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DNS Studies of Burning Fronts in
Passive-Reactive Diffusion
Problem 3: Quenching by Shear Flow, Le=1
The initial band of hot material is distorted by sinusoidal shear.
Reaction term is step function of temperature, so that no burning occur
if temperature is lower than critical. For temperatures higher than
critical burning rate is constant and chosen to make laminar speed
equal to unity.
quicktime movie(1.2M), avi movie(0.8M)
Evolution of flame with the amplitude of shear above critical.
quicktime movie (1.1M),
avi movie (0.7M)
Note the reconnection of upper and lower flame fronts. First, the shear breaks the band into isolated islands of flame. While drifted by convection, these islands grow or shrink, depending on U. If shear velocity is larger than critical the islands reconnect.
The images above are made for L=4. Larger wavelengths of shear break the band to larger, and more sparsely separated, islands. It takes longer for flame to spread across separating cold fluid, and islands get shifted on further distances away from the center. On the other hand, bigger islands have more of the surface area, where burning occurs, and generate more heat, which gives them more chances to reconnect. The movie (1.5M quicktime, 1.1M avi) shows island reconnection for L=8, W=6 and U=12. More examples below.
for different shear velocities.
|
U = 10 movies: qt (1.7M), avi (1.3M) |
U = 20 movies: qt (0.8M), avi (0.5M) |
U = 40 movies: qt (0.4M), avi (0.2M) |
Critical velocity of the shear as function of initial band width
Critical amplitude of the shear flow velocity is proportional to the 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 intensity of shear flow, which leads to the critical velocity proportional to the initial band width.
as function of shear wavelength
The proportionality coefficient in
Ucr = 
(L)W,
is a function of shear wavelength L.
Quenching is more efficient when the shear wave length is of the order of laminar flame thickness. For our test case, L = 4.
The inverse proportionality to the wavelength, Ucr ~ W/L, for small wavelengths can be confirmed using standard homogenization procedure.
Advection-reaction-diffusion problems:
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
