HOPPING MOTION IN ORDERED STATES OF CELLULAR FLAMES

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M. Gorman (gorman@uh.edu),
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. 71-78.

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Abstract

Introduction

A Representative Experiment

Other Experiments

Relevant Theoretical Studies

Discussion

References

Our previous experiments have shown that cellular flames form ordered states consisting of concentric rings of cells. The numbers of cells in the inner and outer rings change independently in integer steps as the flow rate is increased. In this paper we report the observation of states characterized by a hopping motion in which cells abruptly change their angular position in the ring. This hopping proceeds sequentially to the other cells in the ring. The hopping states are typically observed in isobutane-air cellular flames at parameter values between those corresponding to two consecutive ordered states. The physical characteristics of these states are similar to those of modulated traveling waves found by Bayliss, Matkowsky and Riecke in numerical simulations of the full equations that describe the thermodiffusive instability. The similarities and differences between our experimental results and their theoretical predictions are discussed.

Ordered states of cellular flames consist of concentric rings of cells. Our previously reported observations (Gorman, Hamill, el-Hamdi and Robbins, 1993) of periodic dynamics associated with the ring structure of ordered states include rotating rings of cells and the spiral motion of a single cell. These two types of phenomena are also found in other physical systems (Gorman and Swinney, 1982; Winfree, 1972), and the characteristics of the observed states and their solutions have been extensively studied and documented in a number of studies (Farr and Golubitsky, 1992; Keener and Tyson, 1986).

This paper presents experimental observations of an unusual hopping motion which is observed when an individual cell abruptly changes its angular position in the ring. The physical characteristics of this motion are similar to those of modulated traveling waves found by Bayliss, Matkowsky and Riecke (1992).

Our experimental setup has been described elsewhere (el-Hamdi, Gorman and Robbins, 1993). A 5.62 cm water-cooled circular porous plug burner is placed in a combustion chamber made of process glass pipe. A mirror at the top of the chamber allows a Silicon Intensified Target camera to view the flame from the top. In a typical experiment the flow rate and the equivalence ratio are varied while the type of fuel and the ambient pressure remain constant. At this pressure a steady flat flame is 0.5 mm thick and sits 5 mm above the porous plug. All these experiments were conducted at an atmospheric pressure of 1/2 atmosphere because ordered states of cellular flames are easily stabilized there. Hopping states have been observed in isobutane-air flames but not propane-air cellular flames.

The motion of the flame front is recorded on videotape using a VHS videotape recorder. A time code generator is used to label each frame. Certain frames are selected, captured by a frame grabber, digitized, enhanced, and printed using a laser printer. Measurements of the angular positions of the cells are made by superposing a 360 degree angular grid over the image of the flame on a 25" video monitor.

Ordered states of cellular flames are labeled according to the number of cells in each ring, moving from outside to inside. A 12/6/1 state corresponds to an outer ring of twelve cells surrounding an inner ring of six cells which encircles a one central cell. Hopping states are designated in a similar manner with the letter, H, placed next to the ring corresponding to the hopping motion. A state with eleven fixed outer cells and three hopping inner cells is labeled 11/3H.

In the section 2 the characteristics of the 11/3H state are presented. In section 3 three additional hopping states are described. In section 4 a direct comparison is made between our experimental results of the 13/4H state and the numerical simulations of Bayliss, Matkowsky and Riecke of a ring of four cells. In section 5 the similarities and differences between the numerical results and the experimental observations are discussed.

A representative example of a hopping state of cellular flames is shown in Figure 1. Twelve consecutive frames of digitized videotape of the 11/3H state are presented. The first frame is a view of the entire cellular flame, with eleven outer cells and three inner cells; the other frames are close-ups of the inner cells. This sequence depicts three consecutive hops by the inner cells; it also illustrates the basic characteristics of hopping states. When viewed on videotape, the hopping motion is quite striking. In the 11/3H state the hopping frequency is 6 Hz, and the visual appearance is that the cells hop in rapid fire succession, one after another.

In frame 1 the cell marked with dot begins a hop which is completed in frame 6. The cell behind it begins a hop in frame 5 which is completed in frame 8. The cell ahead of it begins a hop in frame 7 which is completed in frame 11. The cell marked with a dot is at rest in frames 6-8 and begins a second hop in frame 9. This mode was originally designated the "square dance" mode (el-Hamdi, Gorman and Robbins, 1990) because it appeared as if three cells were moving among four sites arranged in a square.

The quantities that characterize the hopping motion are: the time or duration of a hop, the angular displacement of a hop, and the relative phase of successive hops. The video frame rate of 30 frames/sec is too slow relative to the typical hopping time (~5 frames) for a precise determination of the time at which the cell begins or finishes its hop. For instance, in the description above, the duration of each hop for the three cells is six, four and five frames respectively. The relative phase and the duration of the hop are, therefore, eliminated as measureable quantities. The hopping motion is such that a given cell is at rest for at least one frame The angular displacement in consecutive hops can be determined by a direct measurement from the frames of videotape.

The hopping states are observed in transitions from ordered states or other periodic states, both as the flow rate is increased and decreased. A typical point in parameter space has multiple possible stable states depending on initial conditions, but there is usually a region of parameter space in which only a hopping state is stable.

Four hopping states are shown in Figure 2. Five consecutive frames of videotape are shown for each of these states. Representative parameter values correspondingto each state are listed in Table 1.

In Figure 2b the hopping motion among the cells in the 13/5H/1 state is shown using consecutive alternate frames. In frame 1 the marked cell is at rest. This cell begins its hop in frame 2 and completes the motion by frame 5. The time between frames has been chosen so that each pair of successive cells are in motion in each frame. In this sequence the hopping motion moves clockwise from 11 o'clock to 7 o'clock.

Hopping motion in the outer ring occurs only when a very stable structure is present in the inner ring. Otherwise, the motion in the outer ring perturbs the inner ring and a highly complicated motion of both rings ensues. A single inner cell and six cells surrounding one central cell are the only arrangements of inner cells in which stable hopping motion has been observed in the outer cells.

Figure 2c shows the hopping motion of the llH/6/1 state at intervals of 1/6 second in which eleven cells in the outer ring sequentially change their position around seven fixed inner cells. Figure 4d shows five frames, 1/15 second apart, depicting the hopping motion of the 7H/1 state. In this example the seven cells in the outer ring move around one central cell. The motion has a qualitatively different visual appearance from the other four examples. appearing more wave-like than hopping.

Bayliss, Matkowsky and Riecke (1993) performed numerical studies of the full equations describing the thermodiffusive model. They considered a ribbon flame, in which a one-dimensional flame front in the shape of a circular ring, is concentric with the source of premixed gas. Ordered states with a small number of cells, traveling waves, and modulated traveling waves occurred in the parameter range considered. A comparable study of a one-dimensional flame front in a constrained geometry has not been undertaken using the hydrodynamical model.

At parameter values between those corresponding to a 4-cell state and a 5-cell state a modulated traveling wave, labeled MTW1, was found. In this state a cell expands. compressing the cell in front of it. At a later time a second cell, immediately behind the first cell, expands, compressing the first cell. This process proceeds sequentially around the ring of cells.

In the numerical simulations of this state four cells fill the ring, as shown in Figure 3. The boundaries between the cells are demarked by humps which correspond to decreases in the local temperature or, equivalently, the distance of the flame front from the burner. This graph depicts the sequential motion of cells 2, 3, and 4. The periodic nature of the state can be seen in the constant relative phase of the cell motion and the uniform angular displacement of the four cells between the bottom and top of Figure 3.

The 13/4H state from our experiments can be used for a direct comparison with these results. Twelve consecutive frames of this state are displayed in Figure 4. Frame 1 is a view of the entire flame front, and the other eleven frames are close-ups of the inner cells. In frame 2 the four cells are labeled with numbers corresponding to those in Figure 3.

The motion of the cells in Figure 4 can be correlated with the corresponding motion depicted in Figure 3. The frame numbers associated with each hop are indicated next to the corresponding motion in the simulation. Cells 2, 3, and 4 sequentially change their angular positions in frames 1-6, 3-8, and 6-11 respectively.

There is a subtle difference between the motion depicted in the simulations and that observed in most of the experiments. The hopping states shown in Figure 1 and Figure 2a-c show a single motion of a cell jumping from one site to another. This one-step motion appears like that in the game of musical chairs, with N cells hopping among N+1 sites. In the simulations there are two parts to each angular displacement: the expansion of a given cell, followed by the compression from the cell behind it. This motion is like that of a caterpillar in which one part expands, compresses the part in front of it, then gets compressed by the part in back. Bayliss, Matkowsky and Riecke refer to this mode as pushmipullyu in order to convey the sense of this motion.

This discretization of the hopping motion in Figure 1, Figure 2a-c, and Figure 4 may be caused by the presence of another ring of cells which creates a modulated structure in (or around) which the hopping motion takes place. In Figure 2d the single central cell is circularly symmetric. In this case the wave-like motion of the cells most closely resembles the motion found in the simulations.

The absence of a modulated structure associated with the inner ring allows the angular displacement to take on a much wider range of values. Hopping states with a single inner cell can have large angular displacements relative to the cell size. For instance the 7H/1 state in Figure 2d can have angular displacements as large as 90ø which is almost two cell widths.

Hopping states with the same number of inner cells appear to be periodic in visual observation. The full complexity of this dynamics only becomes apparent with a quantitative analysis of extended lengths of videotape. The principal manifestation of the differences between theory and experiment is the variation in angular displacement with time. In MTW1 of Bayliss et al. the angular displacement, the relative phase of the cells, and the duration of the hopping motion each remain constant during the motion. In Figure 5 a histogram of measured values of the angular diplacement for the 11/3H state is presented, demonstrating the 30 degree range of measured displacements. These results imply that the relative phase and the hopping time also vary. Hopping states with different numbers of outer cells, such as 12/3H and 11/3H, should have different hopping dynamics.

The numerical simulations do not give any guidance concerning the magnitude of the angular displacement. One interpretation of the hopping states is that they are traveling waves which result from an interaction between an ordered ring with N cells and one with N+1 cells. Such a view is supported by the occurrence of these states, in both the simulations and the experiments, at parameter values in between those of the corresponding N and N+1 ordered states.

This research was supported by a grant, N00014-K-0613, from the office of Naval Research. Multimedia versions of this paper with embedded video are available from the ftp site vip.cs.utsa.edu. We thank Bernard Matkowsky, Alvin Bayliss and Hermann Riecke for useful conversations concerning their numerical results on the hopping states. Manickam Neelakandan prepared the figures for this paper.

References

[1] Bayliss, A., Matkowsky, B. J and Riecke, H. (1993) Modulated Traveling Waves in Combustion, in Numerical Methods for PDE's with Critical Parameters, edited by H. Kaper and M. Garbey, Klower Academic Publishers, to appear.

[2] el-Hamdi, M., Gorman, M., and Robbins, K.A. (1990) A Picturebook of Dynamical Modes of Flat, Laminar Premixed Flames, Technical Report #2, University of Houston.

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

[4] Farr, W. W. and Golubitsky, M. (1992) Rotating Chemical Waves in the Gray-Scott Model, SIAM J. Appl. Math. 52, 181.

[5] Gorman, M., el-Hamdi, M. and Robbins, K. A. (1994) Experimental Observation of Ordered States in Cellular Flames, Comb. Sci. and Technol., 98 37.

[6] Gorman, M., Hamill, C. F., el-Hamdi, M. and Robbins, K. A. (1993) Rotating and Modulated Rotating States in Cellular Flames, Comb. Sci. and Technol..

[7] Gorman, M. and Swinney, H. L. (1982). Spatial and Temporal Characteristics of Modulated Waves in Circular Couette Flow, J. Fluid Mech. 117, 123.

[8] Keener, J. P. and Tyson, J. J. (1986). Spiral Waves in the BelousovZhabotinskii Reaction, Physica D 21, 307.

[9] Winfree, A. T. (1972) Spiral Waves of Chemical Activity, Science 175, 634.