The Lagrangian solution method solves the composition PDF transport equation by stochastically tracking Lagrangian particles through the domain. It is computationally expensive since a large number of particles are required to represent the PDF, and a large number of iterations are necessary to reduce statistical errors and explicitly convect the particles through the domain. The Eulerian PDF transport model overcomes these limitations by assuming a shape for the PDF, which allows Eulerian transport equations to be derived. Stochastic errors are eliminated and the transport equations are solved implicitly, which is computationally economical. The multi-dimensional PDF shape is assumed as a product of delta functions. As with the Lagrangian PDF model, the highly nonlinear chemical source term is closed. However, the turbulent scalar flux and molecular mixing terms must be modeled, and are closed with the gradient diffusion and the IEM models, respectively.
The composition PDF of 
dimension (
species and
enthalpy) is represented as a collection of 
delta functions
(or modes). This presumed PDF has the following form:
 | (7–158) |
where 
is the probability in each mode, 
is the conditional mean composition of species 
in the 
th mode, 
is the composition space variable of species 
, and 
is the delta function.
The Eulerian PDF transport equations are derived by substituting
Equation 7–158 into the closed composition
PDF transport equation (Equation 7–148 with Equation 7–149 and Equation 7–155). The unknown terms, 
and 
, are determined by forcing lower moments of this
transported PDF to match the RANS lower moment transport equations,
using the Direct Quadrature Method of Moments (DQMOM) approach [187], [412]. The resulting
transport equations are:
where 
is the probability of the 
th mode, and 
is the 
th species probability weighted conditional mean composition of the 
th mode. 
is the effective turbulent diffusivity. The terms 
, 
and 
represent mixing, reaction
and correction terms respectively. Note that only 
probability
equations are solved and the 
th probability is calculated
as one minus the sum of the 
solved probabilities.
For more information, see the following sections:
The reaction source term 
in Equation 7–160 for the 
th composition and the 
th mode is calculated
as,
 | (7–161) |
where 
is the net reaction rate for the 
th component.
The micro-mixing term 
is modeled with the IEM mixing model:
 | (7–162) |
where 
is the turbulence time-scale and 
is
the mixing constant.
Hence, for the two-mode DQMOM-IEM model, the mixing terms for
component 
are,
 | (7–163) |
The default value of 
is 2, which is appropriate
for gas-phase combustion. For reactions in liquids, where the diffusivities
are much smaller than gases, the Liquid Micro-Mixing option interpolates 
from model turbulence [530] and scalar [187] spectra.
Using assumptions to ensure realizability and boundedness, the
correction terms 
in Equation 7–160 for the 
th composition are determined from the linear system,
 | (7–164) |
where 
are the non-negative integer
lower moments (1–
) for each component 
. Note that the condition
of the matrix decreases with increasing 
, which reduces
the stability of higher mode simulations.
The dissipation term 
in Equation 7–164 is calculated
as,
 | (7–165) |
For the two-mode DQMOM-IEM model, the correction terms for the 
th component are,
 | (7–166) |
