The dispersion of particles due to turbulence in the fluid phase can be predicted using the stochastic tracking model. The stochastic tracking (random walk) model includes the effect of instantaneous turbulent velocity fluctuations on the particle trajectories through the use of stochastic methods (see Stochastic Tracking). For stochastic tracking a model is available to account for the generation or dissipation of turbulence in the continuous phase (see Coupling Between the Discrete and Continuous Phases).
Important: Turbulent dispersion of particles cannot be included if the Spalart-Allmaras turbulence model is used.
When the flow is turbulent, Ansys Fluent will predict the trajectories
of particles using the mean fluid phase velocity, 
, in the trajectory equations
(Equation 12–1). Optionally, you can include the
instantaneous value of the fluctuating gas flow velocity,
 | (12–22) |
to predict the dispersion of the particles due to turbulence.
In the stochastic tracking approach, Ansys Fluent predicts
the turbulent dispersion of particles by integrating the trajectory
equations for individual particles, using the instantaneous fluid
velocity, 
, along
the particle path during the integration. By computing the trajectory
in this manner for a sufficient number of representative particles
(termed the "number of tries"), the random effects of
turbulence on the particle dispersion can be included.
Ansys Fluent uses a stochastic method (random walk model) to determine the instantaneous gas velocity. In the discrete random walk (DRW) model, the fluctuating velocity components are discrete piecewise constant functions of time. Their random value is kept constant over an interval of time given by the characteristic lifetime of the eddies.
The DRW model may give nonphysical results in strongly nonhomogeneous diffusion-dominated flows, where small particles should become uniformly distributed. Instead, the DRW will show a tendency for such particles to concentrate in low-turbulence regions of the flow. In addition, this model is known to give poor prediction of the wall impaction rate of particles with a diameter less than a few microns due to turbulence.
Prediction of particle dispersion makes use of the concept of
the integral time scale, 
:
 | (12–23) |
The integral time is proportional to the particle dispersion
rate, as larger values indicate more turbulent motion in the flow.
It can be shown that the particle diffusivity is given by 
.
For small "tracer" particles that move with the
fluid (zero drift velocity), the integral time becomes the fluid Lagrangian
integral time, 
. This time scale can be approximated
as
 | (12–24) |
where 
is to be determined as it is
not well known. By matching the diffusivity of tracer particles, 
, to the scalar diffusion rate predicted by the turbulence
model, 
, one can obtain
 | (12–25) |
for the 
- 
model and its variants, and
 | (12–26) |
when the Reynolds stress model (RSM) is used [132]. For the 
- 
models, substitute 
into Equation 12–24. The LES model uses the
equivalent LES time scales.
In the discrete random walk (DRW) model, or "eddy lifetime" model, the interaction of a particle with a succession of discrete stylized fluid phase turbulent eddies is simulated [215]. Each eddy is characterized by
a Gaussian distributed random velocity fluctuation,
, 
, and 
a time scale,
The values of 
, 
, and 
that prevail during
the lifetime of the turbulent eddy are sampled by assuming that they
obey a Gaussian probability distribution, so that
 | (12–27) |
where 
is a normally distributed random number,
and the remainder of the right-hand side is the local RMS value of
the velocity fluctuations. Since the kinetic energy of turbulence
is known at each point in the flow, these values of the RMS fluctuating
components can be defined (assuming isotropy) as
 | (12–28) |
for the 
- 
model, the 
- 
model, and their variants.
When the RSM is used, nonisotropy of the stresses is included in the
derivation of the velocity fluctuations:
 | (12–29) |
 | (12–30) |
 | (12–31) |
when viewed in a reference frame in which the second moment of the turbulence is diagonal [731]. For the LES model, the velocity fluctuations are equivalent in all directions. See Inlet Boundary Conditions for Scale Resolving Simulations for details.
The characteristic lifetime of the eddy is defined either as a constant:
 | (12–32) |
where 
is given by Equation 12–24 in general (Equation 12–25 by default), or as
a random variation about 
:
 | (12–33) |
where 
is a uniform random number greater than zero and
less than 1 and 
is given by Equation 12–25. The option of random
calculation of 
yields a more realistic
description of the correlation function.
The particle eddy crossing time is defined as
 | (12–34) |
where 
is the particle relaxation time, 
is the eddy
length scale, and 
is the magnitude
of the relative velocity.
The particle is assumed to interact with the fluid phase eddy
over the smaller of the eddy lifetime and the eddy crossing time.
When this time is reached, a new value of the instantaneous velocity
is obtained by applying a new value of 
in Equation 12–27.
The only inputs required for the DRW model are the value for
the integral time-scale constant, 
(see Equation 12–24 and Equation 12–32) and the choice of the method used
for the prediction of the eddy lifetime. You can choose to use either
a constant value or a random value by selecting the appropriate option
in the Set Injection Properties Dialog Box for each injection,
as described in Stochastic Tracking in the User’s
Guide.
Important: Turbulent dispersion of particles cannot be included if the Spalart-Allmaras turbulence model is used.