Upon combining the steady state flamelet solution with a

characteristic timescale, the time rate of change of species i can be written as:

where indicates filtering at the grid level  and

is thewhere Y˜*

is the value of the mass fraction obtained  from

LES Favre averaging。 ˜ij  is the viscous term and   often

approximated as [14]:

the flamelet library while Yi    is the resolved mass fraction

used   by   CFD   code,   and

chem

is   a   characteristic

timescale obtained by expanding the reaction rate about the flamelet solution。

The characteristic time scale chem

represents the rate at

The subgrid stress tensorij

is defined as:

which each species proceeds towards its steady flamelet

This term represents the subgrid scale mixing effects。 Since it can not be determined from the resolved field,  it

needs  to  be  modeled。  The  term  is  modeled  using  a  j   2



tensor coefficient。 Using the dynamic procedure

proposed in [10], this found to be:

In Equation (16), the mixture fraction flux is defined as:

where Lij and L kk are the Leonard stress tensor and its trace respectively。 The Leonard stress tensor is:

The mixture fraction variance flux term in Equation   (17)

is similarly given by:

The Leonard stress tensor can be determined from    the

resolved field and a second filtered field, which is the resolved  field  filtered  again  by  a  test  filter  and    this

process is denoted by

。 k is the subgrid kinetic energy:

The subgrid scale scalar dissipation rate is defined as:

The subgrid kinetic energy is solved from a transport equation, the modeled form of which is given by [10]:

The subgrid scale scalar dissipation rate sgs is modeled by a zero-equation model:

The model for the subgrid kinetic energy dissipation is also given in [10]:

It  is  straightforward  to relate

to  the  mean scalar

dissipation rate used by the combustion model according to (2) and (20):

where is a coefficient determined by the filter [10]。 LES SCALAR TRANSPORT

The mean mixture fraction, ˜ , mixture fraction variance,

~"2 , and mean scalar dissipation rate, ˜ , represent mixing and need to be obtained from solving LES transport equations。 It should be mentioned that    the

The diffusion term in Equation (16) can be approximated through a simple gradient assumption。

The approach to the dynamical structure model for subgrid stress is extended to modeling the subgrid mixture fraction flux term appearing in the Equation (16) and (17) [15]:

subgrid   mixture   fraction,   defined   by