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DNS Studies of Burning Fronts in
Passive-Reactive Diffusion

Problem 1: Burning in Shear Flow, Le=1, KPP

There are two limiting regimes for the propagation of the burning front depending on the ratio of the flow wavelength L to the thickness of the laminar flame. If this ratio is large, flame propagates normal to itself, like the light in geometric optics. Smaller wavelengths correspond to the diffusive regime. As a characteristic lengthscale of the laminar flame we will use l_o=k/v_o (on some plots it's also called "delta").

Geometric Optics	Diffusive Regime
U/vo=4,    L/lo=256	U/vo=2,    L/lo=32,16,8,4

In the geometric optics limit, the local flame structure is laminar and the flame propagates normal to itself with the speed v_o, so that the speed of the flame tip is U+v_o . The rest of the flame only adjusts its angle to catch up.

In the diffusive regime local flame structure is influenced by external flow. Isotherms for different temperatures have the same shape, determined by the shape of the shear velocity. Amplitude of the isotherms is proportional to UL², while the interface thickness is larger than laminar and related to burning velocity, l/l_o=v/v_o. It was found that the velocity is proportional to the length of the isotherm.

Dependence of Flame Velocity on Flow Parameters

Flame acceleration v is bounded between v_o and U+v_o in shear flows. For large velocities U, flame acceleration depends linearly on U (confirmed theoretically with equation scaling). The maximum acceleration is achieved for geometric optics regime, where v=U+v_o. Acceleration deceases as flow wavelength get smaller; the slope approaches zero in the limit. Flame velocity v is in agreement with linear lower bound found by Peter Constantin group.

In the diffusive regime flame acceleration approaches laminar flame velocity v_o as (UL)². The modification of the laminar front velocity can be understood in terms of shear enhanced diffusion (Taylor 1953, Rhines and Young 1983), and is in agreement with the expression proposed by Clavin and Williams (1978).

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

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