Four Types of Chaotic Dynamics
in Cellular Flames
In order to demonstrate the characteristics of this dynamics more clearly, fifteen sequential frames of the intermittently ordered 9/3 state have been printed in Figure 4. An ordered 9/3 state is visible in frame 1. In frames 2 through 12 the ring structure is not identifiable, and it is only the outer boundary of the luminous area that suggests the circular geometry of the burner. In frames 13 and 14 the outer ring can be seen, but there is no hint of the presence of the three inner cells which abruptly reappear in frame 15.
The visual impressions of the disordered and intermittently ordered states are quite different. In the disordered state an approximately ordered state is seen all the time; but there is never a time at which the ordered state threatens to reappear. In the intermittently ordered state a rapidly changing, highly irregular cellular pattern, which does not even remotely suggest a ring structure, is present most of the time. An ordered state seems to suddenly materialize out of nowhere and remains for varying lengths of time. Most of the residence times of the ordered states are short, on the order of ten frames, but some are quite long, persisting for hundreds of frames. A histogram of the residence times has an exponential distribution, with a characteristic time of about 2 seconds (60 frames).
The disordered states are always observed at a flow rates larger than those of the ordered states. Once this critical value of the flow rate for the onset of disordered states has been passed, an ordered state never occurs again. The intermittently ordered states are observed at flow parameters that are interleaved between those corresponding to the ordered states. Their relative occurrences are shown in Figure 5 and Table 1.
Figure 5 is a composite picture depicting the occurrence of the intermittently ordered states with increasing flow rate in propane-air mixtures. These states are prevalent in propane-air cellular flames; they have not been observed in isobutane-air flames. In this figure a temporal sequence of frames, corresponding to the motion of each observed state, is displayed horizontally and the succession of states observed with increasing flow rate is displayed vertically. Ordered states with 1, 3, 4, and 5 inner cells can be seen respectively in rows a, c, e, and g. In alternate rows--b, d, f, h-- the corresponding intermittently ordered states are observed.
At flow rates above 7.35 lit/min, corresponding to the 9/3 ordered state, an intermittently ordered state is observed in which the 9/3 state abruptly appears, for varying lengths of time. Other ordered states (not shown) also appear. Ordered states in which the number of inner and/or outer cells changes by one cell, such as the 10/4, 9/4, or 10/3 states, are also observed. Different relative orientations of the inner and outer rings in the ordered states also occur, as can be seen by comparing frames 1 and 5 in row d of Figure 5. As the flow rate is increased, a state with four inner cells appears more often and for greater lengths of time, until at a flow rate of 8.39 lit/min the ordered 4-cell state becomes stable.
| Figure | State | Flow Rate | Flow Velocity | Eq. Ratio |
|---|---|---|---|---|
| 5a | 9/1 | 6.32 | 4.16 | 1.83 |
| 5b | 1HC | 6.49 | 4.27 | 1.85 |
| 5c | 9/3 | 7.35 | 4.84 | 1.85 |
| 5d | 3HC | 8.27 | 5.44 | 1.85 |
| 5e | 10/4 | 8.39 | 5.52 | 1.85 |
| 5f | 4HC | 8.62 | 5.67 | 1.85 |
| 5 | 11/5 | 8.80 | 5.79 | 1.85 |
| 5h | 5HC | 9.38 | 6.17 | 1.85 |
| 6a | Pulsate | 5.16 | 3.39 | 2.22 |
| 6b | Puls-Cel | 5.52 | 3.63 | 2.22 |
| 6c | Cellular | 6.10 | 4.01 | 2.21 |
The absence of an ordered state with two inner cells in this sequence of states in propane-air flames is related to our previously reported observation of the absence of an ordered state with two inner cells in isobutane-air flames. It would be expected that such an ordered state would be observed between those parameter values corresponding to ordered states with one and three inner cells. For a reason that has yet to be clearly identified, ordered states of cellular flames with two inner cells are not stable states of the system.
