Theory of wrinkled laminar flames

At least for weak strain and weak curvature, the influences of these strain and curvature phenomena on the flame structure can be characterized in terms of a single quantity, an effective curvature of the flame with respect to the flow or the total stretch of the flame surface produced by the flow with respect to the moving, curved flame. By evaluating b from the rate-of-strain tensor in the products just behind the flame, the last of these quantities may be expressed nondimensionally as 

Although strain and curvature effects can be combined as in equation (55), it cannot be concluded that they are of equal importance for wrinkled laminar flames in turbulent flows. If it is assumed that the flame shape is affected mainly by the large eddies, then in terms of the flame thickness 3 and the integral scale /, the nondimensional curvature is of order 3/i This may be compared with the corresponding relevant nondimensional strain 

The formulation of Section 9.5.1 has served to remove the chemistry from the field equations, replacing it by suitable jump conditions across the reaction sheet. The expansion for small S/l, subsequently serves to separate the problem further into near-field and far-field problems. The domains of the near-field problems extend over a characteristic distance of order S on each side of the reaction sheet. The domains of the far-field problems extend upstream and downstream from those of the near-field problems over characteristic distances of orders from to /. Thus the near-field problems pertain to the entire wrinkled flame, and the far-field problems pertain to the regions of hydrodynamic adjustment on each side of the flame in essentially constant-density turbulent flow. Either matched asymptotic expansions or multiple-scale techniques are employed to connect the near-field and far-field problems. The near-field analysis has been completed for a one-reactant system with allowance made for a constant Lewis number differing from unity (by an amount of order l/P) for ideal gases with constant specific heats and constant thermal conductivities and coefficients of viscosity [122], [124], [125] the results have been extended to ideal gases with constant specific heats and constant Lewis and Prandtl numbers but thermal conductivities that vary with temperature [126]. The far-field analysis has been 

The near-field analysis provides expressions for the wrinkled-flame motion and the reaction-sheet temperature in terms of the flame shape and the gas velocities at the edge of the wrinkled flame. To the first order in the small parameter bj, the equation for the flame motion may be written, in the nondimensional notation of Section 9.5.1, as 

Here k is given by equation (55), and just as in equation (55), the nondimensional turbulent velocity components f ii to be evaluated locally 

The meaning of the results typified by equation (56) can be explained by addressing the various terms sequentially. If the turbulence velocities are large compared with the flame speeds, then equation (56) reduces to df/dz + Vj / = U, which defines the motion of a sheet attached to fluid elements. In the coordinates adopted, the velocity with which the sheet moves normal to itself with respect to the fluid locally is given by 

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