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Problem 4: Quenching by Cellular Flow, Le=1
The initial band of hot material is distorted by cellular flow.
Reaction term is a step function of temperature, so that no burning occurs
if the temperature is lower than critical. For temperatures higher than
critical, the burning rate is constant and chosen to make laminar speed
equal to unity.
Some Visual Observations
Quenching by cellular flow requires much higher velocities compared to shear flow, but it is possible.
The process of flame propagation is as follows. In any unburned row, hot material from a neighboring row is quickly dispersed, uniformly preheating the cell. Before ignition in that cell, preheating of the next cell begins.
Quenching occurs if a row of unburned cells loses too much heat to its cooler neighbors before igniting.
It takes significantly longer to ignite the first row, because it has not been preheated (note the logarithmic time scale for images below).
Quenching is not observed if the first unburned row ignites.
We did not investigate the large cell size limit, corresponding to the geometric optics regime. In that regime for shear flow, the band may break into isolated flamelets, which may or may not reconnect later, depending on U. In cellular flow we did not observe isolated flamelets.
Evolution of flame with the amplitude of shear above critical.
| Temperature movie | dt=0.0100, tmax=3.00: | quicktime (1.3M), |
avi
(1.2M) |
| Reaction rate movie | dt=0.0100, tmax=3.00: | quicktime (1.4M), | avi (1.0M) |
| Temperature movie | dt=0.0001, tmax=0.03: | quicktime (1.6M), | avi (1.4M) |
| Reaction rate movie | dt=0.0001, tmax=0.03: | quicktime (1.5M), | avi (1.2M) |
Evolution of flame with the amplitude of shear below critical.
| Temperature movie | dt=0.0100, tmax=3.00: | quicktime (0.7M), | avi (0.6M) |
| Reaction rate movie | dt=0.0100, tmax=3.00: | quicktime (0.7M), | avi (0.5M) |
| Temperature movie | dt=0.0001, tmax=0.03: | quicktime (1.6M), | avi (1.5M) |
| Reaction rate movie | dt=0.0001, tmax=0.03: | quicktime (1.4M), | avi (1.0M) |
Critical velocity as function of initial band width
Critical amplitude of the shear flow velocity is proportional to the forth power of initial band width. The computer simulations were suggested by P. Constantin, A. Kiselev and L. Ryzhik, and show good agreement with their theory.
The dependence of critical velocity on initial band width can be explained from the point of view of the effective flame. For the flame to be quenched, band width should be of the order of effective flame thickness. Effective flame thickness is proportional to the U1/4, which leads to the critical velocity proportional to the forth power of initial band width.
Problem 1:
Burning in Shear Flow, Le=1, KPP
Problem 2:
Burning in Cellular Flow, Le=1, KPP
Problem 3:
Quenching by Shear Flow, Le=1, step function reaction
Problem 4:
Quenching by Cellular Flow, Le=1, step function reaction
Problem 5:
Quenching by Shear Flow, Le>1, step function reaction
