The mixture-fraction variable

Since the reaction zone is thin, most of the analysis of its structure can be performed without reference to a particular configuration. To introduce a general approach of this type, consider a two-stream problem having uniform properties over one portion of the boundary, called the fuel stream (subscript F, 0, possibly at infinity), and different uniform properties over the rest of the boundary, called the oxidizer stream (subscript O, 0, also possibly at infinity) assume that there is no oxidizer in the fuel stream and no fuel in the oxidizer stream. For a one-step reaction, the form given in equation (1) may be adopted, and in terms of the oxidizer-fuel coupling function jS, appearing in equation (6), the mixture fraction may be defined as 

A conservation equation for Z may be derived from the general conservation equations. Under assumptions 1, 2, and 3 of Section 1.3, if the binary diffusion coefficients are equal then equation (1-37) is obtained, and from equation (1-4) it is readily found that 

The chemical-source term does not appear in equation (71) because (like P or Zj) Z is a conserved scalar, a scalar quantity that is neither created nor destroyed by chemical reactions. 

Under additional assumptions, all conserved scalars are expressible in terms of Z. For example, if initial conditions and boundary conditions are appropriate, then formulas for all Zj in terms of Z are obtained by solving equation (70) for p, replaced by Zj. Energy conservation warrants special consideration. The sum of the thermal and chemical enthalpies is h Substitution of this into equation (1-10) and the result into equation (1-3) may be shown by use of equations (1-1) and (1-2) to provide as a general form of energy conservation 

For flows at low Mach numbers with negligible radiant heat flux, it may be shown from equation (1-6) under the assumptions introduced above, that equation (72) may be written as 

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For gaseous flames, the LES/FMDF can be implemented via two combustion models (1) a finite-rate, reduced-chemistry model for nonequilibrium flames and (2) a near-equilibrium model employing detailed kinetics. In (1), a system of nonlinear ordinary differential equations (ODEs) is solved together with the FMDF equation for all the scalars (mass fractions and enthalpy). Finite-rate chemistry effects are explicitly and exactly" included in this procedure since the chemistry is closed in the formulation. In (2). the LES/FMDF is employed in conjunction with the equilibrium fuel-oxidation model. This model is enacted via fiamelet simulations, which consider a laminar counterflow (opposed jet) flame configuration. At low strain rates, the flame is usually close to equilibrium. Thus, the thermochemical variables are determined completely by the mixture fraction variable. A fiamelet library is coupled with the LES/FMDF solver in which transport of the mixture fraction is considered. It is useful to emphasize here that the PDF of the mixture fraction is not assumed a priori (as done in almost all other flamelet-based models), but is calculated explicitly via the FMDF. The LES/FMDF/flamelet solver is computationally less expensive than that described in (1) thus, it can be used for more complex flow configurations. 

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