A Lagrangian Monte Carlo method is used to solve for the 
dimensional PDF
Transport equation. Monte Carlo methods are appropriate for high-dimensional
equations since the computational cost increases linearly with the
number of dimensions. The disadvantage is that statistical errors
are introduced, and these must be carefully controlled.
To solve the modeled PDF transport equation, an analogy is made with a stochastic differential equation (SDE) that has identical solutions. The Monte Carlo algorithm involves notional particles that move randomly through physical space due to particle convection, and also through composition space due to molecular mixing and reaction. The particles have mass and, on average, the sum of the particle masses in a cell equals the cell mass (cell density times cell volume). Since practical meshes have large changes in cell volumes, the particle masses are adjusted so that the number of particles in a cell is controlled to be approximately constant and uniform.
The processes of convection, diffusion, and reaction are treated in fractional steps as described in the sections that follow. For information on the fractional step method, refer to [92].
Information about this method is described in the following sections:
A spatially second-order-accurate Lagrangian method is used in Ansys Fluent, consisting of two steps. At the first convection step, particles are advanced to a new position
 | (7–150) |
| where | |
 = particle
position vector | |
 = Favre mean fluid-velocity
vector at the particle position | |
 = particle time step |
For unsteady flows, the particle time step is the physical time step. For steady-state flows, local time steps are calculated for each cell as
 | (7–151) |
| where | |
 = convection number  / (cell fluid velocity) | |
 = diffusion number  / (cell turbulent diffusivity) | |
 = mixing number  turbulent time scale | |
 = characteristic cell length
=  where  is the problem dimension |
After the first convection step, all other sub-processes, including diffusion and reaction are treated. Finally, the second convection step is calculated as
 | (7–152) |
| where | |
 = mean cell fluid density | |
 = mean fluid-velocity
vector at the particle position | |
 =
effective viscosity | |
 = turbulent Schmidt number | |
 = standardized
normal random vector |
Molecular mixing of species and heat must be modeled and is usually the source of the largest modeling error in the PDF transport approach. Ansys Fluent provides three models for molecular diffusion: the Modified Curl model [278], [480], the IEM model (which is sometimes called the LSME model) [146] and the EMST model [635].
For the Modified Curl model, a few particle pairs are selected at random from all the particles in a cell, and their individual compositions are moved toward their mean composition. For the special case of equal particle mass, the number of particle pairs selected is calculated as
 | (7–153) |
| where | |
 = total number of particles in the cell | |
 =
mixing constant (default = 2) | |
 = turbulent time scale
(for the  -  model this is  ) |
The algorithm in [480] is used for the general case of variable particle mass.
For each particle pair, a uniform random number 
is selected and each
particle’s composition 
is moved toward the pair’s mean composition
by a factor proportional to 
:
 | (7–154) |
where 
and 
are
the composition vectors of particles 
and 
, and 
and 
are the masses
of particles 
and 
.
For the Interaction by Exchange with the Mean (IEM) model, the composition of all particles in a cell are moved a small distance toward the mean composition:
 | (7–155) |
where 
is the composition before mixing, 
is the composition after mixing, and 
is the Favre mean-composition vector at the particle’s location.
Physically, mixing occurs between fluid particles that are adjacent to each other. The Modified Curl and IEM mixing models take no account of this localness, which can be a source of error. The Euclidean Minimum Spanning Tree (EMST) model mixes particle pairs that are close to each other in composition space. Since scalar fields are locally smooth, particles that are close in composition space are likely to be close in physical space. The particle pairing is determined by a Euclidean Minimum Spanning Tree, which is the minimum length of the set of edges connecting one particle to at least one other particle. The EMST mixing model is more accurate than the Modified Curl and IEM mixing models, but incurs a slightly greater computational expense. Details on the EMST model can be found in reference [635].
The particle composition vector is represented as
 | (7–156) |
where 
is the 
th species mass fraction, and 
is the temperature.
For the reaction fractional step, the reaction source term is integrated as
 | (7–157) |
where 
is the chemical source term. Most realistic chemical mechanisms consist of
tens of species and hundreds of reactions. Typically, a reaction does not occur until an
ignition temperature is reached, but then proceeds very quickly until reactants are consumed.
Hence, some reactions have very fast time scales, in the order of
10-10 s, while others have much slower time scales, on the order of
1 s. This time-scale disparity results in numerical stiffness, which means that extensive
computational work is required to integrate the chemical source term in Equation 7–157. In Ansys Fluent, the reaction step (that is, the calculation of
) can be performed either by Direct Integration or by In-Situ Adaptive
Tabulation (ISAT), as described in the following paragraphs.
A typical steady-state PDF transport simulation in Ansys Fluent may
have 50000 cells, with 20 particles per cell, and requires 1000 iterations
to converge. Hence, at least 
stiff
ODE integrations are required. Since each integration typically takes
tens or hundreds of milliseconds, on average, the direct integration
of the chemistry is extremely CPU-demanding.
For a given reaction mechanism, Equation 7–157 may be considered as a mapping. With
an initial composition vector 
, the final
state 
depends only
on 
and the mapping
time 
. In theory, if a table could
be built before the simulation, covering all realizable 
states and time steps, the integrations could be
avoided by table look-ups. In practice, this a priori tabulation is not feasible since a full table in 
dimensions (
species, temperature, pressure
and time step) is required. To illustrate this, consider a structured
table with 
points in each dimension. The required table size
is 
, and for a conservative
estimate of 
discretization
points and 
species, the table would contain 
entries.
On closer examination, the full storage of the entire realizable
space is very wasteful because most regions are never accessed. For
example, it would be unrealistic to find a composition of 
and 
in a real combustor. In fact, for steady-state,
3D laminar simulations, the chemistry can be parameterized by the
spatial position vector. Thus, mappings must lie on a three dimensional
manifold within the 
dimensional composition space.
It is, hence, sufficient to tabulate only this accessed region of
the composition space.
The accessed region, however, depends on the particular chemical mechanism, molecular transport properties, flow geometry, and boundary conditions. For this reason, the accessed region is not known before the simulation and the table cannot be preprocessed. Instead, the table must be built during the simulation, and this is referred to as in-situ tabulation. Ansys Fluent employs ISAT [526] to dynamically tabulate the chemistry mappings and accelerate the time to solution. ISAT is a method to tabulate the accessed composition space region "on the fly" (in-situ) with error control (adaptive tabulation). When ISAT is used correctly, accelerations of two to three orders of magnitude are typical. However, it is important to understand how ISAT works in order to use it optimally.