When enabled by the user in Rocky under the CFD coupling settings, the turbulent dispersion acts as a diffusion mechanism in dispersed flows, resulting in a transference of particles from high volume fraction regions to low volume fraction regions due to turbulent fluctuations.
The model of turbulent dispersion detailed in this chapter assumes that the instantaneous velocity of the fluid phase is a combination of a mean and a fluctuating value:
 | (3–78) |
where 
is the mean velocity of the fluid phase and 
is the fluctuation caused by turbulent effects. The turbulent force then
arises as a consequence of the presence of 
in Equation 3–78 , as it will
generate an additional drag force when 
is substituted in Equation 3–5.
Rocky takes an approach similar to Gosman and Ioannides [20] in
order to model 
, where the dispersion of particles due to turbulence in the fluid phase is
predicted stochastically.
Note: Unless mentioned otherwise, all equations of this section follow the model proposed by Gosman and Ioannides.
In this model, the turbulence is randomly sampled during each particles trajectory and allowed to influence its motion. The gross behavior of the turbulence in the simulated system emerges as a consequence of the averaging that naturally occurs when the random sampling is performed for a statistically significant number of particles.
Specifically, the influence of the turbulence over a particle is simulated by means of
the interaction with a succession of discrete fluid phase turbulent eddies [44]. Each eddy is characterized by a fluid velocity fluctuation
and a time scale 
(the eddy lifetime). When a particle is interacting with a turbulent eddy,
influences the particle by means of an additional drag force during the
particle-eddy interaction time 
.
The following sections detail how Rocky estimates 
, 
and .
In order to estimate 
, it is assumed that the turbulence is isotropic. The fluid fluctuating
velocity can be decomposed as a magnitude scalar and a direction unit vector:
 | (3–79) |
in which both the magnitude and the direction unit vector contain random components as
described ahead. At the start of the lifetime of a turbulent eddy, 
and 
are sampled independently, and Equation 3–79 is used to estimate the eddys
characteristic value 
.
The magnitude of the fluctuating velocity is estimated as:
 | (3–80) |
where 
is a random variable distributed normally around zero:
 | (3–81) |
The standard deviation 
of 
is given by:
 | (3–82) |
where 
is the kinetic energy of the turbulence associated with the flow.
Note: is automatically provided by Fluent when its k-epsilon
viscous model is employed for two- and one-way coupled simulations. For constant
one-way coupled simulations, must be directly set by the user under the Constant
One-Way settings in Rocky.
The direction of the fluctuating velocity is assumed to be a random variable given by:
 | (3–83) |
where 
is a uniform distribution of points over the surface of a unit sphere.
The lifetime 
is another characteristic value of turbulent eddies. An estimate of
is made under the further assumption that the characteristic size of the
eddy is equal to the dissipation length scale of the system, given by:
 | (3–84) |
where 
is the dissipation rate of the turbulent kinetic energy associated with
the flow.
Note: is automatically provided by Fluent when its k-epsilon viscous model
is employed for two- and one-way coupled simulations. For constant one-way coupled
simulations, must be directly set by the user under the Constant One-Way settings
in Rocky.
The eddy lifetime is then approximated as:
Note: In Gosman and Ioannides's paper, equation Equation 3–85 has the denominator replaced by the magnitude of the fluid fluctuating velocity. This was found to cause unrealistically long eddy lifetimes however.
 | (3–85) |
A further assumption of this turbulent model is that each particle of the simulation
has a one-to-one association with a turbulent eddy during an interaction time interval
. For estimating this particle-eddy interaction time, two possible
outcomes are considered:
The particle moves sufficiently slowly relative to the fluid in order to remain within the influence of the eddy during its whole lifetime
; The relative velocity between the particle and the fluid is high enough to allow the particle to transverse the eddy in a transit time
shorter than 
.
The particle-eddy interaction time is therefore defined as the minimum of the above, i.e.:
 | (3–86) |
The transit time 
is estimated from the following solution of a simplified form of the
motion equation of a small particle in a fluid medium:
 | (3–87) |
where 
is the particle relaxation time defined as:
 | (3–88) |
In cases where 
, Equation 3–87 has no
solution. This can be interpreted as the particle being "captured" by the
turbulent eddy, in which case 
in Equation 3–88 and
consequently 
.