

Slip condition at the walls.

concerning the momentum component tangential to the wall we have compared the effects of a slip condition and of a wall friction.

Computational Conditions The calculations have been performed for a L x h = 9 x 3 cm rectangular box. The gridding is uniform and is made with square ceils. In order to detect nonsymmetrical effects (if any), we computed the reacting flow in the whole vessel; the existence of a plane of symmetry has thereby not been accounted for. The timestep is At = 2 A tCFL. The box is filled with a fuel-air mixture and the initial fuel mass fraction is 0.05. Initially, the temperature inside the box is 300 K, the pressure is 105 Pa and the flow velocity is zero. Ignition is simulated by setting, at t = 0 s and in some chosen cells, the fuel mass fraction to zero.

The same ignition zone as in case I. The wall friction is taken into account.

Case l l h Effect of Aspect Ratio Ignition in two 6 x 6 cells square zones at one end wall (Fig. 3). Slip condition at the walls.

Case IV: Study of Darrieus-Landau Instability

In cases I, II, and IV the gridding is 90 × 30 and AtCF L = 1.5 X 10 -6 s; of course, tests have been made in order to be sure that the 90 x 30 gridding was sufficient to provide a solution which is everywhere grid independent within 5%. In case III, a 180 x 60 gridding (and AtcF L = 7.5 10 -7 s) has been used.

Analysis of Case I Description. Figures lb and lc, and l d - l g , respectively, display the isovalues of fuel mass

RESULTS A N D D I S C U S S I O N

The Cases Studied Four simulations have been carried out.

t ignition zone

3] 400At

600At

Fig. lb. Isovalues o f fuel m a s s fraction at t = 2 0 0 A t , 4 0 0 A t , and 6 0 0 A t (case I).

206

M. GONZALEZ ET AL.

.

800At

.

.

.

.

.

.

.

.

J

/

.

.

.

.

.

.

2000At

1000A t Fig. lc. Isovalues of fuel mass fraction at t = 800 At, 1000 A t, and 2000 A t (case I).

Fig. If. Velocity field in the vicinity of the flame front at t = 600 A t (case I). Maximum vector, 26.3 m s - t.

Fig. ld. Velocity field in the vicinity of the flame front at t = 200 A t (case I). Maximum vector, 43.1 m s - t.

Fig. le. Velocity field in the vicinity of the flame front at t = 400 A t (case I). Maximum vector, 29.7 m s - t .

fraction and the velocity field in the vicinity of the flame at various instants after ignition. In Figs. lb and lc the appearance of the wellknown tulip-shaped flame is clearly shown. In the very first phase of propagation, the front is almost cylindrical. Then, due to the effect of the side wails of the box, the lateral parts of the flame are flattened while the frontal part remains curved. When the lateral parts of the flame front disappear at the walls, the frontal part becomes fiat. During the last stage of propagation, the front gets out of shape and displays a growing curvature concave

Fig. lg. Velocity field in the vicinity of the flame front at t = 2000 At (case I). Maximum vector, 5.89 m s - I .

FLAME F R O N T / F L O W F I E L D INTERACTION IN A CLOSED VESSEL toward the fresh gas that finally leads to a fully developed tulip flame. Another outstanding feature is the strong deceleration of the front after the lateral parts vanish at the walls; indeed, since the surface area of the lateral parts represents the most of the total surface area of the flame front, its disappearance results in a significant reduction of the amount of expanding burned gases and, thereby, in a noticeable deceleration of the flame. The velocity field generated by the flame is very complex: At t = 200 At the flame front has not yet reached the side walls (Fig. lb). The fresh gases are pushed downstream but the flow in the burned gases is more complicated (Fig. ld); indeed the gases are also pushed away from the flame front, but the lateral parts of the flame act in such a way that the burned gases they produce push the frontal part more strongly toward the fresh mixture. At t = 400 A t the lateral parts of the flame front reach the side walls (Fig. Ib); looking at the flowfield (Fig. le) one can note the following: 1. The flow in the burned region seems to be formed by two symmetrical vortices: the flow coming from the central part now overshadows the flow generated by the lateral parts and drives the gases toward the left end wall; the burned gases do not push the central part of the flame front anymore. 2. In the unburned region, the velocity is still directed toward the right end wall but its magnitude is a little smaller in the central region than in the side wall region.

207

4. One can also note the decelerating effect of the front between t = 4 0 0 At and t = 600 At due to the disappearance of the main volume source after the lateral parts have vanished at the walls. At t = 800 A t a radical alteration occurs in the flame shape since the lateral parts of the flame now propagate faster than the central part; the front is folded, with a backward directed dome (Fig. lo). The flow pattern remains almost the same (Fig. lo) but the velocity is now larger in the burned gases than in the fresh mixture; in the burned region, the flow near the walls is directed toward the front. After that, from t = 1000 At to t = 2000 At, the folding of the front increases and a tulip flame clearly develops. The magnitude of the velocity in the unburned region tends to zero (Fig. lg). At t = 2000/x t the burned gas flow directed toward the flame front in the wall region disappears (Fig. lg); the fresh gases flow toward the central region of the front (Fig. lh). After the simulation of case I it can be concluded that the approximated equations and the numerical scheme we have used are able to reproduce the tulip phenomenon. The simple assumptions that are made (specially no thermodiffusive instabilities) are then compatible with this effect. The fact that our simulated laminar flame is thicker than actual ones is also acceptable in this

All those features are more marked at t = 600 At, when the lateral parts of the front have vanished and the remaining leading part is nearly completely flattened (Fig. lf): 1. The magnitude of the velocity in the central part, just ahead of the front, is very low and the magnitude of the reversal flow velocity in the burned region has increased because the frontal part is now the only volume source. 2. An accelerated flow is generated in the unburned region wedged by the flame front and the side wall, just ahead of the front; some authors use the expression "squish flow" when referring to that phenomenon [4, 6]. 3. The fresh gases are pushed both ahead and toward the center of the box.

Fig. lh. Velocity field in the fresh mixture at t = 2000 At; only vectors having an intensity lower than 0.3 m s i are plotted (case I).

208 respect. Note also that our results corroborate the fact already pointed out by other authors [4, 7, 8] that a tulip flame may be simulated without heat losses or viscous effects at the walls.

Comparison with Previous Studies. Some numerical studies have put forward purely aerodynamical effects in order to explain the tulip flame phenomenon [7, 9]. According to Rotman and Oppenheim [7], a recirculation motion of unburned gas appearing at the closed back end would generate the tulip shape by providing a flame stretch. Indeed, it seems to be obvious that in the last stage of propagation, when the flame comes close to the end wall, the fresh gas pushed ahead of the front is driven to recirculate as a consequence of the closed end. Figure lh (on which only velocity vectors with an intensity lower than 0.3 m s have been plotted) shows that at t = 2000 A t, when tulip shape is fully developed, our computed flowfield clearly displays the exact features that have been pointed out by Rotman and Oppenheim, that is, a recirculating flow at the corners that makes the gas flow toward the flame front. Nevertheless, the interpretation of Rotman and Oppenheim does not seem to be corroborated by experimental results. Gu6noche [1] has reported some experiments in which cusped flames are formed in tubes with an open downstream end. More recently, Starke and Roth [5] have carried out visualization and measurements of the propagation of a flame front in closed cylindrical vessels of various aspect ratios. The appearance of a tulip flame is shown by their photographic records for an aspect ratio equal to 3.8. In case of the greater aspect ratio ( L / d = 7.6) a tulip flame is obtained but completely vanishes when the flame front approaches the end wall. Hwang et al. suggest that tulip flame is generated by the simultaneous effects of the front middle part slow propagation (due to reversal flow in the burned region and gas velocity reduction by thrust reduction in the compressed unburned region) and the "squish flow" of unburned gas wedged by the flame front and the side wall, which finally drive the flame to propagate faster at the wall than in the central region. Our results are qualitatively similar to those of Hwang et al. [9]; as a matter of fact, the main

M. GONZALEZ ET AL. features of the flame front-flowfield interaction obtained by the two simulations are comparable: slow propagation of the front central part, and "squish flow" generated between the flame front and the side wails. Hwang et al. [9] have also noted the formation of a reversal flow and two vortices in the burned gases but only after the lateral parts of the front have vanished at the walls. In our case, a reversal flow and vortices are created in burned region before the disappearance of the lateral parts of the flame (Fig. le). The difference between the two simulations can be explained owing to the fact that Hwang et al. [9] studied the propagation in a cylindrical vessel and, in their case, the surface area of the lateral part of the flame increases with the radius, unlike in our planar geometry. Thus, in their simulation, the burned gas flow pushed by the lateral part of the flame front always overshadows the burned gas flow generated by its leading part. On the contrary, in our simulation, since the geometry is planar, the burned gas flows originating, respectively, from the lateral parts and the leading part can counterbalance each other. Anyway, the reversal flow after the disappearance of the lateral parts is obtained at t = 600 At (Fig. If).

The Oscillatory Regime. The time evolution from t = 0 A t to t = 2000 A t of the longitudinal component of velocity at x = 7.55 cm away from the left end wall and y = h / 2 (i.e., on the middle plane of the box) has been plotted on Fig. li; the point (x = 7.55 cm, y = h i 2 ) is always located in the unburned region during the time interval we consider. From Figure li it clearly appears that as soon as the lateral parts of the flame vanish at the walls and the front becomes flat (at t = 600 At), the velocity in the unburned gas exhibits damped periodical oscillations with negative values. The time evolution of pressure during the same time interval (Fig. l j) demonstrates that the oscillatory regime occurring at t = 600 At is acoustic in nature. The same feature has been experimentally put into evidence by Starke and Roth [5]; they explain that the oscillations of the velocity in the fresh gases are caused by the sudden reduction of the amount of expanding burned gas (resulting in a rapid decrease in heat release) when the lateral parts of the flame front disappear at the walls. We can

FLAME FRONT/FLOWFIELD INTERACTION IN A CLOSED VESSEL

LONGITUDINAL

VELOCITY

,

X=7.55CH.Y=H/2

L

n

.

209

.

18 ~6

14

10

10 2 d,

8

2

-6

.

L

0

,

I

,

200

I

,

400

I

i

600

I

i 10100

800

I

1

A

2I

. . . . . .

i 14k00

00

i

1

61

00

i

I

i

1800

i

2000

TIMESTEP

Fig. lk. Plot of velocity displaying the flowfield in the whole unburned region at t = 600 A t (case I). M a x i m u m vector,

Fig. l i. Time evolution of the longitudinal component of velocity at x = 7.55 cm and y = h/2 (case I).

23.3 m s

note that despite the oscillating unburned gas velocity, the fiat front becomes continuously more concave and finally turns into a tulip shape; it can then be thought that the oscillatory regime is only superimposed on the tulip phenomenon and interacts weakly with it. Figures l k - 1 o show the flowfield in the whole unburned region at t = 600 z~t, 650 A t, 700 z~ t, 750 At, and 800 At. At t = 600 At, as already mentioned, our computed flowfield in the vicinity

of the front displays a "squish flow" at the walls and a very low velocity in the central region. Nevertheless, at t = 650 At, the "squish flow" almost disappears. But on the other hand, the velocity in the central region has increased in magnitude and is directed against the front, driving the central part of the flame to propagate more slowly; consequently, the front continues to fold. Figures l m - l o show that the features pointed out at t = 600 At and t = 650 At

PRESSURE 5500:0

,

F '

I

'

I

'

: I

t.

X=7.55CM,Y=H/2 '

I

'

I

'

t

'

i

,

i

,

500010

450010

400010

35000~

300009

ii!iiiiili

25000q

200009

150010

100010

200

400

600

800

1000

1000

1400

1600

1800

2000

TINESTEP

Fig. lj. Time evolution o f pressure at x = 7.55 cm and y = h / 2 (case I).

Fig. 11. Plot of velocity displaying the flowfleld in the whole unburned region at t = 650 A t (case I). M a x i m u m vector, 26.8 m s i.

210

M. G O N Z A L E Z ET AL.

ZZZZZ~

.r~

Fig. lm. Plot of velocity displaying the flowfield in the whole unburned region at t = 700 At (case I). Maximum vector, 19.8 m s- i.

periodically recur: at t = 700 A t and t = 800 A t , the "squish f l o w " is weaker but the velocity in the central region is greater and backward directed. C o n c l u s i o n o f Case I. The simulation of case I shows that, simultaneously to the tulip formation process, the following features occur: deceleration of the flame front (particulary in the central part), "squish f l o w , " unburned gas flow directed against the central part o f the front. It is also

Fig. lo. Plot of velocity displaying the flowfield in the whole unburned region at t = 800 At (case I). Maximum vector, 12.1 ms -I.

found that right before the front begins to fold, the burned gases located near the side walls flow toward the flame. All those phenomena are so intertwined that it is difficult to distinguish causes from consequences. W e believe that it is more important to address now two questions: first, the effect of the aspect ratio (case III), and second, the differences between the D a r r i e u s - L a n d a u instability o f an initially perturbed fiat front propagating in the box (case IV) and the tulip flame classical case. Moreover, since the calculation o f case I has been performed with slip condition at the walls, the effect of wall friction deserves also to be studied, which is the question the case II addresses.

Analysis

_ _

,', ?,,

Fig. In. Plot of velocity displaying the flowfield in the whole unburned region at t = 750 At (case I). Maximum vector, 20.6 m s -z.

of the Influence

of Wall Friction

This case differs from the previous one only by allowance for wall friction. It should be pointed out that since the value o f our dynamical viscosity (/x = 3 × 10 -3 kg m -1 s - I ) is two orders o f magnitude greater than the actual molecular one, the simulation is not realistic; as a matter o f fact, the zero wall friction of case I is probably closer to reality. However, the comparison between cases I and II makes it possible to highlight the role o f the flowfield in the formation of the tulip shape.

F L A M E F R O N T / F L O W F I E L D I N T E R A C T I O N IN A C L O S E D VESSEL

PAt ignition zone Fig. 2a. Isovalues of fuel mass fraction at t = 200 At, 400 At, and 600 A t (case II).

Description. Looking at Fig. 2a we note that at t = 400 A t the flame shape does not differ greatly from that of case I (Fig. lb); indeed, the lateral parts of the flame front reach the side walls at t = 400 A t only and, before that, are slightly affected by wall friction. At t = 600 At, the respective flame shapes and flowfields of cases I and II exhibit larger differences (Figs. lb and lf, 2a and 2c). Concerning the flame front, it can be seen from Figs. 2a that although the leading region is flat, a nonnegligible lateral part remains as a consequence of the boundary layer effect. Figure 2c shows the reversal flow in the burned region, but contrary to the case I (Fig. 1f) no vortices are formed near the wall. In the fresh gas, the velocity in the central region is lower than it is at each end of the frontal part, but, due to the wall friction, no "squish flow" is generated in the zone wedged by the front and the side wall; instead, we note that two vortices have appeared very close to the walls, in the unburned region just ahead of the front. At t = 1000 At, the central region of the flame is convex toward the burned side (Fig. 2b) and a tulip shape develops, but the lateral parts have not yet completely disappeared. The velocity

211

field in the burned region is entirely directed toward the left end wall and there is no burned gas flow directed toward the front near the side walls (Fig. 2d). In the unburned region the gas is pushed ahead by the flame. After that, the folding of the front is smoothed and at t = 1500 At, the tulip shape has almost disappeared (Fig. 2b). The velocity of the unburned gas is now directed toward the front (Fig. 2e). At t = 3600 At the flame reaches the end wall; the central part of the front is fiat and the lateral parts have vanished (Fig. 2b). Due to the viscosity effect the front propagates more slowly at the wall than in the middle region.

Discussion From that simulation it is now obvious that the generation of the tulip shape strongly depends on the flowfield in the enclosure. It also seems that vortices in burned gases do not play a crucial role in the tulip formation since the flame begins to fold without such a flow structure. Another interesting feature is the appearance of the folding after the leading part of the front has taken a flat shape and undergone a transversal velocity gradient on the fresh side (Fig. 2c). However, no "squish flow" is present at the side walls in the unburned region. The transversal velocity gradient is detected on Fig. 2c; nevertheless, it disap-

-DI J

I 1000A t

I 3600A t

1500A t Fig. 2b. Isovalues of fuel mass fraction at t = 1000 At, 1500 At, and 3600 At (case II).

Fig. 2c. Velocity field in the vicinity of the flame front at t = 600 At (case II). Maximum vector, 28.7 m s- 1.

212

M. G O N Z A L E Z ET AL. LONGITUDINAL

VELOCITY

,

X=7.55CM,Y=H/2

22

16 14 12

iii!il i;-i'i'ii!i iii

' / i

'2

i:

i ....... 0

Fig. 2d. Velocity field in the vicinity of the flame front at t = I000 At (case II). Maximum vector, 19.3 m s- ~. pears on Fig. 2d and the tulip shape vanishes also before the flame comes near the right end wall. The influence of the latter is shown by Fig. 2d, on which it can be noted that the transversal velocity gradient ahead of the flame finally vanishes. Furthermore, the reversal unburned gas flow depicted on Fig. 2e emphasizes the smoothing o f the curvature of the central region of the front.

The Oscillatory Regime. The time evolution of the longitudinal velocity at point ( x = 7.55 cm, y = h / 2 ) (Fig. 2f) gives further information.

2~0

40~

60B

BOB

1BBO 1201a 1100

1600

1 B 0 ~ 20B0

IIMESTEP

Fig. 2f. Time evolution of the longitudinal component of velocity at x = 7.55 cm and y = h/2 (case II).

It can be seen that, similarly to case I, an oscillatory regime occurs in the fresh mixture after the lateral parts of the flame have almost vanished and the leading part has become fiat (t = 600 At; Fig. 2a). W e note that although the oscillations are noticeably more violent than in case I (Fig. lc), the backward directed folding of the front is weaker. That feature corroborates the fact that, as already pointed out in case I, the velocity oscillations in the unburned medium do not interfere with the tulip-shaped flame process.

Analysis of the Influence of Aspect Ratio In case o f the ignition in two zones, the gridding of the computation domain has to be refined by a factor of 2. Thereby, the timestep given by the C F L criterion is half that o f the previous cases; the integration timestep (still stated to twice the C F L timestep), now denoted by At' is, consequently, divided by a factor of 2.

Fig. 2e. Velocity field in the vicinity of the flame front at t = 1500 At (case II). Maximum vector, 8.81 m s ~.

Description. Figures 3a and 3b show that two tulips are obtained in the same way as the tulip of case I: first, a propagation of two fronts, each consisting of an hemispherical leading part and two lateral parts (Fig. 3a, t = 400 A t ' and t = 600 A t ' ) ; then, the disappearance of the lateral parts and the formation o f two flat fronts (Fig. 3a, t = 700 A t ' ) , which, in the following, are

FLAME FRONT/FLOWFIELD INTERACTION IN A CLOSED VESSEL

400At' ignition

600A t' 700A t'

zones

Fig. 3a. Isovalues of fuel mass fraction at t = 400 A t', 600 A t ' , and 700 A t ' (case Ill).

213

1 6 0 0 A t ' I 3000At' 2000A t' Fig. 3c. Isovalues of fuel mass fraction at t = 1600 A t ' , 2000 A t ' and 3000 A t ' (case III).

800A t' 1200A t ' Fig. 3b. lsovalues of fuel mass fraction at t = 800 A t ' and 1200 A t ' (case Ili).

folded (Fig. 3b, t = 800 At') and finally turn into two tulips (Fig. 3b, t = 1200 At'). At t = 1600 At' (Fig. 3c), each tulip collapses and then the flame front displays a smoother shape until the end of propagation (Fig. 3c, t = 2000 At' and t = 3000 At'). Figure 3d displays the flowfield in the vicinity of the flame at t = 700 At'; the flow pattern just before the front begins to fold is very similar to that of case I (Fig. lf, t = 600 At): strong "squish flow" at the walls and backward directed velocity in the flat regions of the front. At t = 1200 At' (Figs. 3e), the flowfield in the unburned medium is everywhere directed against the flame front; it is worth noting that in the side wall regions the backward-directed velocity is greater than in the central region. The Multitulip-Shaped Flame. Pocheau and Kwon [22] have recently carried out an experimental study of the propagation of a flame in a rectangular enclosure and visualized the flame front by means of a laser tomography technique. They have specially studied the multiignition configurations and have shown that multitulipshaped flames are obtained when the ignition is performed on several lines of sparks; in particular, with two ignition lines (a case that corresponds to

S S ?

Fig. 3d. Velocity field in the vicinity of the flame front at t = 700 A t ' (case III). Maximum vector, 42.2 m s ~.

Fig. 3e. Velocity field in the vicinity of the flame front at t = 1200 A t ' (case III). Maximum vector, 33.7 m s i

214 our simulation), they have clearly put into evidence the appearance of a double tulip flame. Their records do not show a collapse of the tulips in the following of the propagation, but they only display the propagation on a downstream distance equal to the width of the box, while it appears from our simulation that the tulips collapse after the flame has covered a distance that is roughly twice the width of the enclosure. An outstanding feature of the double tulip flame is that the two tulips are not symmetrical with respect to the planes y = h / 4 and y = 3 h / 4 . Figure 3a shows that, at t = 600 At' and t = 700 At' the lateral parts of the flame that are close to the walls still remain while the lateral parts located on both sides of the middle plane of the box have almost disappeared. The dissymetry with respect to the planes y = h / 4 and y = 3 h / 4 remains in the following of propagation until the two tulips are formed; as a matter of fact, at t = 1200 At' (Fig. 3b) the convex parts of the front close to the side walls are noticeably ahead of the central convex region. The same feature is displayed by the visualization of Pocheau and Kwon. Indeed, it can be seen from our numerical simulation that from the beginning of propagation, the unburned gases located between the lateral parts of the flame and the side walls are less rapidly consumed than those that are in the central region of the box, that is, between the two lateral parts propagating toward one another. Consequently, the flattened front (t = 700 At', Fig. 3a) displays important lateral parts in the side wall regions while lateral parts of the central region have vanished; for that reason, the two tulips that develop from the flattened parts of the flame are not symmetrical with respect to the planes y = h / 4 and y = 3 h / 4 . As the simulation has been performed with zero heat fluxes at the walls and zero wall friction, that nonsymmetrical behavior cannot be explained simply; it may be due to a lack of stability of the exact symmetrical solution which obviously exists.

Effect of the Aspect Ratio. We have checked that despite its slight dissymetry each half-flame displays shape variations that are almost similar to those of a tulip flame propagating in a 9 x 1.5 cm after ignition in a single zone. We thereby consider case III as providing the essential information on the effect of aspect ratio.

M. GONZALEZ ET AL. It is already well known that the aspect ratio of the enclosure is a crucial parameter in the problem of tulip flame formation since it has been demonstrated [1, 2] that the tulip shape does not appear if the aspect ratio is lower than approximatively 2; that is easily understood considering that the side walls have to be reached by the front sufficiently before the end wall. Concerning the results of the case under study, we first note that the two flattened regions on the front (t = 700 A t ' = 350 At, Fig. 3a) are obtained before the fiat front of case I (t = 600 At, Fig. la); consequently, the two tulips also appear before the tulip of case I. However, that is not due exactly to the aspect ratio L / h but to the vessel width h, which is smaller in case III. The time at which the fiat front appears is actually just the time it takes for the lateral parts of the flame to reach the side walls, and therefore depends only on the flame speed and the box width h. Indeed, in the experiment of Starke and Roth [5], the aspect ratio is varied by increasing the tube length with a constant diameter d; in their study the fiat front appears at a time that does not depend on the aspect ratio L / d . The simulation of case III also puts into evidence some new features occurring for a large aspect ratio. The first outstanding result is the collapse of each tulip flame (Fig. 3c), which shows that the tulip shape does not survive under all conditions. As a point of fact, at t = 1200 At' (Fig. 3b), each tulip is rather slender; the propagation of its lips one toward the other finally causes the collapse (t = 1600 At', Fig. 3c). Although the experiment of Starke and Roth does not clearly display the front shape variations in the last stage of propagation, it agrees qualitatively well with our numerical results. Actually, their visualization shows that for L / d -- 7.6 the tulip flame is noticeably more slender than for L / d --3.8 and that it vanishes well before the end of propagation.

The Oscillatory Regime. Figure 3f displays the time evolution of the longitudinal component of velocity at x = 7.55 cm from the left end wall and y = h / 4 . The oscillating behavior of velocity in the unburned region occurs roughly at t = 700 At', that is, as in case I, when the front becomes flat. However, the velocity oscillations are now much more violent than in case I. That result is

FLAME F R O N T / F L O W F I E L D INTERACTION IN A CLOSED VESSEL LONGITUDINAL

VELOCITY

,

X=7.55CM,Y=H/4

30 25 20

10 5 0

jl

/

-5

15 201

i

I 4OO

~

I 60S

J

8OO lelBo

I i I i I i ~ i I , i I , I 120e 14~(I 16a~ 18~B 20~1~ 2201~ 241~¢J 261~

20~0

TIMESTEP

Fig. 3f. Time evolution of the longitudinal component of velocity at x = 7.55 cm and y = hi4 (case III).

corroborated by the experiment of Starke and Roth [5], who have furthermore measured negative values of the longitudinal velocity in the region near the side wall.

Discussion. Although in the case under study the tulips appear after the flame front has been flattened, with a delay time of approximately 300 At' (150 At), it is found that in case I 800 At is required to obtain a fully developed tulip shape from the flat front. That can be explained by the fact that if the aspect ratio is large, the transversal velocity gradient generated ahead of the flame is also large and, thereby, the flame would undergo a violent warping that results rapidly in a very slender tulip shape. On the contrary, in a box of a smaller aspect ratio, the transversal velocity gradient is smaller and would act more gently on the flattened front; then, the tulip develops more slowly and its final shape is also less elongated than in the case of a large aspect ratio. It seems that the conditions under which the tulip shape remains or not may be understood by means of the same analysis. Indeed, since the tulip appearing in a box of a large aspect ratio is very slender, the propagation of its lips toward one another causes the collapse before it reaches the downstream wall. If the aspect ratio is smaller, the tulip flame can propagate and reach the end of the box keeping a robust shape. Furthermore, when coming close to the end wall, the tulip may

215

probably be strengthened by corner flow structures already described by Rotman and Oppenheim [7]. Lastly, some partial results we have recently obtained from a very refined computation seem to show that, after the collapse of a tulip shape, the front is folded again with a wavelength-that is half that of the first folding and thus leads to a double tulip that further collapses; the front is afterwards folded once more, displaying a quadruple tulip, and so on. If that trend is finally proved by a study that needs a more refined gridding (in progress), one could conclude that the single tulip shape is nothing but the first wavelength of an instability that affects the flame front at an ever decreasing lengthscale during its propagation in a tube.

Comparisons with the Instability of an Initially Flat Flame Front Some authors have suspected the tulip shape to result from an intrinsic instability of the flame front. Indeed, Ghoniem and Knio [23], DunnRankin et al. [4], and Fernandez [8] have suggested such a mechanism. As a matter of fact, the Darrieus-Landau instability is always present, but it is now well known that a non-infinitely thin laminar flame front may, however, remain stable to small wavelength perturbations as a consequence of its local curvature and molecular processes [24]. Nevertheless, long wavelength perturbations are not damped by those mechanisms if no other stabilizing effect exists (for instance, acceleration of gravity for downward-propagating flames [17]) and may thereby affect the flame front stability. In order to test the hypothesis of an intrinsic instability of the flame front, we have performed a simulation that consists of igniting the combustible mixture in a slightly concave zone located all along one end of the vessel (Fig. 4a). The case of the box closed at both ends as well as the case in which only the ignition end is open have been studied.

Tube Closed at Both EndsmDescription. In Fig. 4a the isovalues of fuel mass fraction from ignition to t = 2400 A t are reported. As a matter of fact, the behavior of the flame front is quite different from what it is in the previous cases.

216

I

M. GONZALEZ ET AL.

600At 1200At 2400At 200At 400At ,[800At/160?At/

100At 300At 500At[ 1000At\ initial front 7 0 0 At 2000A t Fig. 4a. Isovalues of fuel mass fraction from t = 400 A t to t = 6800 A t after ignition in a concave zone; the time interval between two plots is 800 A t ; the tube is closed at both ends (case IV).

.

During the first stage of propagation, the initial concave shape is amplified and, at t = 300 At, the flame front exhibits a developed backward directed cusp. Then, from t = 300 At to t = 700 At, the cusp is damped: at t = 700 At, the front is nearly fiat. However, in the last stage of propagation, it takes a shape convex toward the fresh gases. Most likely, the burned gases, which, in the first stage of propagation recirculate toward the flame front, play a crucial role. Indeed, at t = 100 At (Fig. 4b, in which the velocity field in the burned gases only is emphasized), the burned gases, pushed by the central part of the front against the left end wall, recirculate in the side

.

.

.

MI~CTOl

Fig. 4c. Velocity field in the vicinity of the flame front at t = 100 A t ; the tube is closed at both ends (case IV). M a x i m u m vector, 3.4 m s - n.

wall region, pushing in turn the edges of the flame ahead. On the contrary, from t = 300 At (Fig. 4d) to t = 600 At, the burned gases recirculate against the cusp. During the two phases, corresponding respectively to the amplification and damping of the cusp, the unburned gases in the vicinity of the flame are always pushed ahead (Fig. 4c) and do not seem to affect its shape. From t = 700 At to the end of propagation, as the flame front becomes convex toward the

Y ,

M

\

i i ~ ' , ,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

z

. .

.

.

-

~

.

Fig. 4b. Velocity field in the burned gases at t = 100 At; only vectors with an intensity lower than 10 m s i are plotted; the tube is closed at both ends (case IV).

.

\

\

~

-

~

/ z

~

/

/.

/

/

1

~

Fig. 4d. Velocity field in the burned gases at t = 300 A t ; only vectors with an intensity lower than 14.5 m s -1 are plotted; the tube is closed at both ends (case IV).

FLAME FRONT/FLOWFIELD INTERACTION IN A CLOSED VESSEL

217

unburned medium, it is found that the velocity of the burned gases is directed toward the left end wall while the fresh gases flow alternately against the flame and toward the back end of the tube.

Tube Opened at the Ignition End and Closed at the Opposite One--Description. The simulation starts from the same initial slightly concave zone as in the previous case. In order to avoid a burned gases recirculating flow, the ignition end is open. Figure 4e displays the isovalues of fuel mass fraction from t = 400 At to t = 6800 At; the time interval between two plots is 800 zat. It can be seen that the initial perturbation grows and that, at t = 2000 At, the flame front is folded, displaying a tulip shape that remains till the end of propagation. The celerity of the flame is noticeably constant; this is not surprising since the tube is open and the flame front propagates in an uncompressed medium. The flowfield in the burned region is rather simple. The gases are pushed backward by the flame and leak through the open end; the resulting flow is parallel to the tube walls except in the vicinity of the flame where it is affected by the front curvature (Fig. 4f). The unburned gases are pushed ahead by the flame but, as a consequence of the confinement at the right end and the open left end, they are also driven to flow toward the left. Figure 4g, in which only velocity vectors with an intensity lower than 6m s-r have been plotted, emphasizes the unburned gas flowfield. One can note that the flow is directed toward the right end in the wall region and against the flame front in the central region; the flame-generated flow in the unburned medium has thereby a favorable effect on the tulip shape.

.....

........

zt

. . . . . . .

;::°i Fig. 4f. Velocity field in the vicinity of the flame front at t = 3600 A t ; the tube is open at the ignition end (case IV). M a x i m u m vector, 30.6 m s - ~.

(Fig. lc), may be obtained also from an initially perturbed flat front is the outstanding feature of the above mentioned results. It supports the idea that an intrinsic instability of the flat flame front could play a role in the appearance of the tulip flame. It does not, however, allow us to conclude that the tulip is independent of the method of ignition, as suggested by Fernandez [8]. Furthermore, if the tube is closed at both ends, the initial concave front first turns rapidly into a

i i ....

Discussion. The fact that a tulip-shaped flame, although it is less cusped than the one of case I

.... I t .....

I

Fig. 4e. Isovala~ t = 6800 A t a r e interval between t ignition end (caset~ ~.

~nass fraction from t = 400 A t to n in a concave zone; the time is 800 At; the tube is open at the

i

,

Fig. 4g. Velocity field in the fresh mixture at t = 3600 A t ; only vectors with an intensity lower than 6 m s - i are plotted; the tube is open at the ignition end (case IV).

218 tulip but, afterwards, the tulip shape is smoothed and the flame front finally assumes a convex shape. That example, as well as the collapse in tubes of large aspect ratio, suggests that the tulip flame, rather than a robust phenomenon, is a fragile and transient manifestation. The work of Zeldovich [25], demonstrating that the convex shape of a flame front propagating in a tube from an open to a closed end is a manifestation of the Darrieus-Landau instability, is well known. However, the appearance of a tulip-shaped flame under Darrieus-Landau instability in a vessel that is open at the ignition end would not disagree with that analysis since, as already suggested in Ref. 4, the results are analogous when transposing the plane of symmetry with the side walls. Furthermore, simulating the ignition in a slightly convex zone at the open end of the box (Fig. 4h), we have found that the resulting flame assumes and retains a fully convex shape during the whole propagation (Fig. 4i); as a matter of fact, under influence of the Darrieus-Landau instability the flame front may assume a concave as well as a convex shape depending on the initial perturbation. It seems to be fruitful to compare the time it takes in each case to get a tulip flame from a perturbed fiat front. In the case of the initially perturbed flat front in a tube opened at the ignition end, it is found that 1800 timesteps are necessary to obtain a fully developed tulip shape (or a fully convex shape if the initial front is slightly convex) whereas, in case I (central ignition in a closed tube), the growth of the tulip shape from the flattened front that appears at t = 600 At takes 800 timesteps. The difference with the perturbed fiat front starting from an open end lies in the respective velocity fields existing at the onset of the perturbation amplification; in case I, when the front is flattening, a transversal velocity gradient

M. G O N Z A L E Z ET AL.

Fig. 4i. Isovalues of fuel mass fraction from t = 100 At to t = 2400 At; the time interval between two plots is 100 At from t = 100 At to t = 800 At, afterwards, the isovalues are plotted at t = 1000 At, 1200 At, 1600 At, 2000 At and 2400 At; the tube is open at the ignition end (case IV). is generated ahead of the flame, on the fresh side, while the perturbed flat front of case IV is prescribed in a flow initially at rest. From those quantitative observations, it clearly arises that the flame-generated flowfield, affected by confinement, acting alone on the flame front, could generate a tulip shape in a shorter time interval than the Darrieus-Landau instability. Finally, the respective effects of the confined flowfield and of the Darrieus-Landau instability are coexistent; concerning tulip shape formation, they are compatible one with the other but, at least in case of the flame under study, it appears that the characteristic timescale related to the flowfield effect is shorter than the growth characteristic timescale of the Darrieus-Landau instability. Anyhow, trying to distinguish between the respective effects of flowfield and of the intrinsic instability of the flame front seems to be an useless task insofar as those mechanisms are closely coupled. In point of fact, not only does the transversal velocity gradient along the flattened front probably trigger the Darrieus-Landau instability but our simulation clearly shows that when becoming concave from the initial perturbed front, the flame generates a velocity field that sustains and perhaps emphasizes the folding process (Fig. 4g). CONCLUSION

initial front Fig. 4h. Convex ignition zone at the open end of the tube (case IV).

Using numerical simulatio~l~-v~ave investigated the interaction between a pppp a ~ _ . ~ a m e front and its generated flow u n d e r " - .'nt conditions in a rectangular clos~ ~,,.. method is classical, based on finite vi .and time splitting techniques but is fulF ~.order and allows a precise study of the r e ' a c ~ g flow. Its only

FLAME F R O N T / F L O W F I E L D INTERACTION IN A CLOSED VESSEL shortcoming lies in the fact that avoiding a highly refined gridding restricts the investigation to a rather low activation energy, and viscosity and diffusivity not as small as the actual ones. Nevertheless, none of this prevents the detailed study of the flame front/flowfield interaction, and we have specially addressed that point in our investigation of the tulip flame phenomenon. From the numerical simulations, a complete and detailed description of the phenomena occurring during tulip shape formation has been derived, filling to a certain extent gaps in our knowledge of the process and removing uncertainties that were still remaining. In particular, the acceleration and following deceleration of the flame front, its flattening before it begins to fold, the burned gases vortices at the edges of the flame as well as the transversal velocity gradient appearing on the fresh side along the flattened front have clearly been pointed out. In agreement with the experiment of Starke and Roth, it has also been found that an acoustic oscillatory regime begins as the flame front is flattened; the pressure and velocity oscillations are, however, only superimposed to the tulip mechanism and do not interact with it. It has also been demonstrated that an excessive wall friction precludes the tulip phenomenon. As reported in the experimental results of Starke and Roth but never proved numerically, it also clearly appears that with increasing aspect ratio the tulip flame is more slender. In addition, if the aspect ratio is large enough, the tulip shape collapses and vanishes before the end of propagation. Furthermore, the oscillatory regime displays a larger amplitude if the aspect ratio is increased; that result also is corroborated by the experiment of Starke and Roth. The intrinsic Darrieus-Landau instability of the flame has been compared with the tulip phenomenon. The simulation shows the appearance of a tulip shape from an initially slightly concave front, propagating in a tube, from an open to a closed end. However, the mechanism is different. On another hand, if the initial front is slightly convex, the perturbation is found to be amplified into a fully convex shape, in agreement with Zeldovich's analysis. In the case of a tube closed at both ends, consequently to the recirculation of burned gases, the tulip shape generated from the initial concave front is damped and, in the last

219

stage of propagation, turns into a convex front, corroborating the idea that tulip flame is a phenomenon that is very sensitive to various parameters. Furthermore, in the classical tulip case, the tulip shape appears from the flattened front in a shorter delay time than in case of the initial slightly concave front starting from an open end. Hence, the important conclusion may be derived that, although the Darrieus-Landau instability is compatible with the tulip shape, it affects the flattened front less rapidly than the confined flowfield can do during tulip formation process. As previously mentioned, the complexity of the interaction between the propagating flame front and the subsequent generated flowfield makes difficult, or even useless, the detection of a single leading mechanism. However, at least two effects are crucial in tulip flame formation, namely the transversal velocity gradient along the flattened front subsequent to the "squish flow" and the deceleration of the central part of the flame, and the well-known Darrieus-Landau instability, which is always present. REFERENCES I. Gu~noche, H., in Nonsteady Flame Propagation, Pergamon, New York, 1964, pp. 107-137. 2. Ellis, O. C. de C., J. Fuel Sci. 7:502-508 (1928). 3. Salamandra, G. D., Bazhenova, T. V., and Naboko, I. M., Seventh Symposium (International) on Combustion, The CombustionInstitute, Butterworths, 1959,

p. 851. 4. Dunn-Rankin, D., Barr, P. K., and Sawyer, R. F., Twenty-First Symposium (International) on Combustion, The CombustionInstitute, Pittsburgh, 1986, p.

1291. 5. Starke, R., and Roth, P., Combust. Flame 66:249-259 (1986). 6. Jeung, I. S., Cho, K. K., and Jeong, K. S., TwentySeventh Aerospace Sciences Meeting, AIAA, Reno, 1989. 7. Rotman, D. A., and Oppenheim, A. K., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, p. 103. 8. Fernandez,G., Ph.D. thesis, Universit~ de Nice, 1989. 9. Hwang, S. S., Jeung, I. S., and Cho, K. K., in press. 10. Lewis, B., and von Elbe, G., Combustion, Flames, and Explosions o f Gases, 2nd ed., Academic, New York, 1961. 11. Oppenheim,A. K., and Ghoniem, A. F., Twenty-First Aerospace Sciences Meeting, AIAA, Reno, 1983. 12. Dunn-Rankin, D., and Sawyer, R. F., Tenth International Colloquim on Dynamicsof Explosions and Reactive Systems, Berkeley, CA, 1985, p. 115.

220

13.

Strehlow, R. A., Combustion Fundamentals, McGraw Hill, New York, 1984. 14. Sethian, J., J. Comp. Phys. 54:425-456 (1984). 15. Larrouturou, B., Ph.D. thesis, Universit~ de Paris-Nord, 1987 16. Cloutman, L. D. Dukowicz, J. K. Ramshaw, J. D., and Amsden, A. A., Los Alamos National Laboratory Report LA-9294-MS, 1982. 17. Clavin, P., Prog. Ener. Combust. Sci. 11:1-59 (1985). 18. O'Rourke, P. J., and Bracco, F. V., Comp. Phys. 33:185-203 (1979). 19. Dutoya, D., and Michard, P. J., Rech. A&ospat. 1980(2): 123-129 (1980).

M. GONZALEZ

20. 21. 22.

23.

24. 25.

ET AL.

Dupoirieux, F., and Dutoya, D., Rech. AOrospat. 1987(6): 15-22 (1987). Beam, B. M., and Warming, R. F., A I A A J. 16:393-402 (1978). Pocheau, A., and Kwon, C. W., Colloque de I' A. R.C. "Mod~lisation de la Combustion dans les Moteurs ?l Piston", C.N.R.S.-P.I.R.S.E.M., Paris, 1989, p. 62. Ghoniem, A. F., and Knio, O. M., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, p. 1313. Markstein, G. H., in Nonsteady Flame Propagation, Pergamon, New York, 1964, pp. 15-73. Zeldovich, YA.B., Combust. Flame 40:225-234 (1981).