For multiphase turbulent flows using the Eulerian model, Ansys Fluent can include the effects of turbulent dispersion forces which account for the interphase turbulent momentum transfer. The turbulent dispersion force acts as a turbulent diffusion in dispersed flows. For example for a boiling flow in a heated vertical pipe, vapor is generated on heated wall surfaces. The turbulent dispersion force plays a crucial role in driving the vapor away from the vicinity of the wall towards the center of the pipe.
The turbulent dispersion force arises from averaging the interphase drag term. For a
dispersed phase, 
, and a continuous phase, 
, the turbulent drag is modeled as:
 | (14–330) |
The term on the left side is the instantaneous drag. The first term on the right hand side,
, appears in Equation 14–196 and represents the mean
momentum exchange between the two phases. 
is the interphase exchange coefficient, described in Interphase Exchange Coefficients, and 
and 
are the mean phase velocity vectors. The second term, 
, is commonly referred to as the turbulent dispersion force:
 | (14–331) |
where 
is the drift velocity and accounts for the dispersion of the secondary
phases due to transport by turbulent fluid motion. 
is a factor that can be used to impose a limiting
function on the turbulent dispersion force. For clarity, the limiting
factor is omitted from the descriptions of the turbulent dispersion
models that follow. See Limiting Functions for the Turbulent Dispersion Force
for details about
the implementation of the limiting factor.
The following sections describe the models for turbulent dispersion force that are available in Ansys Fluent:
You can also specify the turbulent dispersion force with a User Defined Function. For information on enabling turbulent dispersion force in your model, see Including the Turbulent Dispersion Force in the User’s Guide.
Instead of following Equation 14–331 and modeling the drift velocity, Lopez de Bertodano proposed the following formulation [389]:
 | (14–332) |
where 
is the continuous phase density, 
is the turbulent kinetic energy in the continuous phase, and 
is the gradient of dispersed phase volume fraction, and 
is a user-modifiable constant. By default, 
.
Simonin and Viollet [598]
proposed that
the drift velocity, 
, is calculated from:
 | (14–333) |
In Equation 14–333, 
is the fluid-particulate dispersion tensor.
For the dispersed two-equation turbulence model (k- ε Dispersed Turbulence Model), Ansys Fluent
uses Tchen theory [247] and assumes that 
is a scalar, expressed as given by 
in Equation 14–436 with a dispersion Prandtl
number, 
(
by default). The turbulent dispersion force is then given by:
 | (14–334) |
Where 
is a user-modifiable constant, which is set to 1 by
default.
For the per-phase turbulence model, the turbulent dispersion force is given by:
 | (14–335) |
Where 
and 
are the viscosity and density of the dispersed phase, respectively; and
and 
are the viscosity and density of the continuous phase, respectively.
For the mixture turbulence model (k- ε Mixture Turbulence Model), the dispersion scalar is equal to the mixture turbulent kinematic viscosity.
Burns et al. [88] derived a formulation based on Favre averaging of the drag term. The final expression is similar to Simonin’s model. For the Burns model, the dispersion scalar is estimated by the turbulent viscosity of the continuous phase:
 | (14–336) |
and
 | (14–337) |
By default, 
and 
. As in the Simonin model, when using the mixture turbulence model the
dispersion scalar is equal to the mixture turbulent kinetic viscosity.
Instead of treating the turbulent dispersion as an interfacial momentum force in the phase
momentum equations, you can model it as a turbulent diffusion term in the governing equations
of phase volume fractions [615]. With the turbulent dispersion term,
the governing equation for the volume fraction in phase 
, Equation 14–193, becomes:
 | (14–338) |
where 
is the diffusion
coefficient in the 
phase, and the
term 
is the turbulent dispersion term that must
satisfy the constraint:
 | (14–339) |
In order to satisfy Equation 14–339
the diffusion coefficients
for the secondary phases are estimated from the phase turbulent viscosities, 
as
 | (14–340) |
where 
by default.
For the primary phase, 
, the diffusion term is
 | (14–341) |
In some applications it is desirable to apply the turbulent dispersion force only in
particular flow regimes or conditions. For example, a user may want to include the turbulent
dispersion force only in a bubbly flow regime. To accommodate this, a limiting function can be
applied through the factor, 
, in Equation 14–331.
Ansys Fluent provides three options for determining 
:
- None
By default, no limiting is performed on the turbulent dispersion force (
).- Standard
Ansys Fluent includes a standard limiting function that varies linearly from
0to1based on the volume fraction of the dispersed phase(s),
: 
(14–342)
By default,
If necessary, the values of
and
can be changed using the following
domainsetvarscheme commands:(domainsetvar <phase-id> ‘mp/td/vof-lower-limit < 
—value>)
(domainsetvar <phase-id> ‘mp/td/vof-upper-limit < 
—value>)
with appropriate values substituted for <phase-id> and <
—value>.
Note that the value for <phase-id> for a given phase can be
found from the
Phases
dialog box in the user
interface.
- User-Defined
You can specify your own limiting function by creating a User-Defined Function (UDF) using the
DEFINE_EXCHANGE_PROPERTYmacro (DEFINE_EXCHANGE_PROPERTY). Note that
must have a value between 0and1, and should vary continuously to ensure numerical stability and physical solutions.
