The mixture turbulence model is the default multiphase turbulence model. It represents the first extension of the single-phase -  model, and it is applicable when phases separate, for stratified (or nearly stratified) multiphase flows, and when the density ratio between phases is close to 1. In these cases, using mixture properties and mixture velocities is sufficient to capture important features of the turbulent flow.

The and equations describing this model (and without including buoyancy, dilation, and source terms) are as follows:

(14–397)

and

(14–398)

where the mixture density, , molecular viscosity, , and velocity, , are computed from

(14–399)

(14–400)

and

(14–401)

where , , , and are, respectively, the volume fraction, density, viscosity, and velocity of the i^th phase.

The turbulent viscosity for the mixture, , is computed from

(14–402)

and the production of turbulence kinetic energy, , is computed from

(14–403)

The terms, and are source terms that can be included to model the turbulent interaction between the dispersed phases and the continuous phase (Turbulence Interaction Models):

The turbulent viscosity for phase is computed from

(14–404)

The constants in these equations are the same as those described in Standard k-ε Model for the single-phase -  model.

The dispersed turbulence model is the appropriate model when the concentrations of the secondary phases are dilute, or when using the granular model. Fluctuating quantities of the secondary phases can be given in terms of the mean characteristics of the primary phase and the ratio of the particle relaxation time and eddy-particle interaction time.

The model is applicable when there is clearly one primary continuous phase and the rest are dispersed dilute secondary phases.

 

14.5.18.1.2.2. Turbulence in the Continuous Phase

The eddy viscosity model is used to calculate averaged fluctuating quantities. The Reynolds stress tensor for continuous phase takes the following form:

(14–405)

where is the phase-weighted velocity.

The turbulent viscosity is written in terms of the turbulent kinetic energy of phase :

(14–406)

and a characteristic time of the energetic turbulent eddies is defined as

(14–407)

where is the dissipation rate and .

The length scale of the turbulent eddies is

(14–408)

Turbulent predictions are obtained from the modified -  model. The transport equations (excluding buoyancy, dilation, and user-defined source terms) are:

(14–409)

and

(14–410)

Here, the terms containing and are source terms that can be included to model the influence of the dispersed phases on the continuous phase (Turbulence Interaction Models: is the production of turbulent kinetic energy, as defined in Modeling Turbulent Production in the k-ε Models. All other terms have the same meaning as in the single-phase -  model.

The most general multiphase turbulence model solves a set of and transport equations for each phase. This turbulence model is the appropriate choice when the turbulence transfer among the phases plays a dominant role.

Note that, since Ansys Fluent is solving two additional transport equations for each secondary phase, the per-phase turbulence model is more computationally intensive than the dispersed turbulence model.

 

14.5.18.1.3.1. Transport Equations

The Reynolds stress tensor and turbulent viscosity are computed using Equation 14–405 and Equation 14–406. Turbulence predictions are obtained from

(14–411)

and

(14–412)

The terms and can be approximated as

(14–413)

where is defined by Equation 14–435. The terms and are source terms which can be included to model the influence of the inter-phase turbulence interaction (Turbulence Interaction Models).

The dispersed turbulence model is used when the concentrations of the secondary phase are dilute and the primary phase turbulence is regarded as the dominant process. Consequently, the transport equations for turbulence quantities are only solved for the primary (continuous) phase, while the predictions of turbulence quantities for dispersed phases are obtained using the Tchen theory. The transport equation for the primary phase Reynolds stresses in the case of the dispersed model are:

(14–419)

The variables in Equation 14–419 are per continuous phase and the subscript is omitted for clarity. In general, the terms in Equation 14–419 are modeled in the same way as for the single phase case described in Reynolds Stress Model (RSM). The last term, , takes into account the interaction between the continuous and the dispersed phase turbulence. A general model for this term can be of the form:

(14–420)

where and are unknown coefficients, is the relative velocity, represents the drift or the relative velocity, and is the unknown particulate-fluid velocity correlation. To simplify this unknown term, the following assumption has been made:

(14–421)

where is the Kronecker delta, and represents the modified version of the original Simonin model [599].

(14–422)

where represents the turbulent kinetic energy of the continuous phase, is the continuous-dispersed phase velocity covariance and finally, and stand for the relative and the drift velocities, respectively. In order to achieve full closure, the transport equation for the turbulent kinetic energy dissipation rate () is required. The modeling of together with all other unknown terms in Equation 14–422 are modeled in the same way as in [114].

The main assumption for the mixture model is that all phases share the same turbulence field which consequently means that the term in the Reynolds stress transport equations (Equation 14–419) is neglected. Apart from that, the equations maintain the same form but with phase properties and phase velocities being replaced with mixture properties and mixture velocities. The mixture density, for example, can be expressed as

(14–423)

while mixture velocities can be expressed as

(14–424)

where is the number of species.

In the Simonin et al. model [599], the turbulence interaction is modeled by additional source terms in the turbulence transport equation(s). The Simonin model is only available with Dispersed and Per Phase turbulence models.

 

14.5.18.3.1.1. Formulation in Dispersed Turbulence Models

 

14.5.18.3.1.1.1. Continuous Phase

The term is derived from the instantaneous equation of the continuous phase and takes the following form, where represents the number of secondary phases:

(14–426)

which can be simplified to

(14–427)

where: is a user-modifiable model constant. By default, . is the covariance of the velocities of the continuous phase and the dispersed phase , calculated from Equation 14–436. is the relative velocity. is the drift velocity, calculated from Equation 14–333.

For granular flows, .

Equation 14–427 can be split into two terms as follows:

(14–428)

The second term in Equation 14–428 is related to drift velocity and is referred to as the drift turbulent source. It is an option term in the turbulent kinetic energy source and can be included as described in Including Turbulence Interaction Source Terms in the Fluent User's Guide.

is modeled according to Elgobashi et al. [163] :

(14–429)

where .

 

14.5.18.3.1.1.2. Dispersed Phases

For the dispersed phases, the characteristic particle relaxation time connected with inertial effects acting on a dispersed phase is defined as:

(14–430)

where is the drag function described in Interphase Exchange Coefficients.

The time scale of the energetic turbulent eddies is defined as:

(14–431)

The eddy particle interaction time is mainly affected by the crossing-trajectory effect [125], and is defined as

(14–432)

where , the parameter, , is given by

(14–433)

and

(14–434)

where is the angle between the mean particle velocity and the mean relative velocity. The ratio between these two characteristic times is written as

(14–435)

Following Simonin [599], Ansys Fluent writes the turbulence quantities for dispersed phase as follows:

(14–436)

and is the added-mass coefficient. For granular flows, is negligible. The viscosity for the secondary phase, for granular flows is approximated as:

(14–437)

while for bubbly flows it can be left as .

 

14.5.18.3.1.2. Formulation in Per Phase Turbulence Models

For Per Phase turbulence models, only the second term from Equation 14–428 (the drift turbulence source) is added in the phase turbulence modeling equations.

For the turbulent kinetic energy equations:

Continuous Phase

(14–438)

Dispersed Phases

(14–439)

Turbulence dissipation sources for all phases are computed as:

(14–440)

Troshko and Hassan [659] proposed an alternative model to account for the turbulence of the dispersed phase in the - equations.

 

14.5.18.3.2.1. Troshko-Hassan Formulation in Mixture Turbulence Models

In the Mixture turbulence models, the Troshko-Hassan turbulence interaction terms are:

(14–441)

(14–442)

By default, and . These values are user-modifiable as described in Including Turbulence Interaction Source Terms in the Fluent User's Guide. is the characteristic time of the induced turbulence defined as

(14–443)

is the virtual mass coefficient and is the drag coefficient.

 

14.5.18.3.2.2. Troshko-Hassan Formulation in Dispersed Turbulence Models

 

14.5.18.3.2.2.1. Continuous Phase

In the Dispersed turbulence models, the term is calculated as follows:

(14–444)

and is calculated as

(14–445)

By default, and . These values are user-modifiable as described in Including Turbulence Interaction Source Terms in the Fluent User's Guide. is the characteristic time of the induced turbulence defined as in Equation 14–443.

 

14.5.18.3.2.2.2. Dispersed Phases

For the Troshko-Hassan model, the kinematic turbulent viscosity of the dispersed phase is calculated as

(14–446)

 

14.5.18.3.2.3. Troshko-Hassan Formulation in Per-Phase Turbulence Models

 

14.5.18.3.2.3.1. Continuous Phase

In the Per-Phase turbulence models, the continuous phase equations are modified with the following terms:

(14–447)

(14–448)

By default, and . These values are user-modifiable as described in Including Turbulence Interaction Source Terms in the Fluent User's Guide. is the characteristic time of the induced turbulence defined as in Equation 14–443.

 

14.5.18.3.2.3.2. Dispersed Phases

The dispersed phase equations are also modified with source terms defined as follows:

(14–449)

(14–450)

Unlike the Simonin and Troshko-Hassan models, the Sato model [571] does not add explicit source terms to the turbulence equations. Instead, in an attempt to incorporate the effect of the random primary phase motion induced by the dispersed phase in bubbly flow, Sato et al. proposed the following relation:

(14–451)

where the relative velocity and the diameter of the dispersed phase represent velocity and time scales and .

For the mixture model

where is the primary phase turbulent viscosity before the Sato correction, which is calculated from Equation 14–404.

For the dispersed and per-phase turbulence models, and are the primary phase turbulence intensity and eddy dissipation rate, respectively.

If None is selected, then no source terms are added to account for turbulent interaction. This is appropriate if you prefer to add your own source terms using the DEFINE_SOURCE UDF macro ( DEFINE_SOURCE in the Fluent Customization Manual).