Close coupling solvers

We now comment on methods that solve the equations as coupled ordinary diflFerential equations (these considerations are taken as an extraction from 

We substitute (6.9) into (6.8) and project with a particular basis function, y (f), to obtain the coupled equations 

In fact, if one includes a total of AT-basis functions in the expansion, (6.10) will have 2N linearly independent solutions. 

However, only N of these can be made to be regular at r=0, and these are the physically relevant ones. It is therefore convenient to add a label to the Uim fo signify the linearly independent solution it represents  

Note that I and K are diagonal matrices, and we interpret the Im indices of U lmlomo r) as final state indices and Zq o as initial state indices. In addition to the regularity condition at r = 0 (which for typical potentials in atom-diatom scattering becomes the condition that U lmlomo 0) = 0), we want to impose the asymptotic condition 

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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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