with the boundary conditions (28), the ODE (30) can be analytically solved for mixture fraction as a function of x 。

Rich  Lean  Rich Lean

narrower as the mixture becomes more homogeneous。

Recently, it is proposed that the conditional dissipation rate can be determined by solving a PDF transport equation for mixture fraction [18]。 However, this method would incur additional computational cost and might not be appropriate for engine simulations。

2   2

Furthermore, according to equation (2), the functional form for scalar dissipation rate will be obtained by differentiating equation (31) with respect to x:

Hergart has proposed a new model for the conditional scalar dissipation rate applied to diesel combustion [19]。 This approach fits into a single flamelet RIF model in that    the    model    determines    an    averaged   scalar

dissipation rate over the whole cylinder after the end of the fuel injection。 However, this model is not suitable  for

Substituting Equation (32) into Equation (2):

the flamelet combustion models with spatially resolved flamelets, for example, the Flamelet Time Scale model used here。

(Lean Rich )2





Equation (31) is used to solve for x in terms of :

Rich Lean

These boundaries in mixture vary with time and space and represent the local boundary conditions feeding an instantaneous flamelet。

In  connection  to  the  flamelet  time  scale   combustion

 

The functional form of is subsequently obtained by substituting Equation (34) into Equation (33) for x and organized as:

Lean Rich

model, the flamelet equation can be understood as a result of the asymptotic expansion around the stoichiometric mixture fraction surface [17]。 In  this sense, the instantaneous scalar dissipation rate should be that at stoichiometric position, namely st 。 Thus, st predicted  by  Equation  (39)  should  be  used  as  in

Equation (4) to evaluate species mass fractions。

where 0 is the maximum scalar dissipation and functionf has the form:

The turbulence and combustion models  discussed above   are   implemented  into  the  KIVA-3V  code,   as

f (;Lean,Rich) exp2 erf 1    2

modified by the University of Wisconsin   Engine





Research  Center  [20]。  The  governing  equations    are

 

It should be noted that erf-1 is the inverse error function other than reciprocal of the error function。  can     thus

be related to its value at the stoichiometric condition using Equation (35):

f (; Lean , Rich )

solved by standard KIVA numerical methods including quasi second-order upwinding for convection terms, central differencing for diffusion terms, and time split scheme with inner and outer iterations for implicit terms plus pressure coupling [21]。