Transport equations for reaction rate in laminar and turbulent premixed flames characterized by non-unity Lewis number

Journal article, 2018

Transport equations for (i) the rate W of product creation and (ii) its Favre-averaged value W ̃ are derived from the first principles by assuming that W depends solely on the temperature and mass fraction of a deficient reactant in a premixed turbulent flame characterized by the Lewis number Le different from unity. The right hand side of the transport equation for W ̃ involves seven unclosed terms, with some of them having opposite signs and approximately equal large magnitudes when compared to the left-hand-side terms. Accordingly, separately closing each term does not seem to be a promising approach, but a joint closure relation for the sum (T_Σ ) ̅ of the seven terms is sought. For this purpose, theoretical and numerical investigations of variously stretched laminar premixed flames characterized by Le<1 are performed and the linear relation between T_Σ integrated along the normal to a laminar flame and a product of (i) the consumption velocity u_c and (ii) the stretch rate s ̇_w evaluated in the flame reaction zone is obtained. Based on this finding and simple physical reasoning, a joint closure relation of (T_Σ ) ̅∝(ρWs ̇ ) ̅ is hypothesized, where ρ is the density and s ̇ is the stretch rate. The joint closure relation is tested against 3D DNS data obtained from three statistically 1D, planar, adiabatic, premixed turbulent flames in the case of a single-step chemistry and Le=0.34, 0.6, or 0.8. In all three cases, the agreement between (T_Σ ) ̅ and (ρWs ̇ ) ̅ extracted from the DNS is good with exception of large (c ̅>0.4) values of the mean combustion progress variable c ̅ in the case of Le=0.34. The developed linear relation between (T_Σ ) ̅ and (ρWs ̇ ) ̅ helps to understand why the leading edge of a premixed turbulent flame brush can control its speed.