M. Gorman (gorman@uh.edu),
C. F. Hamill
M. el-Hamdi
(el-hamdi@uhphys.phys.uh.edu)
Department of Physics
University of Houston
Houston, TX 77204-5506
K. A. Robbins (krobbins@runner.utsa.edu)
Division of Computer Science
The University of Texas at San Antonio
San Antonio, Texas 78249
Combustion Science and Technology 98 (1994)pp. 25-35.
Ordered states of cellular flames on circular porous plug burners consist of concentric rings of cells. At certain values of the flow rate and equivalence ratio a transition is made to a state in which entire rings of cells rotate. The direction of rotation depends only on the initial conditions. Our observations of rotating cellular flames include a single rotating ring, an outer ring rotating about a single inner cell, a rotating inner ring surrounded by a fixed outer ring, and two corotating or counterrotating concentric rings. A rotating ring of cells can also make a transition to a modulated rotating state in which the shapes of the cells and the speed ofrotation periodically change. In another rotating mode, a single central cell takes the shape of a spiral which rotates inside a fixed outer ring of cells. The physical characteristics of these modes are described and comparisons with relevant theoretical studies are made.
From the first discovery of a polyhedral flame structure by Smithells and Ingle (1892), cellular flames were observed to rotate. These authors noted that "the inner cone is divided by lines into several petal-like segmentswhich often revolve with great rapidity around the vertical axis". Later, in a more systematic study, Smith and Pickering (1929) found that the rate of rotation could be influenced by changing the composition or the flow rate.
Markstein (1964) also observed rotating cells in his study of cellular flames freely propagating in tubes. "Greatly enlarged cells symmetrically arranged and rotating about the tube axis were observed when the wall temperature temperature was increased further." The single picture presented does not adequately convey the motion of the cells.
We have found a number of rotating states of cellular flames in experiments on circular porous plug burners. These rotating states are observed in transitions from ordered states which consist of concentric rings of cells or from other periodic states. Our observations of rotating states include: a single rotating ring of cells, a rotating ring of outer cells around a single inner cell, a rotating ring of two cells inside an outer ring of fixed cells, two corotating or counterrotating rings of cells, a modulated rotating state, and a spiral-shaped cell rotating inside a fixed outer ring of cells. These states are observed in a range of parameters of flow rate and equivalence ratio where the dynamics of the cellular patterns change significantly, but the traditional issues of combustion science do not.
In section 2 the relevant aspects of the experimental setup are described. In sections 3-5 the characteristics of rotating states, a modulated rotating state and a rotating spiral state are described, respectively. In section 6 the relevant theoretical work on rotating cellular flames is reviewed. In section 7a discussion of the implications of these results is presented.
Our experimental apparatus has been described elsewhere (el-Hamdi, Gorman and Robbins, 1993). Cellular flames of isobutane-air mixtures are stabilized on a 5.62 cm diameter circular porous plug burner which is housed in a combustion chamber made from process glass pipe. The rotating states are observed in bifurcations from ordered states of concentric rings which are easier to stabilize at low pressure. The flow rates of the gases and the pressure of 1/2 atmosphere in the chamber are controlled to 0.1%. At this pressure a steady flame is a luminous disk, 0.5 mm thick sitting 5 mm above the porous plug.
The flame motion is viewed from above, through a mirror mounted at the top of the combustion chamber, and is recorded using a Dage-MTI Silicon Intensified Target camera. This camera has a lag characteristic of 1 1/2 frames which is the time required for the image intensity of a point which suddenly disappears (or moves) in the view of the camera to decay to 1/e of its original value. The high sensitivity and low lag combine to minimize the effects of blurring due to the characteristics of the vidicon; however, there is still substantial blurring from the video frame rate, 30Hz. Standard NTSC video of the camera output is displayed on a monitor and recorded on 1/2" VHS videotape. Measurements of angular position of the cells are made by displaying each frame on a 25" video monitor and superposing a 360 degree angular grid. The leading edge of each cell in the direction of motion is taken as the position of the cell.
Ordered states of cellular flames are labeled according to the numberof cells in each ring moving from the outer ring inward. For instance, the 12/6/1 state has a ring of twelve outer cells concentric with a ring of six inner cells which surrounds one central cell. Rotating states are labeled in a similar fashion, with an R attached to the number corresponding to the rotating ring. A superscript asterisk indicates clockwise rotation; a subscript asterisk, counterclockwise rotation. A state with six cells rotating counterclockwise around one inner cell is designated as a 6R*/1 state.
A. Physical Characteristics
Figure 1 shows four representative examples of rotating cellular states observed in our experiments. Five consecutive frames of digitized videotape are displayed vertically. A dot marks the same cell in each frame sequence. The flow rates, equivalence ratio, and rotation rate corresponding to each rotating state are listed in Table 1. Figure la shows a 6R* state in which six cells rotate clockwise around a bright point in the center of the burner. Some of the physical features displayed here are characteristic of rotating states. There is substantial blurring of the image due to the magnitude of the cell motion relative to the video frame rate. The pattern in Figure la completely rotates in 34 frames. The blurring of the images of the individual cells significantly complicates our ability to measure the spatial characteristics of the motion.
Figure lb shows a 5R*/1 state in which a ring of five cells rotates clockwise around one central cell which maintains a constant diameter. In Figure lc two cells rotate inside a fixed outer ring of ten cells, creating the 10/2R* state.
| Figure 1 | State | Flow Rate | Flow Velocity | Equivalence Ratio | Rotation Rate |
|---|---|---|---|---|---|
| 1a | 6R* | 6.11 | 4.02 | 2.3 | 0.88 |
| 1b | 5R/1 | 5.7 | 3.75 | 2.0 | 0.75 |
| 1c | 10/2R* | 5.8 | 3.82 | 2.0 | 2.7 |
| 1d | 6R*/2R* | 7.24 | 4.76 | 2.3 | 2.0,I;0.90,O |
| 2 | 6R/1 | 6.84 | 4.50 | 2.1 | 0.81 |
| 3 | Mod. 5R* | 5.9 | 3.88 | 1.9 | 0.60 |
| 5 | 13/1SP | 5.6 | 3.68 | 2.1 | 5.7 |
Rotating wave solutions have the property that the direction of rotation depends on the initial conditions. Figure 2 shows six cells rotating around one central cell, in the (a) counterclockwise, 6R*/1, or (b) clockwise, 6R /1, direction. Both of these states are observed at the same values of the flow rates of the gases.
B. Bifurcation structure
In polyhedral flames on Bunsen burners rotating cellular states probably occupy a substantial region of parameter space because they are encountered in even the most casual study (Smithells and Ingle, 1892). In our experiments rotating rings of cells are difficult to find. They occupy relatively small regions in parameter space in isobutane-air flames and have not been observed at all in propane-air flames.
The most commonly encountered rotating cellular state is that of Figure lc in which two cells rotate inside a fixed outer ring. This state occurs in a parameter range between that corresponding to an ordered state with one inner cell and one with three inner cells. It is typically found as the flow rate is decreased. An ordered state with three inner cells becomes unstable. One of the three cells disappears and the other two begin to rotate. This state is stable over a range of equivalence ratios and flow rates.
The other rotating states--- Figure 1 a,b,d and Figure 2---were discovered, somewhat by accident. This general region of parameter space had been visited previously without finding them. These states result from a sequence of bifurcations that happen in a narrow range of parameters. They are observed in a region of parameter space in which a number of states compete for stability.
At the onset of the cellular instability the ordered state with the largest critical wavelength is an 8/1 state in which eight outer cells encircleone inner cell. With increasing flow rate the number of cells typically increases. The number of outer cells can decrease when the pulsating radial state interacts with an ordered cellular state. In this pulsating-cellular state the outer ring of cells is modulated by the radial motion of the pulsations. The number of cells in the outer ring changes with successive maxima of the oscillation, usually decreasing the number from that of the ordered state. In this case the number is reduced from eight to six outer cells. At the maximum amplitude of one cycle the six outer cells begin to rotate about the inner cell, and the radial pulsations disappear. A rotating ring of six cells is a relatively stable structure from which other rotating states can bifurcate. A slight decrease in the flow rate increases the stable wavelength, triggering a collapse of the inner cell producing the state pictured in Figure la. The bright spot at the center of each frame is the remnant of that central cell. If, instead, the flow rate is abruptly increased from the 6R /1 rotating state, the double rotating state, 6R /2R* of Figure ld appears.
The point of this extended discussion is to emphasize the complex nature of the stability boundary diagram associated with the states of cellular flames and to describe some of the techniques for producing these states. Most of our observations have been made with slow, small variations of parameters. The observations of double rotating states show the efficacy of abrupt, large changes.
Modulated states of rotating cellular flames have also been observed. Figure 3 shows twenty consecutive frames of a state with a single ring of five rotating cells whose spatial characteristics change with time. A dot marks the same cell in each frame, and we have placed it at the leading edge to indicate our determination of the cell position.
This particular state has characteristic frequencies of rotation and modulation that are nearly commensurate with the video frame rate. In ten frames the 5-cell pattern rotates 1/5th of a revolution, so that the eleventh frame looks identical to the first, rotated by 72 degrees. A similar correspondence is observed between the second and the twelfth and so on to the tenth and the twentieth.
This 5-cell pattern is modulated every 3 1/3 frames. The effect of the modulation can be seen by noticing frames that differ by four frames look somewhat alike, ones that differ by seven look more alike (2 and 9, 4 and 11, 5 and 12, 6 and 13, etc.) and those that differ by ten are almost identical.
Any four consecutive frames comprise (slightly more than) one modulation cycle. In frame 1 the five cells appear as in the purely rotating state. In frame 2 the intensity of the flame front has diminished, but the overall shape of the blurred cells remains the same. In frame 3 each cell of the flame front contains a small, almost circular, bright region and the overall radial extent of the pattern has decreased. In frame 4 the flame front appears quite similar to that in frame 1.
Here is a plausible scenario for this modulation. As the cells rotate, they increase their curvature and reduce their rotation speed. Their emitted intensity decreases with the increased curvature as shown in frame 2. In frame 3 the intensity increases as the curvature decreases and the cells speed up, returning to their original speed and intensity in frame 4. A camera with a higher frame rate will be necessary to quantitatively measure the characteristics of this modulation.
The angular displacement of the leading edge of the marked cell is plotted in Figure 4 for each of the twenty frames displayed in Figure 3. The modulation of the angular velocity is clearly visible. The uniform rotation of a 5-cell state, observed at a slightly different value of the flow rate, is shown for comparison.
In the parameter region in which one steady inner cell and two rotating inner cells compete for stability, a state in which one inner cell takes the shape of a spiral is also observed. In the two sequences of five consecutive frames shown in Figure 5, this spiral-shaped cell is shown to rotate in either the clockwise or the counterclockwise direction, depending on the initial conditions. This spiral state is the only state in which a cell takes a shape other than a circle (as in Figure lb) or an ovoid (all other cells).
A side view of the spiral indicates that it is three-dimensional. Its tip is closest to the burner and the spiral tail extends upward. As the tail sweeps around, it creates a small amplitude modulation of the inner side of the cells in the outer ring. Spiral motion is usually characterized by the dynamics of the tip. The frames in Figure 5 suggest either periodic or chaotic tip motion. Other observed states show a tip which rotates about a fixed point.
Both the hydrodynamical model and the thermodiffusive model have been used in studies which have found rotating solutions. Buckmaster (1984) considers a model in which a polyhedral flame is represented as a one-dimensional front in the shape of a circle. A state with m cells interacts with one of 2m cells to produce a rotating cellular flame. This model demonstrates that rotating states are solutions of the equations; but the restrictive assumptions of the model imply that the relevant physical situation is a polyhedral flame in which a state with one cell interacts with a state with two cells to produce a rotating state. Such states are not observed on Bunsen burners or on flat flame burners.
Margolis and Sivashinsky (1984) employed a nonlinear stability analysis to describe the transcritical bifurcation of axisymmetric cylindrical cellular flames. They also used the hydrodynamical model (1990) to consider the interaction of a certain class of nonaxisymmetric cellular flames near a double eigenvalue of the linearized problem. Using a higher-order bifurcation analysis they find spinning cellular flames as a nearly vertical secondary or a tertiary infinite-period bifurcation from the steady branch of (ordered) cellular flames. Although the class of nonaxisymmetric flames is restricted in a manner similar to that of Buckmaster's model, the results indicate the kind of bifurcations that are likely to occur.
Recently, Bayliss and Matkowsky (1992) and Bayliss, Matkowsky, and Riecke (1993) have conducted numerical studies of the full equations that describe the thermodiffusive instability. They consider a one-dimensional flame front concentric with a source of premixed gas. As the Lewis number is decreased, an ordered state with four cells is found to undergo an infinite period Hopf bifurcation. At different parameter values other rotating states are observed. These states are labeled TWN, corresponding to a traveling wave with N cells.
A. Rotating States
In an experiment an infinite period bifurcation corresponds to an ordered state which slowly begins to rotate. None of the rotating states reported in this paper have been observed to occur in this manner. These rotating states appear in abrupt transitions from ordered states with different numbers of cells or from other periodic states. The rings of cells rotate rapidly, not slowly, implying that the rotating states are observed at some distance from the point at which they bifurcate from the corresponding ordered states. Such an observation is consistent with a nearly vertical secondary bifurcation found by Margolis and Sivashinsky (1990).
A common characteristic of all the rotating states presented here is that the corresponding ordered states are not observed at any other flow rate. Some of the corresponding ordered states are observed at other pressures; some are never observed. Cellular flames with a single ring of five or six ordered cells are stable at 1/3 atmosphere, but not at 1/2 atmosphere, the working pressure of these experiments. Similar results hold for the 6R/1 and 5R/1 rotating states and their corresponding ordered states, 6/1 and 5/1. An ordered state with two steady inner cells has never been observed-regardless of the pressure. These states are not stable and decay immediately to a state with one inner cell.
B. Modulated Rotating States
Bayliss, Matkowsky and Riecke (1993) discuss three type of modulated traveling waves. A modulated traveling wave, MTW1, corresponds to our observations of hopping states (Gorman, el-Hamdi and Robbins, 1994). Two other modulated traveling waves, MTW2 and MTW, correspond to changes in the relative sizes of adjacent cells. The modulated rotating state reported here in Figure 3 has all the cells modulated in phase, so it is not possible to make a quantitative comparison with their numerical results.
C. Spiral States
Spiral solutions have been seen in models of a variety of chemically reacting systems (Keener and Tyson, 1986), and spirals have been observed in a number of physical systems (Muller, Plesser and Hess, 1987; Winfree, 1972). Numerical studies of cellular flames in constrained geometries have not found spiral solutions. In studies based on the thermodiffusive model Bayliss and Matkowsky (1992) considered a one-dimensional circular flame front, a geometry incapable of exhibiting two-dimensional motion. Nicolenko's (1993) numerical studies of the modified two-dimensional Kuramoto-Sivashinsky equation were implemented with a square geometry and periodic boundary conditions. There have not been any comparable studies using the hydrodynamical model.
Six kinds of rotating states are presented: 1) a single rotating ring, 2) a fixed outer ring and a rotating inner ring, 3) a rotating outer ring surrounding a single, circular inner cell, 4) two rotating rings, 5) modulated rotating rings, and 6) a spiral rotating inside a fixed outer ring. Each of these states allows us to investigate particular characteristics of the ring structure of the ordered states. The double rotating states, such as the 6R/2R states, are particularly interesting because they are a dramatic demonstration that the concentric rings are independent, interacting dynamical systems. These states were discovered in a survey of parameter space and their characteristics have not been extensively measured.
This research was supported by a grant, N00014-K-0613, from the Office of Naval Research. Multimedia versions of this paper with embedded video can be obtained from the ftp site vip.cs.utsa.edu. We appreciate useful conversations with Bernard Matkwosky, Alvin Bayliss, Stephen Margolis, John Buckmaster and Hermann Riecke. Manickam Neelakandan prepared figures for this manuscript.
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