Ordered states of cellular flames, consisting of concentric rings of approximately equally sized cells, are observed over a wide range of parameters in our experiments using heavy hydrocarbon-air mixtures on a circular porous plug burner at low pressure. These ordered states have been found with cell numbers ranging from five to thirty. At a critical value of the flow rate, which depends on the equivalence ratio, a transition to a disordered state is observed, in which the ring structure disappears; the cells change their shape, size and number; and they move about in an irregular manner.

The principal experimental work on the dynamics of cellular flames is Markstein's series of papers (1951,1953a, b) which culminated in his treatise on the subject (1964). Markstein describes two dynamical regimes of cellular flames using a slot burner: steady and unsteady. The one-dimensional steady states were ordered lines of large numbers (from 20 to 50) of cells; the unsteady states were lines of cells in "rapid vibratory or translatory motion". The occurrence of these dynamical regimes was studied as a function of flow rate and equivalence ratio.

Markstein (1953a, b) also studied cellular flames freely propagating in circular tubes. Although pictures of these two-dimensional flames were presented, there was no categorization of the dynamical states and no reported observation of ordered cells comparable to the steady states of the slot burner. Most experimental combustion studies (Markstein,1951; Mitani and Williams, 1980; Vantelon et al., 1986) have been concerned with measuring and understanding the cell size. These experiments were conducted using hydrocarbon-air mixtures at atmospheric pressure with a large number ( > 50) of cells present in order to minimize boundary effects. The dynamics of cellular flames was described as "flames in incessant irregular motion, too fast to be analyzed visually". In discussions of measurements of cell size there is repeated reference to the variation in cell diameter and the necessity of averaging in order to measure this quantity.

Although there is anecdotal evidence of observations of ordered states of cellular flames from a number of researchers (Markstein, Pagni-personal communications), we have found no published experimental study of these states. In this paper the characteristics of ordered states in heavy hydrocarbon-air cellular flames stabilized on a porous plug burner are reported. These states consist of concentric rings of approximately equally sized cells with numbers ranging from five to thirty. The ordered cellular states have been found in propane, isobutane and pentane flames over a range of parameters in which a variety of periodic and chaotic states are also observed. They form a set of primary states from which these other states bifurcate. The details of our experimental setup have been described elsewhere (el-Hamdi et al., 1993). These experiments are conducted using a circular, stainless steel porous plug burner designed by Patrick Pagni (Vantelon etal., 1986) and manufactured by McKenna Products of Pittsburg, CA. The 5.62cm porous plug is cooled by flowing chilled water through a helically wound coil embedded in the plug. The design of this burner contributes significantly to our ability to observe these ordered states. The burner is placed is a combustion chamber made of process glass pipe where a constant pressure is maintained. The flame is viewed from above by a Dage-MTI Silicon Intensified Target (SIT) camera after the image is reflected from a plane mirror mounted at the top of the combustion chamber.

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As shown in Figure 1a, a steady flame front appears as a luminous disk, 0.5 mm thick sitting 5 mm above the porous plug. As the flow rate is increased the flame front curves away from the burner, forming cells. The dark lines are folds along which two cells join; three or more folds intersect to form a cusp. These cusp and folds correspond to regions of low emitted optical intensity, not to regions of extinction. They appear dark because of the limited dynamic range of videotape.

In this parameter range cellular flames form ordered states containing one or more rings of cells. With increasing flow rate there are transitions to other ordered states with different numbers of cells, to periodic states in which the cells move in rings, or to other chaotic states. The parameter values at which these particular ordered states, shown in Figure 1, were observed are listed in Table 1. These values are representative of the flow rate corresponding to each state. The absolute values of the parameters at which the ordered states are observed depend critically on the burner size and pressure. In general, the number of inner cells increases with the total flow rate; however, every ordered state is not necessarily observed at a given equivalence ratio. A complete stability boundary diagram is quite complicated because of the large number of possible states and the hysteresis in the flow rates.

A single ring of cells is not stable at a working pressure of 1/2 atmosphere. In order to observe a single ring the pressure was reduced to 1/3 atmosphere. The smallest number of cells in an ordered state that has been recorded on videotape is five as shown in Figure 1b. (An ordered state with four cells was seen in a coarse survey of parameter space, but not recorded). Five cells of approximately equal size surround a point which corresponds to a single central cusp. At a larger value of the total flow rate, a state with six cells is observed as shown in Figure 1c. If the flow rate were increased further, a second ring of cells would form. A careful survey of parameter space was made to find a state with seven cells around a point; no such state was ever found. Six is the largest number of cells that has been observed to form around a single point. Our investigations at this pressure have only been concerned with observations of a single ring of cells. and these two ordered states were the onlv states studied.

At a pressure of 1/2 atmosphere, the normal working pressure for most of our experiments, the simplest observed state is a ring of cells surrounding a single central cell shown in Figure 1d. As the flow rate is increased further, the number of outer cells and the number of inner cells change independently. Ordered states with a given number of inner (outer) cells can be stabilized with a range of values of the number of outer (inner) cells by suitably adjusting the flow rates and equivalence ratio. Ordered states are labeled according to the number of cells in each ring from the outside in. The ordered state in Figure 1j is called the 13/6/1 state. Most of the interesting dynamics, such as rotating and hopping, takes place in the ring of inner cells while the outer ring of cells remains fixed.

Ordered states with two, three, four, five and six inner cells are formed at successively larger flow rates, as shown in Figure 1e-i. Again, six is the maximum number of cells that are observed to form around a point. The next transition involves the formation of a third ring in which a single cell in the center is surrounded by six cells for a total of seven cells inside the outer ring.

As the flow rate is slowly increased further, a second cell appears in the inner ring and then a third cell as shown in Figure 1k. At this point a new pattern of cells begins to emerge. The three innermost cells act as the nucleating center for the growth of a hexagonal structure. Notice that the second ring of cells has distorted from a circular shape to a triangular shape. Each of the three cells in the interior now has six nearest neighbors arranged in a quasi-hexagonal array. The outer boundaries of this array collide with the ring of cells at the periphery, resulting in a mismatch region near the edge of the burner. At still larger values of the flow rate, cells which form in this region between the (outer) circular and (inner) hexagonal structures are in constant motion, and a recognizable ordered pattern never again appears. This point in parameter space and all points beyond it correspond to those of most other experimental studies of cellular flames.

There are other significant aspects of these observations of ordered states. Both the state with two inner cells, Figure 1d, and the state with six inner cells, Figure 1h, are unstable. They form, but after a relatively short time, on the order of seconds, they decay to another ordered state-the former to a state with one inner cell and the latter to a state with five inner cells.

The ordered cells are not steady; they are chaotic. A time series of the light intensity emitted at a point in a 10/3 ordered state is shown in Figure 2. The power spectrum of this time series has a broad peak whose high frequency fall-off can be fit equally well by a power law or an exponential, implying intermediate dimensional chaotic dynamics (el-Hamdi et al., 1993). The intensity variation of a disordered cellular flame is approximately ten times larger. A plot of a time series from a steady flame shows no intensity variation and the corresponding power spectrum is featureless. Direct visual observation and other measurements indicate that time dependence of cellular flames arises from low amplitude, small scale intensity variations which are strongest near the cusps and the folds.

The ordered states permit quantitative measurements of the emitted light intensity (using a photodetector), the local temperature (using a thermocouple) and the flow field (using a hot-wire anemometer) in the cusp and fold regions. Such measurements could not be made previously because of the constant irregular motion of the cells.

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The flow rates listed in Table 1 are representative values at which ordered states of cells are observed. Burners of different size will form these states at different values of the flow rate and pressure. The important parameter is the aspect ratio, which is the ratio of the burner area to the cell area. It determines the total number of cells on the burner. The ordered states of concentric rings of cells can be observed whenever this number is between five and (approximately) thirty. Burners with different geometries may form ordered states with different arrays of cells.

Although the particular values at which the ordered states are observed depends on the design of the burner, there are certain features of the phenomenology that any theoretical description of cellular flames must account for:

• Ordered states of cells for small values of the cell number,
  
• A transition from order to disorder as the number of cells is increased,
  
• A maximum of six cells around a point,
  
• Two cells around a point are not stable,
  
• Chaotic motion of the ordered states.
  

Two principal mechanisms have been studied extensively in order to describe the formation of cellular flames: the thermodiffusive instability in which mass diffusivity and thermal diffusivity compete; and the hydrodynamic instability in which the flame front interacts with the changes in the flow field that it generates. Williams (1985) and Clavin (1985) each provide an extensive review of the theoretical foundations of both models.

Most of the theoretical studies relevant to our experiments have used the thermodiffusive model. Shtilman and Sivashinsky (1991) have shown that the hexagonal structure of cellular flames is stable for spherical cellular flames provided the radius is sufficiently large. Because a flat flame corresponds to the case of an infinite radius, it would be expected that flat cellular flames would form an ordered hexagonal array in the absence of the boundaries.

The effects of the boundaries can be minimized by a transient experiment in which a parameter, for instance the fuel, is abruptly changed so that the steady state of a flat flame makes a transition to a cellular state with a large numbers of cells (small critical wavelength). The result of such an experiment is shown in Figure lk. This frame of videotape was taken just after the pattern formed. At a later time the inner cells relax to a state in which the boundary has a more pronounced effect. The pattern of interior cells is actually slightly rhombic, a feature explained by Ouyang, Gunaratne, and Swinney (1993).

Two theoretical studies have simulated cellular flames with a relatively small number of cells. In a series of papers Bayliss and Matkowsky (1990,1992) have examined the dynamics of a one-dimensional circular cellular flame front propagating radially inward (a ribbon flame). Their studies are numerical simulations of the full equations which describe the thermodiffusive instability. Bayliss and Matkowsky have selected parameters such that states with three, four and five cells are observed. Most of their results are pertinent to the parameter range in which periodic dynamics of rotation and hopping are observed. Their results include a transition from ordered to disordered states. Their circular line of ordered states are steady and not chaotic. Our measurements of a one-dimensional array of cells on a linear porous plug burner also found steady cells, suggesting that the chaotic motion of the cells is a two-dimensional effect.

Nicolenko (1993) has performed numerical simulations of a small number of cells using a modified form of the two-dimensional Kuramoto-Sivashinsky equation for a square geometry with periodic boundary conditions. His results capture all five of the phenomenological characteristics listed above, although points 4 and 5 were not explicitly addressed.

Numerical studies of cellular flames using the hydrodynamical model have not been undertaken for the constrained geometries comparable to those considered by Bayliss and Matkowsky or by Nicolenko. These five characteristics and other measurements of the properties of ordered states may provide the basis for separating the effects produced by the hydrodynamic and thermodiffusive instabilities in the formation of cellular flames. Our current results cannot be interpreted as favoring one model over the other.

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Shtilman and Sivashinsky (1991) predicted an abrupt transition from an ordered hexagonal state to a disordered state in which the ordered array is replaced by irregularly recombining time-dependent cellular structures. A transition from some of the ordered states, shown in Figure 1, to disordered states is observed over a wide range of parameters. The flow rate at which this transition takes place depends on the equivalence ratio and the total number of inner cells.

This transition to disorder is currently observed in the experiment by visual identification. A more quantitative method will be necessary for a precise measurement of the "melting" curve of the ordered states. A systematic search has not been undertaken to determine the minimum number of cells which must be present before the transition to a disordered state is observed, although experience would suggest that at least two rings of cells must be present.

The spatial and temporal characteristics of a disordered cellular flame is compared with that of an ordered cellular flame in Figure 3. The first column shows four successive frames of the 12/4 state in which four inner cells are surrounded by twelve outer cells. The second column shows the time dependence of the disordered state that occurs at a parameter just above that at which the 13/6/1 state is stable. In the disordered state the ring structure is broken, the cells are no longer of equal size, and they move around in an irregular manner. A dot has been placed to mark the same cell in each successive frame.

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The top view of the ordered states presented in Figure 1 emphasizes the arrangement of the cells but does not adequately convey the sharp structures that protrude up away from the burner surface. The unmistakable impression of each of these states from direct visual observation is one of ordered arrangements of cusps and folds.

One of the many dynamical modes associated with the ordered states is the rotation of the 12/6/1 state in which the entire flame front slowly rotates relative to the burner. Corresponding top and side views (a separate fixed camera is used for each view) at two different angles are presented in Figure 4 in order to demonstrate how the rotation of the cellular pattern is also visible in the motion of the array of cusps. A dot marks the same cell in each frame. There is an optical illusion associated with the side view. Many observers see a pattern similar to water drops on the underside of a plate. The correct view is that the foreground is at the bottom of the figure. The dark cusps stick up and point away from the burner surface.

The measurement of the positions and the amplitudes of the cusps and folds from the side view is complicated by bright cells shining through the darker cusps and folds. A careful comparison of Figure 4a and 4b shows that the bright cells "chop" offthe tops of the cusps in their immediate foreground. The identification of the positions of the cusps becomes even more difficult for states without the symmetry of the 12/6/1 state.

The picture that emerges from our experiments is that the ordered states can be viewed as independent, interacting, rings of cells which subtly change their size and shape. An alternate, but equivalent, view is that they correspond to a lattice of cusps connected by folds, executing small amplitude chaotic vibrations. Steady two-dimensional cellular flames have never been observed. Cellular flames exhibit chaotic dynamics at every point parameter space. The ideas and concepts of nonlinear dynamics and chaos are essential and crucial to a proper description of this dynamics.

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References

[2] Bayliss, A. and Matkowsky, B. J. (1992). Nonlinear Dynamics of Cellular Flames, SIAM J. Appl. Math. 52, 396.

[3] Clavin, P. (1985). Dynamical Behavior of Premixed Flame Fronts in Laminar and Turbulent Flows, Prog. Energy Combust. Sci. 11, 1.

[4] el-Hamdi, M., Gorman, M. and Robbins, K. A. (1993). Deterministic Chaos in Laminar Premixed Flames: Experimental Classification of Chaotic Dynamics, Combust. Sci. and Tech. 94, 87.

[5] Markstein, G. H. (1953a). Instability Phenomena in Combustion Waves, 4th Symposium on Combustion, Williams and Wilkins, Baltimore, 44.

[6] Markstein, G. H. and Somers, L. M. (1953b). Cellular Flame Structure and Vibratory Flame Movement in N-Butane-Methane Mixtures, 4th Symposium on Combustion, 527.

[7] Markstein, G. H. (1951). Experimental and Theoretical Studies of Flame Front Stability, J. Aero. Sci. 18,199. Markstein, G. H. ed. (1964). Non-Steady Flame Propagation, Macmillan, NY.

[8] Mitani, T. and Williams, F. A. (1980). Studies of Cellular Flames in Hydrogen-Oxygen-Nitrogen Mixtures, Comb. and Flame 39, 169.

[9] Nicolenko, B. (1993). Phase Turbulence, Spatiotemporal Intermittency and Coherent Structures, in Dynamics Proceedings at Les Houches, Plenum Press, in press.

[10] Ouyang, Q., Gunaratne, G. and Swinney, H. L. (1993). Formation of Rhombic Patterns, Chaos, 3, 707.

[11] Shtilman, H. and Sivashinsky, G. I. (1990). On the Hexagonal Structure of Cellular Flames, Can. J. of Phys. 68, 768.

[12] Vantelon, J. P., Pagni, P. A. and Dunsky, C. M. (1986). Cellular Flame Structures on a Cooled Porous Plug Burner, in Dynamics of Reactive Systems, Prog. in Astronautics and Aeronautics,

[13] Bowen, J. R., Leyer, J.-C. and Soloukin, R. l. (eds.) 105, 131.

[14] Williams, F. A. (1985). Combustion Theory, Benjamin-Cummings, Menlo Park.