Modelling Multi Phase Flow

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Chapter23. ModelingMultiphaseFlows

This chapter discusses the general multiphase models that are available in FLUENT.

Section 23.1: Introduction provides a brief introduction to multiphase modeling, Chap-

ter 22: Modeling Discrete Phase discusses the Lagrangian dispersed phase model, and

Chapter 24: Modeling Solidification and Melting describes FLUENT’smodel for solidifi-

cation and melting.

• Section 23.1: Introduction

• Section 23.2: Choosing a General Multiphase Model

•

Section 23.3: Volume of Fluid (VOF) Model Theory• Section 23.4: Mixture Model Theory

• Section 23.5: Eulerian Model Theory

• Section 23.6: Wet Steam Model Theory

• Section 23.7: Modeling Mass Transfer in Multiphase Flows

• Section 23.8: Modeling Species Transport in Multiphase Flows

• Section 23.9: Steps for Using a Multiphase Model

•Section 23.10: Setting Up the VOF Model

• Section 23.11: Setting Up the Mixture Model

• Section 23.12: Setting Up the Eulerian Model

• Section 23.13: Setting Up the Wet Steam Model

• Section 23.14: Solution Strategies for Multiphase Modeling

• Section 23.15: Postprocessing for Multiphase Modeling


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ModelingMu ltiphaseFlows

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23.1 Introduction

A large number of flows encountered in nature and technology are a mixture of phases.

Physical phases of matter are gas, liquid, and solid, but the concept of phase in a mul-

tiphase flow system is applied in a broader sense. In multiphase flow, a phase can be

defined as an identifiable class of material that has a particular inertial response to and

interaction with the flow and the potential field in which it is immersed. For example,different-sized solid particles of the same material can be treated as different phases be-

cause each collection of particles with the same size will have a similar dynamical response

to the flow field.

23.1.1 MultiphaseFlow Regimes

Multiphase flow regimes can be grouped into four categories: gas-liquid or liquid-liquid

flows; gas-solid flows; liquid-solid flows; and three-phase flows.

Gas-Liquidor Liquid-LiquidFlowsThe following regimes are gas-liquid or liquid-liquid flows:

• Bubbly flow: This is the flow of discrete gaseous or fluid bubbles in a

continuous fluid.

• Droplet flow: This is the flow of discrete fluid droplets in a continuous gas.

• Slug flow: This is the flow of large bubbles in a continuous fluid.

• Stratified/free-surface flow: This is the flow of immiscible fluids separated

by a clearly-defined interface.

See Figure 23.1.1 for illustrations of these regimes.

Gas-SolidFlows

The following regimes are gas-solid flows:

• Particle-laden flow: This is flow of discrete particles in a continuous gas.

• Pneumatic transport: This is a flow pattern that depends on factors such as

solid loading, Reynolds numbers, and particle properties. Typical patterns are

dune flow, slug flow, packed beds, and homogeneous flow.• Fluidized bed: This consists of a vertical cylinder containing particles, into

which a gas is introduced through a distributor. The gas rising through the bed

suspends the particles. Depending on the gas flow rate, bubbles appear and rise

through the bed, intensifying the mixing within the bed.



23.1 I n t roduction


Liquid-SolidFlows

The following regimes are liquid-solid flows:

• Slurry flow: This flow is the transport of particles in liquids. The

fundamental behavior of liquid-solid flows varies with the properties of the solid

particles relative to those of the liquid. In slurry flows, the Stokes number (see

Equation 23.2-4) is normally less than 1. When the Stokes number is larger than

1, the characteristic of the flow is liquid-solid fluidization.

• Hydrotrans port: This describes densely-distributed solid particles in a

continuous liquid

• Sedimentation: This describes a tall column initially containing a uniform

dispersed mixture of particles. At the bottom, the particles will slow down and

form a sludge layer. At the top, a clear interface will appear, and in the middle aconstant settling zone will exist.


Three-PhaseFlows

Three-phase flows are combinations of the other flow regimes listed in the previous sec-

tions.


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slug flow bubbl y, droplet, or

particle-laden flow

stratified/f ree-surface flow pneumatic transport,

hydrotransport, or slurry flow

sedimentation fluidized bed

Figure 23.1.1: Multiphase Flow Regimes



23.2 Choosinga Ge neralMultiphaseM ode l

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23.1.2 Examplesof MultiphaseSystems

Specific examples of each regime described in Section 23.1.1: Multiphase Flow Regimes

are listed below:

• Bubbly flow examples include absorbers, aeration, air lift pumps, cavitation,

evap- orators, flotation, and scrubbers.

• Droplet flow examples include absorbers, atomizers, combustors, cryogenic

pump- ing, dryers, evaporation, gas cooling, and scrubbers.

• Slug flow examples include large bubble motion in pipes or tanks.

• Stratified/free-surface flow examples include sloshing in offshore separator

devices and boiling and condensation in nuclear reactors.

• Particle-laden flow examples include cyclone separators, air classifiers, dust

collec- tors, and dust-laden environmental flows.

• Pneumatic transport examples include transport of cement, grains, and metal pow- ders.

• Fluidized bed examples include fluidized bed reactors and circulating fluidized

beds.

• Slurry flow examples include slurry trans port and mineral processing

• Hydrotrans port examples include mineral processing and biomedical and

physio- chemical fluid systems

• Sedimentation examples include mineral processing.

23.2 Choosinga GeneralMultiphaseModel

The first step in solving any multiphase problem is to determine which of the regimes

provides some broad guidelines for determining appropriate models for each regime, and

how to determine the degree of interphase coupling for flows involving bubbles, droplets,

or particles, and the appropriate model for different amounts of coupling.





23.2.1 Approachesto MultiphaseModeling

Advances in computational fluid mechanics have provided the basis for further insight

into the dynamics of multiphase flows. Currently there are two approaches for the nu-

merical calculation of multiphase flows: the Euler-Lagrange approach (discussed in Sec-

tion 22.1.1: Overview) and the Euler-Euler approach (discussed in the following section).

Th e Euler-EulerApproach

In the Euler-Euler approach, the different phases are treated mathematically as inter-

penetrating continua. Since the volume of a phase cannot be occupied by the other

phases, the concept of phasic volume fraction is introduced. These volume fractions are

assumed to be continuous functions of space and time and their sum is equal to one.

Conservation equations for each phase are derived to obtain a set of equations, which

have similar structure for all phases. These equations are closed by providing constituti ve

relations that are obtained from empirical information, or, in the case of granular flows,

by application of kinetic theory.

In FLUENT, three different Euler-Euler multiphase models are available: the volume

of fluid (VOF) model, the mixture model, and the Eulerian model.

TheVOFModel

The VOF model (described in Section 23.3: Volume of Fluid (VOF) Model Theory) is

a surface-tracking technique applied to a fixed Eulerian mesh. It is designed for two or

more immiscible fluids where the position of the interface between the fluids is of interest.

In the VOF model, a single set of momentum equations is shared by the fluids, and the

volume fraction of each of the fluids in each computational cell is tracked throughout the

domain. Applications of the VOF model include stratified flows, free-surface flows,filling, sloshing, the motion of large bubbles in a liquid, the motion of liquid after a dam

break, the prediction of jet breakup (surface tension), and the steady or transie nt

tracking of any liquid-gas interface.

TheMixtureModel

The mixture model (described in Section 23.4: Mixture Model Theory) is designed for two

or more phases (fluid or particulate). As in the Eulerian model, the phases are treated as

interpenetrating continua. The mixture model solves for the mixture momentum

equation and prescribes relative velocities to describe the dispersed phases.

Applications of the mixture model include particle-laden flows with low loading, bubblyflows, sedimentation, and cyclone separators. The mixture model can also be used

without relative velocities for the dispersed phases to model homogeneous multiphase

flow.




TheEulerianModel

The Eulerian model (described in Section 23.5: Eulerian Model Theory) is the most com-

plex of the multiphase models in FLUENT. It solves a set of n momentum and continuity

equations for each phase. Coupling is achieved through the pressure and interphase ex-

change coefficients. The manner in which this coupling is handled depends upon the type

of phases involved; granular (fluid-solid) flows are handled differently than nongranular (fluid-fluid) flows. For granular flows, the properties are obtained from application of ki-

netic theory. Momentum exchange between the phases is also dependent upon the type

of mixture being modeled. FLUENT’suser-defined functions allow you to customize the

calculation of the momentum exchange. Applications of the Eulerian multiphase model

include bubble columns, risers, particle suspension, and fluidized beds.

23.2.2 ModelComparisons

In general, once you have determined the flow regime that best represents your multiphase

system, you can select the appropriate model based on the following guidelines:

• For bubbly, droplet, and particle-laden flows in which the phases mix

and/or dispersed-phase volume fractions exceed 10%, use either the mixture

model (described in Section 23.4: Mixture Model Theory) or the Eulerian model

(described in Section 23.5: Eulerian Model Theory ).

• For slug flows, use the VOF model. See Section 23.3: Volume of Fluid (VOF)

Model

Theory for more information about the VOF model.

• For stratified/free-surface flows, use the VOF model. See Section 23.3: Volume

of

Fluid (VOF) Model Theory for more information about the VOF model.

• For pneumatic transport, use the mixture model for homogeneous flow

(described in Section 23.4: Mixture Model Theory) or the Eulerian model for

granular flow (described in Section 23.5: Eulerian Model Theory ).

• For fluidized beds, use the Eulerian model for granular flow. See Section 23.5:

Eu- lerian Model Theory for more information about the Eulerian model.

• For slurry flows and hydrotrans port, use the mixture or Eulerian model

(described, respectively, in Sections 23.4 and 23.5).

• For sedimentation, use the Eulerian model. See Section 23.5: Eulerian

ModelTheory for more information about the Eulerian model.

• For general, complex multiphase flows that involve multiple flow regimes,

select the aspect of the flow that is of most interest, and choose the model that

is most appropriate for that aspect of the flow. Note that the accuracy of results

will not be as good as for flows that involve just one flow regime, since the model

you use will be valid for only part of the flow you are modeling.





As discussed in this section, the VOF model is appropriate for stratified or free-surface

flows, and the mixture and Eulerian models are appropriate for flows in which the phases

mix or separate and/or dispersed-phase volume fractions exceed 10%. (Flows in which

the dispersed-phase volume fractions are less than or equal to 10% can be modeled using

the discrete phase model described in Chapter 22: Modeling Discrete Phase .)

To choose between the mixture model and the Eulerian model, you should consider thefollowing guidelines:

• If there is a wide distribution of the dispersed phases (i.e., if the particles

vary in size and the largest particles do not separate from the primary flow

field), the mixture model may be preferable (i.e., less computationally

expensive). If the

dispersed phases are concentrated just in portions of the domain, you should use

the Eulerian model instead.

• If interphase drag laws that are applicable to your system are available

(either within FLUENT or through a user-defined function), the Eulerian model can

usually provide more accurate results than the mixture model. Even though you

can apply the same drag laws to the mixture model, as you can for a nongranular

Eulerian simulation, if the interphase drag laws are unknown or their applicability

to your system is questionable, the mixture model may be a better choice. For

most cases with spherical particles, then the Schiller-Naumann law is more than

adequate. For cases with nonspherical particles, then a user-defined function can

be used.

• If you want to solve a simpler problem, which requires less computational effort,

the mixture model may be a better option, since it solves a smaller number of

equations than the Eulerian model. If accuracy is more important than

computational effort, the Eulerian model is a better choice. Keep in mind,

however, that the complexity of the Eulerian model can make it less

computationally stable than the mixture model.

FLUENT’s multiphase models are compatible with FLUENT’s dynamic mesh modeling

feature. For more information on the dynamic mesh feature, see Section 11: Modeling

Flows Using Sliding and Deforming Meshes. For more information about how other FLU-

ENT models are compatible with FLUENT’smultiphase models, see Appendix A: FLUENT

Model Compatibili ty.

DetailedGuidelines

For stratified and slug flows, the choice of the VOF model, as indicated in Section 23.2.2:

Model Comparisons, is straightforward. Choosing a model for the other types of flows is

less straightforward. As a general guide, there are some parameters that help to identify the

appropriate multiphase model for these other flows: the particulate loading, β, and the

Stokes number, St. (Note that the word “particle” is used in this discussion to refer to

a particle, droplet, or bubble.)



γ

d

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TheEffectof ParticulateLoading

Particulate loading has a major impact on phase interactions. The particulate loading is

defined as the mass density ratio of the dispersed phase (d) to that of the carrier phase

(c):

The material density ratio

β = αdρd

αcρc

(23.2-1)

γ =ρd

ρc

(23.2-2)

is greater than 1000 for gas-solid flows, about 1 for liquid-solid flows, and less than 0.001

for gas-liquid flows.

Using these parameters it is possible to estimate the average distance between the indi-

vidual particles of the particulate phase. An estimate of this distance has been given by

Crowe et al. [68]:

L

π 1 + κ 1/3

= (23.2-3)dd 6 κ

where κ =β

. Information about these parameters is important for determining how the

dispersed phase should be treated. For example, for a gas-particle flow with a particulate

loading of 1, the interparticle spaceL

dis about 8; the particle can therefore be treated

as isolated (i.e., very low particulate loading).

Depending on the particulate loading, the degree of interaction between the phases can

be divided into the following three categories:

• For very low loading, the coupling between the phases is one-way (i.e., the

fluid carrier influences the particles via drag and turbulence, but the particles

have no influence on the fluid carrier). The discrete phase (Chapter 22: Modeling

Discrete Phase), mixture, and Eulerian models can all handle this type of problem

correctly. Since the Eulerian model is the most expensive, the discrete phase or

mixture model is recommended.

• For intermediate loading, the coupling is two-way (i.e., the fluid carrier

influences the particulate phase via drag and turbulence, but the particles in turninfluence the carrier fluid via reduction in mean momentum and turbulence). The

discrete phase(Chapter 22: Modeling Discrete Phase) , mixture, and Eulerian

models are all applicable in this case, but you need to take into account other

factors in order to decide which model is more appropriate. See below for

information about using the Stokes number as a guide.



d

Vs

• For high loading, there is two-way coupling plus particle pressure and

viscous stresses due to particles (four-way coupling). Only the Eulerian model will

handle this type of problem correctly.

The Significanceof the Stokes Number

For systems with intermediate particulate loading, estimating the value of the Stok es

number can help you select the most appropriate model. The Stokes number can be

defined as the relation between the particle response time and the system response time:

τdSt =

ts

(23.2-4)

where τd = ρd

d2

18µc

and ts is based on the characteristic length (Ls ) and the characteristic

velocity (Vs ) of the system under investigation: ts = Ls

.

For St 1.0, the particle will follow the flow closely and any of the three models

(discrete phase(Chapter 22: Modeling Discrete Phase) , mixture, or Eulerian) is

applicable; you can therefore choose the least expensive (the mixture model, in most

cases), or the most appropriate considering other factors. For St > 1.0, the particles will

move independently of the flow and either the discrete phase model (Chapter 22:

Modeling Discrete Phase ) or the Eulerian model is applicable. For St ≈ 1.0, again

any of the three models is applicable; you can choose the least expensive or the most

appropriate considering other factors.

Examples

For a coal classifier with a characteristic length of 1 m and a characteristic velocity of

10 m/s, the Stokes number is 0.04 for particles with a diameter of 30 microns, but 4.0

for particles with a diameter of 300 microns. Clearly the mixture model will not be

applicable to the latter case.

For the case of mineral processing, in a system with a characteristic length of 0.2 m and a

characteristic velocity of 2 m/s, the Stokes number is 0.005 for particles with a diameter

of 300 microns. In this case, you can choose between the mixture and Eulerian models.

(The volume fractions are too high for the discrete phase model (Chapter 22: Modeling

Discrete Phase), as noted b

elow.)

OtherConsiderations

Keep in mind that the use of the discrete phase model (Chapter 22: Modeling Discrete

Phase) is limited to low volume fractions. Also, the discrete phase model is the only mul-

tiphase model that allows you to specify the particle distribution or include com bustion

modeling in your simulation.



∆t

23.2.3 TimeSchemes in MultiphaseFlow

In many multiphase applications, the process can vary spatially as well as temporally. In

order to accurately model multiphase flow, both higher-order spatial and time discretiza-

tion schemes are necessary. In addition to the first-order time scheme in FLUENT, the

second-order time scheme is available in the Mixture and Eulerian multiphase models,

and with the VOF Implicit Scheme.

i The second-order time scheme cannot be used with the VOF Explicit

Schemes.

The second-order time scheme has been adapted to all the transport equations, includ-

ing mixture phase momentum equations, energy equations, species trans port equations,

turbulence models, phase volume fraction equations, the pressure correction equation,

and the granular flow model. In multiphase flow, a general transport equation (similar

to that of Equation 25.3-15) may be written as

∂(αρφ)

∂t+ ∇ · (αρV~ φ) = ∇ · τ + Sφ (23.2-5)

Where φ is either a mixture (for the mixture model) or a phase variable, α is the phase

volume fraction (unity for the mixture equation), ρ is the mixture phase density, V~

is

the mixture or phase velocity (depending on the equations), τ is the diffusion term, and

Sφ is the source term.

As a fully implicit scheme, this second-order time-accurate scheme achieves its accuracy

by using an Euler backward approximation in time (see Equation 25.3-17). The general

transport equation, Equation 23.2-5 is discretized as

3(α pρ pφ pV ol)n+1 − 4(α pρ pφ pV ol)n + (α pρ pφ p)n−1

2∆t= (23.2-6)

X

[Anb(φnb − φ p)]n+1

+ SUn+1 − S p

n+1 φ p

n+1

Equation 23.2-6 can be written in simpler form:

A pφ p =

X

An bφn b + Sφ (23.2-7)

where

A p =P

Anbn+1

+ S pn+1

+1.5(α p ρ p V

ol)

n

n+1

n−1

Sφ = SUn+1

+2(α p ρ p φ p V

ol)

−0.5(α p ρ p φ p V ol)

∆t



This scheme is easily implemented based on FLUENT’sexisting first-order Euler scheme.

It is unconditionally stable, however, the negative coefficient at the time level tn−1 , of

the three-time level method, may produce oscillatory solutions if the time steps are

large.

This problem can be eliminated if a bounded second-order scheme is introduced. How-

ever, oscillating solutions are most likely seen in compressible liquid flows. Therefore,in this version of FLUENT, a bounded second-order time scheme has been implemented

for compressible liquid flows only. For single phase and multiphase compressible liquid

flows, the second-order time scheme is, by default, the bounded scheme.

23.2.4 Stabilityand Convergence

The process of solving a multiphase system is inherently difficult, and you may encounter

some stability or convergence problems. If a time-dependent problem is being solved, and

patched fields are used for the initial conditions, it is recommended that you perform a

few iterations with a small time step, at least an order of magnitude smaller than the

characteristic time of the flow. You can increase the size of the time step after performing

a few time steps. For steady solutions it is recommended that you start with a small

under-relaxation factor for the volume fraction, it is also recommended not to start with

a patch of volume fraction equal to zero. Another option is to start with a mixture

multiphase calculation, and then switch to the Eulerian multiphase model.

Stratified flows of immiscible fluids should be solved with the VOF model (see Sec-

tion 23.3: Volume of Fluid (VOF) Model Theory). Some problems involving small

volume fractions can be solved more efficiently with the Lagrangian discrete phase

model (see Chapter 22: Modeling Discrete Phase ).

Many stability and convergence problems can be minimized if care is taken during thesetup and solution processes (see Section 23.14.4: Eulerian Model).

23.3 Volumeof Fluid(VOF)ModelTheory

23.3.1 Overviewand Limitationsof the VOFModel

Overview

The VOF model can model two or more immiscible fluids by solving a single set of

momentum equations and tracking the volume fraction of each of the fluids throughout

the domain. Typical applications include the prediction of jet breakup, the motion of large bubbles in a liquid, the motion of liquid after a dam break, and the steady or

transie nt tracking of any liquid-gas interface.



23.3 Volum eof Fluid(VOF)ModelTheory

Limitations

The following restrictions apply to the VOF model inFLUENT:

• You must use the pressure-based solver. The VOF model is not available

with either of the density-based solvers.

• All control volumes must be filled with either a single fluid phase or a

com bination of phases. The VOF model does not allow for void regions where no

fluid of any type is present.

• Only one of the phases can be defined as a compressible ideal gas. There is

no limitation on using compressible liquids using user-defined functions.

• Streamwise periodic flow (either specified mass flow rate or specified pressure

drop)

cannot be modeled when the VOF model is used.

• The second-order implicit time-stepping formulation cannot be used with the

VOF

explicit scheme.

• When tracking particles in parallel, the DPM model cannot be used with the

VOF model if the shared memory option is enabled (Section 22.11.9: Parallel

Processing for the Discrete Phase Model). (Note that using the message passing

option, when running in parallel, enables the compatibili ty of all multiphase flow

models with the DPM model.)

Steady-State and TransientVOFCalculations

The VOF formulation in FLUENT is generally used to compute a time-dependent solution,

but for problems in which you are concerned only with a steady-state solution, it is

possible to perform a steady-state calculation. A steady-state VOF calculation is sensible

only when your solution is independent of the initial conditions and there are distinct

inflow boundaries for the individual phases. For example, since the shape of the free

surface inside a rotating cup depends on the initial level of the fluid, such a problem

must be solved using the time-dependent formulation. On the other hand, the flow of

water in a channel with a region of air on top and a separate air inlet can be solved with

the steady-state formulation.

The VOF formulation relies on the fact that two or more fluids (or phases) are not

interpenetrating. For each additional phase that you add to your model, a variable is

introduced: the volume fraction of the phase in the computational cell. In each control

volume, the volume fractions of all phases sum to unity. The fields for all variables and

properties are shared by the phases and represent volume-averaged values, as long as

the volume fraction of each of the phases is known at each location. Thus the variables

and properties in any given cell are either purely representative of one of the phases, or




representative of a mixture of the phases, depending upon the volume fraction values.

In other words, if the qth fluid’s volume fraction in the cell is denoted as αq , then the

following three conditions are possible:

• αq = 0: The cell is empty (of the qth fluid).

• αq = 1: The cell is full (of the qth fluid).

• 0 < αq < 1: The cell contains the interface between the qth fluid and one or

more other fluids.

Based on the local value of αq , the appropriate properties and variables will be assigned

to each control volume within the domain.

23.3.2 VolumeFractionEquation

The tracking of the interface(s) between the phases is accomplished by the solution of acontinuity equation for the volume fraction of one (or more) of the phases. For the qth

phase, this equation has the followingform:

1

∂(αq ρq ) + ∇ · (αq ρq~vq ) = Sαq

+

n

X

(m˙ pq − m˙ qp) (23.3-1)

ρq

∂t

p=1

where m˙ qp is the mass transfer from phase q to phase p and m˙ pq is the mass transfer

from phase p to phase q. By default, the source term on the right-hand side of Equation

23.3-1,S

αq , is zero, but you can specify a constant or user-defined mass source for each phase. See Section 23.7: Modeling Mass Transfer in Multiphase Flows for more

information on the modeling of mass transfer in FLUENT’sgeneral multiphase models.

The volume fraction equation will not be solved for the primary phase; the primary-phase

volume fraction will be computed based on the following constrai nt:

nXαq = 1 (23.3-2)

q=1

The volume fraction equation may be solved either through implicit or explicit time

discretization.



Th e Im plicitScheme

When the implicit scheme is used for time discretization, FLUENT’s standard finite-

difference interpolation schemes, QUICK, Second Order Upwind and First Order

Upwind, and the Modified HRIC schemes, are used to obtain the face fluxes for all cells,

including those near the interface.

αn+1n+1 n n

n

q ρq − αq ρq

V +X

(ρn+1 n+1 n+1X

∆tf

q Uf αq,f ) = Sαq+

p=1

(m˙ pq − m˙ qp) V (23.3-3)

Since this equation requires the volume fraction values at the current time step (rather

than at the previous step, as for the explicit scheme), a standard scalar trans port equation

is solved iteratively for each of the secondary-phase volume fractions at each time step.

The implicit scheme can be used for both time-dependent and steady-state calculations.

See Section 23.10.1: Choosing a VOF Formulation for details.

The ExplicitScheme

In the explicit approach, FLUENT’sstandard finite-difference interpolation schemes are

applied to the volume fraction values that were computed at the previous time step.

αn+1n+1 n n

n

q ρq − αq ρq

V +X

(ρ U nα

n)

=

X

(m˙

m˙ ) + S V (23.3-4)

∆tf

q f q,f

p=1

pq − qp αq

where n + 1

n

=

=

index for new (curre nt) time step

index for previous time step

αq,f = face value of the qth volume fraction, computed from the first-

or second-order upwind, QUICK, modified HRIC, or CICSAM scheme

V = volume of cell

Uf = volume flux through the face, based on normal velocity

This formulation does not require iterative solution of the transport equation during each

time step, as is needed for the implicit scheme.

i When the explicit scheme is used, a time-dependent solution must becom- puted.

When the explicit scheme is used for time discretization, the face fluxes can be interpo-

lated either using interface reconstruction or using a finite volume discretization scheme

(Section 23.3.2: Interpolation near the Interface). The reconstruction based schemes

available in FLUENTare Geo-Reconstruct and Donor-Acceptor. The discretization schemes

available with explicit scheme for VOF are First Order Upwind, Second Order Upwind,

CICSAM,Modified HRIC, and QUICK.



Interpolationnear the Interface

FLUENT’scontrol-volume formulation requires that convection and diffusion fluxes through

the control volume faces be computed and balanced with source terms within the control

volume itself.

In the geometric reconstruction and donor-acceptor schemes, FLUENT applies a spe-cial interpolation treatme nt to the cells that lie near the interface between two phases.

Figure 23.3.1 shows an actual interface shape along with the interfaces assumed during

computation by these two methods.

actual interface shape

interface shape represented by

the geometric r econstruction

(piecewise-linear) scheme

interface shape represented by

the dono r-acceptor scheme

Figure 23.3.1: Interface Calculations



The explicit scheme and the implicit scheme treat these cells with the same interpo-

lation as the cells that are completely filled with one phase or the other (i.e., using

the standard upwind (Section 25.3.1: First-Order Upwind Scheme), second-order (Sec-

tion 25.3.1: Second-Order Upwind Scheme), QUICK (Section 25.3.1: QUICK Scheme,

modified HRIC (Section 25.3.1: Modified HRIC Scheme), or CICSAM scheme), rather

than applying a special treatme nt.

The GeometricReconstructionScheme

In the geometric reconstruction approach, the standard interpolation schemes that are

used in FLUENT are used to obtain the face fluxes whenever a cell is completely filled

with one phase or another. When the cell is near the interface between two phases, the

geometric reconstruction scheme is used.

The geometric reconstruction scheme represents the interface between fluids using a

piecewise-linear approach. InFLUENT this scheme is the most accurate and is applicable

for general unstructured meshes. The geometric reconstruction scheme is generalized

for unstructured meshes from the work of Youngs [411]. It assumes that the interface between two fluids has a linear slope within each cell, and uses this linear shape for

calculation of the advection of fluid through the cell faces. (See Figure 23.3.1.)

The first step in this reconstruction scheme is calculating the position of the linear in-

terface relative to the center of each partially-filled cell, based on information about

the volume fraction and its derivatives in the cell. The second step is calculating the

advecting amount of fluid through each face using the computed linear interface repre-

sentation and information about the normal and tangential velocity distribution on the

face. The third step is calculating the volume fraction in each cell using the balance of

fluxes calculated during the previous step.

i When the geometric reconstruction scheme is used, a time-dependentsolu- tion must be computed. Also, if you are using a conformal grid (i.e.,if thegrid node locations are identical at the boundaries where two subdomains

meet), you must ensure that there are no two-sided (zero-thickness) walls

within the domain. If there are, you will need to slit them, as described in

Section 6.8.6: Slitting Face Zones.



Th e Donor-AcceptorScheme

In the donor-acceptor approach, the standard interpolation schemes that are used in

FLUENT are used to obtain the face fluxes whenever a cell is completely filled with

one phase or another. When the cell is near the interface between two phases, a “donor-

acceptor” scheme is used to determine the amount of fluid advected through the face

[144]. This scheme identifies one cell as a donor of an amount of fluid from one phase

and another (neighbor) cell as the acceptor of that same amount of fluid, and is

used to prevent numerical diffusion at the interface. The amount of fluid from one

phase that can be convected across a cell boundary is limited by the minimum of two

values: the filled volume in the donor cell or the free volume in the acceptor cell.

The orientation of the interface is also used in determining the face fluxes. The interface

orientation is either horizontal or vertical, depending on the direction of the volume

fraction gradient of the qth phase within the cell, and that of the neighbor cell that shares

the face in question. Depending on the interface’s orientation as well as its motion, flux

values are obtained by pure upwinding, pure downwinding, or some combination of the

two.

i When the donor-acceptor scheme is used, a time-dependent solutionmust be computed. Also, the donor-acceptor scheme can be usedonly withquadrilateral or hexahedral meshes. In addition, if you are using a con-

formal grid (i.e., if the grid node locations are identical at the boundaries

where two subdomains meet), you must ensure that there are no two-sided

(zero-thickness) walls within the domain. If there are, you will need to slit

them, as described in Section 6.8.6: Slitting Face Zones.

The CompressiveInterfaceCa ptu ringSchem e for A rbitra ryMeshes (CICSAM )

The compressive interface capturing scheme for arbitrary meshes (CICSAM), based on

the Ubbink’s work [376], is a high resolution differencing scheme. The CICSAM scheme is

particularly suitable for flows with high ratios of viscosities between the phases. CICSAM

is implemented in FLUENT as an explicit scheme and offers the advantage of producing

an interface that is almost as sharp as the geometric reconstruction scheme.



∂t

23.3.3 MaterialProperties

The properties appearing in the transport equations are determined by the presence of

the component phases in each control volume. In a two-phase system, for example,

if the phases are represented by the subscripts 1 and 2, and if the volume fraction of

the second of these is being tracked, the density in each cell is given by

ρ = α2ρ2 + (1 − α2)ρ1 (23.3-5)

In general, for an n-phase system, the volume-fraction-averaged density takes on the

following form:

ρ =X

αq ρq (23.3-6)

All other properties (e.g., viscosity) are computed in this manner.

23.3.4 MomentumEquation

A single momentum equation is solved throughout the domain, and the resulting velocity

field is shared among the phases. The momentum equation, shown below, is dependent

on the volume fractions of all phases through the properties ρ and µ.

∂(ρ~v) + ∇ · (ρ~v~v) = −∇ p + ∇ ·

h

µ ∇~v + ∇~vT i

+

ρ~g + F~

(23.3-7)

One limitation of the shared-fields approximation is that in cases where large velocitydifferences exist between the phases, the accuracy of the velocities computed near the

interface can be adversely affected.

Note that if the viscosity ratio is more than 1x103, this may lead to convergence diffi-

culties. The compressive interface capturing scheme for arbitrary meshes (CICSAM)

(Section 23.3.2: The Compressive Interface Capturing Scheme for Arbitrary Meshes

(CICSAM)) is suitable for flows with high ratios of viscosities between the phases, thus

solving the problem of poor convergence.



n

23.3.5 Energy Equation

The energy equation, also shared among the phases, is shown below.

∂

∂t

(ρE) + ∇ · (~v(ρE + p)) = ∇ · (k eff ∇T ) + Sh (23.3-8)

The VOF model treats energy, E, and temperature, T , as mass-averaged variables:

nXαq ρq Eq

E =q=1

Xαq ρq

q=1

(23.3-9)

where Eq for each phase is based on the specific heat of that phase and the shared

temperature.The properties ρ and k eff (effective thermal conductivi ty) are shared by the phases. The

source term, Sh , contains contributions from radiation, as well as any other volumetric

heat sources.

As with the velocity field, the accuracy of the temperature near the interface is limited in

cases where large temperature differences exist between the phases. Such problems also

arise in cases where the properties vary by several orders of magnitude. For example, if a

model includes liquid metal in combination with air, the conductivities of the materials

can differ by as much as four orders of magnitude. Such large discrepancies in properties

lead to equation sets with anisotropic coefficients, which in turn can lead to convergence

and precision limitations.

23.3.6 AdditionalScalar Equations

Depending upon your problem definition, additional scalar equations may be involved in

your solution. In the case of turbulence quantities, a single set of transport equations is

solved, and the turbulence variables (e.g., k and or the Reynolds stresses) are shared

by the phases throughout the field.

23.3.7 TimeDependence

For time-dependent VOF calculations, Equation 23.3-1 is solved using an explicit time-

marching scheme. FLUENT automatically refines the time step for the integration of the

volume fraction equation, but you can influence this time step calculation by modifying

the Courant number. You can choose to update the volume fraction once for each time

step, or once for each iteration within each time step. These options are discussed in

more detail in Section 23.10.4: Setting Time-Dependent Parameters for the VOF Model.



R R

23.3.8 Surface Tension and Wall

Adhesion

The VOF model can also include the effects of surface tension along the interface between

each pair of phases. The model can be augmented by the additional specification of the

contact angles between the phases and the walls. You can specify a surface tension

coefficient as a consta nt, as a function of temperature, or through a UDF. The solver

will include the additional tangential stress terms (causing what is termed as Marangoni

convection) that arise due to the variation in surface tension coefficient. Variable surface

tension coefficient effects are usually important only in zero/near-zero gravity conditions.

SurfaceTension

Surface tension arises as a result of attracti ve forces between molecules in a fluid. Con-

sider an air bubble in water, for example. Within the bubble, the net force on a molecule

due to its neighbors is zero. At the surface, however, the net force is radially inward, and

the combined effect of the radial components of force across the entire spherical surfaceis to make the surface contract, thereby increasing the pressure on the concave side of

the surface. The surface tension is a force, acting only at the surface, that is required

to maintain equilibrium in such instances. It acts to balance the radially inward inter-

molecular attracti ve force with the radially outward pressure gradient force across the

surface. In regions where two fluids are separated, but one of them is not in the form

of spherical bubbles, the surface tension acts to minimize free energy by decreasing the

area of the interface.

The surface tension model in FLUENT is the continuum surface force (CSF) model pro-

posed by Brackbill et al. [39]. With this model, the addition of surface tension to the

VOF calculation results in a source term in the momentum equation. To understand theorigin of the source term, consider the special case where the surface tension is constant

along the surface, and where only the forces normal to the interface are considered. It can

be shown that the pressure drop across the surface depends upon the surface tension co-

efficient, σ, and the surface curvature as measured by two radii in orthogonal directions,

R 1 and R 2:

1 1 p2 − p1 = σ +

1 2

(23.3-10)

where p1 and p2 are the pressures in the two fluids on either side of the interface.

In FLUENT, a formulation of the CSF model is used, where the surface curvature is

computed from local gradients in the surface normal at the interface. Let n be the

surface normal, defined as the gradient of αq , the volume fraction of the qth phase.

n = ∇αq (23.3-11)



1

The curvature, κ, is defined in terms of the divergence of the unit normal, nˆ [39]:

κ = ∇ · nˆ (23.3-12)

where

nˆ =n

|n|(23.3-13)

The surface tension can be written in terms of the pressure jump across the surface. The

force at the surface can be expressed as a volume force using the divergence theorem. It

is this volume force that is the source term which is added to the momentum equation.

It has the following form:

Fvol =X

pairs ij, i<j

σij

αiρiκ j ∇α j + α j ρ j κ i ∇αi

2 (ρi + ρ j )

(23.3-14)

This expression allows for a smooth superposition of forces near cells where more than

two phases are present. If only two phases are present in a cell, then κ i = −κ jand

∇αi = −∇α j , and Equation 23.3-14 simplifies to

ρκ i ∇αiFvol = σij 12(ρi + ρ j )

(23.3-15)

where ρ is the volume-averaged density computed using Equation 23.3-6. Equation 23.3-15

shows that the surface tension source term for a cell is proportional to the average density

in the cell.

Note that the calculation of surface tension effects on triangular and tetrahedral meshes

is not as accurate as on quadrilateral and hexahedral meshes. The region where surface

tension effects are most important should therefore be meshed with quadrilaterals or

hexahedra.



WhenSurfaceTension EffectsAre Im portant

The importance of surface tension effects is determined based on the value of two di-

mensionless quantities: the Reynolds number, Re, and the capillary number, Ca; or the

Reynolds number, Re, and the Weber number, We. For Re 1, the quantity of

interest is the capillary num ber:

µUCa =

σ(23.3-16)

and for Re 1, the quantity of interest is the Weber num ber:

We =ρLU 2

σ(23.3-17)

where U is the free-stream velocity. Surface tension effects can be neglected if Ca 1

or We 1.

Several surface tension options are provided through the text user interface (TUI) usingthe solve/set/surface-tension command:solve −→ set −→surface-tension

The surface-tension command prompts you for the following information:

• whether you require node-based smoothing

The default value is no indicating that cell-based smoothing will be used for the

VOF calculations.

• the number of smoothings

The default value is 1. A higher value can be used in case of tetrahedraland triangular meshes in order to reduce any spurious velocities.

• the smoothing relaxation factor

The default is 1. This is useful in the cases where VOF smoothing causes a problem

(e.g., liquid enters through the inlet with wall adhesion on).

• whether you want to use VOF gradients at the nodes for curvature calculations

With this option, FLUENT uses VOF gradients directly from the nodes to calculatethe curvature for surface tension forces. The default is yes which produces better results with surface tension compared to gradients that are calculated at the cell

centers.



WallAdhesion

An option to specify a wall adhesion angle in conjunction with the surface tension model

is also available in the VOF model. The model is taken from work done by Brackbill et

al. [39]. Rather than impose this boundary condition at the wall itself, the contact angle

that the fluid is assumed to make with the wall is used to adjust the surface normal in

cells near the wall. This so-called dynamic boundary condition results in the adjustme ntof the curvature of the surface near the wall.

If θw is the contact angle at the wall, then the surface normal at the live cell next to the

wall is

nˆ = nˆw cos θw + tˆw sin θw (23.3-18)

where nˆw and tˆw are the unit vectors normal and tangential to the wall,

respectively. The combination of this contact angle with the normally calculated

surface normal one cell away from the wall determine the local curvature of the surface,and this curvature is used to adjust the body force term in the surface tension

calculation.

The contact angle θw is the angle between the wall and the tangent to the interface

at the wall, measured inside the phase listed in the left column under Wall Adhesion in

the Momentum tab of the Wall panel. For example, if you are setting the contact angle

between the oil and air phases in the Wall panel shown in Figure 23.3.2, θw is measured

inside the oil phase, as seen in Figure 23.3.3.



Figure 23.3.2: The Wall Panel for a Mixture in a VOF Calculation with Wall

Adhesion



o

interface

AIR

OIL

OR

θW = 30

AIR

θ = 30o

interface

OIL

wall

W

wall

Figure 23.3.3: Measuring the Contact Angle

23.3.9 Open ChannelFlow

FLUENT can model the effects of open channel flow (e.g., rivers, dams, and surface-

piercing structures in unbounded stream) using the VOF formulation and the open chan-

nel boundary condition. These flows involve the existence of a free surface between the

flowing fluid and fluid above it (generally the atmosphere). In such cases, the wave prop-

agation and free surface behavior becomes important. Flow is generally governed by the

forces of gravity and inertia. This feature is mostly applicable to marine applications

and the analysis of flows through drainage systems.

Open channel flows are characterized by the dimensionless Froude Number, which is

defined as the ratio of inertia force and hydrostatic force.

VF r = √

gy(23.3-19)

where V is the velocity magnitude, g is gravity, and y is a length scale, in this case,

the distance from the bottom of the channel to the free surface. The denominator in

Equation 23.3-19 is the propagation speed of the wave. The wave speed as seen by the

fixed observer is defined as

Vw = V ±√gy (23.3-20)



Based on the Froude number, open channel flows can be classified in the following three

categories:

• When F r < 1, i.e., V <√gy (thus Vw < 0 or Vw > 0), the flow is known to

be subcritical where disturbances can travel upstream as well as downstream. In

this case, downstream conditions might affect the flow upstream.

• When F r = 1 (thus Vw = 0), the flow is known to be critical, where

upstream propagating waves remain stationar y. In this case, the character of the

flow changes.

• When F r > 1, i.e., V >√gy (thus Vw > 0), the flow is known to be

super critical where disturbances cannot travel upstream. In this case,

downstream conditions do not affect the flow upstream.

UpstreamBoundaryConditions

There are two options available for the upstream boundary condition for open channel

flows:

• pressure inlet

• mass flow rate

PressureInlet

The total pressure p0 at the inlet can be given as

12 p0 =

2 (ρ − ρ0)V + (ρ − ρ0)|−→g |(gˆ · (

−→ b − −→a )) (23.3-

21)

where−→

b and −→a are the position vectors of the face centroid and any point on

the free

surface, respectively, Here, free surface is assumed to be horizontal and normal to thedirection of gravity. −→g is the gravity vector, |−→g | is the gravity magnitude, gˆ isthe unitvector of gravity, V is the velocity magnitude, ρ is the density of the mixture in the cell,

and ρ0 is the reference density.From this, the dynamic pressure q is

and the static pressure ps is

q =ρ − ρ0

2V

2(23.3-22)

ps = (ρ − ρ0)|−→g |(gˆ · (−→

b − −→a )) (23.3-23)



which can be further expanded to

ps = (ρ − ρ0 )|−→g |((gˆ ·

−→ b ) + ylocal ) (23.3-24)

where the distance from the free surface to the reference position, y local , is

ylocal = −(−→a · gˆ) (23.3-25)

MassFlow Rate

The mass flow rate for each phase associated with the open channel flow is defined by

m˙ phase = ρ phase (Ar ea phase )(V elocity) (23.3-26)

VolumeFractionSpecification

In open channel flows, FLUENT internally calculates the volume fraction based on the

input parameters specified in the Boundary Conditions panel, therefore this option has

been disabled.

For subcritical inlet flows (Fr < 1), FLUENT reconstructs the volume fraction values on

the boundary by using the values from the neighboring cells. This can be accomplished

using the following procedure:

• Calculate the node values of volume fraction at the boundary using the cell

values.

• Calculate the volume fraction at the each face of boundary using theinterpolated node values.

For supercritical inlet flows (Fr > 1), the volume fraction value on the boundary can

be calculated using the fixed height of the free surface from the bottom.

DownstreamBoundaryConditions

PressureOutlet

Determining the static pressure is dependent on the Pressure Specification Method. Usingthe Free Surface Level, the static pressure is dictated by Equation 23.3-23 and Equa-

tion 23.3-25, otherwise you must specify the static pressure as the Gauge Pressure.

For subcritical outlet flows (Fr < 1), if there are only two phases, then the pressure is

taken from the pressure profile specified over the boundary, otherwise the pressure is

taken from the neighboring cell. For supercritical flows (Fr >1), the pressure is always

taken from the neighboring cell.



23.4MixtureModelTheory

Outflow Boundary

Outflow boundary conditions can be used at the outlet of open channel flows to model

flow exits where the details of the flow velocity and pressure are not known prior to

solving the flow problem. If the conditions are unknown at the outflow boundaries, then

FLUENT will extrapolate the required information from the interior.

It is important, however, to understand the limitations of this boundary type:

• You can only use single outflow boundaries at the outlet, which is achieved byset- ting the flow rate weighting to 1. In other words, outflow splitting is not

permittedin open channel flows with outflow boundaries.

• There should be an initial flow field in the simulation to avoid convergence

issues due to flow reversal at the outflow, which will result in an unreliable

solution.

• An outflow boundary condition can only be used with mass flow inlets. It is

not compatible with pressure inlets and pressure outlets. For example, if you

choose the inlet as pressure-inlet, then you can only use pressure-outlet at the outlet.

If you choose the inlet as mass-flow-inlet, then you can use either outflow or

pressure-outlet boundary conditions at the outlet. Note that this only holds true

for open channel flow.

• Note that the outflow boundary condition assumes that flow is fully

developed in the direction perpendicular to the outflow boundary surface.

Therefore, such surfaces should be placed accordingly.

Backflow VolumeFractionSpecification

FLUENT internally calculates the volume fraction values on the outlet boundary by using

the neighboring cell values, therefore, this option is disabled.

23.4 M ixtureM odelTheory

23.4.1 Overviewand Limitationsof the MixtureModel

Overview

The mixture model is a simplified multiphase model that can be used to model multiphaseflows where the phases move at different velocities, but assume local equilibrium

over short spatial length scales. The coupling between the phases should be strong. It

can also be used to model homogeneous multiphase flows with very strong coupling and

the phases moving at the same velocity. In addition, the mixture model can be

used to calculate non-Newtonian viscosity.




The mixture model can model n phases (fluid or particulate) by solving the momentum,

continuity, and energy equations for the mixture, the volume fraction equations for the

secondary phases, and algebraic expressions for the relative velocities. Typical applica-

tions include sedimentation, cyclone separators, particle-laden flows with low loading,

and bubbly flows where the gas volume fraction remains low.

The mixture model is a good substitute for the full Eulerian multiphase model in severalcases. A full multiphase model may not be feasible when there is a wide distribution of

the particulate phase or when the interphase laws are unknown or their reliability can

be questioned. A simpler model like the mixture model can perform as well as a full

multiphase model while solving a smaller number of variables than the full multiphase

model.

The mixture model allows you to select granular phases and calculates all properties of

the granular phases. This is applicable for liquid-solid flows.

Limitations

The following limitations apply to the mixture model in FLUENT:

• You must use the pressure-based solver. The mixture model is not available

with either of the density-based solvers.

• Only one of the phases can be defined as a compressible ideal gas. There is


• Streamwise periodic flow with specified mass flow rate cannot be modeled

when the mixture model is used (the user is allowed to specify a pressure drop).

•Solidification and melting cannot be modeled in conjunction with the

mixture model.

• The LES turbulence model cannot be used with the mixture model if the

cavitation model is enabled.

• The relative velocity formulation cannot be used in combination with the MRF

and mixture model (see Section 10.3.1: Limitations ).

• The mixture model cannot be used for inviscid flows.

• The shell conduction model for walls cannot be used with the mixture model.

•

When tracking particles in parallel, the DPM model cannot be used with themixture model if the shared memory option is enabled (Section 22.11.9: Parallel

Pro- cessing for the Discrete Phase Model). (Note that using the message passing

option, when running in parallel, enables the compatibili ty of all multiphase flow

models with the DPM model.)



23.4MixtureModelTheory

The mixture model, like the VOF model, uses a single-fluid approach. It differs from the

VOF model in two respects:

• The mixture model allows the phases to be interpenetrating. The volume

fractions

αq and α p for a control volume can therefore be equal to any value between 0 and1, depending on the space occupied by phase q and phase p.

• The mixture model allows the phases to move at different velocities, using

the concept of slip velocities. (Note that the phases can also be assumed to

move at the same velocity, and the mixture model is then reduced to a

homogeneous multiphase model.)

The mixture model solves the continuity equation for the mixture, the momentum equa-

tion for the mixture, the energy equation for the mixture, and the volume fraction equa-

tion for the secondary phases, as well as algebraic expressions for the relative velocities

(if the phases are moving at different velocities).

23.4.2 ContinuityEquation

The continuity equation for the mixture is

∂

∂t (ρm) + ∇ · (ρm~vm) = 0 (23.4-1)

where ~vm is the mass-averaged velocity:

Pn αk ρk ~vk

and ρm is the mixture density:

~vm = k =1

ρm

(23.4-2)

n

ρm =X

αk ρk (23.4-3)k =1

αk is the volume fraction of phase k .



m

k k ρ

k

∂

∂t

23.4.3 MomentumEquation

The momentum equation for the mixture can be obtained by summing the individual

momentum equations for all phases. It can be expressed as

(ρm~vm) + ∇ · (ρm~vm~vm) = −∇ p + ∇ ·

h

µm ∇~vm + ∇~vT i

+

ρm~g + F~ + ∇ ·

n

!X αk ρk ~vdr ,k ~vdr ,k k =1 (23.4-4)

where n is the number of phases, F~ is a body force, and µm is the viscosity of the

mixture:

n

µm =X

αk µk (23.4-5)k =1

~vdr,k is the drift velocity for secondary phase k:

~vdr,k = ~vk − ~vm (23.4-6)

23.4.4 Energy Equation

The energy equation for the mixture takes the following form:

∂ n nX

(αk ρk Ek ) + ∇ ·X

(αk ~vk (ρk Ek + p)) = ∇ · (k eff ∇T ) + SE (23.4-7)∂t

k =1 k =1

where k eff is the effective conductivity (P

αk (k k + k t)), where k t is the turbule nt thermal

conductivi ty, defined according to the turbulence model being used). The first term on

the right-hand side of Equation 23.4-7 represents energy transfer due to conduction. SE

includes any other volumetric heat sources.

In Equation 23.4-7,

E = h − p

k

v 2

+ (23.4-8)2

for a compressible phase, and Ek = hk for an incompressible phase, where hk is the

sensible enthalpy for phase k .



p

23.4.5 Relative(Slip)Velocityandthe DriftVelocity

The relative velocity (also referred to as the slip velocity) is defined as the velocity of a

secondary phase (p) relative to the velocity of the primary phase (q):

~v pq = ~v p − ~vq (23.4-9)

The mass fraction for any phase (k) is defined as

ck =αk ρk

ρm

(23.4-10)

The drift velocity and the relative velocity (~vqp) are connected by the following

expression:

n

~vdr,p = ~v pq −

X

ck ~vqk (23.4-11)k =1

FLUENT’smixture model makes use of an algebraic slip formulation. The basic assump-

tion of the algebraic slip mixture model is that to prescribe an algebraic relation for the

relative velocity, a local equilibrium between the phases should be reached over

short spatial length scale. Following Manninen et al. [229], the form of the relative

velocity is given by:

τ p~v pq =

f

(ρ p − ρm)

ρ~a (23.4-12)

drag p

where τ p is the particle relaxation time

τ p =ρ pd2

18µq

(23.4-13)

d is the diameter of the particles (or droplets or bubbles) of secondary phase p, ~a isthe secondary-phase particle’s acceleration. The default drag function f drag is taken

from Schiller and Naumann [320]:

f drag =

(

1 + 0.15 Re0.687 Re≤ 10000.0183 Re Re > 1000

(23.4-14)

and the acceleration ~a is of the form

~a = ~g − (~vm ·

∇)~vm −

∂~vm(23.4-15)

∂t



p~a

∂∂t

The simplest algebraic slip formulation is the so-called drift flux model, in which the ac-

celeration of the particle is given by gravity and/or a centrifugal force and the particulate

relaxation time is modified to take into account the presence of other particles.

In turbule nt flows the relative velocity should contain a diffusion term due to the

dispersion appearing in the momentum equation for the dispersed phase. FLUENT

adds this dispersion to the relative velocity:

~v pq =(ρ p − ρm)d2

18µq f drag

νm

α pσD

∇αq (23.4-16)

where (νm) is the mixture turbule nt viscosity and (σD ) is a Prandtl dispersion coefficient.

When you are solving a mixture multiphase calculation with slip velocity, you can directly

prescribe formulations for the drag function. The following choices are available:

•

Schiller-Naumann (the default formulation)• Morsi-Alexander

• symmetric

• constant

• user-defined

See Section 23.5.4: Interphase Exchange Coefficients for more information on these drag

functions and their formulations, and Section 23.11.1: Defining the Phases for the Mixture

Model for instructions on how to enable them.

Note that, if the slip velocity is not solved, the mixture model is reduced to a homogeneous

multiphase model. In addition, the mixture model can be customized (using user-defined

functions) to use a formulation other than the algebraic slip method for the slip velocity.

See the separate UDF Manual for details.

23.4.6 VolumeFractionEquationfor the SecondaryPhases

From the continuity equation for secondary phase p, the volume fraction equation for

secondary phase p can be obtained:

n

(α pρ p) + ∇ · (α pρ p~vm) = −∇ · (α pρ p~vdr,p) +X

(m˙ qp − m˙ pq ) (23.4-17)q=1



5

s

23.4.7 GranularProperties

Since the concentration of particles is an important factor in the calculation of the effec-

tive viscosity for the mixture, we may use the granular viscosity (see section on Eulerian

granular flows) to get a value for the viscosity of the suspension. The volume weighted

averaged for the viscosity would now contain shear viscosity arising from particle mo-

mentum exchange due to translation and collision.

The collisional and kinetic parts, and the optional frictional part, are added to give the

solids shear viscosity:

µs = µs,col + µs,kin + µs,fr (23.4-18)

CollisionalViscosity

The collisional part of the shear viscosity is modeled as [119, 363]

KineticViscosity

4µs,col =

5 αs ρs ds g0,ss (1 + ess )

Θs1/2

π(23.4-19)

FLUENT provides two expressions for the kinetic viscosity.

The default expression is from Syamlal et al. [363]:

αs ds ρs

√Θs π 2

µs,kin =6 (3 − ess )

1 + (1 + ess ) (3ess − 1) αs

g0,ss

(23.4-20)

The following optional expression from Gidaspow et al. [119] is also available:

10ρs ds

√Θs π 4

2

µs,kin =96α (1 + ess) g0,ss

1 + g0,ss αs (1 + ess )5

(23.4-21)



ss

23.4.8 GranularTemperature

The viscosities need the specification of the granular temperature for the sth solids phase.

Here we use an algebraic equation derived from the transport equation by neglecting

convection and diffusion and takes the form [363]

where

0 = (− ps I + τ s ) : ∇~vs − γΘs + φls (23.4-22)

(− ps I + τ s ) : ∇~vs = the generation of energy by the solid stress tensor

γΘs = the collisional dissipation of energy

φls = the energy exchange between the lth

fluid or solid phase and the sth solid phase

The collisional dissipation of energy, γΘs , represents the rate of energy dissipation within

the sth solids phase due to collisions between particles. This term is represented by the

expression derived by Lun et al. [221]

γΘm =12(1 − e2 )g0,ss

ρs α2Θ

3/2(23.4-23)

ds

√π

s s

The transfer of the kinetic energy of random fluctuations in particle velocity from the sth

solids phase to the lth fluid or solid phase is represented by φls [119]:

φls = −3K ls Θs (23.4-24)

FLUENT allows you to solve for the granular temperature with the following options:

• algebraic formulation (the default)

This is obtained by neglecting convection and diffusion in the transport equation

(Equation 23.4-22) [363].

• constant granular temperature

This is useful in very dense situations where the random fluctuations are small.

• UDF for granular temperature

23.4.9 Solids Pressure

The total solid pressure is calculated and included in the mixture momentum equations:

N

Ps,total =X

pq (23.4-25)q=1

where pq is presented in the section for granular flows by equation Equation 23.5-48



23.5 EulerianModelTheory


Details about the Eulerian multiphase model are presented in the following subsections:

• Section 23.5.1: Overview and Limitations of the Eulerian Model

•Section 23.5.2: Volume Fractions

• Section 23.5.3: Conservation Equations

• Section 23.5.4: Interphase Exchange Coefficients

• Section 23.5.5: Solids Pressure

• Section 23.5.6: Maximum Packing Limit in Binary Mixtures

• Section 23.5.7: Solids Shear Stresses

• Section 23.5.8: Granular Temperature

• Section 23.5.9: Description of Heat Transfer

• Section 23.5.10: Turbulence Models

• Section 23.5.11: Solution Method in FLUENT

23.5.1 Overviewand Limitationsof the EulerianModel

Overview

The Eulerian multiphase model in FLUENT allows for the modeling of multiple sepa-

rate, yet interacting phases. The phases can be liquids, gases, or solids in nearly anycombination. An Eulerian treatme nt is used for each phase, in contrast to the Eulerian-

Lagrangian treatme nt that is used for the discrete phase model.

With the Eulerian multiphase model, the number of secondary phases is limited only

by memory requirements and convergence behavior. Any number of secondary phases

can be modeled, provided that sufficient memory is available. For complex multiphase

flows, however, you may find that your solution is limited by convergence behavior.

See Section 23.14.4: Eulerian Model for multiphase modeling strategies.

FLUENT’sEulerian multiphase model does not distinguish between fluid-fluid and fluid-

solid (granular) multiphase flows. A granular flow is simply one that involves at least

one phase that has been designated as a granular phase.




The FLUENT solution is based on the following:

• A single pressure is shared by all phases.

• Momentum and continuity equations are solved for each phase.

• The following parameters are available for granular phases:

– Granular temperature (solids fluctuating energy) can be calculated for each

solid phase. You can select either an algebraic formulation, a consta nt, a

user-defined function, or a partial differential equation.

– Solid-phase shear and bulk viscosities are obtained by applying kinetic the-

ory to granular flows. Frictional viscosity for modeling granular flow is also

available. You can select appropriate models and user-defined functions for

all properties.

• Several interphase drag coefficient functions are available, which are

appropriate for various types of multiphase regimes. (You can also modify the

interphase drag coefficient through user-defined functions, as described in theseparate UDF Man- ual.)

• All of the k- turbulence models are available, and may apply to all phases or

to the mixture.

Limitations

All other features available in FLUENT can be used in conjunction with the Eulerian

multiphase model, except for the following limitations:

•The Reynolds Stress turbulence model is not available on a per phase basis.

• Particle tracking (using the Lagrangian dispersed phase model) interacts only

with the primary phase.

• Streamwise periodic flow with specified mass flow rate cannot be modeled

when the Eulerian model is used (the user is allowed to specify a pressure drop).

• Inviscid flow is not allowed.

• Melting and solidification are not allowed.

• When tracking particles in parallel, the DPM model cannot be used with the

Eule- rian multiphase model if the shared memory option is enabled (Section22.11.9: Par- allel Processing for the Discrete Phase Model). (Note that using the

message passing option, when running in parallel, enables the compatibili ty of

all multiphase flow models with the DPM model.)




To change from a single-phase model, where a single set of conservation equations for

momentum, continuity and (optionally) energy is solved, to a multiphase model, addi-

tional sets of conservation equations must be introduced. In the process of introduc-

ing additional sets of conservation equations, the original set must also be modified.

The modifications involve, among other things, the introduction of the volume fractions

α1 , α2 , . . . αn for the multiple phases, as well as mechanisms for the exchange of

momentum, heat, and mass between the phases.

23.5.2 VolumeFractions

The description of multiphase flow as interpenetrating continua incorporates the concept

of phasic volume fractions, denoted here by αq . Volume fractions represent the

space occupied by each phase, and the laws of conservation of mass and momentum are

satisfied by each phase individually. The derivation of the conservation equations can

be done by ensemble averaging the local instantaneous balance for each of the phases

[10] or by using the mixture theory approach [36].

The volume of phase q, Vq , is defined by

Z

Vq =V

αq dV (23.5-1)

where

nXαq = 1 (23.5-2)

q=1

The effective density of phase q is

ρˆq = αq ρq (23.5-3)

where ρq is the physical density of phase q.



q

∂

∂t

23.5.3 ConservationEquations

The general conservation equations from which the equations solved by FLUENT are

derived are presented in this section, followed by the solved equations themselves.

Equationsin GeneralFormConservationof Mass

The continuity equation for phase q is

n

(αq ρq ) + ∇ · (αq ρq~vq ) =X

(m˙ pq − m˙ qp) + Sq (23.5-4) p=1

where ~vq is the velocity of phase q and m˙ pq characterizes the mass transfer from the

pth to qth phase, and m˙ qp characterizes the mass transfer from phase q to phase p, and

you are able to specify these mechanisms separatel y.By default, the source term Sq on the right-hand side of Equation 23.5-4 is zero, but you

can specify a constant or user-defined mass source for each phase. A similar term appears

in the momentum and enthalpy equations. See Section 23.7: Modeling Mass Transfer in

Multiphase Flows for more information on the modeling of mass transfer in FLUENT’s

general multiphase models.

Conservationof Momentum

The momentum balance for phase q yields

∂

∂t (αq ρq~vq ) + ∇ · (αq ρq~vq~vq ) = −αq ∇ p + ∇ · τ q + αq ρq~g+

nX(R ~

pq + m˙ pq~v pq − m˙ qp~vqp) + (F~q + F~

lift,q + F~vm,q )

(23.5-5) p=1

where τ q is the qth phase stress-strain tensor

τ q = αq µq ( ∇~vq + ∇~vT

) + αq (λ q

−

2

3 µq ) ∇ · ~vq I (23.5-6)

Here µq and λ q are the shear and bulk viscosity of phase q, F~q is an external body force,

F~lift,q is a lift

force,

F~vm,q is a virtual mass

force,

R ~ pq is an interaction force

between

phases, and p is the pressure shared by all phases.

~v pq is the interphase velocity, defined as follows. If m˙ pq > 0 (i.e., phase p mass is

being transferred to phase q), ~v pq = ~v p; if m˙ pq < 0 (i.e., phase q mass is being

transferred to phase p), ~v pq = ~vq . Likewise, if m˙ qp > 0 then vqp = vq , if m˙ qp < 0



then vqp = v p.



Equation 23.5-5 must be closed with appropriate expressions for the interphase force R ~

pq . This force depends on the friction, pressure, cohesion, and other effects, and is

sub ject to the conditions that R ~ pq = −R ~ qp and R ~

qq = 0.

FLUENT uses a simple interaction term of the following form:

n nXR ~

pq =X

K pq (~v p − ~vq ) (23.5-7) p=1 p=1

where K pq (= K qp) is the interphase momentum exchange coefficient (described in Sec-

tion 23.5.4: Interphase Exchange Coefficients).

Lift Forces

For multiphase flows, FLUENT can include the effect of lift forces on the secondary phase

particles (or droplets or bubbles). These lift forces act on a particle mainly due to velocity

gradients in the primary-phase flow field. The lift force will be more significant for larger particles, but the FLUENT model assumes that the particle diameter is much smaller

than the interparticle spacing. Thus, the inclusion of lift forces is not appropriate for

closely packed particles or for very small particles.

The lift force acting on a secondary phase p in a primary phase q is computed from [88]

F~lift = −0.5ρq α p(~vq − ~v p) × ( ∇ × ~vq ) (23.5-8)

The lift force F~lift will be added to the right-hand side of the momentum equation

for both phases (F~lift,q = −F~

lift ,p).

In most cases, the lift force is insignificant compared to the drag force, so there is noreason to include this extra term. If the lift force is significant (e.g., if the phases separate

quickly), it may be appropriate to include this term. By default, F~lift is not

included. The lift force and lift coefficient can be specified for each pair of phases, if desired.

i It is important that if you include the lift force in your calculation,you need not include it everywhere in the computational domain sinceit iscomputationally expensive to converge. For example, in the wall boundary

layer for turbule nt bubbly flows in channels, the lift force is significantwhen the slip velocity is large in the vicinity of high strain rates for the

primary phase.



dt

Virtual Mass Force

For multiphase flows, FLUENT includes the “virtual mass effect” that occurs when a

secondary phase p accelerates relative to the primary phase q. The inertia of the primary-

phase mass encountered by the accelerating particles (or droplets or bubbles) exerts a

“virtual mass force” on the particles [88]:

F~vm = 0.5α pρq

dq~vq

dt

d p~v p!

−dt

(23.5-9)

The termdq

denotes the phase material time derivative of the form

dq (φ)=

∂(φ)+

(~v

· ∇)φ (23.5-10)

dt ∂tq

The virtual mass force F~vm will be added to the right-hand side of the momentum

equation for both phases (F~vm,q = −F~

vm,p).

The virtual mass effect is significant when the secondary phase density is much smaller

than the primary phase density (e.g., for a transie nt bubble column). By default, F~vm

is not included.

Conservationof Energy

To describe the conservation of energy in Eulerian multiphase applications, a separate

enthalpy equation can be written for each phase:

∂

∂t (αq ρq hq ) + ∇ · (αq ρq ~uq hq ) =

−αq

∂pq

∂t

n

+ τ q : ∇~uq − ∇ ·~qq + Sq + X

(Q pq + m˙ pq h pq −

m˙ q phq p) p=1

(23.5-11)

where hq is the specific enthalpy of the qth phase, ~qq is the heat flux, Sq is a source

term that includes sources of enthalpy (e.g., due to chemical reaction or radiation),

Q pq is the intensity of heat exchange between the pth

and qth

phases, and h pq is theinterphase enthalpy (e.g., the enthalpy of the vapor at the temperature of the droplets,

in the case of evaporation). The heat exchange between phases must comply with the

local balance conditions Q pq = −Qqp and Qqq = 0.



EquationsSolved by FLUENT

The equations for fluid-fluid and granular multiphase flows, as solved by FLUENT, are

presented here for the general case of an n-phase flow.

ContinuityEquation

The volume fraction of each phase is calculated from a continuity equation:

1

∂ n

ρr q

∂t(αq ρq ) + ∇ · (αq ρq~vq ) =

X(m˙ pq − m˙ qp) (23.5-12)

p=1

where ρr q is the phase reference density, or the volume averaged density of the qth phase

in the solution domain.

The solution of this equation for each secondary phase, along with the condition that the

volume fractions sum to one (given by Equation 23.5-2), allows for the calculation of the primary-phase volume fraction. This treatme nt is common to fluid-fluid and granular

flows.

Fluid -Flu idM om entumEquations

The conservation of momentum for a fluid phase q is

∂

∂t (αq ρq~vq ) + ∇ · (αq ρq~vq~vq ) = −αq ∇ p + ∇ · τ q + αq ρq~g +

n

X(K pq (~v p − ~vq ) + m˙ pq~v pq − m˙ qp~vqp) + p=1

(F~q + F~

lift,q + F~vm,q ) (23.5-13)

Here ~g is the acceleration due to gravity and τ q , F~q , F~

lift,q , and F~vm,q are as

defined for Equation 23.5-5.

Fluid-So lidMom entumEquations

Following the work of [7, 51, 79, 119, 198, 221, 267, 363], FLUENT uses a multi-fluidgranular model to describe the flow behavior of a fluid-solid mixture. The solid-phase

stresses are derived by making an analogy between the random particle motion arising

from particle-particle collisions and the thermal motion of molecules in a gas, taking into

account the inelasticity of the granular phase. As is the case for a gas, the intensity of the

particle velocity fluctuations determines the stresses, viscosity, and pressure of the solid

phase. The kinetic energy associated with the particle velocity fluctuations isrepresented



p

by a “pseudothermal” or granular temperature which is proportional to the mean square

of the random motion of particles.

The conservation of momentum for the fluid phases is similar to Equation 23.5-13, and

that for the sth solid phase is

∂

∂t (αs ρs~vs ) + ∇ · (αs ρs~vs~vs ) = −αs ∇ p − ∇ ps + ∇ · τ s + αs ρs~g +

NX(K ls (~vl − ~vs ) + m˙ ls~vls − m˙ sl~vsl ) +

l=1

(F~s + F~

lift ,s + F~vm,s ) (23.5-14)

where ps is the sth solids pressure, K ls = K sl is the momentum exchange coefficient

between fluid or solid phase l and solid phase s, N is the total number of phases, and

F~q , F

~lift,q , and F~

vm,q are as defined for Equation 23.5-5.

Conservationof Energy

The equation solved by FLUENT for the conservation of energy is Equation 23.5-11.

23.5.4 InterphaseExchange Coefficients

It can be seen in Equations 23.5-13 and 23.5-14 that momentum exchange between the phases is based on the value of the fluid-fluid exchange coefficient K pq and, for granular

flows, the fluid-solid and solid-solid exchange coefficients K ls .

Fluid-F luidExchangeCoefficient

For fluid-fluid flows, each secondary phase is assumed to form droplets or bubbles. This

has an impact on how each of the fluids is assigned to a particular phase. For example,

in flows where there are unequal amounts of two fluids, the predomina nt fluid should be

modeled as the primary fluid, since the sparser fluid is more likely to form droplets or

bubbles. The exchangecoefficient for these types of bubbly, liquid-liquid or gas-liquid

mixtures can be written in the following general form:

K pq =αq α pρ pf

τ p(23.5-15)

where f , the drag function, is defined differently for the different exchange-

coefficient models (as described below) and τ p, the “particulate relaxation time”, is

defined as

τ p =ρ pd2

18µq

(23.5-16)



where d p is the diameter of the bubbles or droplets of phase p.

Nearly all definitions of f include a drag coefficient (CD ) that is based on the relative

Reynolds number (Re). It is this drag function that differs among the exchange-coefficientmodels. For all these situations, K pq should tend to zero whenever the primary phase is

not present within the domain. To enforce this, the drag function f is always multiplied

by the volume fraction of the primary phase q, as is reflected in Equation 23.5-15.

• For the model of Schiller and Naumann [320]

where

f =CD Re

24(23.5-17)

CD =

(24(1 + 0.15 Re

0.687 )/Re Re≤ 1000

0.44 Re > 1000(23.5-18)

and Re is the relative Reynolds number. The relative Reynolds number for the

primary phase q and secondary phase p is obtained from

Re =ρq |~v p − ~vq

|d p

µq

(23.5-19)

The relative Reynolds number for secondary phases p and r is obtained from

Re = ρr p|~vr − ~v p|dr p

µr p

(23.5-20)

where µr p = α pµ p + αr µr is the mixture viscosity of the phases p and r .

The Schiller and Naumann model is the default method, and it is acceptable for

general use for all fluid-fluid pairs of phases.

• For the Morsi and Alexander model [252]

where

f =

CD Re

24(23.5-21)

a2 a3CD = a1 +Re

+Re

2(23.5-22)



0

−

2

and Re is defined by Equation 23.5-19 or 23.5-20. The a’s are defined as follows:

a1, a2, a3 =

0, 24, 0 0 < Re < 0.1

3.690, 22.73, 0.0903 0.1 < Re < 1 1.222, 29.1667, −3.8889 1 < Re < 10 0.6167, 46.50, −116.67 10 < Re < 1000.3644, 98.33, 2778 100 < Re < 1000

0.357, 148.62, −47500 1000 < Re < 5000 0.46, −490.546, 578700 5000 < Re < 10000

.5191, −1662.5, 5416700 Re≥

10000

(23.5-23)

The Morsi and Alexander model is the most complete, adjusting the function def-

inition frequently over a large range of Reynolds numbers, but calculations with

this model may be less stable than with the other models.

• For the symmetric model

where

K pq =α p(α pρ p + αq ρq )f

τ pq

(23.5-24)

and

(α pρ p + αq ρq )(d p +dq

)2

τ pq = 2

18(α pµ p + αq µq )(23.5-25)

where

f =CD Re

24(23.5-26)

CD =

(24(1 + 0.15 Re

0.687 )/Re Re≤ 1000

0.44 Re > 1000(23.5-27)

and Re is defined by Equation 23.5-19 or 23.5-20. Note that if there is only one

dispersed phase, then d p = dq in Equation 23.5-25.

The symmetric model is recommended for flows in which the secondary (dispersed) phase in one region of the domain becomes the primary (continuous) phase in

another. Thus for a single dispersed phase, d p = dq and(d p +dq )

= d p. For

example, if air is injected into the bottom of a container filled halfway with

water, the air is the dispersed phase in the bottom half of the container; in the

top half of the container, the air is the continuous phase. This model can also

be used for the interaction between secondary phases.



s

24v2

You can specify different exchange coefficients for each pair of phases. It is also possible

to use user-defined functions to define exchange coefficients for each pair of phases. If the

exchange coefficient is equal to zero (i.e., if no exchange coefficient is specified), the flow

fields for the fluids will be computed independently, with the only “interaction” being

their complementary volume fractions within each computational cell.

Fluid-SolidExchangeCoefficient

The fluid-solid exchange coefficient K sl can be written in the following general form:

K sl =αs ρs f

τs

(23.5-28)

where f is defined differently for the different exchange-coefficient models (as described

below), and τs , the “particulate relaxation time”, is defined as

ρs d2

τs =18µl

where ds is the diameter of particles of phase s.

(23.5-29)

All definitions of f include a drag function (CD ) that is based on the relative Reynolds

number (Res ). It is this drag function that differs among the exchange-coefficient models.

• For the Syamlal-O’Brien model [362]

f =

CD Res αl

r ,s(23.5-30)

where the drag function has a form derived by Dalla Valle [73]

4.8

2

CD = 0.63 + q Res /vr ,s

(23.5-31)

This model is based on measurements of the terminal velocities of particles in

fluidized or settling beds, with correlations that are a function of the volume fraction

and relative Reynolds number [305]:

Res =ρl ds |~vs −

~vl |

µl

(23.5-32)

where the subscript l is for the lth fluid phase, s is for the sth solid phase, and ds is

the diameter of the sth solid phase particles.



r ,s

A = α4.14

l

B = α2.65

α

D

The fluid-solid exchange coefficient has the form

K sl =3αs αl ρl

C4v2 ds

Res

!

vr ,s

|~vs − ~vl | (23.5-33)

where vr ,s is the terminal velocity correlation for the solid phase [113]:

vr ,s = 0.5 A − 0.06 Res +q

2(0.06 Res ) + 0.12 Res (2B − A) +

A2(23.5-34)

with

l (23.5-35)

and

for αl ≤ 0.85,

and

B = 0.8α1.28 (23.5-36)

for αl > 0.85.

l (23.5-37)

This model is appropriate when the solids shear stresses are defined according toSyamlal et al. [363] (Equation 23.5-64).

• For the model of Wen and Yu [396], the fluid-solid exchange coefficient is of

the following form:

where

3K sl =

4 CD

s αl ρl |~vs −

~vl |

ds

−2.65l (23.5-38)

24 h

1 + 0.15(α Re )0.687

i

(23.5-39)CD = α Rel s

l s

and Res is defined by Equation 23.5-32.

This model is appropriate for dilute systems.



α

2

• The Gidaspow model [119] is a combination of the Wen and Yu model [396]

and the Ergun equation [96].

When αl > 0.8, the fluid-solid exchange coefficient K sl is of the following form:

where

3K

sl=

4 C

D

s αl ρl |~vs −

~vl |ds

−2.65

l (23.5-40)

24 h

1 + 0.15(α Re )0.687

i

(23.5-41)CD =α Re

l sl s

When αl ≤ 0.8,

K sl = 150αs (1 − αl )µl

α d2+ 1.75

ρl αs |~vs −

~vl |

d

(23.5-42)

l s s

This model is recommended for dense fluidized beds.

Solid-SolidExchangeCoefficient

The solid-solid exchange coefficient K ls has the following form [361]:

3 (1 + els ) π + Cfr,lsπ α ρ α ρ

(d + d )2

g2 8 s s l l l s 0,ls

K ls =

2π (ρ d

3

+ ρ d

3

)

|~vl − ~vs | (23.5-43)

l l s s

where

els = the coefficient of restitution

Cfr,ls = the coefficient of friction between the lth and sth

solid-phase particles (Cfr,ls = 0)

dl = the diameter of the particles of solid l

g0,ls = the radial distribution coefficient

Note that the coefficient of restitution is described in Section 23.5.5: Solids Pressure

and the radial distribution coefficient is described in Section 23.5.5: Radial Distribution

Function .



s

s

qp

23.5.5 Solids Pressure

For granular flows in the compressible regime (i.e., where the solids volume fraction is less

than its maximum allowed value), a solids pressure is calculated independently and used

for the pressure gradient term, ∇ ps , in the granular-phase momentum equation. Because

a Maxwellian velocity distribution is used for the particles, a granular temperature is

introduced into the model, and appears in the expression for the solids pressure and

viscosities. The solids pressure is composed of a kinetic term and a second term due to

particle collisions:

ps = αs ρs Θs + 2ρs (1 + ess )α2g0,ss Θs (23.5-44)

where ess is the coefficient of restitution for particle collisions, g0,ss is the radial distribu-

tion function, and Θs is the granular temperature. FLUENT uses a default value of 0.9

for ess , but the value can be adjusted to suit the particle type. The granular temperature

Θs is proportional to the kinetic energy of the fluctuating particle motion, and will be

described later in this section. The function g0,ss (described below in more detail) is adistribution function that governs the transition from the “compressible” condition with

α < αs,max , where the spacing between the solid particles can continue to decrease, to

the “incompressible” condition with α = αs,max , where no further decrease in the spacing

can occur. A value of 0.63 is the default for αs,max , but you can modify it during the

problem setup.

Other formulations that are also available in FLUENT are [363]

ps = 2ρs (1 + ess )α2g0,ss Θs (23.5-45)

and [226]

1 ps = αs ρs Θs [(1 + 4αs g0,ss ) +

2 [(1 + ess )(1 − ess + 2µf r ic)]] (23.5-46)

When more than one solids phase are calculated, the above expression does not take into

account the effect of other phases. A derivation of the expressions from the Boltzman

equations for a granular mixture are beyond the scope of this manual, however there is

a need to provide a better formulation so that some properties may feel the presence of

other phases. A known problem is that N solids phases with identical properties should be

consistent when the same phases are described by a single solids phase. Equations derived

empirically may not satisfy this property and need to be changed accordingly withoutdeviating significantly from the original form. From [118], a general solids pressure

formulation in the presence of other phases could be of the form

N π pq = αq ρq Θq +

X

g0,pq d3

nq n p(1 + eq p)f (m p, mq , Θ p, Θq ) (23.5-47) p=1

3



2

d3

d3

where d pq =d p +dq

is the average diameter, n p, nq are the number of particles, m p and mq

are the masses of the particles in phases p and q, and f is a function of the masses of the

particles and their granular temperatures. For now, we have to simplify this expression

so that it depends only on the granular temperature of phase q

N pq = αq ρq Θq +

X

2 p=1

d3 pq(1 + e pq )g0,pq αq α pρq Θq (23.5-48)

q

Since all models need to be cast in the general form, it follows that

N

pq = αq ρq Θq + (X

p=1

d3 pq

pc,qp)ρq Θq (23.5-49)q

where pc,qp is the collisional part of the pressure between phases q and p.

The above expression reverts to the one solids phase expression when N = 1 and q = p

but also has the property of feeling the presence of other phases.

RadialDistributionFunction

The radial distribution function, g0 , is a correction factor that modifies the probabili ty

of collisions between grains when the solid granular phase becomes dense. This function

may also be interpreted as the nondimensional distance between spheres:

g0 =

s + d p

(23.5-50)s

where s is the distance between grains. From Equation 23.5-50 it can be observed that

for a dilute solid phase s → ∞, and therefore g0 → 1. In the limit when the solid phase

compacts, s → 0 and g0 → ∞. The radial distribution function is closely connected

to the factor χ of Chapman and Cowling’s [51] theory of nonuniform gases. χ is equal

to 1 for a rare gas, and increases and tends to infinity when the molecules are so close

together that motion is not possible.



1

1

αs,max)

1 −

1

s

2

k

d

N

In the literature there is no unique formulation for the radial distribution function. FLU-

ENT has a number of options:

• For one solids phase, use [267]:

α ! −3

s

g0 = 1 −α s,max

(23.5-51)

This is an empirical function and does not extends easily to n phases. For two

identical phases with the property that αq = α1 + α2, the above function is not

consistent for the calculation of the partial pressures p1 and p2, pq = p1 + p2. In

order to correct this problem, FLUENT uses the following consistent formulation:

α

! −11 N α

g0,ll = 1 −α

3

+ dl

X(23.5-52)

s,maxk=1

k

where

and k are solids phases only.

N

αs =X

αk (23.5-53)k =1

• The following expression is also available [151]:

1g0,ll = (1 − αs

3+2

dlX αk

k=1dk

(23.5-54)

• Also available [226], slightly modified for n solids phases, is the following:

1 + 2.5αs + 4.59α2 +4.52α3 1 N αk

g0,ll =s s

l

X

α s

αs,max

3 0.678

+2

d

k=1dk

(23.5-55)

• The following equation [363] is

available:

1

3(P N αk )

k=1 dk g0,kl =(1 − α )

+(1 − α )2(d

dk dl (23.5-56)+ d )s s j k



X1 =

(α

d1

α

d1

When the number of solid phases is greater than 1, Equation 23.5-52, Equation 23.5-54

and Equation 23.5-55 are extended to

dmg0,ll + dl g0,mmg0,lm =dm + dl

(23.5-57)

It is interesting to note that equations Equation 23.5-54 and Equation 23.5-55 compare

well with [6] experimental data, while Equation 23.5-56 reverts to the [47] derivation.

23.5.6 Maximum Packing L imitin BinaryMixtures

The packing limit is not a fixed quantity and may change according to the number of

particles present within a given volume and the diameter of the particles. Small particles

accumulate in between larger particles increasing the packing limit. For a binary mixture

FLUENT uses the correlations proposed by [99].

For a binary mixture with diameters d1

> d2

, the mixture composition is defined as α1

α1 +α2

where

X1 <=1,max

α1,max

+ (1 − α1,max)α2,max

(23.5-58))

The maximum packing limit for the mixture is given by

αs,max = (α1,max − α2,max + [1−

s

d2](1 − α1,max )α2,max ) (23.5-59)

X1∗(α1,max + (1 − α1,max )α2,max )1,max

+α2,max (23.5-60)

otherwise, the maximum packing limit for the binary mixture is

[1−

s

d2](α1,max + (1 − α1,max )α2,max )(1 − X1 ) + α1,max (23.5-61)

The packing limit is used for the calculation of the radial distribution function.



5

s

23.5.7 Solids ShearStresses

The solids stress tensor contains shear and bulk viscosities arising from particle momen-

tum exchange due to translation and collision. A frictional component of viscosity can

also be included to account for the viscous-plastic transition that occurs when particles

of a solid phase reach the maximum solid volume fraction.

The collisional and kinetic parts, and the optional frictional part, are added to give the

solids shear viscosity:

µs = µs,col + µs,kin + µs,fr (23.5-62)

CollisionalViscosity

The collisional part of the shear viscosity is modeled as [119, 363]

KineticViscosity

4µs,col =5

αs ρs ds g0,ss (1 + ess ) Θs

1/2

π(23.5-63)

FLUENT provides two expressions for the kinetic part.

The default expression is from Syamlal et al. [363]:

αs ds ρs

√Θs π 2

µs,kin =6 (3 − ess ) 1 + (1 + ess ) (3ess − 1) αs

g0,ss

(23.5-64)

The following optional expression from Gidaspow et al. [119] is also available:

10ρs ds

√Θs π 4

2

µs,kin =96α (1 + ess) g0,ss

1 + g0,ss αs (1 + ess )5

(23.5-65)

Bulk Viscosity

The solids bulk viscosity accounts for the resistance of the granular particles to compres-

sion and expansion. It has the following form from Lun et al. [221]:

4λ s =

3 αs ρs ds g0,ss (1 + ess )

Θs1/2

π(23.5-66)

Note that the bulk viscosity is set to a constant value of zero, by default. It is also

possible to select the Lun et al. expression or use a user-defined function.



τf riction = −Pf r ~~

f riction s s

FrictionalViscosity

In dense flow at low shear, where the secondary volume fraction for a solid phase nears

the packing limit, the generation of stress is mainly due to friction between particles.

The solids shear viscosity computed by FLUENT does not, by default, account for the

friction between particles.

If the frictional viscosity is included in the calculation, FLUENT uses Schaeffer’s [318]

expression:

µs,fr = ps sin φ

2√I2D

(23.5-67)

where ps is the solids pressure, φ is the angle of internal friction, and I2D is the second

invariant of the deviatoric stress tensor. It is also possible to specify a constant or user-

defined frictional viscosity.

In granular flows with high solids volume fraction, instantaneous collisions are less important. The application of kinetic theory to granular flows is no longer relevant since

particles are in contact and the resulting frictional stresses need to be taken into account.

FLUENT extends the formulation of the frictional viscosity and employs other models, as

well as providing new hooks for UDFs. See the separate UDF Manual for details.

The frictional stresses are usually written in Newtonian form:

icti onI + µ ( ∇~u + ( ∇~u )T

) (23.5-68)

The frictional stress is added to the stress predicted by the kinetic theory when the solids

volume fraction exceeds a critical value. This value is normally set to 0.5 when the flowis three-dimensional and the maximum packing limit is about 0.63. Then

PS = Pkinetic + Pf riction (23.5-69)

µS = µkinetic + µf riction (23.5-70)

The derivation of the frictional pressure is mainly semi-empirical, while the frictional vis-

cosity can be derived from the first principles. The application of the modified Coulom b

law leads to an expression of the form

µf riction =Pf riction sin φ

2√I2D

(23.5-71)

Where φ is the angle of internal friction and I2D is the second invariant of the deviatoric

stress tensor.



(α

Two additional models are available in FLUENT: the Johnson and Jackson [165] model

for frictional pressure and Syamlal et al [363].

The Johnson and Jackson [165] model for frictional pressure is defined as

(αs − αs,mi n)n

Pf riction = F r s,max(23.5-72)− αs ) p

With coefficient Fr = 0.05, n=2 and p = 3 [266]. The critical value for the solids volume

fraction is 0.5. The coefficient Fr was modified to make it a function of the volume

fraction:

F r = 0.1αs (23.5-73)

The frictional viscosity for this model is of the form

µf riction = Pf riction sin φ (23.5-74)

The second model that is employed is Syamlal et al [363], described in Equation 23.5-64.

Comparing the two models results in the frictional normal stress differing by orders of

magnitude.

The radial distribution function is an important parameter in the description of the solids

pressure resulting from granular kinetic theory. If we use the models of Lun et al. [221] or

Gidaspow [118] the radial function tends to infinity as the volume fraction tends to the

packing limit. It would then be possible to use this pressure directly in the calculation

of the frictional viscosity, as it has the desired effect. This approach is also available in

FLUENT by default.

i The introduction of the frictional viscosity helps in the description of frictional flows, however a complete description would require theintroductionof more physics to capture the elastic regime with the calculation of the

yield stress and the use of the flow-rule. These effects can be added by the

user via UDFs to model static regime. Small time steps are required to get

good convergence behavior.



ss

ss

23.5.8 GranularTemperature

The granular temperature for the sth solids phase is proportional to the kinetic energy of

the random motion of the particles. The trans port equation derived from kinetic theory

takes the form [79]

3"

∂#

(ρs αs Θs ) + ∇ · (ρs αs~vs Θs

)= (− ps I +τ s ) : ∇~vs + ∇·(k Θs

∇Θs )−γΘs +φls (23.5-75)

2 ∂t

where

(− ps I + τ s ) : ∇~vs = the generation of energy by the solid stress tensor

k Θs ∇Θs = the diffusion of energy (k Θs is the diffusion coefficient)

γΘs= the collisional dissipation of energy

φls = the energy exchange between the lth

fluid or solid phase and the sth solid phase

Equation 23.5-75 contains the term k Θs ∇Θs describing the diffusive flux of granular

energy. When the default Syamlal et al. model [363] is used, the diffusion coefficient for granular energy, k Θs

is given by

15ds ρs αs

√Θs π 2

216

k Θs = 4(41 − 33η)1 + η

5(4η − 3)αs g0,ss +

15π(41 − 33η)ηαs g0,ss

)

(23.5-76)

where

1η =

2 (1 + ess )

FLUENT uses the following expression if the optional model of Gidaspow et al. [119] is

enabled:

150ρs ds

q

(Θπ) 6 2

2

s

Θs

k Θs = 384(1 + e )g0,ss 1 + αs g0,ss (1 + es )5 + 2ρs αs ds (1 + ess )g0,ss (23.5-77)π

The collisional dissipation of energy, γΘs , represents the rate of energy dissipation within

the sth solids phase due to collisions between particles. This term is represented by the

expression derived by Lun et al. [221]

γΘm =12(1 − e2 )g0,ss

ρs α2Θ

3/2(23.5-78)

ds

√π

s s



s6

~

2

−

The transfer of the kinetic energy of random fluctuations in particle velocity from the sth

solids phase to the lth fluid or solid phase is represented by φls [119]:

φls = −3K ls Θs (23.5-79)

FLUENT allows the user to solve for the granular temperature with the following options:

• algebraic formulation (the default)

It is obtained by neglecting convection and diffusion in the transport equation,

Equation 23.5-75 [363].

• partial differential equation

This is given by Equation 23.5-75 and it is allowed to choose different options for

it properties.

• constant granular temperature

This is useful in very dense situations where the random fluctuations are small.

• UDF for granular temperature

For a granular phase s, we may write the shear force at the wall in the following form:

τ~ =π √

3φαs

αρs g0

q Θs Us,|| (23.5-80)

s,max

Here U~s,|| is the particle slip velocity parallel to the wall, φ is the specularity

coefficient between the particle and the wall, αs,max is the volume fraction for the

particles at maximum packing, and g0 is the radial distribution function that is model

dependent.The general boundary condition for granular temperature at the wall takes the form

[165]

π √ αsq π √ αs 2

3

qs =6

3φαs,max

ρs g0 Θs )U~s,|| · U~

s,||

−4

3αs,max

(1 − esw )ρs g0Θs (23.5-81)

23.5.9 Descriptionof HeatTransfer

The internal energy balance for phase q is written in terms of the phase enthalpy, Equa-tion 23.5-11, defined by

Z

Hq = c p,q dTq (23.5-82)

where c p,q is the specific heat at constant pressure of phase q. The thermal boundary

conditions used with multiphase flows are the same as those for a single-phase flow. See

Chapter 7: Boundary Conditions for details.



p

The Heat ExchangeCoefficient

The rate of energy transfer between phases is assumed to be a function of the temperature

difference

Q pq = h pq (T p − Tq ) (23.5-83)

where h pq (= hqp) is the heat transfer coefficient between the pth phase and the qth phase.

The heat transfer coefficient is related to the pth phase Nusselt number, Nu p, by

h pq =6κ q α pαq Nu p

d p2

(23.5-84)

Here κ q is the thermal conductivity of the qth phase. The Nusselt number istypically

determined from one of the many correlations reported in the literature. In the case of

fluid-fluid multiphase, FLUENT uses the correlation of Ranz and Marshall [295, 296]:

Nu p = 2.0 + 0.6Re1/2

Pr 1/3

(23.5-85)

where Re p is the relative Reynolds number based on the diameter of the pth phase and

the relative velocity |u~ p − u~q |, and Pr is the Prandtl number of the qth phase:

c pq µq

Pr =κ q

(23.5-86)

In the case of granular flows (where p = s), FLUENT uses a Nusselt number correlation by Gunn [128], applicable to a porosity range of 0.35–1.0 and a Reynolds number of up

to 105:

Nus = (7 − 10αf + 5α2

)(1 + 0.7Re0.2

Pr 1/3) + (1.33 − 2.4αf + 1.2α

2)Re

0.7Pr

1/3 (23.5-

87) f s f s

The Prandtl number is defined as above with q = f . For all these situations, h pq

should tend to zero whenever one of the phases is not present within the domain. To

enforce this, h pq is always multiplied by the volume fraction of the primary phase q, as

reflected in Equation 23.5-84.



23.5.10 TurbulenceModels

To describe the effects of turbule nt fluctuations of velocities and scalar quantities in

a single phase, FLUENT uses various types of closure models, as described in Chap-

ter 12: Modeling Turbulence. In comparison to single-phase flows, the number of terms

to be modeled in the momentum equations in multiphase flows is large, and this mak es

the modeling of turbulence in multiphase simulations extremely complex.

FLUENT provides three methods for modeling turbulence in multiphase flows within the

context of the k- models. In addition, FLUENT provides two turbulence options within

the context of the Reynolds stress models (RSM).

The k- turbulence model options are:

• mixture turbulence model (the default)

• dispersed turbulence model

• turbulence model for each phase

i Note that the descriptions of each method below are presented basedon the standard k- model. The multiphase modifications to the RNGandrealizable k- models are similar, and are therefore not presented explicitly.

The RSM turbulence model options are:

• mixture turbulence model (the default)

• dispersed turbulence model

For either category, the choice of model depends on the importance of the secondary-

phase turbulence in your application.

k - TurbulenceModels

FLUENT provides three turbulence model options in the context of the k- models: the

mixture turbulence model (the default), the dispersed turbulence model, or a per-phase

turbulence model.



N

k - M ixtureTurbulenceModel

The mixture turbulence model is the default multiphase turbulence model. It represents

the first extension of the single-phase k- 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 turbule nt flow.

The k and equations describing this model are as follows:

∂

∂t (ρmk) + ∇ · (ρm~vmk) = ∇

·

µt,mk σk

∇+ Gk,m − ρm (23.5-88)

and

∂∂t (ρ

m ) + ∇ · (ρm~vm ) = ∇·

µt,m σ

∇ +k

(C1 Gk,m − C2 ρm ) (23.5-89)

where the mixture density and velocity, ρm and ~vm, are computed from

N

ρm =X

αiρi (23.5-90)i=1

and

NXαiρi~vi

~vm =i=1

X

αiρi

i=1

(23.5-91)

the turbule nt viscosity, µt,m, is computed from

k 2µt,m = ρmCµ (23.5-92)

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

Gk,m = µt,m( ∇~vm +

( ∇~vm)T

) : ∇~vm (23.5-93)

The constants in these equations are the same as those described in Section 12.4.1: Stan-

dard k- Model for the single-phase k- model.



k - Dispersed TurbulenceModel

The dispersed turbulence model is the appropriate model when the concentrations of the

secondary phases are dilute. In this case, interparticle collisions are negligible and the

dominant process in the random motion of the secondary phases is the influence of the

primary-phase turbulence. Fluctuating quantities of the secondary phases can therefore

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.

Assumptions

The dispersed method for modeling turbulence in FLUENT assumes the following:

• a modified k- model for the continuous phase

Turbulent predictions for the continuous phase are obtained using the standard

k- model supplemented with extra terms that include the interphase turbule nt

momentum transfer.

• Tchen-theory correlations for the dispersed phases

Predictions for turbulence quantities for the dispersed phases are obtained using the

Tchen theory of dispersion of discrete particles by homogeneous turbulence [141].

• interphase turbule nt momentum transfer

In turbule nt multiphase flows, the momentum exchange terms contain the cor-

relation between the instantaneous distribution of the dispersed phases and theturbule nt fluid motion. It is possible to take into account the dispersion of the

dispersed phases transported by the turbule nt fluid motion.

• a phase-weighted averaging process

The choice of averaging process has an impact on the modeling of dispersion in

turbule nt multiphase flows. A two-step averaging process leads to the appearance

of fluctuations in the phase volume fractions. When the two-step averaging pro-

cess is used with a phase-weighted average for the turbulence, however, turbule nt

fluctuations in the volume fractions do not appear. FLUENT uses phase-weighted

averaging, so no volume fraction fluctuations are introduced into the continuity

equations.



k

k

q

3

∂

∂

k

Turbulence in the Co ntinuous Phase

The eddy viscosity model is used to calculate averaged fluctuating quantities. The

Reynolds stress tensor for continuous phase q takes the following form:

τ 0 0 2 T

q = − 3 (ρq k q + ρq µt,q ∇ · U~ q )I + ρq µt,q ( ∇U~ q

+ ∇U~ q

where U~ q is the phase-weighted velocity.

) (23.5-94)

The turbule nt viscosity µt,q is written in terms of the turbule nt kinetic energy of phase

q:

2

µt,q = ρq Cµq

q

(23.5-95)

and a characteristic time of the energetic turbule nt eddies is defined as

3 k qτt,q = 2

Cµ (23.5-96)

where q is the dissipation rate and Cµ = 0.09.

The length scale of the turbule nt eddies is

Lt,q =

s

3/2

Cµ q2 q

(23.5-97)

Turbulent predictions are obtained from the modified k- model:

∂t (αq ρq k q ) + ∇ · (αq ρq U

~ q k q ) = ∇ ·

(αq

µt,q

σk

∇k q ) + αq Gk,q − αq ρq q + αq ρq Πk q (23.5-98)

and

∂t (αq ρq q ) + ∇ · (αq ρq U

~ q q ) = ∇ ·

(αq

µt,q

σ

∇ q ) + αq q

(C1 Gk,q − C2 ρq q ) + αq ρq Π q

q

(23.5-99)

Here Πk q and Π qrepresent the influence of the dispersed phases on the continuous phase

q, and Gk,q is the production of turbule nt kinetic energy, as defined in Section 12.4.4:

Mod- eling Turbulent Production in the k- Models. All other terms have the same

meaning as in the single-phase k- model.



k Π = C

X

q p

X

The term Πk q can be derived from the instantaneous equation of the continuous phase

and takes the following form, where M represents the number of secondary phases:

M

Πk q =K pq

(< ~v0 0· ~v

0 0 > +( U~

p − U~ q ) · ~vdr ) (23.5-100)

p=1αq ρq

which can be simplified to

M

Πk q =K pq

(k pq − 2k q + ~v pq · ~vdr ) (23.5-101) p=1

αq ρq

where k lq is the covariance of the velocities of the continuous phase q and the dispersed

phase l (calculated from Equation 23.5-109 below), ~v pq is the relative velocity, and ~vdr

is the drift velocity (defined by Equation 23.5-114 below).

Π q is modeled according to Elgobashi et al. [95]:

where C3 = 1.2.

q q 3

q

Πk q (23.5-102)

Turbulence in the Dis persed Phase

Time and length scales that characterize the motion are used to evaluate dispersion

coefficients, correlation functions, and the turbule nt kinetic energy of each dispersed

phase.The characteristic particle relaxation time connected with inertial effects acting ona

dispersed phase p is defined as ρ p

!

τF,pq = α pρq K −1

+ CV (23.5-103) pqρq

The Lagrangian integral time scale calculated along particle trajectories, mainly affected

by the crossing-trajectory effect [69], is defined as

τt,pq = q τt,q

(1 + Cβ ξ2)(23.5-104)

where

ξ =|~v pq |

τt,q

Lt,q

(23.5-105)



τη =

3 3

σ α

ρ

and

Cβ = 1.8 − 1.35 cos2

θ (23.5-106)

where θ is the angle between the mean particle velocity and the mean relative velocity.

The ratio between these two characteristic times is written as

τt,pq pq

F,pq(23.5-107)

Following Simonin [333], FLUENT writes the turbulence quantities for dispersed phase p

as follows:

k p = k q

b2 + η pq

!

1 + η pq b + η pq

!

(23.5-108)

k pq = 2k q

1

1 + η pq

(23.5-109)

Dt,pq = k pq τt,pq (23.5-110)3

2 1D p = Dt,pq + k p − b k pq τF,pq (23.5-111)

1 ρ p

!−

b = (1 + CV ) + CVq

(23.5-112)

and CV = 0.5 is the added-mass coefficient.

Interphase Turbule nt Mome ntum Transfer

The turbule nt drag term for multiphase flows (K pq (~v p −~vq ) in Equation 23.5-7) is

modeled as follows, for dispersed phase p and continuous phase q:

K pq (~v p − ~vq ) = K pq (U~ p − U~

q ) − K pq~vdr (23.5-113)

The second term on the right-hand side of Equation 23.5-113 contains the drift velocity:

D p Dq

!

~vdr = −σ pq α p

∇α p − pq q

∇αq (23.5-114)

Here D p and Dq are diffusivities, and σ pq is a dispersion Prandtl number. When using

Tchen theory in multiphase flows, FLUENT assumes D p = Dq = Dt,pq and the default

value for σ pq is 0.75.



multiphase-

yesyes

∂

The drift velocity results from turbule nt fluctuations in the volume fraction. When

multiplied by the exchange coefficient K pq , it serves as a correction to the momentum

exchange term for turbule nt flows. This correction is not included, by default, but you

can enable it during the problem setup.

You can enable the effect of drift velocity by performing the

following:

1. If it is not already done, set the k-epsilon Multiphase Model to Dispersed in the

Viscous panel.

2. Enter the multiphase-options text command in the console window.

define −→ models −→ viscous −→ multiphase-turbulence −→

multiphase-options

/define/models/viscous/multiphase-turbulence>

Enable dispersion force in momentum?[no] Enable interphase k-epsilon source?

[no]

The effect of the drift velocity is influenced both by the momentum equation and,to a lesser extent, the turbulence equation. Therefore, you should answer yes to both questions to take into account the effect of drift velocity.

k - TurbulenceModelfor EachPhase

The most general multiphase turbulence model solves a set of k and trans port equationsfor each phase. This turbulence model is the appropriate choice when the turbulence

transfer among the phases plays a dominant role.

Note that, since FLUENT is solving two additional transport equations for each sec-

ondary phase, the per-phase turbulence model is more computationally intensive than

the dispersed turbulence model.

Trans port Equations

The Reynolds stress tensor and turbule nt viscosity are computed using Equations 23.5-94

and 23.5-95. Turbulence predictions are obtained from

∂t (αq ρq k q ) + ∇ · (αq ρq U

~ q k q ) = ∇ ·

(αq

µt,q

σk

∇k q ) + (αq Gk,q − αq ρq q ) +

N N µt,l N

µt,qX

K lq (Clq k l − Cql k q ) −X

K lq (U~ l

− U~ q ) ·

∇αl + X

K lq (U~ l −

U~ q ) ·

∇αq (23.5-115)

l=1 l=1αl σl l=1

αq σq



∂

and

∂t (αq ρq q ) + ∇ · (αq ρq U

~ q q ) = ∇ ·

(αq

µt,q

σ

∇ q ) + q

"

k qC1 αq Gk,q − C2 αq ρq q +

N N µt,l N

µt,q

!#

C3

X

K lq (Clq k l − Cql k q ) −X

K lq (U~ l −

U~ q ) ·

∇αl +X

K lq (U~ l − U~

q ) · ∇αq

l=1 l=1αl σl l=1

αq σq

(23.5-116)

The terms Clq and Cql can be approximated as

Clq = 2, Cql = 2

ηlq

!

1 + ηlq

(23.5-117)

where ηlq is defined by Equation 23.5-107.

Interphase Turbule nt Mome ntum Transfer

The turbule nt drag term (K pq (~v p − ~vq ) in Equation 23.5-7) is modeled as follows,

wherel is the dispersed phase (replacing p in Equation 23.5-7) and q is the continuous phase:

N N NXK lq (~vl − ~vq ) =

XK lq (U~

l − U~ q ) −

XK lq~vdr,lq (23.5-118)

l=1 l=1 l=1

Here U~ l and U~

q are phase-weighted velocities, and ~vdr,lq is the drift velocity for

phasel (computed using Equation 23.5-114, substituting l for p). Note that FLUENT will

compute the diffusivities Dl and Dq directly from the transport equations, rather than

using Tchen theory (as it does for the dispersed turbulence model).

As noted above, the drift velocity results from turbule nt fluctuations in the volume

fraction. When multiplied by the exchange coefficient K lq , it serves as a correction to

the momentum exchange term for turbule nt flows. This correction is not included, by

default, but you can enable it during the problem setup.

The turbulence model for each phase in FLUENT accounts for the effect of the turbulence

field of one phase on the other(s). If you want to modify or enhance the interaction of the

multiple turbulence fields and interphase turbule nt momentum transfer, you can supply

these terms using user-defined functions.



˜

∂

c =

RSMTurbulence Models

Multiphase turbulence modeling typically involves two equation models that are

based on single-phase models and often cannot accurately capture the underlying flow

physics. Additional turbulence modeling for multiphase flows is diminished even more

when the basic underlying single-phase model cannot capture the complex physics of the

flow. In such situations, the logical next step is to combine the Reynolds stress model

with the multiphase algorithm in order to handle challenging situations in which both

factors, RSM for turbulence and the Eulerian multiphase formulation, are a

precondition for accurate predictions [65].

The phase-averaged continuity and momentum equations for a continuous phase are:

∂ (αc ρc ) + ∇ · (αcρc Uc ) = 0 (23.5-119)

∂t

(αc ρr mcU˜c) + ∇ · (αc ρr mcU˜c

O

U˜c ) = −αc ∇ p˜ + ∇ · τ˜t + FDc (23.5-120)∂tc

For simplicity, the laminar stress-strain tensor and other body forces such as gravityhave been omitted from Equations 23.5-119-23.5-120. The tilde denotes phase-averaged

variables while an overbar (e.g., αc) reflects time-averaged values. In general, any variable

Φ can have a phase-average value defined as

Φ˜ αc Φc

αc

(23.5-121)

Considering only two phases for simplicity, the drag force between the continuousand the dispersed phases can be defined as:

" αd u0

c u0 !#

FDc = K dc (U˜d − U˜

c )−

d c − (23.5-122)

αd αc

where K dc is the drag coefficient. Several terms in the Equation 23.5-122 need to be

modeled in order to close the phase-averaged momentum equations. Full descriptions of

all modeling assumptions can be found in [64]. This section only describes the different

modeling definition of the turbule nt stresses τ˜t that appears in Equation 23.5-120.

The turbule nt stress that appears in the momentum equations need to be defined on a

per-phase basis and can be calculated as:

τ˜tk = −αk ρk R ̃ k ,ij (23.5-123)



˜

where the subscript k is replaced by c for the primary (i.e., continuous) phase or by d for

any secondary (i.e., dispersed) phases. As is the case for single-phase flows, the currentmultiphase Reynolds stress model (RSM) also solves the transport equations for Reynolds

stresses R ij. FLUENT includes two methods for modeling turbulence in multiphase flows

within the context of the RSM model: the dispersed turbulence model, and the mixture

turbulence model.

RSMDispersed TurbulenceModel

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. Conse-

quently, 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:

∂ (αρR ij) +

∂t∂

∂xk

(αρU˜k R ̃ ij) =

−αρ

R ̃ i

k

∂U˜

j

∂xk

+

R ̃ jk

∂U˜i

!

+∂xk

∂

∂xk

" ∂αµ

∂xk

#

(R ̃ ij)

∂ ∂u0∂u0

[αρu0u0 u0 ] + αp(i+

j)−

∂xk i j k ∂x j ∂xi

− αρ ˜ij + ΠR,ij (23.5-124)

The variables in Equation 23.5-124 are per continuous phase c and the subscript is omitted

for clarity. The last term of Equation 23.5-124, ΠR,ij, takes into account the interaction

between the continuous and the dispersed phase turbulence. A general model for thisterm can be of the form:

ΠR,ij = K dc C1,dc (R dc,ij − R c,ij) + K dc C2,dc adc,i bdc,j (23.5-125)

where C1 and C2 are unknown coefficients, adc,i is the relative velocity, bdc,j

represents the drift or the relative velocity, and R dc,ij is the unknown particulate-

fluid velocity correlation. To simplify this unknown term, the following assumption has

been made:

2ΠR,ij =

3 δijΠk (23.5-126)

where δij is the Kronecker delta, and Πk represents the modified version of the original

Simonin model [333].

Πkc = K dc (k ̃dc − 2k ̃ c + V˜rel · V˜

drift ) (23.5-127)



1

where K ̃ c represents the turbule nt kinetic energy of the continuous

phase,

k ̃ dc is

the

continuous-dispersed phase velocity covariance and finally, V˜rel and V˜

drift stand for

the

relative and the drift velocities, respectively. In order to achieve full closure, the transport

equation for the turbule nt kinetic energy dissipation rate ( ˜) is required. Themodeling of ˜ together with all other unknown terms in Equation 23.5-127 are modeled

in the same way as in [64].

RSMM ixtur eTurbulenceModel

The main assumption for the mixture model is that all phases share the same turbu-

lence field which consequently means that the term ΠR in the Reynolds stress transport

equations (Equation 23.5-124) 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

N

ρm =X

αiρi (23.5-128)i=1

while mixture velocities can be expressed as

P N

αiρiU˜

iU˜

m = i=1

(23.5-129)P Ni=1 αiρi

where N is the number of species.

23.5.11 SolutionMethodin FLUENT

For Eulerian multiphase calculations, FLUENT uses the phase coupled SIMPLE (PC-

SIMPLE) algorithm [379] for the pressure-velocity coupling. PC-SIMPLE is an extension

of the SIMPLE algorithm [276] to multiphase flows. The velocities are solved coupled by

phases, but in a segregated fashion. The block algebraic multigrid scheme used by the

density-based solver described in [394] is used to solve a vector equation formed by the

velocity components of all phases simultaneously. Then, a pressure correction equation

is built based on total volume continuity rather than mass continuity. Pressure and

velocities are then corrected so as to satisfy the continuity constrai nt.

The Pressure-CorrectionEquation

For incompressible multiphase flow, the pressure-correction equation takes the form

n(

∂X

α ρ + ∇ · α ρ~v0

+ ∇ · αn

ρ ~v∗ −(X

(m˙−m˙

)

)) = 0 (23.5-130)

k=1ρrk ∂t

k k k k k k k k lk k l

l=1




k k

where ρrk is the phase reference density for the k th phase (defined as the total volume

average density of phase k),~v0

is the velocity correction for the k th phase, and ~v∗ isthe

value of ~vk at the current iteration. The velocity corrections are themselves expressed

as functions of the pressure corrections.

VolumeFractions

The volume fractions are obtained from the phase continuity equations. In discretized

form, the equation of the k th volume fraction is

a p,k αk =X

(anb,k αnb,k ) + bk = R k (23.5-131)nb

In order to satisfy the condition that all the volume fractions sum to one,

nXαk = 1 (23.5-132)

k =1

23.6 Wet Steam ModelTheory

23.6.1 Overview and Limitations of the Wet Steam

Model

Overview

During the rapid expansion of steam, a condensation process will take place shortlyafter the state path crosses the vapor-saturation line. The expansion process causes the

super-heated dry steam to first subcool and then nucleate to form a two-phase mixture

of saturated vapor and fine liquid droplets known as wet steam.

Modeling wet steam is very important in the analysis and design of steam turbines. The

increase in steam turbine exit wetness can cause severe erosion to the turbine blades at

the low-pressure stages, and a reduction in aerodynamic efficiency of the turbine stages

operating in the wet steam region [250].

FLUENT has adopted the Eulerian-Eulerian approach for modeling wet steam flow. The

flow mixture is modeled using the compressible Navier-Stokes equations, in addition to

two transport equations for the liquid-phase mass-fraction (β), and the number of liquid-droplets per unit volume (η). The phase change model, which involves the formation of

liquid-droplets in a homogeneous nonequilibrium condensation process, is based on the

classical nonisothermal nucleation theory.

This section describes the theoretical aspects of the wet steam model. Information about

enabling the model and using your own property functions and data with the wet steam

model is provided in Section 23.13: Setting Up the Wet Steam Model. Solution settings



and strategies for the wet steam model can be found in Section 23.14.5: Wet Steam Model.

Postprocessing variables are described in Section 23.15.1: Model-Specific Variables .

Limitations

The following restrictions and limitations currently apply to the wet steam model inFLUENT:

• The wet steam model is available for the density-based solvers only.

• Pressure inlet, mass-flow inlet, and pressure outlet are the only inflow and

outflow boundary conditions available.

• When the wet steam model is active, the access to the Materials panel is

restricted because the fluid mixture properties are determined from the built in

steam property functions or from the user-defined wet steam property functions.

Therefore, if solid properties need to be set and adjusted, then it must be done in

the Materials panel before activating the wet steam model.

23.6.2 Wet Steam Flow Equations

The wet steam is a mixture of two-phases. The primary phase is the gaseous-phase

consisting of water-vapor (denoted by the subscript v) while the secondary phase is the

liquid-phase consisting of condensed-water droplets (denoted by the subscript l).

The following assumptions are made in this model:

•

The velocity slip between the droplets and gaseous-phase is negligible.• The interactions between droplets are neglected.

• The mass fraction of the condensed phase, β (also known as wetness factor),

is small (β < 0.2).

• Since droplet sizes are typically very small (from approximately 0.1 microns

to approximately 100 microns), it is assumed that the volume of the condensed

liquid phase is negligible.

From the preceding assumptions, it follows that the mixture density (ρ) can be related

to the vapor density (ρv ) by the following equation:

ρ =ρv

(1 − β)(23.6-1)



V

In addition, the temperature and the pressure of the mixture will be equivalent to the

temperature and pressure of the vapor-phase.

The mixture flow is governed by the compressible Navier-Stokes equations given in vector

form by Equation 25.5-4:

∂W ∂ Z

∂Q ∂t V

IQ dV +

Z[F − G] · dA = H dV (23.6-2)

where Q=( P,u,v,w,T) are mixture quantities. The flow equations are solved using the

same density-based solver algorithms employed for general compressible flows.

To model wet steam, two additional trans port equations are needed [152]. The first

transport equation governs the mass fraction of the condensed liquid phase (β):

∂ρβ

∂t+ ∇ · (ρ−→v β) = Γ (23.6-3)

where Γ is the mass generation rate due to condensation and evaporation (kg per unit

volume per second). The second transport equation models the evolution of the num ber

density of the droplets per unit volume:

∂ρη

∂t+ ∇ · (ρ−→v η) = ρI (23.6-4)

where I is the nucleation rate (number of new droplets per unit volume per second).

To determine the number of droplets per unit volume, Equation 23.6-1 and the average

droplet volume Vd are combined in the following expression:

βη =(1 − β)Vd(ρl /ρv )(23.6-5)

where ρl is the liquid density and the average droplet volume is defined as

3

where r d is the droplet radius.

Vd =3

πr d (23.6-6)

Together, Equation 23.6-2, Equation 23.6-3, and Equation 23.6-4 form a closed system

of equations which, along with Equation 23.6-1, permit the calculation of the wet steam

flow field.



l

23.6.3 Phase ChangeModel

The following is assumed in the phase change model:

• The condensation is homogeneous (i.e., no impurities present to form nuclei).

• The droplet growth is based on average representative mean radii.

• The droplet is assumed to be spherical.

• The droplet is surrounded by infinite vapor space.

• The heat capacity of the fine droplet is negligible compared with the latent

heat released in condensation.

The mass generation rate Γ in the classical nucleation theory during the nonequilibrium

condensation process is given by the sum of mass increase due to nucleation (the formation

of critically sized droplets) and also due to growth/demise of these droplets [152].

Therefore, Γ is written as:

43Γ =

3 πρlI r ∗ + 4πρlηr 2

∂r

∂t(23.6-7)

where r is the average radius of the droplet, and r ∗ is the Kelvin-Helmholtz critical

droplet radius, above which the droplet will grow and below which the droplet will

evaporate. An expression for r ∗ is given by [410].

2σ

r ∗ = ρ RT ln S (23.6-8)

where σ is the liquid surface tension evaluated at temperature T , ρl is the condensed

liquid density (also evaluated at temperature T ), and S is the super saturation ratiodefined as the ratio of vapor pressure to the equilibrium saturation pressure:

Ps =

Psat (T )(23.6-9)

The expansion process is usually very rapid. Therefore, when the state path crosses the

saturated- vapor line, the process will depart from equilibrium, and the supersaturationratio S can take on values greater than one.



v

e

−3∗

θ =

The condensation process involves two mechanisms, the transfer of mass from the vapor

to the droplets and the transfer of heat from the droplets to the vapor in the form of

latent heat. This energy transfer relation was presented in [408] and used in [152] and

can be written as:

∂r =∂t hlv ρl

P√2πRT

γ + 12γ

C p (T0 − T ) (23.6-10)

where T0 is the droplet temperature.

The classical homogeneous nucleation theory describes the formation of a liquid-phase

in the form of droplets from a supersaturated phase in the absence of impurities or

foreign particles. The nucleation rate described by the steady-state classical homogeneous

nucleation theory [410] and corrected for nonisothermal effects, is given by:

I =

qc

ρ 2!s

2σ

4π r 2

σ K

b

T

(23.6-11)

(1 + θ) ρl Mm3π

where qc is evaporation coefficient, k b is the Boltzmann constant, Mm is mass of one

molecule, σ is the liquid surface tension, and ρl is the liquid density at temperature T .

A nonisothermal correction factor, θ, is given by:

2(γ − 1) hlv

!hlv

!

− 0.5 (23.6-12)(γ + 1) RT RT

where hlv is the specific enthalpy of evaporation at pressure p and γ is the ratio of

specific heat capacities.

23.6.4 Built-inTherm odynamicWet Steam Properties

There are many equations that describe the thermodynamic state and properties of steam.

While some of these equations are accurate in generating property tables, they are not

suitable for fast CFD computations. Therefore, FLUENT uses a simpler form of the

thermodynamic state equations [409] for efficient CFD calculations that are accurate

over a wide range of temperatures and pressures. These equations are described below.



T

647.286

ρ

#

Equationof State

The steam equation of state used in the solver, which relates the pressure to the vapor

density and the temperature, is given by [409]:

P = ρv RT (1 + Bρv + C ρv

2

) (23.6-13)

where B, and C are the second and the third virial coefficients given by the following

empirical functions:

B = a (1 +τ

)−1

1α

+ a2eτ(1 −

e−τ

5

) 2 + a3 τ (23.6-14)

where B is given in m3/kg, τ = 1500

a2= -0.000942, and a3= -0.0004882.with T given in Kelvin, α = 10000.0, a1= 0.0015,

C = a(τ − τ0)e−ατ

+ b (23.6-15)

where C is given in m6/k g2, τ = T with T given in Kelvin, τo= 0.8978, α=11.16,

a= 1.772, and b= 1.5E-06.

The two empirical functions that define the virial coefficients B and C cover the temper-

ature range from 273 K to 1073 K.

The vapor isobaric specific heat capacity C pv is given by:

C pv = C p0(T ) +

R

[(1 − αv T )(B − B1) − B2] ρv

+

(1 − 2αv T )C + αv T C1

−

C 2 2

2v

The vapor specific enthalpy, hv is given by:

(23.6-16)

hv = h0(T ) +

RT

(B − B1)ρv + (C

−

1 )ρ

2

2v (23.6-17)

The vapor specific entropy, sv is given by:

sv = s0(T ) − R

"

ln ρv + (B + B1)ρv +( C + C1)

ρ2

2v

(23.6-18)

The isobaric specific heat at zero pressure is defined by the following empirical equation:



6

C p0(T ) =X

aiT i−2

(23.6-19)i=1



2 2

where C p0 is in KJ/kg K, a1 = 46.0, a2 = 1.47276, a3 = 8.38930E-04, a4 = -2.19989E-07,

a5 = 2.46619E-10, and a6 = -9.70466E-14.

and

B1 = T dB , C1 = T dC , B2 = T 2 dB , and C = T 2dC

.dT dT dT 2 2 dT 2

Both h0(T ) and s0(T ) are functions of temperature and they are defined by:

Z

h0 (T ) = C p0dT + hc (23.6-20)

s0(T ) =

where hc and sc are arbitrary constants.

ZC p0

dT + s (23.6-21)T

c

The vapor dynamic viscosity µv and thermal conductivity K tv are also functions of

temperature and were obtained from [408].

SaturatedVaporLine

The saturation pressure equation as a function of temperature was obtained from [301].

The example provided in Section 23.13.5: UDWSPF Example contains a function called

wetst satP() that represents the formulation for the saturation pressure.

SaturatedLiquidLine

At the saturated liquid-line, the liquid density, surface tension, specific heat C p,

dynamic viscosity, and thermal conductivity must be defined. The equation for liquiddensity, ρl, was obtained from [301]. The liquid surface tension equation was obtained

from [408]. While the values of C pl , µl and K tl were curve fit using published data

from [91] and then written in polynomial forms. The example provided in Section

23.13.5: UDWSPF

Example contains functions called wetst cpl(), wetst mul(), and wetst ktl() that

represent formulations for C pl , µl and K tl

.

MixtureProperties

The mixture properties are related to vapor and liquid properties via the wetness factor

using the following mixing law:

φm = φl β + (1 − β)φv (23.6-22)

where φ represents any of the following thermodynamic properties: h, s,C p, C v, µ

or

K t.



23.7 ModelingMassTransferin MultiphaseFlows

This section describes the modeling of mass transfer in the framework of FLUENT’sgen-

eral multiphase models (i.e., Eulerian multiphase, mixture multiphase, VOF multiphase).

There are numerous kinds of mass transfer processes that can be modeled in FLUENT.You

can use models available in FLUENT (e.g. FLUENT’s cavitation model), or define your

own mass transfer model via user-defined functions. See Section 23.7.3: UDF-Prescri bed

Mass Transfer and the separate UDF Manual for more information about the modeling

of mass transfer via user-defined functions.

Information about mass transfer is presented in the following subsections:

• Section 23.7.1: Source Terms due to Mass Transfer

• Section 23.7.2: Unidirectional Consta nt Rate Mass Transfer

• Section 23.7.3: UDF-Prescri bed Mass Transfer

•

Section 23.7.4: Cavitation Models

i Note that FLUENT’scurrent cavitation model can only be used inthe framework of the mixture multiphase model.

23.7.1 Source Terms due to MassTransfer

FLUENT adds contributions due to mass transfer only to the momentum, species, and

energy equations. No source term is added for other scalars such as turbulence or user-

defined scalars.

Let m pi q j be the mass transfer rate per unit volume from the ith species of phase p to

the jth species of phase q. In case a particular phase does not have a mixture material

associated with it, the mass transfer will be with the bulk phase.

Mass Equation

The contribution to the mass source for phase p in a cell is

m p = −m pi q j (23.7-1)

and for phase q is

mq = m pi q j (23.7-2)



23.7 ModelingMassTransferin Multiphase

p

p

p q

p

mi

m j

i

MomentumEquation

For VOF or mixture models, there is no momentum source.

For the Eulerian model, the momentum source in a cell for phase p is

m p~u p = −m pi q j ~u p (23.7-3)

and for phase q is

mq ~uq = m pi q j ~u p (23.7-4)

Energy Equation

For all multiphase models, the following energy sources are added.

The energy source in a cell for phase p is

H p = −m pi q j (hi ) (23.7-5)

and for phase q is

Hq = m pi q j (hi+ h

f p− hf

j

q ) (23.7-6)

where hf i and hf j are the formation enthalpies of species i of phase p and species j of

phase q respectively and hi is the enthalpy of species i of phase p (with reference to the

formation enthalpy).

Species Equation

The species source in a cell for species i of phase p is

p = −m pi q j (23.7-7)

and for species j of phase q is

q = m pi q j (23.7-8)

OtherScalarEquations

No source/sink terms are added for turbulence quantities and other scalars. The transfer

of these scalar quantities due to mass transfer could be modeled using user-defined source

terms.



23.7.2 UnidirectionalConstantRate Mass Transfer

The unidirectional mass transfer model defines a positive mass flow rate per unit volume

from phase p to phase q:

m ̇pq = max[0, λ pq ] − max[0, −λ pq ] (23.7-9)

where

λ pq = r ̇α pρq (23.7-10)

and r ̇ is a constant rate of particle shrinking or swelling, such as the rate of burning

of a liquid droplet. This is not available for the VOF model.

If phase p is a mixture material and a mass transfer mechanism is defined for species i

of phase p, then

λ pq = r ̇α py p,iρq (23.7-11)

where y p,i is the mass fraction of species i in phase p.

23.7.3 UDF-PrescribedMass Transfer

Because there is no universal model for mass transfer, FLUENT provides a UDF that you

can use to input models for different types of mass transfer, e.g. evaporation, condensa-

tion, boiling, etc. Note that when using this UDF, FLUENT will automatically add the

source contribution to all relevant momentum and scalar equations. Thiscontribution is based on the assumption that the mass “created” or “destroyed” will

have the same momentum and energy of the phase from which it was created or

destroyed. If you would like to input your source terms directly into momentum, energy,

or scalar equations, then the appropriate path is to use UDFs for user-defined sources for

all equations, rather than the UDF for mass transfer. See the separate UDF Manual

for more information about UDF-based mass transfer in multiphase.



23.7.4 CavitationModels

This section provides information about the cavitation model used in FLUENT. You can

use FLUENT’scurrent cavitation model to include cavitation effects in two-phase flows

when the mixture model is used.

Overviewof the CavitationModel

A liquid at constant temperature can be subjected to a decreasing pressure, which may

fall below the saturated vapor pressure. The process of rupturing the liquid by a decrease

of pressure at constant temperature is called cavitation. The liquid also contains the

micro-bubbles of noncondensable (dissolved or ingested) gases, or nuclei, which under

decreasing pressure may grow and form cavities. In such processes, very large and steep

density variations happen in the low-pressure/cavitating regions.

The cavitation model implemented here is based on the so-called “full cavitation model”,

developed by Singhal et al. [334]. It accounts for all first-order effects (i.e., phase change,

bubble dynamics, turbule nt pressure fluctuations, and noncondensable gases). However,unlike the original approach [334] assuming single-phase, isothermal, variable fluid density

flows, the cavitation model in FLUENT is under the framework of multiphase flows. It

has the capability to account for multiphase (N-phase) flows or flows with multiphase

species transport, the effects of slip velocities between the liquid and gaseous phases, and

the thermal effects and compressibility of both liquid and gas phases. The cavitation

model can be used with the mixture multiphase model (with or without slip velocities).

The complete cavitation model capability in FLUENT can be presented in two parts:

• the basic cavitation model

This includes a description of the fundame ntal modeling approach and the standard

two-phase cavitation model.

• the extended cavitation model capabili ty

This includes a description of the extension of the cavitation model for multiphase

(N-phase) flows, or flows with multiphase species trans port applications.



Basic CavitationModel

In the standard two-phase cavitation model, the following assumptions are made:

• The system under investigation involves only two phases (a liquid and its

vapor), and a certain fraction of separately modeled noncondensable gases.

• Both bubble formation (evaporation) and collapse (condensation) are taken

into account in the model.

• The mass fraction of noncondensable gases is known in

advance. The cavitation model offers the following capabilities:

• The cavitation model accounts for the mass transfer between a single liquid and

its vapor.

• It is compatible with all the available turbulence models in FLUENT.

•It can be solved with the mixture energy equation.

• It is fully compatible with dynamic mesh and nonconformal interfaces.

• Both liquid and vapor phases can be incompressible or compressible. The

noncon- densible gases are assumed to always be compressible. For compressible

liquids, the density can be described using a user-defined function. See the separate

UDF Manual for more information on user-defined density functions.

• The parameters used in the mass transfer model for cavitation (vaporization

pressure, liquid surface tension coefficient) can be either a constant or a

function of temperature.

The following limitations apply to the cavitation model in FLUENT:

• The cavitation model cannot be used with the VOF model, because the

surface tracking schemes for the VOF model are incompatible with the

interpenetrating continua assumption of the cavitation model.

• The cavitation model can be used only for multiphase simulations that use

the mixture model. It is always preferable to solve for cavitation using the

mixture model without slip velocity; slip velocities can be turned on if the problem

suggests that there is significant slip between phases.

•

The cavitation model can only used for cavitating flow occurring in a singleliquid fluid;

• With the cavitation model, the primary phase must be liquid, the secondary

phase must be vapor.



VaporM assFractio nand VaporTransport

The working fluid is assumed to be a mixture of liquid, vapor and noncondensable gases.

Standard governing equations in the mixture model and the mixture turbulence model

describe the flow and account for the effects of turbulence. A vapor transport equation

governs the vapor mass fraction, f , given by:

∂

∂t (ρf ) + ∇(ρv~v f ) = ∇(γ ∇f ) + R e − R c (23.7-12)

where ρ is the mixture density, v~v is the velocity vector of the vapor phase, γ is

theeffective exchange coefficient, and R e and R c are the vapor generation and condensation

rate terms (or phase change rates). The rate expressions are derived from the Rayleigh-

Plesset equations, and limiting bubble size considerations (interface surface area per

unit volume of vapor) [334]. These rates are functions of the instantaneous, local static

pressure and are given by:when p < psat

when p > psat

R e = Ce

Vch ρ ρ

σl v

s

2(psat − p)

3ρl

(1 − f ) (23.7-13)

R c = Cc

Vch ρ ρ

σl v

s

2(p − psat )

3ρl

f (23.7-14)

where the suffixes l and v denote the liquid and vapor phases, Vch is a characteristicvelocity, which is approximated by the local turbulence intensity, (i.e. Vch =

√k), σ is

the surface tension coefficient of the liquid, psat is the liquid saturation vapor pressure at

the given temperature, and Ce and Cc are empirical constants. The default values are

Ce = 0.02 and Cc = 0.01.



ρ

Turbulence-InducedPressu reFluctuations

Significant effect of turbulence on cavitating flows has been reported [311]. FLUENT’s

cavitation model accounts for the turbulence-induced pressure fluctuations by simply

raising the phase-change threshold pressure from psat to

where

1 pv =

2 (psat + pturb ) (23.7-15)

pturb = 0.39ρk (23.7-16)

where k is the local turbulence kinetic energy.

Effectsof NoncondensableGases

The operating liquid usually contains small finite amounts of noncondensable gases (e.g.,

dissolved gases, aeration). Even a very small amount (e.g., 10 ppm) of noncondensable

gases can have significant effects on the cavitating flow field due to expansion at low

pressures (following the ideal gas law). In the present approach, the working fluid is

assumed to be a mixture of the liquid phase and the gaseous phase, with the gaseous

phase comprising of the liquid vapor and the noncondensable gases. The density of the

mixture, ρ, is calculated as

ρ = αv ρv + αg ρg + (1 − αv − αg )ρl (23.7-17)

where ρl , ρv , and ρg are the densities of the liquid, the vapor, and the noncondensable

gases, respectively, and αl , αv , and αg are the respective volume fractions. The relation-

ship between the mass fraction (f i) in Equations 23.7-12 – 23.7-14 and the volume fraction

(αi) in Equation 23.7-17 is

ραi = f i i(23.7-18)

The combined volume fraction of vapor and gas (i.e., αv + αg ) is commonly referred to

as the void fraction (α).

It may be noted that the noncondensable gas is not defined as a phase or a material.When using the ideal gas law to compute the noncondensable gas density, the molecular

weight and temperature are required. By default, the gas is assumed to be air and the

molecular weight is set to 29.0. However, if the noncondensable gas is not air, then

the molecular weight can be changed by using a text command. For more information,

contact your FLUENT support engineer.



As for the temperature, the default value is set to 300 K when the energy equation is not

activated. If the temperature is different, but still a constant (i.e., isothermal flow), you

can change the temperature in FLUENT in the following way:

• Activate the energy equation.

Define −→ Models−→Energy...

• Open the Solution Initialization panel.

Solve −→ Initialize−→Initialize...

• In the Solution Initialization, set the initial value as a desired temperature.

• Open the Solution Controls panel.

Solve −→ Controls

−→Solution...

•In the Solution Controls panel, under Equations, turn off the energy equation

by deselecting Energy in the list.. By so doing, FLUENT uses the initial values for

the temperature.

Phase ChangeRates

After accounting for the effects of turbulence-induced pressure fluctuations and noncon-

densable gases, the final phase rate expressions are written as:

when p < pv

when p > pv

R e = Ce

√k

ρl ρvσ

s

2(pv − p)

3ρl

(1 − f v − f g ) (23.7-19)

√k s

2(p − pv )R c = Cc ρl ρl

σ 3ρl

f v (23.7-20)



AdditionalGuidelinesfor the CavitationModel

In practical applications of the cavitation model, several factors greatly influence nu-

merical stability. For instance, high pressure difference between the inlet and exit, large

ratio of liquid to vapor density, and near zero saturation pressure all cause unfavorable

effects on solution convergence. In addition, poor initial conditions very often lead to an

unrealistic pressure field and unexpected cavitating zones, which, once present, are thenusually very difficult for the model to correct. The following is a list of factors that must

be considered when using the cavitation model, along with tips to help address potential

numerical problems:

• relaxation factors

In general, small relaxation factors are advised for momentum equations, usually,

between 0.05 – 0.4; The relaxation factor for the pressure-correction equation should

usually be larger than those for momentum equations, say in the range 0.2 – 0.7.

The density and the vaporization mass (source term in the vapor equation) can also

be relaxed to improve convergence, Typically, the relaxation factor for density is set between the values of 0.3 and 1.0, while for the vaporization mass values between

0.1 and 1.0 may be appropriate. For some extreme cases, even smaller relaxation

factors may be required for all the equations.

• initial conditions

Though no special initial condition settings are required, it is suggested that the

vapor fraction is always set to inlet values. The pressure is set close to the high-

est pressure among the inlets and outlets to avoid unexpected low pressure and

cavitating spots. Also, in complicated cases, it may be beneficial to obtain a real-

istic pressure field before substa ntial cavities are formed. This can be achieved by

performing the followingsteps:

1. Set near zero relaxation factors for the vaporization mass and for density, and

increase them to reasonable values after a sufficient number of iterations.

2. Obtain a converged / near-converged solution for a single phase liquid

flow, and then switch on the cavitation model.

• noncondensable gases

Noncondensable gases are usually present in liquids. Even a small amount(e.g.,

15 ppm) of noncondensable gases can have significant effects on both the physicalrealism and the convergence characteristics of the solution. A value of zero for the

mass fraction of noncondensable gases should generally be avoided. In some cases, if

the liquid is purified of noncondensable gases, a much smaller value (e.g., 10−8 )

may be used to replace the default value of 1.5×10−5. In fact, higher mass

fractions of the noncondensable gases may, in many cases, enhance numerical

stability and lead to more realistic results. In particular, when the saturation

pressure of a liquid at a



certain temperature is zero or very small, noncondensable gases will play a crucial

role both numerically and physically.

• limits for dependent variables

In many cases, setting the pressure upper limit to a reasonable value can help

convergence greatly at the early stage of the solution. It is advised to always limit

the maximum pressure when it is possible. By default, FLUENT sets the maximum

pressure limit to 5.0×1010 Pascal.

• the relaxation factor for the pressure correction equation

For cavitating flows, a special relaxation factor is introduced for the pressure cor-

rection equation. By default, this factor is set to 0.7, which should work well for

most of the cases. For some very complicated cases, however, you may

experience the divergence of the AMG solver. Under those circumstances, this

value may be reduced to no less than 0.4. You can set the value of this relaxation

factor by typing a text command. For more information, contact your FLUENT

support engineer.

• pressure discretization schemes

As for many multiphase flows, it is more desirable to use the following pressure

discretization schemes in cavitation applications:

– body forceweighted

– secondorder

– PRESTO!

The standard and linear schemes generally are not very effective in complex cavi-

tating flows.

ExtendedCavitationModelCapability

In many practical applications, when cavitation occurs, there exist other gaseous species

in the systems investigated. For instance, in a ventilated supercavitating vehicle, air is

injected into a liquid to stabilize or increase the cavitation along the vehicle surfaces. Also

in some cases, the incoming flow is a mixture of a liquid and some gaseous species. In

order to be able to predict those type of cavitating flows, the basic two-phase cavitation

model needs to be extended to a multiphase (N-phase) flows, or a multiphase species

transport cavitation model.



Multiph aseCavita tionModel

The multiphase cavitation model is an extension of the basic two-phase cavitation model

to multiphase flows. In addition to the primary liquid and secondary vapor phase, more

secondary gaseous phases can be included into the computational system under the fol-

lowing assumptions/limitations:

• Mass transfer (cavitation) only occurs between the first and the second phases.

• The basic cavitation model is still used to model the phase changes between

the liquid and vapor.

• Only one secondary phase can be defined as compressible gas phase, while a

user- defined density may be applied to all the phases.

• The predescribed noncondensable gases can still be included in the system.

To exclude noncondensable gases from the system, the mass fraction needs to be

set

to 0, and the noncondensable gas needs to be modeled by a separate compressible

gas phase.

• For an noncavitating phase i, the general transport equation governing the

mass fraction f i given by:

∂

∂t (ρf i) + ∇(ρv~v f i) = ∇(γf i ∇f i) + Sf i (23.7-21)

where Sf i is a (user-defined) source term. By default, Sf i = 0.

MultiphaseSpecies Transport Cavita tio nModel

In some cases, there are several gas phase components in a system. It is desirable to

consider them all compressible. Since only one compressible gas phase is allowed in the

general multiphase approach, the multiphase species transport approach offers an option

to handle these type of applications by assuming that there is one compressible gas phase

with multiple species.

The detailed description of the multiphase species transport approach can be found in

Section 23.8: Modeling Species Transport in Multiphase Flows. The multiphase species

transport cavitation model can be summarized as follows:

• All the assumptions/limitations for the multiphase cavitation model also

apply here.

• The primary phase can only be a single liquid.

• All the secondary phases allow more than one species.



23.8 Mo delingSpecies Transport in Mult iph ase

∂

∂t

q

• The vapor, either as a phase or a species, must be in the second phase.

• The mass transfer between a liquid and a vapor phase/species is modeled by

the basic cavitation model.

• The mass transfer between other phases or species are modeled with the

standard mass transfer approach. In the standard model, the zero constant rate

should be chosen.

• For the phases with multiple species, the phase shares the same pressure as

the other phases, but each species has its own pressure (i.e., partial pressure).

As a result, the vapor density and the pressure used in Equation 23.7-19 are the

partial density and pressure of the vapor.

23.8 ModelingSpecies Transport in MultiphaseFlows

Species transport, as described in Chapter 14: Modeling Species Transport and Finite-

Rate Chemistry, can also be applied to multiphase flows. You can choose to solve theconservation equations for chemical species in multiphase flows by having FLUENT, for

each phase k, predict the local mass fraction of each species, Yik

, through the solution

of a convection-diffusion equation for the ith species. The generalized chemical species

conservation equation (Equation 14.1-1), when applied to a multiphase mixture can berepresented in the following form:

(ρq α

qYi

q )+ ∇·(ρq α

q~v

q Yi

q ) = −∇·α

q

J~i

n

+αq R i

q +α

q Si

q +X

(m˙ pi q j − m˙ q j pi )+R

(23.8-1)

p=1

where R iq

is the net rate of production of homogeneous species i by chemical reaction for

phase q, m˙ q j pi is the mass transfer source between species i and j from phase q to p,

and R is the heterogeneous reaction rate. In addition, αq is the volume fraction for

phase q and Siq

is the rate of creation by addition from the dispersed phase plus any

user-defined sources.

FLUENT treats homogeneous gas phase chemical reactions the same as a single-phase

chemical reaction. The reacta nts and the products belong to the same mixture material

(set in the Species Model panel), and hence the same phase. The reaction rate is scaled

by the volume fraction of the particular phase in the cell.The set-up of a homogeneous gas phase chemical reaction in FLUENT is the same as it is

for a single phase. For more information, see Chapter 14: Modeling Species Transport

and Finite-Rate Chemistry. For most multiphase species transport problems, boundary

conditions for a particular species are set in the associated phase boundary condition

panel (see Chapter 23.9.8: Defining Multiphase Boundary Conditions), and

postprocessing and



reporting of results is performed on a per-phase basis (see Section 23.15: Postprocessing

for Multiphase Modeling).

For multiphase species transport simulations, the Species Model panel allows you to in-

clude Volumetric, Wall Surface, and Particle Surface reactions. FLUENT treats multiphase

surface reactions as it would a single-phase reaction. The reaction rate is scaled with

the volume fraction of the particular phase in the cell. For more information, seeChap-ter 14: Modeling Species Transport and Finite-Rate Chemistry .

i To turn off reactions for a particular phase, while keeping thereactions active for other phases. turn on Volumetric under Reactions inthe SpeciesModel panel. Then, in the Materials panel, select none from the Reactions

drop-down list.

The species of different phases is entirely independent. There is no implicit relationship

between them even if they share the same name. Explicit relationships between species of

different phases can be specified through mass transfer and heterogeneous reactions. For more information on mass transfer and heterogeneous reactions, see Section 23.9.7: In-

cluding Mass Transfer Effects and Section 23.9.6: Specifying Heterogeneous Reactions ,

respectively.

Some phases may have a fluid material associated with them instead of a mixture ma-

terial. The species equations are solved in those phases that are assigned a mixture

material. The species equation above is solved for the mass fraction of the species in a

particular phase. The mass transfer and heterogeneous reactions will be associated with

the bulk fluid for phases with a single fluid material.

23.8.1 Limitations

The following limitations exist for the modeling of species transport for multiphase

flows:

• The nonpremixed, premixed, partially-premixed combustion, or the

composition PDF transport species trans port models are not available for

multiphase species reactions.

• The stiff chemistry solver is not available for multiphase species reactions.

• Only the laminar finite-rate, finite-rate/eddy-dissipation and eddy-dissipation

turbulence- chemistry models of homogeneous reactions are available for multiphasespecies transport.

• The discrete phase model (DPM) is not compatible with multiphase species

transport.



23.8.2 Mass and MomentumTransferwith MultiphaseSpecies Transport

The FLUENT multiphase mass transfer model accommodates mass transfer between

species belonging to different phases. Instead of a matrix- type input, multiple mass

transfer mechanisms need to be input. Each mass transfer mechanism defines the mass

transfer phenomenon from one entity to another entity. An entity is either a particular

species in a phase, or the bulk phase itself if the phase does not have a mixture mate-

rial associated with it. The mass transfer phenomenon could be specified either through

the inbuilt unidirectional “consta nt-rate” mass transfer (Section 23.7.2: Unidirectional

Consta nt Rate Mass Transfer ) or through user-defined functions.

FLUENT loops through all the mass transfer mechanisms to compute the net mass

source/sink of each species in each phase. The net mass source/sink of a species is

used to compute species and mass source terms. FLUENT will also automatically add

the source contribution to all relevant momentum and energy equations based on that

assumption that the momentum and energy carried along with the transferred mass. For

other equations, the transport due to mass transfer needs to be explicitly modeled by

the user.

Source Termsdue to Heterogeneous Reactions

Consider the following reaction:

aA + bB → cC + dD (23.8-2)

Let as assume that A and C belong to phase 1 and B and D to phase 2.

MassTransfer

Mass source for the phases are given by:

S1 = R(cMc − aMa ) (23.8-3)

S2 = R(dMd − bM b) (23.8-4)

where S is the mass source, M is the molecular weight, and R is the reaction rate.

The general expression for the mass source for the ith phase is

Xr r Sr i = −R

r i

γ j M j (23.8-5)

X p pS pi = R pi

γ j M j (23.8-6)

Si = S pi + Sr i (23.8-7)

where γ is the stoichiometric coefficient, p represents the product, and r represents the

reacta nt.



= R(cMc~unet − aMa~u1) (23.8-10)

= R(dMd~unet − bM b~u2) (23.8-11)

P

S

S

r j

r

M

p

p

i

k k

j

i

MomentumTransfer

Momentum transfer is more complicated, but we can assume that the reactants mix

(conserving momentum) and the products take momentum in the ratio of the rate of

their formation.

The net velocity, ~unet, of the reacta nts is given by:

aMa~u1 + bM b~u2

~unet =aMa + bM b

(23.8-8)

The general expression for the net velocity of the reacta nts is given by:

r γr M r

~ur ~unet =

j j jP

γr r (23.8-9)

r j M j

where j represents the jth item (either a reacta nt or a product).

Momentum transfer for the phases is then given by:

~u1

~u2

The general expression is

S~u X r r i = S pi ~unet − R

r i

γ j M j ~ui (23.8-12)

If we assume that there is no momentum transfer, then the above term will be zero.

Species Transfer

The general expression for source for k th species in the jth phase is

S k = −RX

γr r

k i

(23.8-13)

pk pk S k = R X γ j M j (23.8-14)k i

Sk i = S pk + Sr k (23.8-15)

i i



P

SH

SH

c

d

P

i

r

r

P

HeatTransfer

For heat transfer, we need to consider the formation enthalpies of the reactants and

products as well:

The net enthalpy of the reacta nts is given by:

aMa (Ha + hf ) + bM b(H b + hf )

Hnet = a b

aMa + bM b

(23.8-16)

where hf represents the formation enthalpy, and H represents the enthalpy.

The general expression for Hnet is:

r γr M r (H r + hf

)Hnet =

j j j jP (23.8-17)

r γr r j M j

If we assume that this enthalpy gets distributed to the products in the ratio of their mass

production rates, heat transfer for the phases are given by:

1 = R(cMcHnet − aMa Ha

2 = R(dMdHnet − bM bH b

− cMchf ) (23.8-18)

− dMdhf ) (23.8-19)

The last term in the above equations appears because our enthalpy is with reference to

the formation enthalpy.

The general expression for the heat source is: SH

Xr r r

!X

p p f p

i = S pi Hnet −R

γ j M j H j +r i

γ j M j h j pi

(23.8-20)

If we assume that there is no heat transfer, we can assume that the different species only

carry their formation enthalpies with them. Thus the expression for Hnet will be:

r γr M r hf

Hnet = j j j

(23.8-21)r γr r j M j

The expression SHwill be

SH

X p p f p

i = S pi Hnet − R

pi

γ j M j h j (23.8-22)



23.9 Steps for Using a MultiphaseModel

The procedure for setting up and solving a general multiphase problem is outlined below,

and described in detail in the subsections that follow. Remember that only the steps that

are pertinent to general multiphase calculations are shown here. For information about

inputs related to other models that you are using in conjunction with the multiphase

model, see the appropriate sections for those models.

See also Section 23.12.1: Additional Guidelines for Eulerian Multiphase Simulations for

guidelines on simplifying Eulerian multiphase simulations.

1. Enable the multiphase model you want to use (VOF, mixture, or Eulerian) and

specify the number of phases. For the VOF model, specify the VOF formulation

as well.

Define −→ Models

−→Multiphase...

See Sections 23.9.1 and 23.10.1 for details.

2. Copy the material representing each phase from the materials database.

Define−→Materials...

If the material you want to use is not in the database, create a new material.

See Section 8.1.2: Using the Materials Panel for details about copying from the

database and creating new materials. See Sections 23.10.5 and 23.11.3 for additional

information about specifying material properties for a compressible phase (VOF

and mixture models only). It is possible to turn off reactions in some materials

by selecting none in the Reactions drop-down list under Properties in the Materials

panel.

i If your model includes a particulate (granular) phase, you will needto create a new material for it in the fluid materials category (not thesolidmaterials category).

3. Define the phases, and specify any interaction between them (e.g., surface tension

if you are using the VOF model, slip velocity functions if you are using the mixture

model, or drag functions if you are using the Eulerian model).

Define−→Phases...

See Sections 23.9.3 – 23.12.2 for details.

4. (Eulerian model only) If the flow is turbule nt, define the multiphase turbulence

model.


−→Viscous...

See Section 23.12.3: Modeling Turbulence for details.



23.9Steps for Usinga MultiphaseM ode l

5. If body forces are present, turn on gravity and specify the gravitational acceleration.

Define −→Operating Conditions...

See Section 23.9.4: Including Body Forces for details.

6. Specify the boundary conditions, including the secondary-phase volume fractions

at flow boundaries and (if you are modeling wall adhesion in a VOF simulation)the contact angles at walls.

Define −→Boundary Conditions...

See Section 23.9.8: Defining Multiphase Boundary Conditions for details.

7. Set any model-specific solution parameters.

Solve −→ Controls−→Solution...

See Sections 23.10.4 and 23.14 for details.

8. Initialize the solution and set the initial volume fractions for the secondary phases.

Solve −→ Initialize−→

Patch...See Section 23.14.1: Setting Initial Volume Fractions for details.

9. Calculate a solution and examine the results. Postprocessing and reporting of

results is available for each phase that is selected.

See Sections 23.14 and 23.15 for details.

This section provides instructions and guidelines for using the VOF, mixture, and Eule-

rian multiphase models.

Information is presented in the following subsections:

• Section 23.9.1: Enabling the Multiphase Model

• Section 23.9.2: Solving a Homogeneous Multiphase Flow

• Section 23.9.3: Defining the Phases

• Section 23.9.4: Including Body Forces

• Section 23.9.5: Modeling Multiphase Species Transport

• Section 23.9.6: Specifying Heterogeneous Reactions

• Section 23.9.7: Including Mass Transfer Effects

• Section 23.9.8: Defining Multiphase Boundary Conditions




23.9.1 Enablingthe MultiphaseModel

To enable the VOF, mixture, or Eulerian multiphase model, select Volume of Fluid,

Mix- ture, or Eulerian as the Model in the Multiphase Model panel (Figure 23.9.1).

Define −→ Models−→Multiphase...

Figure 23.9.1: The Multiphase Model Panel

The panel will expand to show the relevant inputs for the selected multiphase model.




If you selected the VOF model, the inputs are as follows:

• number of phases

• VOF formulation (see Section 23.10.1: Choosing a VOF Formulation )

•

(optional) implicit body force formulation (see Section 23.9.4: IncludingBody

Forces)

If you selected the mixture model, the inputs are as follows:


• whether or not to compute the slip velocities (see Section 23.9.2: Solving a

Homo- geneous Multiphase Flow)

•

(optional) implicit body force formulation (see Section 23.9.4: IncludingBody

Forces)

If you selected the Eulerian model, the input is the following:


To specify the number of phases for the multiphase calculation, enter the appropriate

value in the Number of Phases field. You can specify up to 20 phases.

23.9.2 Solving a HomogeneousMultiphaseFlow

If you are using the mixture model, you have the option to disable the calculation of

slip velocities and solve a homogeneous multiphase flow (i.e., one in which the phases all

move at the same velocity). By default, FLUENT will compute the slip velocities for the

secondary phases, as described in Section 23.4.5: Relative (Slip) Velocity and the Drift

Velocity. If you want to solve a homogeneous multiphase flow, turn off Slip Velocity

under Mixture Parameters.




23.9.3 Definingthe Phases

To define the phases (including their material properties) and any interphase interaction

(e.g., surface tension and wall adhesion for the VOF model, slip velocity for the mixture

model, drag functions for the mixture and the Eulerian models), you will use the Phases

panel (Figure 23.9.2).


Figure 23.9.2: The Phases Panel

Each item in the Phase list in this panel is one of two types, as indicated in the Type list:

primary-phase indicates that the selected item is the primary phase, and secondary-phase

indicates that the selected item is a secondary phase. To specify any interaction between

the phases, click the Interaction... button.Instructions for defining the phases and interaction are provided in Sections 23.10.3,

23.11.1, and 23.12.2 for the VOF, mixture, and Eulerian models, respectively.




23.9.4 IncludingBody Forces

When large body forces (e.g., gravity or surface tension forces) exist in multiphase flows,

the body force and pressure gradient terms in the momentum equation are almost in

equilibrium, with the contributions of convective and viscous terms small in comparison.

Segregated algorithms converge poorly unless partial equilibrium of pressure gradient and

body forces is taken into account. FLUENT provides an optional “implicit body force”treatme nt that can account for this effect, making the solution more robust.

The basic procedure involves augmenting the correction equation for the face flow rate,

Equation 25.4-13, with an additional term involving corrections to the body force. This

results in extra body force correction terms in Equation 25.4-11, and allows the flow to

achieve a realistic pressure field very early in the iterative process.

To include this body force, turn on Gravity in the Operating Conditions panel and

specify the Gravitational Acceleration.


For VOF calculations, you should also turn on the Specified Operating Density option

in the Operating Conditions panel, and set the Operating Density to be the density of

the lightest phase. (This excludes the buildup of hydrostatic pressure within the lightest

phase, improving the round-off accuracy for the momentum balance.) If any of the phases

is compressible, set the Operating Density to zero.

i For VOF and mixture calculations involving body forces, it isrecom- mended that you also turn on the Implicit Body Force treatme ntfor theBody Force Formulation in the Multiphase Model panel. This treatme nt

improves solution convergence by accounting for the partial

equilibrium of the pressure gradient and body forces in the momentum

equations. See Section 23.9.4: Including Body Forces for details.




23.9.5 ModelingMultiphaseSpecies Transport

FLUENT lets you describe a multiphase species transport and volumetric reaction (Sec-

tion 23.8: Modeling Species Transport in Multiphase Flows) in a fashion that is similar

to setting up a single-phase chemical reaction using the Species Model panel (e.g., Fig-

ure 23.9.3).

Define −→ Models −→ Species −→Transport &

Reaction...

Figure 23.9.3: The Species Model Panel with a Multiphase Model Enabled

1. Select Species Transport under Model.

2. Turn on Volumetric under Reactions.

3. Select a specific phase using the Phase drop-down list under Phase Properties.

4. Click the Set... button to display the Phase Properties panel (Figure 23.9.4).




Figure 23.9.4: The Phase Properties Panel

In the Phase Properties panel, the material for each phase is listed in the Material

drop-down list. From this list, you can choose the material that you want to use

for a specific phase. The drop-down list contains all of the materials that have been

defined for your simulation. If you want to inspect or edit any of the properties

of any of the materials, then you need to open the Materials panel by clicking the

Edit... button.

5. In the Species Model panel, choose the Turbulence-Chemistry Interaction model.Three models are available:

Laminar Finite-Rate computes only the Arrhenius rate (see Equation 14.1-8) and

neglects turbulence- chemistry interaction.

Eddy-Dissipation (for turbule nt flows) computes only the mixing rate (see Equa-

tions 14.1-26 and 14.1-27).

Finite-Rate/Eddy-Dissipation (for turbule nt flows) computes both the Arrhenius

rate and the mixing rate and uses the smaller of the two.

When modeling multiphase species transport, additional inputs may also be requireddepending on your modeling needs. See, for example, Section 23.9.6: Specifying Het-

erogeneous Reactions for more information defining heterogeneous reactions, or Sec-

tion 23.9.7: Including Mass Transfer Effects for more information on mass transfer effects.




23.9.6 Specifying Heterogeneous Reactions

You can use FLUENT to define multiple heterogeneous reactions and stoichiometry using

the Phase Interaction panel (e.g., Figure 23.9.5).


1. In the Phases panel (Figure 23.9.2), click the Interaction... button to open the Phase

Interaction panel.

Figure 23.9.5: The Phase Interaction Panel for Heterogeneous Reactions

2. Click the Reactions tab in the Phase Interaction panel.

3. Set the total number of reactions (volumetric reactions, wall surface reactions, and

particle surface reactions) in the Total Number of Heterogeneous Reactions field.

(Use the arrows to change the value, or type in the value and press<Enter>.)

4. Specify the Reaction Name of each reaction that you want to define.

5. Set the ID of each reaction you want to define. (Again, if you type in the value besure to press <Enter>.)



i,r or ν


6. For each reaction, specify how many reacta nts and products are involved in the

reaction by increasing the value of the Number of Reactants and the Number of

Products. Select each reacta nt or product in the Reaction tab and then set its stoi-

chiometric coefficient in the Stoich. Coefficient field. (The stoichiometric coefficient

is the constant ν 0 0 0i,r in Equation 14.1-6.)

7. For each reaction, indicate the Phase and Species and the stoichiometric coefficientfor each of your reacta nts and products.

8. For each reaction, indicate an applicable user-defined function using the Reaction

Rate Function drop-down list.

i The heterogeneous reaction rates can only be specified using a user-defined function. A UDF is available for an Arrhenius- type reactionwith rateexponents that are equivalent to the stoichiometric coefficients.

For more information, see the separate UDF Manual.

i FLUENT assumes that the reacta nts are mixed thoroughly beforereacting together, thus the heat and momentum transfer is based on thisassump-tion. This assumption can be deactivated using a text command. For more

information, contact your FLUENT support engineer.

23.9.7 Including Mass Transfer

Effects

As discussed in Section 23.7: Modeling Mass Transfer in Multiphase Flows, mass transfer effects in the framework of FLUENT’s general multiphase models (i.e., Eulerian multi-

phase, mixture multiphase, or VOF multiphase) can be modeled in one of three ways:

• unidirectional constant rate mass transfer (not available for VOF calculations)

• UDF-prescribed mass transfer

• mass transfer through cavitation (only valid for the mixture multiphase model)

Because of the different procedures and limitations involved, defining mass transfer

through cavitation is described separately in Section 23.11.2: Including Cavitation Effects.



To define mass transfer in a multiphase simulation, either as unidirectional constant or

using a UDF, you will need to use the Phase Interaction panel (e.g., Figure 23.9.6).


1. In the Phases panel (Figure 23.9.2), click the Interaction... button to open the Phase

Interaction panel.

Figure 23.9.6: The Phase Interaction Panel for Mass Transfer

2. Click the Mass tab in the Phase Interaction panel.

3. Specify the Number of Mass Transfer Mechanisms. You can include any number

of mass transfer mechanisms in your simulation. Note also that the same pair of

phases can have multiple mass transfer mechanisms and you have the ability to

activate and deactivate the mechanisms of your choice.

4. For each mechanism, specify the phase of the source material under From Phase.

5. If species transport is part of the simulation, and the source phase is composed of a mixture material, then specify the species of the source phase mixture material

in the corresponding Species drop-down list.

6. For each mechanism, specify the phase of the destination material phase under To

Phase.



7. If species trans port is part of the simulation, and the destination phase is composed

of a mixture material, then specify the species of the destination phase mixture

material in the corresponding Species drop-down list.

8. For each mass transfer mechanism, select the desired mass transfer correlation

under Mechanism. The following choices are available:

constant-rate enables a consta nt, unidirectional mass transfer.

user-defined allows you to implement a correlation reflecting a model of your

choice, through a user-defined function.

FLUENT will automatically include the terms needed to model mass transfer in all

relevant conservation equations. Another option to model mass transfer between

phases is through the use of user-defined sources and their inclusion in the rele-

vant conservation equations. This approach is a more involved but more powerful,

allowing you to split the source terms according to a model of your choice.

i Note that momentum, energy, and turbulence are also transported withthe mass that is transferred. FLUENT assumes that the reacta nts aremixedthoroughly before reacting together, thus the heat and momentum transfer

is based on this assumption. This assumption can be deactivated using

a text command. For more information, contact your FLUENT support

engineer.

When your model involves the transport of multiphase species, you can define a mass

transfer mechanism between species from different phases. If a particular phase doesnot

have a species associated with it, then the mass transfer throughout the system will be performed by the bulk fluid material.

i Note that including speciestrans port effects in the mass transport of multiphase simulation requires that Species Transport be turned on in theSpeciesModel panel.

Define −→ Models −→ Species −→Transport &

Reaction...



23.9.8 Defining Multiphase Boundary

Conditions

Multiphase boundary conditions are set in the Boundary Conditions panel (Figure 23.9.7),

but the procedure for setting multiphase boundary conditions is slightly different than

for single-phase models. You will need to set some conditions separately for individual

phases, while other conditions are shared by all phases (i.e., the mixture), as described

in detail below.

Define −→Boundary

Conditions...

Figure 23.9.7: The Boundary Conditions Panel

BoundaryConditionsforthe M ixtureand the IndividualPhases

The conditions you need to specify for the mixture and those you need to specify for the

individual phases will depend on which of the three multiphase models you are using.

Details for each model are provided below.



VOFModel

If you are using the VOF model, the conditions you need to specify for each type of zone

are listed below and summarized in Table 23.9.1.

• For an exhaust fan, inlet vent, intake fan, outlet vent, pressure inlet, pressure

outlet, or velocity inlet, there are no conditions to be specified for the primary

phase. For each secondary phase, you will need to set the volume fraction as a

consta nt, a profile (see Section 7.26: Boundary Profiles), or a user-defined

function (see the separate UDF Manual). All other conditions are specified for the

mixture.

• For a mass flow inlet, you will need to set the mass flow rate or mass flux for

each individual phase. All other conditions are specified for the mixture.

i Note that if you read a VOF case that was set up in a version of FLUENT

prior to 6.1, you will need to redefine the conditions at the mass flow inlets.

• For an axis, fan, outflow, periodic, porous jump, radiator, solid, symmetry, or

wall zone, all conditions are specified for the mixture. There are no conditions to

be set for the individual phases.

• For a wall zone, you can specify the contact angle if wall adhesion option is

enabled.

• For a fluid zone, mass sources are specified for the individual phases, and all

other sources are specified for the mixture.

If the fluid zone is not porous, all other conditions are specified for the mixture.

If the fluid zone is porous, you will enable the Porous Zone option in the Fluid panelfor the mixture. The porosity inputs (if relevant) are also specified for the mixture.

The resistance coefficients and direction vectors, however, are specified separately

for each phase. See Section 7.19.6: User Inputs for Porous Media for details about

these inputs. All other conditions are specified for the mixture.

See Chapter 7: Boundary Conditions for details about the relevant conditions for each

type of boundary. Note that the pressure far-field boundary is not available with the

VOF model.



Table 23.9.1: Phase-Specific and Mixture Conditions for the VOF Model

Type Primary Phase Secondary Phase Mixture

exhaust fan;

inlet vent;

intake fan;

outlet vent; pressure inlet;

pressure outlet;

velocity inlet

nothing volume fraction all others

mass flow inlet mass flow/flux mass flow/flux all others

axis;

fan; outflow;

periodic;

porous jump;

radiator;

solid;

symmetry;wall

nothing nothing all others

pressure far-field not available not available not available

fluid mass source;

other porous inputs

mass source;

other porous inputs

porous zone;

porosity;

all others



MixtureModel

If you are using the mixture model, the conditions you need to specify for each type of

zone are listed below and summarized in Table 23.9.2.

• For an exhaust fan, outlet vent, or pressure outlet, there are no conditions to

be specified for the primary phase. For each secondary phase, you will need to setthe volume fraction as a consta nt, a profile (see Section 7.26: Boundary

Profiles), or a user-defined function (see the separate UDF Manual). All other

conditions are specified for the mixture.

• For an inlet vent, intake fan, or pressure inlet, you will specify for the mixture

which direction specification method will be used at this boundary (Normal to

Boundary or Direction Vector). If you select the Direction Vector specification

method, you will specify the coordinate system (3D only) and flow-direction

components for the individual phases. For each secondary phase, you will need to

set the volume fraction (as described above). All other conditions are specified for

the mixture.• For a mass flow inlet, you will need to set the mass flow rate or mass flux for

each individual phase. All other conditions are specified for the mixture.

i Note that if you read a mixture multiphase case that was set up in aversion of FLUENT previous to 6.1, you will need to redefine the conditionsat themass flow inlets.

• For a velocity inlet, you will specify the velocity for the individual phases. For

each secondary phase, you will need to set the volume fraction (as described

above). All other conditions are specified for the mixture.

• For an axis, fan, outflow, periodic, porous jump, radiator, solid, symmetry, or

wall zone, all conditions are specified for the mixture. There are no conditions to

be set for the individual phases. Outflow boundary conditions are not available

for the cavitation model.

• For a fluid zone, mass sources are specified for the individual phases, and all

other sources are specified for the mixture.


If the fluid zone is porous, you will enable the Porous Zone option in the Fluid panel

for the mixture. The porosity inputs (if relevant) are also specified for the mixture.

The resistance coefficients and direction vectors, however, are specified separatelyfor each phase. See Section 7.19.6: User Inputs for Porous Media for details about

these inputs. All other conditions are specified for the mixture.


type of boundary. Note that the pressure far-field boundary is not available with the

mixture model.



Table 23.9.2: Phase-Specific and Mixture Conditions for the Mixture Model


exhaust fan;

outlet vent;

pressure outlet

nothing volume fraction all others

inlet vent;

intake fan;

pressure inlet

coord. system;

flow direction

coord. system;

flow direction;

volume fraction

dir. spec.

method;

all othersmass flow inlet mass flow/flux mass flow/flux all others

velocity inlet velocity velocity;

volume fraction

all others

axis;

fan;

outflow (n/a for

cavitation

model);

periodic;

porous jump;radiator;

solid;

symmetry;

wall


pressure far-field not available not available not available

fluid mass source;

other porous inputs

mass source;

other porous inputs

porous zone;

porosity;

all others



EulerianModel

If you are using the Eulerian model, the conditions you need to specify for each type

of zone are listed below and summarized in Tables 23.9.3, 23.9.4, 23.9.5, and 23.9.6.

Note that the specification of turbulence parameters will depend on which of the three

multiphase turbulence models you are using, as indicated in Tables 23.9.4 – 23.9.6. See

Sections 23.5.10 and 23.12.3 for more information about multiphase turbulence models.

• For an exhaust fan, outlet vent, or pressure outlet, there are no conditions to

be specified for the primary phase if you are modeling laminar flow or using the

mixture turbulence model (the default multiphase turbulence model), except for

backflow total temperature if heat transfer is on.

For each secondary phase, you will need to set the volume fraction as a constant,

a profile (see Section 7.26: Boundary Profiles), or a user-defined function (see the

separate UDF Manual). If the phase is granular, you will also need to set its

granular temperature. If heat transfer is on, you will also need to set the backflow

total temperature.

If you are using the mixture turbulence model, you will need to specify the turbu-

lence boundary conditions for the mixture. If you are using the dispersed turbulence

model, you will need to specify them for the primary phase. If you are using the

per-phase turbulence model, you will need to specify them for the primary phase

and for each secondary phase.

All other conditions are specified for the mixture.

• For an inlet vent, intake fan, or pressure inlet, you will specify for the mixture

which direction specification method will be used at this boundary (Normal to

Boundary or Direction Vector). If you select the Direction Vector specificationmethod, you will specify the coordinate system (3D only) and flow-direction

components for the individual phases. If heat transfer is on, you will also need

to set the total temperature for the individual phases.

For each secondary phase, you will need to set the volume fraction (as described

above). If the phase is granular, you will also need to set its granular temperature.




per-phase turbulence model, you will need to specify them for the primary phase

and for each secondary phase.All other conditions are specified for the mixture.

• For a mass flow inlet, you will need to set the mass flow rate or mass flux for

each individual phase. You will also need to specify the temperature of each phase,

since the energy equations are solved for each phase.



For mass flow inlet boundary conditions, you can specify the slip velocity between

phases. When you select a mass flow inlet boundary for the secondary phase, two

options will be available for the Slip Velocity Specification Method, as shown in

Figure 23.9.8:

– Velocity Ratio

The value for the phase velocity ratio is the secondary phase to primary phase

velocity ratio. By default, it is 1.0, which means velocities are the same (no

slip). By entering a ratio that is greater than 1.0, you are indicating a larger

secondary phase velocity. Otherwise, you can enter a ratio that is less than

1.0 to indicate a smaller secondary phase velocity.

– Volume Fraction

If you specify the volume fraction at an inlet, FLUENT will calculate the phase

velocities.

i If a secondary phase has zero mass flux (i.e., the Eulerian model is usedto run a single phase case), neither Phase Velocity Ratio nor VolumeFractionwill affect the solution.

Figure 23.9.8: Mass-Flow Inlet Boundary Condition Panel

• For a velocity inlet, you will specify the velocity for the individual phases. If

heat transfer is on, you will also need to set the total temperature for the

individual phases.



For each secondary phase, you will need to set the volume fraction (as described

above). If the phase is granular, you will also need to set its granular temperature.




per-phase turbulence model, you will need to specify them for the primary phaseand for each secondary phase.

All other conditions are specified for the mixture.

• For an axis, outflow, periodic, solid, or symmetry zone, all conditions are

specified for the mixture. There are no conditions to be set for the individual

phases.

• For a wall zone, shear conditions are specified for the individual phases. All

other conditions are specified for the mixture, including thermal boundary

conditions, if heat transfer is on.

• For a fluid zone, all source terms and fixed values are specified for theindividual phases, unless you are using the mixture turbulence model or the

dispersed turbulence model. If you are using the mixture turbulence model,

source terms and fixed values for turbulence are specified instead for the mixture.

If you are using the dispersed turbulence model, they are specified only for the

primary phase.


If the fluid zone is porous, you will enable the Porous Zone option in the Fluid panel

for the mixture. The porosity inputs (if relevant) are also specified for the mixture.

The resistance coefficients and direction vectors, however, are specified separately

for each phase. See Section 7.19.6: User Inputs for Porous Media for details aboutthese inputs. All other conditions are specified for the mixture.


type of boundary. Note that the pressure far-field, fan, porous jump, radiator, and mass

flow inlet boundaries are not available with the Eulerian model.



Table 23.9.3: Phase-Specific and Mixture Conditions for the Eulerian Model

(for Laminar Flow)


exhaust fan;

outlet vent;

pressure outlet

(tot. temperature) volume fraction;

gran. temperature

(tot. temperature)

all others

inlet vent;

intake fan;

pressure inlet

coord. system;

flow direction

(tot. temperature)

coord. system;

flow direction;

volume fraction;

gran. temperature(tot. temperature)

dir. spec.

method;

all others

velocity inlet velocity

(tot. temperature)

velocity;

volume fraction;

gran. temperature

(tot. temperature)

all others

axis;

outflow;

periodic;

solid;

symmetry


wall shear condition shear condition all others

pressure

far-field;

fan;

porous jump;

radiator;

mass flow inlet

not available not available not available

fluid all source terms;

all fixed values;

other porous inputs

all source terms;

all fixed values;

other porous inputs

porous zone;

porosity;

all others




(with the Mixture Turbulence Model)


exhaust fan;

outlet vent;

pressure outlet

(tot. temperature) volume fraction;

gran. temperature

(tot. temperature)

all others

inlet vent;

intake fan;

pressure inlet

coord. system;

flow direction

(tot. temperature)

coord. system;

flow direction;

volume fraction;

gran. temperature

(tot. temperature)

dir. spec.

method;

all others

velocity inlet velocity(tot. temperature)

velocity;volume fraction;

gran. temperature

(tot. temperature)

all others

axis;

outflow;

periodic;

solid;

symmetry



pressure

far-field;

fan;

porous jump;

radiator;

mass flow inlet


fluid other source terms;

other fixed values;

other porous inputs

other source terms;

other fixed values;

other porous inputs

source terms for

turbulence;

fixed values for

turbulence;

porous zone;

porosity;all others



Table 23.9.5: Phase-Specific and Mixture Conditions for the Eulerian Model(with the Dispersed Turbulence Model)


exhaust fan;

outlet vent;

pressure outlet

turb. parameters

(tot. temperature)

volume fraction;

gran. temperature

(tot. temperature)

all others

inlet vent;

intake fan;

pressure inlet

coord. system;

flow direction;

turb. parameters;

(tot. temperature)

coord. system;

flow direction;

volume fraction;

gran. temperature

(tot. temperature)

dir. spec.

method;

all others

velocity inlet velocity;

turb. parameters

(tot. temperature)

velocity;

volume fraction;

gran. temperature

(tot. temperature)

all others

axis;

outflow;

periodic;

solid;

symmetry



pressure

far-field;

fan;

porous jump;

radiator;

mass flow inlet


fluid momentum, mass,

turb. sources;

momentum, mass,

turb. fixed values;

other porous inputs

momentum and mass

sources;

momentum and mass

fixed values;

other porous inputs

porous zone;

porosity;

all others




(with the Per-Phase Turbulence Model)


exhaust fan;

outlet vent;

pressure outlet

turb. parameters

(tot. temperature)

volume fraction;

turb. parameters;

gran. temperature

(tot. temperature)

all others

inlet vent;

intake fan;

pressure inlet

coord. system;

flow direction;

turb. parameters

(tot. temperature)

coord. system;

flow direction;

volume fraction;

turb. parameters;

gran. temperature

(tot. temperature)

dir. spec.

method;

all others

velocity inlet velocity;

turb. parameters

(tot. temperature)

velocity;

volume fraction;

turb. parameters;

gran. temperature

(tot. temperature)

all others

axis;

outflow;

periodic;

solid;

symmetry



pressure

far-field;

fan;

porous jump;

radiator;

mass flow inlet


fluid momentum, mass,

turb. sources;

momentum, mass,

turb. fixed values;

other porous inputs

momentum, mass,

turb. sources;

momentum, mass,

turb. fixed values;

other porous inputs

porous zone;

porosity;

all others



Steps for SettingBoundaryConditions

The steps you need to perform for each boundary are as follows:

1. Select the boundary in the Zone list in the Boundary Conditions panel.

2. Set the conditions for the mixture at this boundary, if necessary. (See above for

information about which conditions need to be set for the mixture.)

(a) In the Phase drop-down list, select mixture.

(b) If the current Type for this zone is correct, click Set... to open the

corresponding panel (e.g., the Pressure Inlet panel); otherwise, choose the

correct zone type in the Type list, confirm the change (when prompted), and

the corresponding panel will open automaticall y.

(c) In the corresponding panel for the zone type you have selected (e.g., the Pres-

sure Inlet panel, shown in Figure 23.9.9), specify the mixture boundary condi-

tions.

Figure 23.9.9: The Pressure Inlet Panel for a Mixture

Note that only those conditions that apply to all phases, as described above,

will appear in this panel.

i For a VOF calculation, if you enabled the Wall Adhesion option in

the

Phase Interaction panel, you can specify the contact angle at the wall for

each pair of phases as a constant (as shown in Figure 23.9.10) or a UDF

(see the UDF manual for more information).

The contact angle (θw in Figure 23.3.3) is the angle between the wall and the

tangent to the interface at the wall, measured inside the phase listed in theleft column under Wall Adhesion in the Momentum tab of the Wall panel. For



example, if you are setting the contact angle between the oil and air phases in

the Wall panel shown in Figure 23.9.10, θw is measured inside the oil phase.

Figure 23.9.10: The Wall Panel for a Mixture in a VOF Calculation with

Wall Adhesion

The default value for all pairs is 90 degrees, which is equivalent to no wall

adhesion effects (i.e., the interface is normal to the adjacent wall). A contact

angle of 45◦, for example, corresponds to water creeping up the side of

a container, as is common with water in a glass.

(d) Click OK when you are done setting the mixture boundary conditions.



3. Set the conditions for each phase at this boundary, if necessary. (See above for

information about which conditions need to be set for the individual phases.)

(a) In the Phase drop-down list, select the phase (e.g., water ).

i Note that, when you select one of the individual phases (rather thanthe mixture), only one type of zone appears in the Type list. It is not possibleto assign phase-specific zone types at a given boundary; the zone type is

specified for the mixture, and it applies to all of the individual phases.

(b) Click Set... to open the panel for this phase’s conditions (e.g., the Pressure

Inlet panel, shown in Figure 23.9.11).

Figure 23.9.11: The Pressure Inlet Panel for a Phase

(c) Specify the conditions for the phase. Note that only those conditions that

apply to the individual phase, as described above, will appear in this panel.(d) Click OK when you are done setting the phase-specific boundary conditions.



23.10SettingUp the VOFModel

Steps for Copying BoundaryConditions

The steps for copying boundary conditions for a multiphase flow are slightly different

from those described in Section 7.1.5: Copying Boundary Conditions for a single-phase

flow. The modified steps are listed below:

1. In the Boundary Conditions panel, click the Copy... button. This will open the Copy

BCs panel.

2. In the From Zone list, select the zone that has the conditions you want to copy.

3. In the To Zones list, select the zone or zones to which you want to copy the condi-

tions.

4. In the Phase drop-down list, select the phase for which you want to copy the

conditions (either mixture or one of the individual phases).

i Note that copying the boundary conditions for one phase does notauto- matically result in the boundary conditions for the other phasesand themixture being copied as well. You need to copy the conditions for each

phase on each boundary of interest.

5. Click Copy. FLUENT will set all of the selected phase’s (or mixture’s) boundary

conditions on the zones selected in the To Zones list to be the same as that phase’s

conditions on the zone selected in the From Zone list. (You cannot copy a subset

of the conditions, such as only the thermal conditions.)

See Section 7.1.5: Copying Boundary Conditions for additional information about copying boundary conditions, including limitations.

23.10 SettingUp the VOFModel

23.10.1 Choosinga VOFFormulation

To specify the VOF formulation to be used, select the appropriate VOF Scheme under

VOF Parameters in the Multiphase Model panel.

The VOF formulations that are available in FLUENTare the Explicitand Implicitschemes.




ExplicitSchemes

• Time-dependent with the explicit interpolation scheme: Since the donor-

acceptor scheme is available only for quadrilateral and hexahedral meshes, it cannot

be used for a hybrid mesh containing twisted hexahedral cells. For such cases,

you should use the time-dependent explicit scheme. This formulation can also

be used for other cases in which the geometric reconstruction scheme does not givesatisfactory results, or the flow calculation becomes unstable. Note that the

CICSAM scheme or the modified HRIC scheme can be computationally

inexpensive when compared to the geometric reconstruction scheme and improves

the robustness and stability of the calculations. The Volume Fraction

discretizations, Modified HRIC and CICSAM, are available in the Solution Controls

panel when the explicit VOF scheme is selected.

Note that FLUENT will automatically turn on the unsteady formulation with first-

order discretization for time in the Solver panel.

• Time-dependent with the geometric reconstruction interpolation scheme: This

formulation should be used whenever you are interested in the time-accuratetransie nt behavior of the VOF solution.

To use this formulation, make sure Explicit is selected as the VOF Scheme in the

Multiphase panel, then select Geo-Reconstruct as the Volume Fraction Discretization

scheme in the Solution Controls panel.

• Time-dependent with the donor-acceptor interpolation scheme: This

formulation should be used instead of the time-dependent formulation with the

geometric reconstruction scheme if your mesh contains highly twisted hexahedral

cells. For such cases, the donor-acceptor scheme may provide more accurate

results.

The Donor-Acceptor scheme is used when Explicit is selected as the VOF Scheme

in the Multiphase panel. Initially, this formulation is not available in the GUI. To

make it available, use the following text command:

solve −→ set−→expert

You will be asked a series of questions, one of which is

Allow selection of all applicable discretization schemes? [no]

If your response is yes, then many more discretization schemes will be available

for your selection. You can now use this formulation by selecting Donor-Acceptor

as the Volume Fraction Discretization in the Solution Controls panel.

• The CICSAM scheme gives interface sharpness of the same level as the

geometric reconstruction scheme and is particularly suitable for flows with high

viscosity ratios between the phases.




To use this formulation, select Explicit as the VOF Scheme in the Multiphase panel,

then select CICSAM as the Volume Fraction Discretization in the Solution Controls

panel.

While the explicit time-dependent formulation is less computationally expensive

than the geometric reconstruction scheme, the interface between phases will not be

as sharp as that predicted with the geometric reconstruction scheme. To reduce thisdiffusivity, it is recommended that you use the second-order discretization scheme

for the volume fraction equations. In addition, you may want to consider turning

the geometric reconstruction scheme back on after calculating a solution with the

implicit scheme, in order to obtain a sharper interface.

i For the geometric reconstruction and donor-acceptor schemes, if youare using a conformal grid (i.e., if the grid node locations are identical atthe boundaries wheretwo subdomains meet), you must ensure that there are

no two-sided (zero-thickness) walls within the domain. If there are, you

will need to slit them, as described in Section 6.8.6: Slitting Face Zones.

i The issues discussed above for the explicit time-dependentformulation also apply to the implicit steady-state and time-dependentformulations,described below. You should take the precautions described above to im-

prove the sharpness of the interface.

ImplicitSchemes

• Time-dependent with the implicit interpolation scheme: This formulation can

be used if you are looking for a steady-state solution and you are not interested inthe intermediate transie nt flow behavior, but the final steady-state solution is

dependent on the initial flow conditions and/or you do not have a distinct

inflow boundary for each phase.

To use this formulation, select Implicit as the VOF Scheme, and enable an Unsteady

calculation in the Solver panel (opened with the Define/Models/Solver... menu

item).

• Steady-state with the implicit interpolation scheme: This formulation can be

used if you are looking for a steady-state solution, you are not interested in the

intermediate transie nt flow behavior, and the final steady-state solution is not

affected by the initial flow conditions and there is a distinct inflow boundary for

each phase. Note that the implicit modified HRIC scheme can be used as a robust

alternati ve to the explicit geometric reconstruction scheme.

To use this formulation, select Implicit as the VOF Scheme in the Multiphase panel,

then select Modified HRIC as the Volume Fraction Discretization in the Solution Con-

trols panel.




Examples

To help you determine the best formulation to use for your problem, examples that use

different formulations are listed below:

• jet breakup

Use the explicit scheme (time-dependent with the geometric reconstruction scheme

or the donor-acceptor) if problems occur with the geometric reconstruction scheme.

• shape of the liquid interface in a centrifuge

Use the time-dependent with the implicit interpolation scheme.

• flow around a ship’s hull

Use the steady-state with the implicit interpolation scheme.

23.10.2 ModelingOpen ChannelFlows

Using the VOF formulation, open channel flows can be modeled in FLUENT. To start

using the open channel flow boundary condition, perform the following:

1. Turn on gravity.

(a) Open the Operating Conditions panel.


(b) Turn on Gravity and set the gravitational acceleration fields.

2. Enable the volume of fluid model.

(a) Open the Multiphase Model panel.


−→Multiphase...

(b) Under Model, turn on Volume of Fluid.

(c) Under VOF Scheme, select either Implicit, Explicit.

3. Under VOF Parameters, select Open Channel Flow.




In order to set specific parameters for a particular boundary for open channel flows,

turn on the Open Channel Flow option in the corresponding boundary condition panel.

Table 23.10.1 summarizes the types of boundaries available to the open channel

flow boundary condition, and the additional parameters needed to model open channel

flow. For more information on setting boundary condition parameters, see Chapter 7:

Boundary Conditions .

Table 23.10.1: Open Channel Boundary Parameters for the VOF Model

Boundary Type Parameter

pressure inlet Inlet Group ID;

Secondary Phase for Inlet;

Flow Specification Method;

Free Surface Level, Bottom Level;

Velocity Magnitude

pressure outlet Outlet Group ID;

Pressure Specification Method;

Free Surface Level; Bottom Level

mass flow inlet Inlet Group ID;

Secondary Phase for Inlet;

Free Surface Level;

Bottom Level

outflow Flow Rate Weighting

DefiningInletGroups

Open channel systems involve the flowing fluid (the secondary phase) and the fluid above

it (the primary phase).

If both phases enter through the separate inlets (e.g., inlet-phase2 and inlet-phase1), these two inlets form an inlet group. This inlet group is recognized by the

parameter Inlet Group ID, which will be same for both the inlets that make up the inlet

group. On

the other hand, if both the phases enter through the same inlet (e.g., inlet-combined),

then the inlet itself represents the inlet group.

i In three-phase flows, only one secondary phase is allowed to passthrough one inlet group.




DefiningOutletGroups

Outlet-groups can be defined in the same manner as the inlet groups.

i In three-phase flows, the outlet should represent the outlet group,

i.e., separate outlets for each phase are not recommended in three-phaseflows.

Settingthe InletGroup

For pressure inlets and mass flow inlets, the Inlet Group ID is used to identify the

different inlets that are part of the same inlet group. For instance, when both phases

enter through the same inlet (single face zone), then those phases are part of one inlet

group and you would set the Inlet Group ID to 1 for that inlet (or inlet group).

In the case where the same inlet group has separate inlets (different face zones) for each

phase, then the Inlet Group ID will be the same for each inlet of that group.

When specifying the inlet group, use the following guidelines:

• Since the Inlet Group ID is used to identify the inlets of the same inlet group,

general information such as Free Surface Level, Bottom Level, or the mass flow rate

for each phase should be the same for each inlet of the same inlet group.

• You should specify a different Inlet Group ID for each distinct inlet group.

For example, consider the case of two inlet groups for a particular problem. The

first inlet group consists of water and air entering through the same inlet (a single

face zone). In this case, you would specify an inlet group ID of 1 for that inlet (or

inlet group). The second inlet group consists of oil and air entering through thesame inlet group, but each uses a different inlet (oil-inlet and air-inlet) for

each phase. In this case, you would specify the same Inlet Group ID of 2 for both of

the inlets that belong to the inlet group.

Settingthe OutletGroup

For pressure outlet boundaries, the Outlet Group ID is used to identify the different outlets

that are part of the same outlet group. For instance, when both phases enter through

the same outlet (single face zone), then those phases are part of one outlet group and

you would set the Outlet Group ID to 1 for that outlet (or outlet group).

In the case where the same outlet group has separate outlets (different face zones) for

each phase, then the Outlet Group ID will be the same for each outlet of that group.




When specifying the outlet group, use the following guidelines:

• Since the Outlet Group ID is used to identify the outlets of the same outlet

group, general information such as Free Surface Level or Bottom Level should be

the same for each outlet of the same outlet group.

• You should specify a different Outlet Group ID for each distinct outlet group.

For example, consider the case of two outlet groups for a particular problem. The

first inlet group consists of water and air exiting from the same outlet (a single face

zone). In this case, you would specify an outlet number of 1 for that outlet (or

outlet group). The second outlet group consists of oil and air exiting through the

same outlet group, but each uses a different outlet (oil-outlet and air-outlet)

for each phase. In this case, you would specify the same Outlet Group ID of 2 for

both of the outlets that belong to the outlet group.

i For three-phase flows, when all the phases are leaving through thesame outlet, the outlet should consist only of a single face zone.

Determiningthe Free SurfaceLevel

For the appropriate boundary, you need to specify the Free SurfaceLevelvalue. This pa-

rameter is available for all relevant boundaries, including pressure outlet, mass flow inlet,

and pressure inlet. The Free Surface Level, is represented by ylocal in Equation 23.3-25.

ylocal = −(−→a · gˆ) (23.10-1)

where −→a is the position vector of any point on the free surface, and gˆ is the unit

vector in the direction of the force of gravity. Here we assume a horizontal free surface

that is normal to the direction of gravity.

We can simply calculate the free surface level in two steps:

1. Determine the absolute value of height from the free surface to the origin in the

direction of gravity.

2. Apply the correct sign based on whether the free surface level is above or below

the origin.

If the liquid’s free surface level lies above the origin, then the Free Surface Levelis positive

(see Figure 23.10.1). Likewise, if the liquid’s free surface level lies below the origin, then

the Free Surface Level is negative.




Determiningthe BottomLevel

For the appropriate boundary, you need to specify the Bottom Level value. This

parameter is available for all relevant boundaries, including pressure outlet, mass flow

inlet, and pressure inlet. The Bottom Level, is represented by a relation similar to

Equation 23.3-25.

y bottom = −(−→

b · gˆ) (23.10-2)

where−→

b is the position vector of any point on the bottom of the channel, and gˆ

is the unit vector of gravity. Here we assume a horizontal free surface that is normal

to the direction of gravity.

We can simply calculate the bottom level in two steps:

1. Determine the absolute value of depth from the bottom level to the origin in the

direction of gravity.

2. Apply the correct sign based on whether the bottom level is above or below the

origin.

If the channel’s bottom lies above the origin, then the Bottom Level is positive (see

Figure 23.10.1). Likewise, if the channel’s bottom lies below the origin, then the Bottom

Level is negative.

Free Surface Level (positive)

g

Bottom Level (positive)

Origin

Reference Level

Figure 23.10.1: Determining the Free Surface Level and the Bottom Level



Specifyingthe TotalHeight

The total height, along with the velocity, is used as an option for describing the

flow. The total height is given as

V 2

ytot = ylocal + 2g (23.10-3)

where V is the velocity magnitude and g is the gravity magnitude.

Determiningthe VelocityMagnitude

Pressure inlet boundaries require the Velocity Magnitude for calculating the dynamic

pressure at the boundary. This is to be specified as the magnitude of the upstream inlet

velocity in the flow.

Determiningthe SecondaryPhase for the Inlet

For pressure inlets and mass flow inlets, the Secondary Phase for Inlet field is significan t

in cases of three-phase flows.

i Note that only one secondary phase is allowed to pass through oneinlet group.

Consider a problem involving a three-phase flow consisting of air as the primary phase,

and oil and water as the secondary phases. Consider also that there are two inlet groups:

•

water andair

• oil and air

For the former inlet group, you would choose water as the secondary phase. For the

latter inlet group, you would choose oil as the secondary phase.

Choosingthe Pressure SpecificationMethod

For a pressure outlet boundary, the outlet pressure can be specified in one of two ways:

• by prescribing the local height (i.e., a hydrostatic pressure profile)

• by specifying the constant pressure

i This option is not available in the case of three-phase flows since the pressure on the boundary is taken from the neighboring cell.



Limitations

The following list summarizes some issues and limitations associated with the open chan-

nel boundary condition.

• The conservation of the Bernoulli integral does not provide the conservation

of mass flow rate for the pressure boundary. In the case of a coarser mesh, there

can be a significant difference in mass flow rate from the actual mass flow rate.

For finer meshes, the mass flow rate comes closer to the actual value. So, for

problems having constant mass flow rate, the mass flow rate boundary condition

is a better option. The pressure boundary should be selected when steady and

nonoscillating drag is the main ob jective.

• Specifying the top boundary as the pressure outlet can sometimes lead to a

divergent solution. This may be due to the corner singularity at the pressure

boundary in the air region or due to the inability to specify local flow direction

correctly if the air enters through the top locally.

• Only the heavier phase should be selected as the secondary phase.

• In the case of three-phase flows, only one secondary phase is allowed to

enter through one inlet group (i.e., the mixed inflow of different secondary phases

is not allowed).

Recommendationsfor Setting Up an Open ChannelFlow Problem

The following list represents a list of recommendations for solving problems using the

open channel flow boundary condition:

• In the cases where the inlet group has a different inlet for each phase of fluid,

then the parameter values (such as Free Surface Level,Bottom Level,and Mass Flow

Rate) for each inlet should correspond to all other inlets that belong to the inlet

group.

• The solution begins with an estimated pressure profile at the outlet boundar y.

In general, you can start the solution by assuming that the level of liquid at the

outlet corresponds to the level of liquid at the inlet. The convergence and solution

time is very dependent on the initial conditions. When the flow is completely

subcritical (upstream and downstream), in marine applications for instance, theabove approach is recommended.

If the final conditions of the flow can be predicted by other means, the solution

time can be significantly reduced by using the proper boundary condition.



• The initialization procedure is very critical in the open channel analysis.

If you are interested in the final steady state solution, then perform the following

initialization procedure:

1. Initialize the domain by setting the volume fraction of the secondary phase to

0, and providing the inlet velocity.

2. Patch the domain using a volume fraction value of the secondary phase to 1,

up to the Free Surface Level specified at the inlet.

3. Patch the inlet velocity again in the full domain.

If the Free Surface Level values are different at the inlet and outlet, then patching

some regions with inlet Free Surface Level values and some regions with outlet Free

Surface Level values could be useful for some problems.

The same steps for initialization are also recommended for unsteady flows, but now

the initial conditions are dependent on the user.

• For the initial stability of the solution, a smaller time step is recommended.You can increase the time step once the solution becomes more stable.

23.10.3 Definingthe Phases for the VOFModel

Instructions for specifying the necessary information for the primary and secondary

phases and their interaction in a VOF calculation are provided below.

i In general, you can specify the primary and secondary phaseswhichever way you prefer. It is a good idea, especially in more complicated

problems,to consider how your choice will affect the ease of problem setup. For

example, if you are planning to patch an initial volume fraction of 1 for

one phase in a portion of the domain, it may be more convenient to mak e

that phase a secondary phase. Also, if one of the phases is a compressible

ideal gas, it is recommended that you specify it as the primary phase to

improve solution stabili ty.

i Recall that only one of the phases can be a compressible ideal gas. Besure that you do not select a compressible ideal gas material (i.e., amaterial

that uses the compressible ideal gas law for density) for more than one of the phases. See Sections 23.10.5 and 23.11.3 for details.



Definingthe PrimaryPhase

To define the primary phase in a VOF calculation, perform the following steps:

1. Select phase-1 in the Phase list.

2. Click Set... to open the Primary Phase panel (Figure 23.10.2).

Figure 23.10.2: The Primary Phase Panel

3. In the Primary Phase panel, enter a Name for the phase.

4. Specify which material the phase contains by choosing the appropriate material in

the Phase Material drop-down list.

5. Define the material properties for the Phase Material.

(a) Click Edit..., and the Material panel will open.

(b) In the Material panel, check the properties, and modify them if necessary. (SeeChapter 8: Physical Properties for general information about setting material

properties, Section23.10.5: Modeling Compressible Flows for specific informa-

tion related to compressible VOF calculations, and Section 23.10.6: Modeling

Solidification/Melting for specific information related to melting/solidification

VOF calculations.)

i If you make changes to the properties, remember to click Change before closing the Material panel.

6. Click OK in the Primary Phase panel.



Defininga SecondaryPhase

To define a secondary phase in a VOF calculation, perform the following steps:

1. Select the phase (e.g., phase-2) in the Phase list.

2. Click Set... to open the Secondary Phase panel (Figure 23.10.3).

Figure 23.10.3: The Secondary Phase Panel for the VOF Model

3. In the Secondary Phase panel, enter a Name for the phase.



5. Define the material properties for the Phase Material, following the procedure out-

lined above for setting the material properties for the primary phase.

6. Click OK in the Secondary Phase panel.



IncludingSurfaceTension and Wall AdhesionEffects

As discussed in Section 23.3.8: When Surface Tension Effects Are Important, the impor-

tance of surface tension effects depends on the value of the capillary number, Ca (defined

by Equation 23.3-16), or the Weber number, We (defined by Equation 23.3-17). Surface

tension effects can be neglected if Ca 1 or We 1.

i Note that the calculation of surface tension effects will be moreaccurate if you use a quadrilateral or hexahedral mesh in the area(s) of thecompu-tational domain where surface tension is significant. If you cannot use a

quadrilateral or hexahedral mesh for the entire domain, then you should

use a hybrid mesh, with quadrilaterals or hexahedra in the affected areas.

FLUENT also offers an option to use VOF gradients at the nodes for cur-

vature calculations on meshes when more accuracy is desired. For more

information, see Section 23.3.8: Surface Tension and Wall Adhesion.

If you want to include the effects of surface tension along the interface between one or more pairs of phases, as described in Section 23.3.8: Surface Tension and Wall Adhesion,

click Interaction... to open the Phase Interaction panel (Figure 23.10.4).

Figure 23.10.4: The Phase Interaction Panel for the VOF Model (Surface Ten-sion Tab)



Perform the following steps to include surface tension (and, if appropriate, wall adhesion)

effects along the interface between one or more pairs of phases:

1. Click the Surface Tension tab.

2. If you want to include wall adhesion, turn on the Wall Adhesion option. WhenWall Adhesion is enabled, you will need to specify the contact angle at each wall

as a boundary condition (as described in Section 23.9.8: Defining Multiphase

Boundary Conditions ).

3. For each pair of phases between which you want to include the effects of surface

tension, specify a constant surface tension coefficient. Alternatively you can spec-

ify a temperature dependent, polynomial, piece-wise polynomial, piecewise linear,

or a user-defined surface tension coefficient. See Section 23.3.8: Surface Tension

and Wall Adhesion for more information on surface tension, and the separate UDF

Manual for more information on user-defined functions. All surface tension coeffi-

cients are equal to 0 by default, representing no surface tension effects along theinterface between the two phases.

i For calculations involving surface tension, it is recommended that youalso turn on the Implicit Body Force treatme nt for the Body ForceFormulationin the Multiphase Model panel. This treatme nt improves solution conver-

gence by accounting for the partial equilibrium of the pressure gradient and

surface tension forces in the momentum equations. See Section 23.9.4: In-

cluding Body Forces for details.

23.10.4 SettingTime-DependentParam eters for the VOFModel

If you are using the time-dependent VOF formulation in FLUENT, an explicit solution

for the volume fraction is obtained either once each time step or once each iteration,

depending upon your inputs to the model. You also have control over the time step used

for the volume fraction calculation.

To compute a time-dependent VOF solution, you will need to enable the Unsteady option

in the Solver panel (and choose the appropriate Unsteady Formulation, as discussed in

Section 25.17.1: User Inputs for Time-Dependent Problems). If you choose the

Explicit scheme, FLUENT will turn on the first-order unsteady formulation for youautomaticall y, so you need not visit the Solver panel yourself.

Define −→ Models−→Solver...



There are two inputs for the time-dependent calculation for the VOF model:

• By default, FLUENT will solve the volume fraction equation(s) once for each

time step. This means that the convective flux coefficients appearing in the other

transport equations will not be completely updated each iteration, since the

volume fraction fields will not change from iteration to iteration.If you want FLUENT to solve the volume fraction equation(s) at every iteration

within a time step, use the text command:

define −→ models −→ multiphase−→

and select vof as the model. When prompted to solve vof every iteration?,enter yes. When FLUENT solves these equations every iteration, the convectivefluxcoefficients in the other transport equations will be updated based on the updated

volume fractions at each iteration. This choice is the less stable of the two, and

requires more computational effort per time step than the default choice.

i If you are using sliding meshes, or dynamic meshes with layeringand/or remeshing, using the solve vof every iteration? optionwill yieldmore accurate results, although at a greater computational cost.

• When FLUENT performs a time-dependent VOF calculation, the time step

used for the volume fraction calculation will not be the same as the time step

used for the rest of the transport equations. FLUENT will refine the time step

for VOF automaticall y, based on your input for the maximum Courant Number

allowed near the free surface. The Courant number is a dimensionless number

that compares the time step in a calculation to the characteristic time of transit of

a fluid element across a control volume:

∆t

∆xcell /vfluid

(23.10-4)

In the region near the fluid interface, FLUENT divides the volume of each cell by

the sum of the outgoing fluxes. The resulting time represents the time it would

take for the fluid to empty out of the cell. The smallest such time is used as

the characteristic time of transit for a fluid element across a control volume, as

described above. Based upon this time and your input for the maximum allowed

Courant Number in the Multiphase Models panel, a time step is computed for use

in the VOF calculation. For example, if the maximum allowed Courantnum ber is 0.25 (the default), the time step will be chosen to be at most one-

fourth the minimum transit time for any cell near the interface.

Note that these inputs are not required when the implicit scheme isused.



23.10.5 ModelingCompressibleFlows

If you are using the VOF model for a compressible flow, note the following:

• Only one of the phases can be defined as a compressible ideal gas (i.e., you

can select the ideal gas law for the density of only one phase’s material). There is


• When using the VOF model, for stability reasons, it is better (although not

required) if the primary phase is a compressible ideal gas.

• If you specify the total pressure at a boundary (e.g., for a pressure inlet or

intak e fan) the specified value for temperature at that boundary will be used

as total temperature for the compressible phase, and as static temperature for

the other phases (which are incompressible).

• For each mass flow inlet, you will need to specify mass flow or mass flux for

each individual phase.

i Note that if you read a case file that was set up in a version of FLUENT

previous to 6.1, you will need to redefine the conditions at the mass flow

inlets. See Section 23.9.8: Defining Multiphase Boundary Conditions for

more information on defining conditions for a mass flow inlet in VOF mul-

tiphase calculations.

See Section 9.6: Compressible Flows for more information about compressible flows.



23.10.6 ModelingSolidification/Melting

If you are including melting or solidification in your VOF calculation, note the following:

• It is possible to model melting or solidification in a single phase or in

multiple phases.

• For phases that are not melting or solidifying, you must set the latent heat

(L), liquidus temperature (Tliquidus ), and solidus temperature (Tsolidus ) to zero.

See Chapter 24: Modeling Solidification and Melting for more information about melting

and solidification.

23.11 SettingUp the MixtureModel

23.11.1 Definingthe Phases for the MixtureModel


phases and their interaction for a mixture model calculation are provided below.

i Recall that only one of the phases can be a compressible ideal gas. Besure that you do not select a compressible ideal gas material (i.e., amaterialthat uses the compressible ideal gas law for density) for more than one of

the phases. See Section 23.11.3: Modeling Compressible Flows for details.


The procedure for defining the primary phase in a mixture model calculation is the same

as for a VOF calculation. See Section 23.10.3: Defining the Primary Phase for details.

Defininga NongranularSecondaryPhase

To define a nongranular (i.e., liquid or vapor) secondary phase in a mixture multiphase

calculation, perform the following steps:





23.11SettingUp the M ixtu reModel

Figure 23.11.1: The Secondary Phase Panel for the Mixture Model




5. Define the material properties for the Phase Material, following the same pro-

cedure you used to set the material properties for the primary phase (see Sec-

tion 23.10.3: Defining the Primary Phase). For a particulate phase (which must

be placed in the fluid materials category, as mentioned in Section 23.9: Steps for

Using a Multiphase Model), you need to specify only the density; you can ignore

the values for the other properties, since they will not be used.

6. In the Secondary Phase panel, specify the Diameter of the bubbles, droplets, or

particles of this phase (d p in Equation 23.4-12). You can specify a constant value,

or use a user-defined function. See the separate UDF Manual for details about

user-defined functions. Note that when you are using the mixture model withoutslip velocity, this input is not necessary, and it will not be available to you.





Defininga GranularSecondaryPhase

To define a granular (i.e., particulate) secondary phase in a mixture model multiphase




Figure 23.11.2: The Secondary Phase Panel for a Granular Phase Using the

Mixture Model






23.11SettingUp the M ixtu reModel



tion 23.10.3: Defining the Primary Phase). For a granular phase (which must be

placed in the fluid materials category, as mentioned in Section 23.9: Steps for Us-

ing a Multiphase Model), you need to specify only the density; you can ignore the

values for the other properties, since they will not be used.

i Note that all properties for granular flows can utilize user-defined functions

(UDFs).

See the separate UDF Manual for details about user-defined functions.

6. Turn on the Granular option.

7. In the Secondary Phase panel, specify the following properties of the particles of

this phase:

Diameter specifies the diameter of the particles. You can select constant in the

drop-down list and specify a constant value, or select user-defined to use a user-defined function. See the separate UDF Manual for details about user-defined

functions.

Granular Viscosity specifies the kinetic part of the granular viscosity of the par-

ticles (µs,kin in Equation 23.4-18). You can select constant (the default) in the

drop-down list and specify a constant value, select syamlal-obrien to compute

the value using Equation 23.4-20, select gidaspow to compute the value using

Equation 23.4-21, or select user-defined to use a user-defined function.

Granular Temperature specifies temperature for the solids phase and is propor-

tional to the kinetic energy of the random motion of the particles. Choose

either the algebraic, the constant, or user-defined option.

Solids Pressure specifies the pressure gradient term, ∇ ps , in the granular-phase

momentum equation. Choose either the lun-et-al, the syamlal-obrien, the ma-ahmadi, or the user-defined option.

Radial Distribution specifies a correction factor that modifies the probability of

collisions between grains when the solid granular phase becomes dense. Choose

either the lun-et-al, the syamlal-obrien, the ma-ahmadi, the arastoopour, or a

user-defined option.

Elasticity Modulus is defined as

with G≥

0.

G =

∂Ps

∂αs (23.11-1)

Choose either the derived or user-defined options.



Packing Limit specifies the maximum volume fraction for the granular phase

(αs,max ). For monodispersed spheres, the packing limit is about 0.63, which is

the default value in FLUENT. In polydispersed cases, however, smaller spheres

can fill the small gaps between larger spheres, so you may need to increase themaximum packing limit.


DefiningDrag BetweenPhases

For mixture multiphase flows with slip velocity, you can specify the drag function to

be used in the calculation. The functions available here are a subset of those discussed

in Section 23.12.2: Defining the Phases for the Eulerian Model. See Section 23.4.5:

Relative (Slip) Velocity and the Drift Velocity for more information.

To specify drag laws, click Interaction... to open the Phase Interaction panel (Figure 23.11.3),

and then click the Drag tab.

Figure 23.11.3: The Phase Interaction Panel for the Mixture Model (Drag

Tab)



Definingthe Slip Velocity

If you are solving for slip velocities during the mixture calculation, and you want to

modify the slip velocity definition, click Interaction... to open the Phase Interaction panel

(Figure 23.11.4), and then click the Slip tab.

Figure 23.11.4: The Phase Interaction Panel for the Mixture Model (Slip Tab)

Under Slip Velocity, you can specify the slip velocity function for each secondary phase

with respect to the primary phase by choosing the appropriate item in the adjacentdrop-down list.

• Select maninnen-et-al (the default) to use the algebraic slip method of Manninen

et al. [229], described in Section 23.4.5: Relative (Slip) Velocity and the Drift

Velocity.

• Select none if the secondary phase has the same velocity as the primary phase

(i.e., no slip velocity).

• Select user-defined to use a user-defined function for the slip velocity.

See theseparate UDF Manual for details.



23.11.2 IncludingCavitationEffects

For mixture model calculations, it is possible to include the effects of cavitation, using

FLUENT’scavitation model described in Section 23.7.4: Cavitation Models. To enable the

cavitation model, turn on the Cavitation option in the Mass tab of the Phase Interaction

panel.

Figure 23.11.5: The Phase Interaction Panel for Mass Transfer with Cavita-

tion Enabled

When you are using FLUENT’scavitation model, you will specify three parameters to be

used in the calculation of mass transfer due to cavitation. Under Cavitation Parameters

in the Phase Interaction panel, set the Vaporization Pressure (psat in Equation 23.7-15),

the Surface Tension Coefficient , and the Non-CondensableGas Mass Fraction . The default

value of psat is 2540 Pa, the vaporization pressure for water at am bient temperature.

Note that psat and the surface tension are properties of the liquid, depending mainly on

temperature. Non-Condensable Gas Mass Fraction is the mass fraction of dissolved gases,

which depends on the purity of the liquid.



When multiple species are included in one or more secondary phases, or the heat transfer

due to phase change needs to be taken into account, the mass transfer mechanism must

be defined before turning on the Cavitation option. This is defined in the same way as

described Section 23.9.7: Including Mass Transfer Effects. It may be noted, however,

that for cavitation problems, at least two mass transfer mechanisms are defined:

• mass transfer from liquid to vapor.

• mass transfer from vapor to liquid.

In the Phase Interaction panel (Figure 23.11.6), you should choose the constant-rate op-tion in the Mechanism drop-down list and leave the value as 0 for both mass transfer mechanisms before turning on the Cavitation option.

Figure 23.11.6: The Phase Interaction Panel for Mass Transfer with Cavita-

tion Disabled



23.11.3 ModelingCompressibleFlows

If you are using the mixture model for a compressible flow, note the following:

• Only one of the phases can be defined as a compressible ideal gas (i.e., you

can select the ideal gas law for the density of only one phase’s material). There is



intak e fan) the specified value for temperature at that boundary will be used

as total temperature for the compressible phase, and as static temperature for

the other phases (which are incompressible).




previous to 6.1, you will need to redefine the conditions at the mass flowinlets. See Section 23.9.8: Defining Multiphase Boundary Conditions for

more information on defining conditions for a mass flow inlet in mixture

multiphase calculations.




23.12 SettingUp the Eu lerianModel

23.12 SettingUp the EulerianModel

23.12.1 Additional Guidelinesfor EulerianMultiphaseSimulations

Once you have determined that the Eulerian multiphase model is appropriate for your

problem (as described in Section 23.2: Choosing a General Multiphase Model), you should

consider the computational effort required to solve your multiphase problem. The re-quired computational effort depends strongly on the number of transport equations be-

ing solved and the degree of coupling. For the Eulerian multiphase model, which has a

large number of highly coupled trans port equations, computational expense will be high.

Before setting up your problem, try to reduce the problem stateme nt to the simplest

form possible.

Instead of trying to solve your multiphase flow in all of its complexity on your first

solution attempt, you can start with simple approximations and work your way up to

the final form of the problem definition. Some suggestions for simplifying a multiphase

flow problem are listed below:

• Use a hexahedral or quadrilateral mesh (instead of a tetrahedral or

triangular mesh).

• Reduce the number of phases.

You may find that even a very simple approximation will provide you with useful infor-

mation about your problem.

See Section 23.14.4: Eulerian Model for more solution strategies for Eulerian multiphase

calculations.

23.12.2 Definingthe Phases for the EulerianModel


phases and their interaction for an Eulerian multiphase calculation are provided below.


The procedure for defining the primary phase in an Eulerian multiphase calculation is

the same as for a VOF calculation. See Section 23.10.3: Defining the Primary Phase for

details.




Defininga NongranularSecondaryPhase

To define a nongranular (i.e., liquid or vapor) secondary phase in an Eulerian multiphase




Figure 23.12.1: The Secondary Phase Panel for a Nongranular Phase






tion 23.10.3: Defining the Primary Phase ).6. In the Secondary Phase panel, specify the Diameter of the bubbles or droplets of

this phase. You can specify a constant value, or use a user-defined function. See

the separate UDF Manual for details about user-defined functions.





Defininga GranularSecondaryPhase

To define a granular (i.e., particulate) secondary phase in an Eulerian multiphase calcu-

lation, perform the following steps:



Figure 23.12.2: The Secondary Phase Panel for a Granular Phase


4. Specify which material the phase contains by choosing the appropriate material inthe Phase Material drop-down list.






tion 23.10.3: Defining the Primary Phase). For a granular phase (which must be

placed in the fluid materials category, as mentioned in Section 23.9: Steps for Us-

ing a Multiphase Model), you need to specify only the density; you can ignore the

values for the other properties, since they will not be used.

i Note that all properties for granular flows can utilize user-defined functions

(UDFs).

See the separate UDF Manual for details about user-defined functions.

6. Turn on the Granular option.

7. (optional) Turn on the Packed Bed option if you want to freeze the velocity field

for the granular phase. Note that when you select the packed bed option for a

phase, you should also use the fixed velocity option with a value of zero for all

velocity components for all interior cell zones for that phase.

8. Specify the Granular Temperature Model. Choose either the default Phase Prop-

erty option or the Partial Differential Equation option. See Section 23.5.8: Granular

Temperature for details.

9. In the Secondary Phase panel, specify the following properties of the particles of

this phase:

Diameter specifies the diameter of the particles. You can select constant in the

drop-down list and specify a constant value, or select user-defined to use a user-

defined function. See the separate UDF Manual for details about user-defined

functions.

Granular Viscosity specifies the kinetic part of the granular viscosity of the par-

ticles (µs,kin in Equation 23.5-62). You can select constant (the default) in the

drop-down list and specify a constant value, select syamlal-obrien to compute

the value using Equation 23.5-64, select gidaspow to compute the value using

Equation 23.5-65, or select user-defined to use a user-defined function.

Granular Bulk Viscosity specifies the solids bulk viscosity (λ q in Equation 23.5-6).

You can select constant (the default) in the drop-down list and specify a con-

stant value, select lun-et-al to compute the value using Equation 23.5-66, or

select user-defined to use a user-defined function.

Frictional Viscosity specifies a shear viscosity based on the viscous-plastic flow

(µs,fr in Equation 23.5-62). By default, the frictional viscosity is neglected,

as indicated by the default selection of none in the drop-down list. If you

want to include the frictional viscosity, you can select constant and specify a

constant value, select schaeffer to compute the value using Equation 23.5-67,

select johnson-et-al to compute the value using Equation 23.5-72, or select

user-defined to use a user-defined function.




Angle of Internal Friction specifies a constant value for the angle φ used in Scha-

effer’s expression for frictional viscosity (Equation 23.5-67). This parameter

is relevant only if you have selected schaeffer or user-defined for the

Frictional Viscosity.

Frictional Pressure specifies the pressure gradient term, ∇Pf rictio n, in the granular-

phase momentum equation. Choose none to exclude frictional pressure fromyour calculation, johnson-et-al to apply Equation 23.5-72, syamlal-obrien to ap-

ply Equation 23.5-30, based-ktgf, where the frictional pressure is defined by the

kinetic theory [79]. The solids pressure tends to a large value near the packing

limit, depending on the model selected for the radial distribution function.

You must hook a user-defined function when selecting the user-defined option.

See the separate UDF manual for information on hooking a UDF.

Frictional Modulus is defined as

G =∂Pf r iction

∂αf r iction

(23.12-1)

with G ≥ 0, which is the derived option. You can also specify a user-

defined

function for the frictional modulus.

Friction Packing Limit specifies the maximum volume fraction for the granular

phase (αs,max ). For monodispersed spheres, the packing limit is about 0.63,

which is the default value in FLUENT. In polydispersed cases, however, smaller

spheres can fill the small gaps between larger spheres, so you may need toincrease the maximum packing limit.

Granular Conductivity specifies the solids granular conductivity (k Θsin

Equation 23.5-75). You can select syamlal-obrien to compute the value usingEquation 23.5-76, select gidaspow to compute the value using Equation 23.5-77,

or select user-defined to use a user-defined function. Note, however, that

FLU- ENT currently uses an algebraic relation for the granular temperature.

This has been obtained by neglecting convection and diffusion in the

transport equation, Equation 23.5-75 [363].

Granular Temperature specifies temperature for the solids phase and is propor-

tional to the kinetic energy of the random motion of the particles. Choose

either the algebraic, the constant, or user-defined option.

Solids Pressure specifies the pressure gradient term, ∇ ps , in the granular-phase

momentum equation. Choose either the lun-et-al, the syamlal-obrien, the ma-ahmadi, none, or a user-defined option.

Radial Distribution specifies a correction factor that modifies the probability of

collisions between grains when the solid granular phase becomes dense. Choose

either the lun-et-al, the syamlal-obrien, the ma-ahmadi, the arastoopour, or a

user-defined option.



Elasticity Modulus is defined as

with G≥

0.

G =∂Ps

∂αs

(23.12-2)

Packing Limit specifies the maximum volume fraction for the granular phase(αs,max ). For monodispersed spheres, the packing limit is about 0.63, which is

the default value in FLUENT. In polydispersed cases, however, smaller spheres

can fill the small gaps between larger spheres, so you may need to increase themaximum packing limit.


Definingthe InteractionBetw eenPhases

For both granular and nongranular flows, you will need to specify the drag function to be used in the calculation of the momentum exchange coefficients. For granular

flows, you will also need to specify the restitution coefficient(s) for particle collisions. It

is also possible to include an optional lift force and/or virtual mass force (described

below) for both granular and nongranular flows.

To specify these parameters, click Interaction... to open the Phase Interaction panel (Fig-

ure 23.12.3).

Figure 23.12.3: The Phase Interaction Panel for the Eulerian Model



Specifyingthe DragFunction

FLUENT allows you to specify a drag function for each pair of phases. Perform the

following steps:

1. Click the Drag tab to display the Drag Function inputs.

2. For each pair of phases, select the appropriate drag function from the corresponding

drop-down list.

• Select schiller-naumann to use the fluid-fluid drag function described by

Equa- tion 23.5-18. The Schiller and Naumann model is the default method,

and it is acceptable for general use in all fluid-fluid multiphase calculations.

• Select morsi-alexander to use the fluid-fluid drag function described by

Equa- tion 23.5-22. The Morsi and Alexander model is the most complete,

adjusting the function definition frequently over a large range of Reynolds

numbers, but calculations with this model may be less stable than with the

other models.

• Select symmetric to use the fluid-fluid drag function described by

Equation 23.5-27. The symmetric model is recommended for flows in which

the secondary (dispersed) phase in one region of the domain becomes the

primary (continuous) phase in another. For example, if air is injected into the

bottom of a container filled halfway with water, the air is the dispersed phase

in the bottom half of the container; in the top half of the container, the air is

the continuous phase.

• Select wen-yu to use the fluid-solid drag function described by Equation 23.5-

39.

The Wen and Yu model is applicable for dilute phase flows, in which thetotal secondary phase volume fraction is significantly lower than that of the

primary phase.

• Select gidaspow to use the fluid-solid drag function described by

Equation 23.5-41. The Gidaspow model is recommended for dense fluidized

beds.

• Select syamlal-obrien to use the fluid-solid drag function described by

Equa- tion 23.5-31. The Syamlal-O’Brien model is recommended for use in

conjunction with the Syamlal-O’Brien model for granular viscosity.

• Select syamlal-obrien-symmetric to use the solid-solid drag function

described by Equation 23.5-43. The symmetric Syamlal-O’Brien model isappropriate for a pair of solid phases.

• Select constant to specify a constant value for the drag function, and

then specify the value in the text field.



• Select user-defined to use a user-defined function for the drag function (see

the separate UDF Manual for details).

• If you want to temporarily ignore the interaction between two phases,

select

none.

Specifyingthe RestitutionCoefficients(GranularFlow Only)

For granular flows, you need to specify the coefficients of restitution for collisions

between particles (els in Equation 23.5-43 and ess in Equation 23.5-44). In addition to

specifying the restitution coefficient for collisions between each pair of granular phases,

you will also specify the restitution coefficient for collisions between particles of the

same phase.

Perform the following steps:

1. Click the Collisionstab to display the Restitution Coefficient inputs.

2. For each pair of phases, specify a constant restitution coefficient. All restitution

coefficients are equal to 0.9 by default.

Includingthe LiftForce

For both granular and nongranular flows, it is possible to include the effect of lift forces

(F~lift in Equation 23.5-8) on the secondary phase particles, droplets, or bubbles.

These lift forces act on a particle, droplet, or bubble mainly due to velocity gradients

in the primary-phase flow field. In most cases, the lift force is insignificant compared

to the drag force, so there is no reason to include it. If the lift force is significant (e.g.,

if the phases separate quickly), you may want to include this effect.

i Note that the lift force will be more significant for larger particles, but the

FLUENTmodel assumes that the particle diameter is much smaller than the

interparticle spacing. Thus, the inclusion of lift forces is not appropriate

for closely packed particles or for very small particles.



To include the effect of lift forces, perform the following steps:

1. Click the Lift tab to display the Lift Coefficient inputs.

2. For each pair of phases, select the appropriate specification method from the cor-

responding drop-down list. Note that, since the lift forces for a particle, droplet, or bubble are due mainly to velocity gradients in the primary-phase flow field, you will

not specify lift coefficients for pairs consisting of two secondary phases; lift coeffi-

cients are specified only for pairs consisting of a secondary phase and the primary

phase.

• Select none (the default) to ignore the effect of lift forces.

• Select constant to specify a constant lift coefficient, and then specify the

value in the text field.

• Select user-defined to use a user-defined function for the lift coefficient (see

the separate UDF Manual for details).

Includingthe VirtualMassForce

For both granular and nongranular flows, it is possible to include the “virtual mass force”

(F~vm in Equation 23.5-9) that is present when a secondary phase accelerates relative to

the primary phase. The virtual mass effect is significant when the secondary phase

density is much smaller than the primary phase density (e.g., for a transie nt bubble

column).

To include the effect of the virtual mass force, turn on the Virtual Mass option in the

Phase Interaction panel. The virtual mass effect will be included for all secondary phases;

it is not possible to enable it just for a particular phase.



23.12.3 ModelingTurbulence

If you are using the Eulerian model to solve a turbule nt flow, you will need to choose

one of turbulence models described in Section 23.5.10: Turbulence Models in the Viscous

Model panel (Figure 23.12.4).

Figure 23.12.4: The Viscous Model Panel for an Eulerian Multiphase Calcu-

lation



The procedure is as follows:

1. Select either k-epsilon or Reynolds Stress under Model.

2. Select the desired k-epsilon Model or RSM Multiphase Model and any other related

parameters, as described for single-phase calculations in Section 12.12: Steps inUsing a Turbulence Model.

3. Under k-epsilon Multiphase Model or RSM Multiphase Model, indicate the desired

multiphase turbulence model (see Section 23.5.10: Turbulence Models for details

about each):

• Select Mixture to use the mixture turbulence model. This is the default model.

• Select Dispersed to use the dispersed turbulence model. This model is

applicable when there is clearly one primary continuous phase and the rest

are dispersed dilute secondary phases.

• Select Per Phase to use a k- turbulence model for each phase. This model

is appropriate when the turbulence transfer among the phases plays a

dominant role.

IncludingSource Terms

By default, the interphase momentum, k, and sources are not included in the calcula-

tion. If you want to include any of these source terms, you can enable them using the

multiphase-options command in the define/models/viscous/m ultiphase-turbulence/text menu. Note that the inclusion of these terms can slow down convergence noticeably.

If you are looking for additional accuracy, you may want to compute a solution firstwithout these sources, and then continue the calculation with these terms included. In

most cases these terms can be neglected.

Customizingthe k - M ultiphaseTurbulentViscosity

If you are using the k- multiphase turbulence model, a user-defined function can be

used to customize the turbule nt viscosity for each phase. This option will enable you to

modify µt in the k- model. For more information, see the separate UDF Manual.

In the Viscous Model panel, under User-Defined Functions, select the appropriate user-

defined function in the Turbulent Viscosity drop-down list.



23.12.4 IncludingHeatTransferEffects

To define heat transfer in a multiphase Eulerian simulation, you will need to visit the

Phase Interaction panel, after you have turned on the energy equation in the Energy panel.


1. Click the Interaction... button to open the Phase Interaction panel (e.g., Fig-

ure 23.12.5).

Figure 23.12.5: The Phase Interaction Panel for Heat Transfer

2. Click on the Heat tab in the Phase Interaction panel.

3. Select the desired correlation for the Heat Transfer Coefficient. Note the following

regarding the available choices:

gunn is frequently used for Eulerian multiphase simulations involving a granular

phase.

ranz-marshall is frequently used for Eulerian multiphase simulations not involv-ing a granular phase.

none allows you to ignore the effects of heat transfer between the two phases

user-defined allows you to implement a correlation reflecting a model of your

choice, through a user-defined function.



4. Set the appropriate thermal boundary conditions. You will specify the thermal

boundary conditions for each individual phase on most boundaries, and for the

mixture on some boundaries. See Chapter 7: Boundary Conditions for more in-

formation on boundary conditions, and Section 23.9.8: Eulerian Model for more

information on specifying boundary conditions for a Eulerian multiphase calcula-

tion.

See Section 23.5.9: Description of Heat Transfer for more information on heat transfer in

the framework of a Eulerian multiphase simulation.

23.12.5 Modeling Compressible

Flows

You can model compressible multiphase flows, and can use it in conjunction with the

energy multiphase equations and available multiphase turbulence models. When using

the Eulerian multiphase model for a compressible flow, note the following:

• While you can specify both compressible gas phases and compressible liquid

phases, you can only define one of the phases as a compressible ideal gas (i.e., you

can select the ideal-gas for the density in the Materials panel of only one phase’s

material). There is no limitation on using compressible liquids using user-defined

functions.

• You can define only one compressible fluid phase.




intak e fan), FLUENT will use the specified value for temperature at that

boundary as total temperature for the compressible phase, and as static

temperature for the other phases (which are incompressible).


previous to 6.1, you will need to redefine the conditions at the mass flow

inlets. See Section 23.9.8: Defining Multiphase Boundary Conditions for

more information on defining conditions for a mass flow inlet in Eulerian

multiphase calculations.




23.13 SettingUp the Wet Steam Model

Once you have enabled either of the density-based solvers in FLUENT,you can activate the

wet steam model (see Section 23.6: Wet Steam Model Theory) by opening the Multiphase

panel and selecting the Wet Steam option.

Define −→ Models−→Multiphase...

Figure 23.13.1: The Multiphase Model Panel with the Wet Steam Model Ac-

tivated

This section includes information about using your own property functions and data with

the wet steam model. Solution settings and strategies for the wet steam model can be

found in Section 23.14.5: Wet Steam Model. Postprocessing variables are described in

Section 23.15.1: Model-Specific Variables.

23.13.1 Using User-DefinedTherm odynamicWet Steam Properties

FLUENT allows you to use your own property functions and data with the wet steam

model. This is achieved with user-defined wet steam property functions (UDWSPF).

These user-defined functions are written in the C programming language and there is a

certain programming format that must be used so that you can build a successful library

that can be loaded into the FLUENT code.



23.13SettingUp the Wet SteamModel

The following is the procedure for using the user-defined wet steam property functions

(UDWSPF):

1. Define the wet steam equation of state and all related thermodynamic and transport

property equations.

2. Create a C source code file that conforms to the format defined in this section.

3. Start FLUENT and set up your case file in the usual way.

4. Turn on the wet steam model.

5. Compile your UDWSPF C functions and build a shared library file using the text

user interface.

define −→ models −→ multiphase −→

wet-steam −→compile-user-defined-we tsteam-functions

6. Load your newly created UDWSPF library using the text user interface.define −→ models −→ multiphase −→

wet-steam −→load-unload-user-defined-wetsteam-library

7. Run your calculation.

i Note that the UDWSPF can only be used when the wet steam modelis activated. Therefore, the UDWSPF are available for use with thedensity- based solver only.

23.13.2 Writingthe User-DefinedWet Steam Property Functions(UDWSPF)

Creating a UDWSPF C function library is reasonably straightforward:

• The code must contain the udf.h file inclusion directive at the beginning of the source code. This allows the definitions for DEFINE macros and other FLUENTfunctions to be accessible during the compilation process.

• The code must include at least one of the UDF’s DEFINE functions

(i.e. DEFINE ON DEMAND) to be able to use the compiled UDFs utili ty.

• Any values that are passed to the solver by the UDWSPF or returned by the

solver to the UDWSPF are assumed to be in SI units.

• You must use the principle set of user-defined wet steam property functions

in your UDWSPF library, as described in the list that follows. These functions

are the mechanism by which your thermodynamic property data istransferred to

the FLUENT solver.




The following lists the user-defined wet steam property function names and arguments,

as well as a short description of their functions. Function inputs from the FLUENT solver

consist of one or more of the following variables: T = temperature (K ), P =

pressure (P a), and ρ = vapor-phase density (k g/m3).

• void wetst init(Domain *domain)

This will be called when you load the UDWSPF. You use it to initialize wet steam

model constants or your own model constants. It returns nothing.

• real wetst satP(real T)

This is the saturated pressure function, which takes on temperature in K and

returns saturation pressure in Pa.

• real wetst satT(real P, real T)

This is the saturated temperature function, which takes on pressure in Pa and a

starting guess temperature in K and returns saturation temperature in K.

• real wetst eosP(real rho, real T)

This is the equation of state, which takes on vapor density in kg/m 3 and Temper-

ature in K and returns pressure in Pa.

• real wetst eosRHO(real P, real T)

This is the equation of state, which takes on pressure in Pa and temperature in K

and returns vapor density in kg/m 3.

• real wetst cpv(real T, real rho)

This is the vapor specific heat at constant pressure, which takes on temperature

in K and vapor density in kg/m

3

and returns specific heat at constant pressure inJ/kg/K.

• real wetst cvv(real T, real rho)

This is the vapor specific heat at constant volume, which takes on temperature

in K and vapor density in kg/m 3 and returns specific heat at constant volume in

J/kg/K.

• real wetst hv(real T,real rho)

This is the vapor specific enthalpy, which takes on temperature in K and vapor

density in kg/m 3 and returns specific enthalpy in J/Kg.

• real wetst sv(real T, real rho)This is the vapor specific entropy, which takes on temperature in K and vapor

density in kg/m 3 and returns specific entropy in J/Kg/K.




• real wetst muv(real T, real rho)

This is the vapor dynamic viscosity, which takes on temperature in K and

vapor density in kg/m 3 and returns viscosity in kg/m/s.

• real wetst ktv(real T, real rho)

This is the vapor thermal conductivi ty, which takes on temperature in K and

vapor density in kg/m 3 and returns thermal conductivity in W/m/K.

• real wetst rhol(real T)

This is the saturated liquid density, which takes on temperature in K and returns

liquid density in kg/m 3.

• real wetst cpl(real T)

This is the saturated liquid specific heat at constant pressure, which takes on tem-

perature in K and returns liquid specific heat in J/kg/K.

• real wetst mul(real T)

This is the liquid dynamic viscosity, which takes on Temperature in K and returnsdynamic viscosity in kg/m/s.

• real wetst ktl(real T)

This is the liquid thermal conductivi ty, which takes on temperature in K and returns

thermal conductivity in W/m/K.

• real wetst surft(real T)

This is the liquid surface tension, which takes on Temperature in K and returns

surface tension N/m.




At the end of the code you must define a structure of type WS Functions whose

mem-

bers are pointers to the principle functions listed previously. The structure is of type

WS Functions and its name is WetSteamFunctionLis t.

UDF_EXPORT WS_Functions WetSteamFunctionList =

{wetst_init, /*initialization function*/wetst_satP, /*Saturation pressure*/wetst_satT, /*Saturation temperature*/wetst_eosP, /*equation of state*/wetst_eosRHO, /*equation of state*/wetst_hv, /*vapor enthalpy*/wetst_sv, /*vapor entropy*/wetst_cpv, /*vapor isobaric specific heat*/wetst_cvv, /*vapor isochoric specific heat*/

wetst_muv, /*vapor dynamic viscosity*/wetst_ktv, /*vapor thermal conductivity*/wetst_rhol, /*sat. liquid density*/wetst_cpl, /*sat. liquid specific heat*/wetst_mul, /*sat. liquid viscosity*/wetst_ktl, /*sat. liquid thermal conductivity*/wetst_surft /*liquid surface tension*/

};

23.13.3 CompilingYour UDW SPF and Building a Shared Library

File

This section presents the steps you will need to follow to compile your UDWSPF C code

and build a shared library file. This process requires the use of a C compiler. Most UNIX

operating systems provide a C compiler as a standard feature. If you are using a PC,

you will need to ensure that a C ++ compiler is installed before you can proceed (e.g.,

Microsoft Visual C ++, v6.0 or higher).

i To use the UDWSPF you will need to first build the UDWSPFlibrary by compiling your UDWSPF C code and then loading the libraryinto the

FLUENT code.

The UDWSPF shared library is built in the same way that the FLUENT executable itself is built. Internally, a script called Makefile is used to invoke the system C compiler to build an object code library that contains the native machine language translation of

your higher-level C source code. This shared library is then loaded into FLUENT (either

at runtime or automatically when a case file is read) by a process called dynamic loading.

The object libraries are specific to the computer architecture being used, as well as to the

particular version of the FLUENT executable being run. The libraries must, therefore,




be rebuilt any time FLUENT is upgraded, when the computer’s operating system level

changes, or when the job is run on a different type of computer.

The general procedure for compiling UDWSPF C code is as follows:

• Place the UDWSPF C code in your working directory (i.e., where your case

file resides).

• Launch FLUENT.

• Read your case file into FLUENT.

• You can now compile your UDWSPF C code and build a shared library file

using the commands provided in the text command interface (TUI):

– Select the define/models/multiphase/wet-steam menu item

define −→ models −→ multiphase −→wet-steam

– Select the compile-user-defined-wetsteam-functions option.

– Enter the compiled UDWSPF library name.

The name given here is the name of the directory where the shared library(e.g., libudf ) will reside. For example, if you hit <Enter> then adirectory should exist with the name libudf , and this directory will containlibrary file called libudf . If, however, you type a new library name such asmywetsteam, then a directory called mywetsteam will be created andit will contain the library libudf .

– Continue on with the procedure when prompted.

– Enter the C source filenames.

i Ideally you should place all of your functions into a single file.

However, you can split them into separate files if desired.

– Enter the header file names, if applicable. If you do not have an extra header file, then press <Enter> when prompted.

FLUENT will then start compiling the UDWSPF C code and put it in the

appropriate architecture director y.




23.13.4 Loadingthe UDW SPFSha red LibraryFile

To load the UDWSPF library, perform the following steps:

• Go to the define/models/multiphase/wet-steam menu item in the text

user interface.

define −→ models −→ multiphase −→wet-steam

• Select the load-unload-user-defined-wetsteam-library option and

follow the procedure when prompted.

If the loading of the UDWSPF library is successful, you will see a message similar

to the following:

Opening user-defined wet steam library "libudf"...Library "libudf/lnx86/2d/libudf.so" opened

Setting material properties to Wet-Steam...

Initializing user defined material properties...




23.13.5 UDWSPFExample

This section describe a simple UDWSPF. You can use this example as a the basis for your

own UDWSPF code. For approximate calculations at low pressure, the simple ideal-gas

equation of state and constant isobaric specific heat is assumed and used. The properties

at the saturated liquid line and the saturated vapor line used in this example are similar

to the one used by FLUENT.

/**********************************************************************/

/* User Defined Wet SteamProperties: EOS : Ideal Gas Eq.Vapor Sat. Line : W.C.Reynolds tables (1979)Liquid Sat. Line: E. Eckert & R. Drake book (1972)

Use ideal-gas EOS with Steam properties

to model wet steam condensation in low pressure nozzle

Author: L. ZoriDate : Jan. 29 2004

*//**********************************************************************/#include "udf.h"#include "stdio.h"#include "ctype.h"#include "stdarg.h"

/*Global Constants for this model*/

real ws_TPP = 338.150 ;real ws_aaa = 0.01 ;real cpg = 1882.0 ;/* Cp-vapor at low-pressure region*/

DEFINE_ON_DEMAND(I_do_n othing)

{/* This is a dummy function to allow us to use *//* the Compiled UDFs utility */

}




voidwetst_init(Domain *domain){

/* You must initialize these material property constants..they will be used in the wet steam model in fluent

*/ws_Tc = 647.286 ;/*Critical Temp. */ws_Pc = 22089000.00 ;/*Critical Pressure */mw_f = 18.016 ;/*fluid droplet molecular weight (water) */Rgas_v = 461.50 ;/*vapor Gas Const*/

}

realwetst_satP(real T){

real psat ;

realSUM=0.0;real pratio;real F ;real a1 = -7.41924200 ;real a2 = 2.97210000E- ;real a3 = -1.15528600E- ;real a4 = 8.68563500E- ;real a5 = 1.09409899E- ;

real a6 = -4.39993000E- ;real a7 = 2.52065800E- ;real a8 = -5.21868400E- ;

if (T > ws_Tc) T = ws_Tc ;F = ws_aaa*(T - ws_TPP) ;SUM = a1 + F*(a2+ F*(a3+ F*(a4+ F*(a5+ F*(a6+ F*(a7+ F*a8)))))) ;

pratio = (ws_Tc/T - 1.0)*SUM;

psat = ws_Pc *exp(pratio) ;

return psat; /*Pa */}




realwetst_satT(real P, real T){

real tsat ;

real dT, dTA,dTM,dP,p1,p2,dPdT;

int i ;for (i=0; i<25; ++i)

{if (T > ws_Tc) T = ws_Tc-0.5;

p1= wetst_satP(T) ;p2= wetst_satP(T+0.1) ;dPdT = (p2-p1)/0.1 ;

dP = P - p1 ;

dT = dP/dPdT ;

dTA =fabs(dT);

dTM = 0.1*T;

if (dTA > dTM)

dT=dT*dTM/dTA ; T = T + dT;if (fabs(dT) < TEMP_eps*T)break;

}tsat = T;

return tsat; /*K */}

realwetst_eosP(real rho, real T){

real P ;

P = rho* Rgas_v * T ;

return P; /*Pa */}




realwetst_eosRHO(real P, real T){

real rho ;

rho = P/(Rgas_v * T) ;

return rho; /*kg/m3 */}

realwetst_cpv(real T, real rho){

real cp;

cp = cpg ;

return cp; /* (J/Kg/K) */}

realwetst_cvv(real T, real rho){

real cv;

cv = wetst_cpv(T,rho) - Rgas_v ;

return cv; /* (J/Kg/K) */}

realwetst_hv(real T,real rho){

real h;

h = T* wetst_cpv(T,rho) ;

return h; /* (J/Kg) */}




realwetst_sv(real T, real rho){

real s ;

real TDatum=288.15;

real PDatum=1.01325e5;

s=wetst_cpv(T,rho)*log(T/TDatum)+Rgas_v*log(PDatum/(Rgas_v*T*rho));

return s; /* (J/Kg/K) */}

realwetst_muv(real T, real rho){

real muv ;

muv=1.7894e-05 ;

return muv; /* (Kg/m/s) */}

real

wetst_ktv(real T, real rho){real ktv ;

ktv=0.0242 ;

return ktv; /* W/m/K */}




realwetst_rhol(real T){

real rhol;

real SUM = 0.0 ;int ii ;int i ;real rhoc = 317.0 ;real D[8] ;

D[0] = 3.6711257 ;D[1] = -2.8512396E+01 ; D[2]=

2.2265240E+02

; D[3] = -8.8243852E+02 ; D[4]=

2.0002765E+03; D[5] = -2.6122557E+03 ; D[6]=

1.8297674E+03; D[7] = -5.3350520E+02 ;if (T > ws_Tc) T = ws_Tc ;

for(ii=0;ii < 8;++ii){

i = ii+1 ;SUM += D[ii] * pow((1.0 - T/ws_Tc), i/3.0) ;

}rhol = rhoc*(1.0+SUM);

return rhol; /* (Kg/m3) */}

realwetst_cpl(real T){

real cpl;

real a1= -36571.6 ;real a2= 555.217 ;real a3= -2.96724 ;real a4=




0.00778551; real a5=-1.00561e-05; reala6= 5.14336E-09;




if (T > ws_Tc) T = ws_Tc ;cpl = a1 + T*(a2+ T*(a3+ T*(a4+ T*(a5+ T*a6)))) ;

return cpl; /* (J/Kg/K) */}

realwetst_mul(real T){

real mul ;

real a1= 0.530784;real a2= -0.00729561;real a3= 4.16604E-05; real a4= -1.26258E-

07; real a5=2.13969E-10;

real a6= -1.92145E-13;real a7= 7.14092E-17;

if (T > ws_Tc) T = ws_Tc ;

mul = a1 + T*(a2+ T*(a3+ T*(a4+ T*(a5+ T*(a6+ T*a7))))) ;

return mul; /* (Kg/m/s) */}

realwetst_ktl(real T){

real ktl ;

real a1= -1.17633;real a2=

0.00791645; real a3=1.48603E-

05; real a4=

-1.31689E-07; reala5= 2.47590E-10; real a6=-1.55638E-13;

if (T > ws_Tc) T = ws_Tc ;

ktl = a1 + T*(a2+ T*(a3+ T*(a4+ T*(a5+ T*a6)))) ;

return ktl; /* W/m/K */}




realwetst_surft(real T){

real sigma ;

real Tr ;real a1= 82.27 ;real a2= 75.612 ;real a3= -256.889 ;real a4= 95.928 ;

if (T > ws_Tc) T = ws_Tc ; Tr = T/ws_Tc;sigma = 0.001*(a1 + Tr*(a2+ Tr*(a3+ Tr*a4))) ;

return sigma ;/* N/m */}

/* do not change the order of the function list */UDF_EXPORT WS_Functions WetSteamFunctionList={

wetst_init, /*initialization function*/wetst_satP, /*Saturation pressure*/wetst_satT, /*Saturation temperature*/

wetst_eosP, /*equation of state*/wetst_eosRHO, /*equation of state*/wetst_hv, /*vapor enthalpy*/wetst_sv, /*vapor entropy*/wetst_cpv, /*vapor isobaric specific heat*/wetst_cvv, /*vapor isochoric specific heat*/wetst_muv, /*vapor dynamic viscosity*/wetst_ktv, /*vapor thermal conductivity*/wetst_rhol, /*sat. liquid density*/wetst_cpl, /*sat. liquid specific heat*/wetst_mul, /*sat. liquid viscosity*/

wetst_ktl, /*sat. liquid thermal conductivity*/wetst_surft /*liquid surface tension*/

};/**********************************************************************/




23.14 SolutionStrategies for MultiphaseModeling

23.14.1 Setting Initia lVolum eFractions

Once you have initialized the flow (as described in Section 25.14: Initializing the Solution),

you can define the initial distribution of the phases. For a transie nt simulation, this dis-

tribution will serve as the initial condition at t = 0; for a steady-state simulation, settingan initial distribution can provide added stability in the early stages of the calculation.

You can patch an initial volume fraction for each secondary phase using the Patch panel.

Solve −→ Initialize

−→Patch...

If the region in which you want to patch the volume fraction is defined as a separate

cell zone, you can simply patch the value there. Otherwise, you can create a cell “reg-

ister” that contains the appropriate cells and patch the value in the register. See Sec-

tion 25.14.2: Patching Values in Selected Cells for details.

Solution strategies for the VOF, mixture, and Eulerian models are provided in Sec-tions 23.14.2, 23.14.3, and 23.14.4, respectively.

23.14.2 VOFModel

Several recommendations for improving the accuracy and convergence of the VOF solu-

tion are presented here.

Settingthe Reference Pressure Location

The site of the reference pressure can be moved to a location that will result in less round-

off in the pressure calculation. By default, the reference pressure location is the center

of the cell at or closest to the point (0,0,0). You can move this location by specifying a

new Reference Pressure Location in the Operating Conditions panel.


The position that you choose should be in a region that will always contain the least

dense of the fluids (e.g., the gas phase, if you have a gas phase and one or more liquid

phases). This is because variations in the static pressure are larger in a more dense

fluid than in a less dense fluid, given the same velocity distribution. If the zero of the

relative pressure field is in a region where the pressure variations are small, less round-off

will occur than if the variations occur in a field of large nonzero values. Thus in systemscontaining air and water, for example, it is important that the reference pressure location

be in the portion of the domain filled with air rather than that filled with water.



23.14SolutionStrateg iesfor MultiphaseModeling

Pressure InterpolationScheme

For all VOF calculations, you should use the body-force-weighted pressure interpolation

scheme or the PRESTO! scheme.


−→Solution...

DiscretizationScheme Selectionfor the Implicitand ExplicitFormulations

When the implicit scheme is used, the available options for Volume Fraction Discretization

are

• First Order Upwind

• Second Order upwind

• Modified HRIC

• QUICK

When the explicit scheme is used, the available options for Volume Fraction Discretization

are

• Geo-Reconstruct

• CICSAM

• Modified HRIC

•

QUICK

When using the explicit scheme, First Order Upwind, Second Order upwind, and Donor-

Acceptor can be made available under Volume Fraction Discretization by using the

following text command:

solve −→ set−→expert

You will be asked a series of questions, one of which is

Allow selection of all applicable discretization schemes? [no]

to which you will respond yes.

i You are encouraged to use the CICSAM scheme, as it gives asharper interface than the modified HRIC scheme.




i In VOF modeling, using a high-order discretization scheme for themomen- tum transport equations may reduce the stability of the solutioncomparedto cases using first-order discretization. In such situations, it is recom-

mended to use a low-order variant of Rhie-Chow face flux interpolation,

which can be turned on using the text command:solve −→ set−→numerics

When asked to disable high order Rhie-Chow flux?[no], enter

yes.

Pressure-Velocity Couplingand Under-Relaxationfor the Time-dependent

Formulations

Another change that you should make to the solver settings is in the pressure-velocity

coupling scheme and under-relaxation factors that you use. The PISO scheme is recom-

mended for transie nt calculations in general. Using PISO allows for increased values onall under-relaxation factors, without a loss of solution stability. You can generally in-

crease the under-relaxation factors for all variables to 1 and expect stability and a rapid

rate of convergence (in the form of few iterations required per time step). For

calculations on tetrahedral or triangular meshes, an under-relaxation factor of 0.7–0.8 for

pressure is recommended for improved stability with the PISO scheme.


−→Solution...

As with any FLUENT simulation, the under-relaxation factors will need to be decreased

if the solution exhibits unstable, divergent behavior with the under-relaxation factors set

to 1. Reducing the time step is another way to improve the stabili ty.

Under-Relaxationfor the Steady-State Formulation

If you are using the steady-state implicit VOF scheme, the under-relaxation factors for

all variables should be set to values between 0.2 and 0.5 for improved stabili ty.

23.14.3 MixtureModel

Settingthe Under-Relaxatio nFactorfor the Slip Velocity

You should begin the mixture calculation with a low under-relaxation factor for the slip

velocity. A value of 0.2 or less is recommended. If the solution shows good convergence

behavior, you can increase this value gradually.




Ca lcula tingan InitialSolution

For some cases (e.g., cyclone separation), you may be able to obtain a solution more

quickly if you compute an initial solution without solving the volume fraction and slip

velocity equations. Once you have set up the mixture model, you can temporarily disable

these equations and compute an initial solution.


−→Solution...

In the Solution Controls panel, deselect Volume Fraction and Slip Velocity in the Equations

list. You can then compute the initial flow field. Once a converged flow field is

obtained, turn the Volume Fraction and Slip Velocity equations back on again, and

compute the mixture solution.

23.14.4 EulerianModel

Ca lcula tingan InitialSolution

To improve convergence behavior, you may want to compute an initial solution before

solving the complete Eulerian multiphase model. There are three methods you can use

to obtain an initial solution for an Eulerian multiphase calculation:

• Set up and solve the problem using the mixture model (with slip velocities)

instead of the Eulerian model. You can then enable the Eulerian model,

complete the setup, and continue the calculation using the mixture-model solution

as a starting point.

• Set up the Eulerian multiphase calculation as usual, but compute the flow for

only the primary phase. To do this, deselect Volume Fraction in the Equations list

in the Solution Controls panel. Once you have obtained an initial solution for the

primary phase, turn the volume fraction equations back on and continue the

calculation for all phases.

• Use the mass flow inlet boundary condition to initialize the flow conditions. It

is recommended that you set the value of the volume fraction close to the value

of the volume fraction at the inlet.

i You should not try to use a single-phase solution obtained withoutthe mixture or Eulerian model as a starting point for an Eulerianmultiphasecalculation. Doing so will not improve convergence, and may make it even

more difficult for the flow to converge.




Temporarily Ign oringLiftan d VirtualMass Forces

If you are planning to include the effects of lift and/or virtual mass forces in a steady-state

Eulerian multiphase simulation, you can often reduce stability problems that sometimes

occur in the early stages of the calculation by temporarily ignoring the action of the lift

and the virtual mass forces. Once the solution without these forces starts to converge,

you can interrupt the calculation, define these forces appropriatel y, and continue thecalculation.

Using W-Cycle Multigrid

For problems involving a packed-bed granular phase with very small particle sizes (on

the order of 10 µm), convergence can be obtained by using the W-cycle multigrid for the

pressure. Under FixedCycle Parameters in the Multigrid Controls panel, you may need to

use higher values for Pre-Sweeps, Post-Sweeps, and Max Cycles. When you are choosing

the values for these parameters, you should also increase the Verbosity to 1 in order to

monitor the AMG performance; i.e., to make sure that the pressure equation is solved to

a desired level of convergence within the AMG solver during each global iteration. See

Section 23.12.2: Defining the Phases for the Eulerian Model for more information about

granular phases, and Sections 25.6.2 and 25.22.3 for details about multigrid cycles.

23.14.5 Wet Steam Model

BoundaryConditions,Initialization,and Patching

When you use the wet steam model (described in Section 23.6: Wet Steam Model Theory

and Section 23.13: Setting Up the Wet Steam Model), the following two field variables

will show up in the inflow, outflow boundary panels, and in the Solution Initialization andPatch panels.

• Liquid Mass Fraction (or the wetness factor)

In general, for dry steam entering flow boundaries the wetness factor is zero.

• Log10(Droplets Per Unit Volume)

In general this value is set to zero, indicating zero droplets entering the domain.




SolutionLimitsfor the Wet Steam Model

When you activate the wet steam model for the first time, a message is displayed indicat-

ing that the Minimum Static Temperature should be adjusted to 273 K since the accuracy

of the built-in steam data is not guaranteed below a value of 273 K. If you use your own

steam property functions, you can adjust this limit to whatever is permissible for your

data.

To adjust the temperature limits, go to the Solution Limits panel.


−→Limits...

The default maximum wetness factor or liquid mass fraction (β) is set to 0.1. In general,

during the convergence process, it is common that this limit will be reached, but even-

tually the wetness factor will drop below the value of 0.1. However, in cases where the

limit must be adjusted, you can do so using the text user interface.

define −→ models −→ multiphase −→ wet-steam −→

set −→max-liquid-mass-fraction

i Note that the maximum wetness factor should not be set beyond 0.2since the present model assumes a low wetness factor. When the wetnessfactor isgreater than 0.1, the solution tends to be less stable due to the large source

terms in the transport equations. Thus, the maximum wetness factor has

been set to a default value of 0.1, which corresponds to the fact that most

nozzle and turbine flows will have a wetness factor less that 0.1.

SolutionStrategiesfor the Wet Steam Model

If you face convergence difficulties while solving wet steam flow, try to initially lower

the CFL value and use first-order discretization schemes for the solution. If you are still

unable to obtain a converged solution, then try the following solver settings:

1. Lower the under-relaxation factor for the wet steam equation below the current set

value. The under-relaxation factor can be found in the Solution Controls panel.

Solve −→ Controls−→Solution...

2. Solve for an initial solution with no condensation. Once you have obtained a proper initial solution, turn on the condensation.

To turn condensation on or off, go to the Solution Controls panel.


−→Solution...

In the Solution Controls panel, deselect Wet Steam in the Equations list. When

doing so, you are preventing condensation from taking place while still computing

the flow based on steam properties. Once a converged flow field is obtained, turn

the Wet Steam equation back on again and compute the mixture solution.




23.15 Postprocessing for MultiphaseModeling

Each of the three general multiphase models provides a number of additional field func-

tions that you can plot or report. You can also report flow rates for individual phases

for all three models, and display velocity vectors for the individual phases in a mixture

or Eulerian calculation.

Information about these postprocessing topics is provided in the following subsections:

• Section 23.15.1: Model-Specific Variables

• Section 23.15.2: Displaying Velocity Vectors

• Section 23.15.3: Reporting Fluxes

• Section 23.15.4: Reporting Forces on Walls

• Section 23.15.5: Reporting Flow Rates

23.15.1 Model-SpecificVariables

When you use one of the general multiphase models, some additional field functions

will be available for postprocessing, as listed in this section. Most field functions that

are available in single phase calculations will be available for either the mixture or each

individual phase, as appropriate for the general multiphase model and specific options

that you are using. See Chapter 30: Field Function Definitions for a complete list of field

functions and their definitions. Chapters 28 and 29 explain how to generate graphics

displays and reports of data.

VOFModel

For VOF calculations you can generate graphical plots or alphanumeric reports of the

following additional item:

• Volume fraction (in the Phases...

category) This item is available for each

phase.

The variables that are not phase specific are available (e.g., variables in the Pressure... andVelocity... categories) represent mixture quantities. Thermal quantities will be

available only for calculations that include the energy equation.



23.15 Postprocessing for Multip haseModeling

MixtureModel

For calculations with the mixture model, you can generate graphical plots or alphanu-

meric reports of the following additional items:

• Diameter (in the Properties... category)

This item is available only for secondary phases.



phase.

The variables that are not phase specific are available (e.g., variables in the Pressure...

category) represent mixture quantities. Thermal quantities will be available only for

calculations that include the energy equation.

EulerianModel

For Eulerian multiphase calculations you can generate graphical plots or alphanumeric

reports of the following additional items:

• Diameter (in the Properties... category)

This item is available only for secondary phases.

• Granular Conductivity (in the Properties...

category) This item is available only for granular

phases.

• Granular Pressure (in the Granular Pressure...

category) This item is available only for granular

phases.

• Granular Temperature (in the Granular Temperature...

category) This item is available only for granular phases.


category) This item is available for each phase.

The availability of turbulence quantities will depend on which multiphase turbulence

model you used in the calculation. Thermal quantities will be available (on a per-phase

basis) only for calculations that include the energy equation.




MultiphaseSpecies Transport

For calculations using species transport with either of the multiphase models, you can

generate graphical plots or alphanumeric reports of the following additional items:

• Mass Fraction of species-n (in the Species...

category) This item is available for each species.

• Mole Fraction of species-n (in the Species...


• Molar Concentration of species-n (in the Species...


• Lam. Diff Coeff of species-n (in the Species...


• Eff. Diff. Coeff. of species-n (in the Species...category) This item is available for each species.

• Enthalpy of species-n (in the Species...


species.

• Relative Humidity (in the Species... category).

• Turbulent Rate of Reaction-n (in the Reactions...


• Rate of Reaction (in the Reactions... category).

• Mass Transfer Rate n (in the Phase Interaction... category)

This item is available for each mass transfer mechanism that you defined.

Thermal quantities will be available only for calculations that include the energy equation.




Wet Steam Model

FLUENT provides a wide range of postprocessing information related to the wet steam

model.

The wet steam related items can be found in Wet Steam.... category of the variable

selection drop-down list that appears in the postprocessing panels.

• Liquid Mass Fraction

• Liquid Mass Generation Rate

• Log10(Droplets Per Unit Volume)

• Log10(Droplets Nucleation Rate)

• Steam Density (Gas-Phase)

• Liquid Density (Liquid-Phase)

• Mixture Density

• Saturation Ratio

• Saturation Pressure

• Saturation Temperature

• Subcooled Vapor Temperature

• Droplet Surface Tension

• Droplet Critical Radius (microns)

• Droplet Average Radius (microns)

• Droplet Growth Rate (microns/s)




23.15.2 Displaying VelocityVectors

For mixture and Eulerian calculations, it is possible to display velocity vectors for the

individual phases using the Vectors panel.

Display

−→Vectors...

To display the velocity of a particular phase, select Velocity in the Vectors Of drop-

down list, and then select the desired phase in the Phase drop-down list. You can also

choose Relative Velocity to display the phase velocity relative to a moving reference frame.

To display the mixture velocity ~vm (relevant for mixture model calculations only),

select Velocity (or Relative Velocity for the mixture velocity relative to a moving reference

frame), and mixture as the Phase. Note that you can color vectors by values of any

available variable, for any phase you defined. To do so, make the appropriate

selections in the Color by and following Phase drop-down lists.

23.15.3 ReportingFluxesWhen you use the Flux Reports panel to compute fluxes through boundaries, you will be

able to specify whether the report is for the mixture or for an individual phase.

Report

−→Fluxes...

Select mixture in the Phase drop-down list at the bottom of the panel to report fluxes for

the mixture, or select the name of a phase to report fluxes just for that phase.

23.15.4 Reporting Forces on Walls

For Eulerian calculations, when you use the Force Reports panel to compute forces or

moments on wall boundaries, you will be able to specify the individual phase for which

you want to compute the forces.

Report−→Forces...

Select the name of the desired phase in the Phase drop-down list on the left side of the

panel.




23.15.5 Reporting Flow Rates

You can obtain a report of mass flow rate for each phase (and the mixture) through eachflow boundary using the report/mass-flow text command:report −→mass-flow

When you specify the phase of interest (the mixture or an individual phase), FLUENT

will list each zone, followed by the mass flow rate through that zone for the specified

phase. An example is shown below.

/report> mf (mixture water air)domain id/name [mixture] airzone 10 (spiral-press-outlet): -1.2330244zone 3 (pressure-outlet): -9.7560663zone 11 (spiral-vel-inlet): 0.6150589zone 8 (spiral-wall): 0zone 1 (walls): 0

Documents

Modelling Multi Phase Flow