chapter 20. general multiphase models - · pdf filechapter 20. general multiphase models ......

Chapter 20. General Multiphase

Models

This chapter discusses the general multiphase models that are availablein FLUENT. Chapter 18 provides a brief introduction to multiphase mod-eling, Chapter 19 discusses the Lagrangian dispersed phase model, andChapter 21 describes FLUENT’s model for solidification and melting.

• Section 20.1: Choosing a General Multiphase Model

• Section 20.2: Volume of Fluid (VOF) Model

• Section 20.3: Mixture Model

• Section 20.4: Eulerian Model

• Section 20.5: Cavitation Effects

• Section 20.6: Setting Up a General Multiphase Problem

• Section 20.7: Solution Strategies for General Multiphase Problems

• Section 20.8: Postprocessing for General Multiphase Problems

20.1 Choosing a General Multiphase Model

As discussed in Section 18.4, the VOF model is appropriate for stratifiedor free-surface flows, and the mixture and Eulerian models are appropri-ate for flows in which the phases mix or separate and/or dispersed-phasevolume fractions exceed 10%. (Flows in which the dispersed-phase vol-ume fractions are less than or equal to 10% can be modeled using thediscrete phase model described in Chapter 19.)

To choose between the mixture model and the Eulerian model, youshould consider the following, in addition to the detailed guidelines inSection 18.4:

c© Fluent Inc. November 28, 2001 20-1

General Multiphase Models

• If there is a wide distribution of the dispersed phases, the mixturemodel may be preferable. If the dispersed phases are concentratedjust in portions of the domain, you should use the Eulerian modelinstead.

• If interphase drag laws that are applicable to your system areavailable (either within FLUENT or through a user-defined func-tion), the Eulerian model can usually provide more accurate resultsthan the mixture model. If the interphase drag laws are unknownor their applicability to your system is questionable, the mixturemodel may be a better choice.

• If you want to solve a simpler problem, which requires less com-putational effort, the mixture model may be a better option, sinceit solves a smaller number of equations than the Eulerian model.If accuracy is more important than computational effort, the Eu-lerian model is a better choice. Keep in mind, however, that thecomplexity of the Eulerian model can make it less computationallystable than the mixture model.

Brief overviews of the three models, including their limitations, are pro-vided in Sections 20.1.1, 20.1.2, and 20.1.3. Detailed descriptions of themodels are provided in Sections 20.2, 20.3, and 20.4.

20.1.1 Overview and Limitations of the VOF Model

Overview

The VOF model can model two or more immiscible fluids by solving asingle set of momentum equations and tracking the volume fraction ofeach of the fluids throughout the domain. Typical applications includethe prediction of jet breakup, the motion of large bubbles in a liquid, themotion of liquid after a dam break, and the steady or transient trackingof any liquid-gas interface.

Limitations

The following restrictions apply to the VOF model in FLUENT:

20-2 c© Fluent Inc. November 28, 2001


• You must use the segregated solver. The VOF model is not avail-able with either of the coupled solvers.

• All control volumes must be filled with either a single fluid phaseor a combination of phases; the VOF model does not allow for voidregions where no fluid of any type is present.

• Only one of the phases can be compressible.

• Streamwise periodic flow (either specified mass flow rate or spec-ified pressure drop) cannot be modeled when the VOF model isused.

• Species mixing and reacting flow cannot be modeled when the VOFmodel is used.

• The LES turbulence model cannot be used with the VOF model.

• The second-order implicit time-stepping formulation cannot be usedwith the VOF model.

• The VOF model cannot be used for inviscid flows.

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

Steady-State and Transient VOF Calculations

The VOF formulation in FLUENT is generally used to compute a time-dependent solution, but for problems in which you are concerned onlywith a steady-state solution, it is possible to perform a steady-state cal-culation. A steady-state VOF calculation is sensible only when yoursolution is independent of the initial conditions and there are distinct in-flow boundaries for the individual phases. For example, since the shapeof the free surface inside a rotating cup depends on the initial level of thefluid, such a problem must be solved using the time-dependent formula-tion. On the other hand, the flow of water in a channel with a region ofair on top and a separate air inlet can be solved with the steady-stateformulation.



20.1.2 Overview and Limitations of the Mixture Model

Overview

The mixture model is a simplified multiphase model that can be used tomodel multiphase flows where the phases move at different velocities, butassume local equilibrium over short spatial length scales. The couplingbetween the phases should be strong. It can also be used to modelhomogeneous multiphase flows with very strong coupling and the phasesmoving at the same velocity.

The mixture model can model n phases (fluid or particulate) by solv-ing the momentum, continuity, and energy equations for the mixture,the volume fraction equations for the secondary phases, and algebraicexpressions for the relative velocities. Typical applications include sedi-mentation, cyclone separators, particle-laden flows with low loading, andbubbly flows where the gas volume fraction remains low.

The mixture model is a good substitute for the full Eulerian multiphasemodel in several cases. A full multiphase model may not be feasiblewhen there is a wide distribution of the particulate phase or when theinterphase laws are unknown or their reliability can be questioned. Asimpler model like the mixture model can perform as well as a full mul-tiphase model while solving a smaller number of variables than the fullmultiphase model.

Limitations

The following limitations apply to the mixture model in FLUENT:

• You must use the segregated solver. The mixture model is notavailable with either of the coupled solvers.

• Only one of the phases can be compressible.

• Streamwise periodic flow (either specified mass flow rate or speci-fied pressure drop) cannot be modeled when the mixture model isused.



• Species mixing and reacting flow cannot be modeled when the mix-ture model is used.

• Solidification and melting cannot be modeled in conjunction withthe mixture model.

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

• The second-order implicit time-stepping formulation cannot be usedwith the mixture model.

• The mixture model cannot be used for inviscid flows.

• The shell conduction model for walls cannot be used with the mix-ture model.

20.1.3 Overview and Limitations of the Eulerian Model

Overview

The Eulerian multiphase model in FLUENT allows for the modeling ofmultiple separate, yet interacting phases. The phases can be liquids,gases, or solids in nearly any combination. An Eulerian treatment isused for each phase, in contrast to the Eulerian-Lagrangian treatmentthat is used for the discrete phase model.

With the Eulerian multiphase model, the number of secondary phasesis limited only by memory requirements and convergence behavior. Anynumber of secondary phases can be modeled, provided that sufficientmemory is available. For complex multiphase flows, however, you mayfind that your solution is limited by convergence behavior. See Sec-tion 20.7.3 for multiphase modeling strategies.

FLUENT’s Eulerian multiphase model differs from the Eulerian model inFLUENT 4 in that there is no global distinction between fluid-fluid andfluid-solid (granular) multiphase flows. A granular flow is simply onethat involves at least one phase that has been designated as a granularphase.

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 cal-culated for each solid phase. This is based on an algebraicrelation.

– Solid-phase shear and bulk viscosities are obtained from appli-cation of kinetic theory to granular flows. Frictional viscosityis also available.

• Several interphase drag coefficient functions are available, whichare appropriate for various types of multiphase regimes. (You canalso modify the interphase drag coefficient through user-definedfunctions, as described in the separate UDF Manual.)

• All of the k-ε turbulence models are available, and may apply toall phases or to the mixture.

Limitations

All other features available in FLUENT can be used in conjunction withthe Eulerian multiphase model, except for the following limitations:

• Only the k-ε models can be used for turbulence.

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

• Streamwise periodic flow (either specified mass flow rate or speci-fied pressure drop) cannot be modeled when the Eulerian model isused.

• Compressible flow is not allowed.

• Inviscid flow is not allowed.

• The second-order implicit time-stepping formulation cannot be usedwith the Eulerian model.



• Melting and solidification are not allowed.

• Species transport and reactions are not allowed.

• Heat transfer cannot be modeled.

• The only type of mass transfer between phases that is allowed iscavitation; evaporation, condensation, etc. are not allowed.

Stability and Convergence

The process of solving a multiphase system is inherently difficult, andyou may encounter some stability or convergence problems, althoughthe current algorithm is more stable than that used in FLUENT 4. Ifa time-dependent problem is being solved, and patched fields are usedfor the initial conditions, it is recommended that you perform a fewiterations with a small time step, at least an order of magnitude smallerthan the characteristic time of the flow. You can increase the size of thetime step after performing a few time steps. For steady solutions it isrecommended that you start with a small under-relaxation factor for thevolume fraction.

Stratified flows of immiscible fluids should be solved with the VOF model(see Section 20.2). Some problems involving small volume fractions canbe solved more efficiently with the Lagrangian discrete phase model (seeChapter 19).

Many stability and convergence problems can be minimized if care istaken during the setup and solution processes (see Section 20.7.3).



20.2 Volume of Fluid (VOF) Model

The VOF formulation relies on the fact that two or more fluids (orphases) are not interpenetrating. For each additional phase that youadd to your model, a variable is introduced: the volume fraction of thephase in the computational cell. In each control volume, the volumefractions of all phases sum to unity. The fields for all variables and prop-erties are shared by the phases and represent volume-averaged values,as long as the volume fraction of each of the phases is known at eachlocation. Thus the variables and properties in any given cell are eitherpurely representative of one of the phases, or representative of a mix-ture of the phases, depending upon the volume fraction values. In otherwords, if the qth fluid’s volume fraction in the cell is denoted as αq, thenthe 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 fluidand one or more other fluids.

Based on the local value of αq, the appropriate properties and variableswill be assigned to each control volume within the domain.

20.2.1 The Volume Fraction Equation

The tracking of the interface(s) between the phases is accomplished bythe solution of a continuity equation for the volume fraction of one (ormore) of the phases. For the qth phase, this equation has the followingform:

∂αq

∂t+ ~v · ∇αq =

Sαq

ρq(20.2-1)

By default, the source term on the right-hand side of Equation 20.2-1 iszero, but you can specify a constant or user-defined mass source for eachphase.



The volume fraction equation will not be solved for the primary phase;the primary-phase volume fraction will be computed based on the fol-lowing constraint:

n∑q=1

αq = 1 (20.2-2)

20.2.2 Properties

The properties appearing in the transport equations are determined bythe presence of the component phases in each control volume. In a two-phase system, for example, if the phases are represented by the subscripts1 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 (20.2-3)

In general, for an n-phase system, the volume-fraction-averaged densitytakes on the following form:

ρ =∑

αqρq (20.2-4)

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

20.2.3 The Momentum Equation

A single momentum equation is solved throughout the domain, and theresulting velocity field is shared among the phases. The momentumequation, shown below, is dependent on the volume fractions of all phasesthrough the properties ρ and µ.

∂

∂t(ρ~v) + ∇ · (ρ~v~v) = −∇p+ ∇ ·

[µ(∇~v + ∇~vT

)]+ ρ~g + ~F (20.2-5)



One limitation of the shared-fields approximation is that in cases wherelarge velocity differences exist between the phases, the accuracy of thevelocities computed near the interface can be adversely affected.

20.2.4 The Energy Equation

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

∂

∂t(ρE) + ∇ · (~v(ρE + p)) = ∇ · (keff∇T ) + Sh (20.2-6)

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

E =

n∑q=1

αqρqEq

n∑q=1

αqρq

(20.2-7)

where Eq for each phase is based on the specific heat of that phase andthe shared temperature.

The properties ρ and keff (effective thermal conductivity) are shared bythe 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 inter-face is limited in cases where large temperature differences exist betweenthe phases. Such problems also arise in cases where the properties varyby several orders of magnitude. For example, if a model includes liquidmetal in combination with air, the conductivities of the materials candiffer by as much as four orders of magnitude. Such large discrepanciesin properties lead to equation sets with anisotropic coefficients, which inturn can lead to convergence and precision limitations.



20.2.5 Additional Scalar Equations

Depending upon your problem definition, additional scalar equationsmay be involved in your solution. In the case of turbulence quantities,a single set of transport equations is solved, and the turbulence vari-ables (e.g., k and ε or the Reynolds stresses) are shared by the phasesthroughout the field.

20.2.6 Interpolation Near the Interface

FLUENT’s control-volume formulation requires that convection and dif-fusion fluxes through the control volume faces be computed and bal-anced with source terms within the control volume itself. There are fourschemes in FLUENT for the calculation of face fluxes for the VOF model:geometric reconstruction, donor-acceptor, Euler explicit, and implicit.

In the geometric reconstruction and donor-acceptor schemes, FLUENTapplies a special interpolation treatment to the cells that lie near theinterface between two phases. Figure 20.2.1 shows an actual interfaceshape along with the interfaces assumed during computation by thesetwo methods.

The Euler explicit scheme and the implicit scheme treat these cells withthe same interpolation as the cells that are completely filled with onephase or the other (i.e., using the standard upwind, second-order, orQUICK scheme), rather than applying a special treatment.

The Geometric Reconstruction Scheme

In the geometric reconstruction approach, the standard interpolationschemes that are used in FLUENT are used to obtain the face fluxeswhenever a cell is completely filled with one phase or another. When thecell is near the interface between two phases, the geometric reconstruc-tion scheme is used.

The geometric reconstruction scheme represents the interface betweenfluids using a piecewise-linear approach. In FLUENT this scheme is themost accurate and is applicable for general unstructured meshes. Thegeometric reconstruction scheme is generalized for unstructured meshes



actual interface shape

interface shape represented by the donor-acceptor scheme

interface shape represented by the geometric reconstruction (piecewise-linear) scheme

Figure 20.2.1: Interface Calculations



from the work of Youngs [273]. It assumes that the interface betweentwo fluids has a linear slope within each cell, and uses this linear shapefor calculation of the advection of fluid through the cell faces. (SeeFigure 20.2.1.)

The first step in this reconstruction scheme is calculating the positionof the linear interface relative to the center of each partially-filled cell,based on information about the volume fraction and its derivatives inthe cell. The second step is calculating the advecting amount of fluidthrough each face using the computed linear interface representation andinformation about the normal and tangential velocity distribution on theface. The third step is calculating the volume fraction in each cell usingthe balance of fluxes calculated during the previous step.

When the geometric reconstruction scheme is used, a time-dependent!solution must be computed. Also, if you are using a conformal grid(i.e., if the grid node locations are identical at the boundaries where twosubdomains meet), you must ensure that there are no two-sided (zero-thickness) walls within the domain. If there are, you will need to slitthem, as described in Section 5.7.8.

The Donor-Acceptor Scheme

In the donor-acceptor approach, the standard interpolation schemes thatare used in FLUENT are used to obtain the face fluxes whenever a cellis completely filled with one phase or another. When the cell is nearthe interface between two phases, a “donor-acceptor” scheme is usedto determine the amount of fluid advected through the face [93]. Thisscheme identifies one cell as a donor of an amount of fluid from one phaseand another (neighbor) cell as the acceptor of that same amount of fluid,and is used to prevent numerical diffusion at the interface. The amountof fluid from one phase that can be convected across a cell boundary islimited by the minimum of two values: the filled volume in the donorcell 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 onthe direction of the volume fraction gradient of the qth phase withinthe 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 valuesare obtained by pure upwinding, pure downwinding, or some combinationof the two.

When the donor-acceptor scheme is used, a time-dependent solution!must be computed. Also, the donor-acceptor scheme can be used onlywith quadrilateral or hexahedral meshes. In addition, if you are using aconformal grid (i.e., if the grid node locations are identical at the bound-aries where two subdomains meet), you must ensure that there are notwo-sided (zero-thickness) walls within the domain. If there are, you willneed to slit them, as described in Section 5.7.8.

The Euler Explicit Scheme

In the Euler explicit approach, FLUENT’s standard finite-difference in-terpolation schemes are applied to the volume fraction values that werecomputed at the previous time step.

αn+1q − αn

q

∆tV +

∑f

(Unf α

nq,f ) = 0 (20.2-8)

where n+ 1 = index for new (current) time stepn = index for previous time stepαq,f = face value of the qth volume fraction, computed

from the first- or second-order upwind orQUICK scheme

V = volume of cellUf = volume flux through the face, based on

normal velocity

This formulation does not require iterative solution of the transport equa-tion during each time step, as is needed for the implicit scheme.

When the Euler explicit scheme is used, a time-dependent solution must!be computed.



The Implicit Scheme

In the implicit interpolation method, FLUENT’s standard finite-differenceinterpolation schemes are used to obtain the face fluxes for all cells, in-cluding those near the interface.

αn+1q − αn

q

∆tV +

∑f

(Un+1f αn+1

q,f ) = 0 (20.2-9)

Since this equation requires the volume fraction values at the currenttime step (rather than at the previous step, as for the Euler explicitscheme), a standard scalar transport equation is solved iteratively foreach 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 20.6.4 for details.

20.2.7 Time Dependence

For time-dependent VOF calculations, Equation 20.2-1 is solved using anexplicit time-marching scheme. FLUENT automatically refines the timestep for the integration of the volume fraction equation, but you caninfluence this time step calculation by modifying the Courant number.You can choose to update the volume fraction once for each time step, oronce for each iteration within each time step. These options are discussedin more detail in Section 20.6.12.

20.2.8 Surface Tension and Wall Adhesion

The VOF model can also include the effects of surface tension along theinterface between each pair of phases. The model can be augmented bythe additional specification of the contact angles between the phases andthe walls.

Surface Tension

Surface tension arises as a result of attractive forces between moleculesin a fluid. Consider an air bubble in water, for example. Within the



bubble, the net force on a molecule due to its neighbors is zero. Atthe surface, however, the net force is radially inward, and the combinedeffect of the radial components of force across the entire spherical surfaceis to make the surface contract, thereby increasing the pressure on theconcave side of the surface. The surface tension is a force, acting only atthe surface, that is required to maintain equilibrium in such instances.It acts to balance the radially inward inter-molecular attractive forcewith the radially outward pressure gradient force across the surface. Inregions where two fluids are separated, but one of them is not in the formof spherical bubbles, the surface tension acts to minimize free energy bydecreasing the area of the interface.

The surface tension model in FLUENT is the continuum surface force(CSF) model proposed by Brackbill et al. [25]. With this model, theaddition of surface tension to the VOF calculation results in a sourceterm in the momentum equation. To understand the origin of the sourceterm, consider the special case where the surface tension is constantalong the surface, and where only the forces normal to the interface areconsidered. It can be shown that the pressure drop across the surfacedepends upon the surface tension coefficient, σ, and the surface curvatureas measured by two radii in orthogonal directions, R1 and R2:

p2 − p1 = σ

(1R1

+1R2

)(20.2-10)

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

In FLUENT, a formulation of CSF model is used, where the surface cur-vature is computed from local gradients in the surface normal at theinterface. Let n be the surface normal, defined as the gradient of αq, thevolume fraction of the qth phase.

n = ∇αq (20.2-11)

The curvature, κ, is defined in terms of the divergence of the unit normal,n [25]:



κ = ∇ · n (20.2-12)

where

n =n

|n| (20.2-13)

The surface tension can be written in terms of the pressure jump acrossthe surface. The force at the surface can be expressed as a volume forceusing the divergence theorem. It is this volume force that is the sourceterm which is added to the momentum equation. It has the followingform:

Fvol =∑

pairs ij, i<j

σijαiρiκj∇αj + αjρjκi∇αi

12 (ρi + ρj)

(20.2-14)

This expression allows for a smooth superposition of forces near cellswhere more than two phases are present. If only two phases are present ina cell, then κi = −κj and ∇αi = −∇αj, and Equation 20.2-14 simplifiesto

Fvol = σijρκi∇αi

12 (ρi + ρj)

(20.2-15)

where ρ is the volume-averaged density computed using Equation 20.2-4.Equation 20.2-15 shows that the surface tension source term for a cell isproportional to the average density in the cell.

Note that the calculation of surface tension effects on triangular andtetrahedral meshes is not as accurate as on quadrilateral and hexahedralmeshes. The region where surface tension effects are most importantshould therefore be meshed with quadrilaterals or hexahedra.

When Surface Tension Effects are Important

The importance of surface tension effects is determined based on thevalue of two dimensionless quantities: the Reynolds number, Re, and



the capillary number, Ca; or the Reynolds number, Re, and the We-ber number, We. For Re � 1, the quantity of interest is the capillarynumber:

Ca =µU

σ(20.2-16)

and for Re � 1, the quantity of interest is the Weber number:

We =σ

ρLU2(20.2-17)

where U is the free-stream velocity. Surface tension effects can be ne-glected if Ca � 1 or We � 1.

Wall Adhesion

An option to specify a wall adhesion angle in conjunction with the sur-face tension model is also available in the VOF model. The model istaken from work done by Brackbill et al. [25]. Rather than impose thisboundary condition at the wall itself, the contact angle that the fluid isassumed to make with the wall is used to adjust the surface normal incells near the wall. This so-called dynamic boundary condition resultsin the adjustment of the curvature of the surface near the wall.

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

n = nw cos θw + tw sin θw (20.2-18)

where nw and tw are the unit vectors normal and tangential to the wall,respectively. The combination of this contact angle with the normallycalculated surface normal one cell away from the wall determine the localcurvature of the surface, and this curvature is used to adjust the bodyforce term in the surface tension calculation.

The contact angle θw is the angle between the wall and the tangent tothe interface at the wall, measured inside the first phase of the pair listedin the Wall panel, as shown in Figure 20.2.2.



θw

wall

interface

first phase

second phase

Figure 20.2.2: Measuring the Contact Angle



20.3 Mixture Model

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. Thevolume fractions αq and αp for a control volume can therefore beequal to any value between 0 and 1, depending on the space occu-pied 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 alsobe assumed to move at the same velocity, and the mixture modelis then reduced to a homogeneous multiphase model.)

The mixture model solves the continuity equation for the mixture, themomentum equation for the mixture, the energy equation for the mix-ture, and the volume fraction equation for the secondary phases, as wellas algebraic expressions for the relative velocities (if the phases are mov-ing at different velocities).

20.3.1 Continuity Equation for the Mixture

The continuity equation for the mixture is

∂

∂t(ρm) + ∇ · (ρm~vm) = m (20.3-1)

where ~vm is the mass-averaged velocity:

~vm =∑n

k=1 αkρk~vk

ρm(20.3-2)

and ρm is the mixture density:

ρm =n∑

k=1

αkρk (20.3-3)


20.3 Mixture Model

αk is the volume fraction of phase k.

m represents mass transfer due to cavitation (described in Section 20.5)or user-defined mass sources.

20.3.2 Momentum Equation for the Mixture

The momentum equation for the mixture can be obtained by summingthe individual momentum equations for all phases. It can be expressedas

∂

∂t(ρm~vm) + ∇ · (ρm~vm~vm) = −∇p + ∇ ·

[µm

(∇~vm + ∇~vT

m

)]+

ρm~g + ~F + ∇ ·(

n∑k=1

αkρk~vdr,k~vdr,k

)(20.3-4)

where n is the number of phases, ~F is a body force, and µm is theviscosity of the mixture:

µm =n∑

k=1

αkµk (20.3-5)

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

~vdr,k = ~vk − ~vm (20.3-6)

20.3.3 Energy Equation for the Mixture

The energy equation for the mixture takes the following form:

∂

∂t

n∑k=1

(αkρkEk)+∇·n∑

k=1

(αk~vk(ρkEk + p)) = ∇· (keff∇T )+SE (20.3-7)

where keff is the effective conductivity (k + kt, where kt is the turbulentthermal conductivity, defined according to the turbulence model being



used). The first term on the right-hand side of Equation 20.3-7 representsenergy transfer due to conduction. SE includes any other volumetric heatsources.

In Equation 20.3-7,

Ek = hk − p

ρk+v2k

2(20.3-8)

for a compressible phase, and Ek = hk for an incompressible phase,where hk is the sensible enthalpy for phase k.

20.3.4 Relative (Slip) Velocity and the Drift Velocity

The relative velocity (also referred to as the slip velocity) is defined as thevelocity of a secondary phase (p) relative to the velocity of the primaryphase (q):

~vqp = ~vp − ~vq (20.3-9)

The drift velocity and the relative velocity (~vqp) are connected by thefollowing expression:

~vdr,p = ~vqp −n∑

k=1

αkρk

ρm~vqk (20.3-10)

FLUENT’s mixture model makes use of an algebraic slip formulation. Thebasic assumption of the algebraic slip mixture model is that, to prescribean algebraic relation for the relative velocity, a local equilibrium betweenthe phases should be reached over short spatial length scales. The formof the relative velocity is given by

~vqp = τqp~a (20.3-11)

where ~a is the secondary-phase particle’s acceleration and τqp is the par-ticulate relaxation time. Following Manninen et al. [150] τqp is of theform:


20.3 Mixture Model

τqp =(ρm − ρp)d2

p

18µqfdrag(20.3-12)

where dp is the diameter of the particles (or droplets or bubbles) ofsecondary phase p, and the drag function fdrag is taken from Schillerand Naumann [202]:

fdrag =

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

(20.3-13)

and the acceleration ~a is of the form

~a = ~g − (~vm · ∇)~vm − ∂~vm

∂t(20.3-14)

The simplest algebraic slip formulation is the so-called drift flux model,in which the acceleration of the particle is given by gravity and/or acentrifugal force and the particulate relaxation time is modified to takeinto account the presence of other particles.

Note that, if the slip velocity is not solved, the mixture model is reducedto a homogeneous multiphase model. In addition, the mixture model canbe customized (using user-defined functions) to use a formulation otherthan the algebraic slip method for the slip velocity. See the separateUDF Manual for details.

20.3.5 Volume Fraction Equation for the Secondary Phases

From the continuity equation for secondary phase p, the volume fractionequation for secondary phase p can be obtained:

∂

∂t(αpρp) + ∇ · (αpρp~vm) = −∇ · (αpρp~vdr,p) (20.3-15)



20.4 Eulerian Model

To change from a single-phase model, where a single set of conserva-tion equations for momentum and continuity is solved, to a multiphasemodel, additional sets of conservation equations must be introduced. Inthe process of introducing additional sets of conservation equations, theoriginal set must also be modified. The modifications involve, amongother things, the introduction of the volume fractions α1, α2, . . . αn forthe multiple phases, as well as a mechanism for the exchange of momen-tum between the phases.

Details about the Eulerian multiphase model are presented in the fol-lowing subsections:

• Section 20.4.1: Volume Fractions

• Section 20.4.2: Conservation Equations

• Section 20.4.3: Interphase Exchange Coefficients

• Section 20.4.4: Solids Pressure

• Section 20.4.5: Solids Shear Stresses

• Section 20.4.6: Granular Temperature

• Section 20.4.7: Turbulence Models

• Section 20.4.8: Solution Method in FLUENT

20.4.1 Volume Fractions

The description of multiphase flow as interpenetrating continua incorpo-rates the concept of phasic volume fractions, denoted here by αq. Vol-ume fractions represent the space occupied by each phase, and the lawsof conservation of mass and momentum are satisfied by each phase in-dividually. The derivation of the conservation equations can be doneby ensemble averaging the local instantaneous balance for each of thephases [3] or by using the mixture theory approach [22].

The volume of phase q, Vq, is defined by


20.4 Eulerian Model

Vq =∫

VαqdV (20.4-1)

where

n∑q=1

αq = 1 (20.4-2)

The effective density of phase q is

ρq = αqρq (20.4-3)

where ρq is the physical density of phase q.

20.4.2 Conservation Equations

The general conservation equations from which the equations solved byFLUENT are derived are presented in this section, followed by the solvedequations themselves.

Equations in General Form

Conservation of Mass

The continuity equation for phase q is

∂

∂t(αqρq) + ∇ · (αqρq~vq) =

n∑p=1

mpq (20.4-4)

where ~vq is the velocity of phase q and mpq characterizes the mass transferfrom the pth to qth phase. From the mass conservation one can obtain

mpq = −mqp (20.4-5)

and



mpp = 0 (20.4-6)

Conservation of Momentum

The momentum balance for phase q yields

∂

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

n∑p=1

(~Rpq + mpq~vpq) +

αqρq(~Fq + ~Flift,q + ~Fvm,q) (20.4-7)

where τ q is the qth phase stress-strain tensor

τ q = αqµq(∇~vq + ∇~vTq ) + αq(λq − 2

3µq)∇ · ~vqI (20.4-8)

Here µq and λq are the shear and bulk viscosity of phase q, ~Fq is anexternal body force, ~Flift,q is a lift force, ~Fvm,q is a virtual mass force,~Rpq is an interaction force between phases, and p is the pressure sharedby all phases.

~vpq is the interphase velocity, defined as follows. If mpq > 0 (i.e., phasep mass is being transferred to phase q), ~vpq = ~vp; if mpq < 0 (i.e., phaseq mass is being transferred to phase p), ~vpq = ~vq; and ~vpq = ~vqp.

Equation 20.4-7 must be closed with appropriate expressions for the in-terphase force ~Rpq. This force depends on the friction, pressure, cohesion,and other effects, and is subject to the conditions that ~Rpq = −~Rqp and~Rqq = 0.

FLUENT uses a simple interaction term of the following form:

n∑p=1

~Rpq =n∑

p=1

Kpq(~vp − ~vq) (20.4-9)

where Kpq (= Kqp) is the interphase momentum exchange coefficient(described in Section 20.4.3).


20.4 Eulerian Model

Lift Forces

For multiphase flows, FLUENT can include the effect of lift forces onthe secondary phase particles (or droplets or bubbles). These lift forcesact on a particle mainly due to velocity gradients in the primary-phaseflow field. The lift force will be more significant for larger particles, butthe FLUENT model assumes that the particle diameter is much smallerthan the interparticle spacing. Thus, the inclusion of lift forces is notappropriate for closely packed particles or for very small particles.

The lift force acting on a secondary phase p in a primary phase q iscomputed from [57]

Flift = −0.5ρqαp|~vq − ~vp| × (∇× ~vq) (20.4-10)

The lift force Flift will be added to the right-hand side of the momentumequation for both phases (Flift,q = −Flift,p).

In most cases, the lift force is insignificant compared to the drag force, sothere is no reason to include this extra term. If the lift force is significant(e.g., if the phases separate quickly), it may be appropriate to include thisterm. By default, Flift is not included. The lift force and lift coefficientcan be specified for each pair of phases, if desired.

Virtual Mass Force

For multiphase flows, FLUENT includes the “virtual mass effect” thatoccurs when a secondary phase p accelerates relative to the primaryphase q. The inertia of the primary-phase mass encountered by theaccelerating particles (or droplets or bubbles) exerts a “virtual massforce” on the particles [57]:

Fvm = 0.5αpρq

(dqvq

dt− dpvp

dt

)(20.4-11)

The term dq

dt denotes the phase material time derivative of the form

dq(φ)dt

=∂(φ)∂t

+ (~vq · ∇)φ (20.4-12)



The virtual mass force Fvm will be added to the right-hand side of themomentum equation for both phases (Fvm,q = −Fvm,p).

The virtual mass effect is significant when the secondary phase densityis much smaller than the primary phase density (e.g., for a transientbubble column). By default, Fvm is not included.

Equations Solved by FLUENT

The equations for fluid-fluid and granular multiphase flows, as solved byFLUENT, are presented here for the general case of an n-phase flow.

Continuity Equation

The volume fraction of each phase is calculated from a continuity equa-tion:

∂

∂t(αq) + ∇ · (αq~vq) =

1ρq

n∑

p=1

mpq − αqdqρq

dt

(20.4-13)

The solution of this equation for each secondary phase, along with thecondition that the volume fractions sum to one (given by Equation 20.4-2),allows for the calculation of the primary-phase volume fraction. Thistreatment is common to fluid-fluid and granular flows.

Fluid-Fluid Momentum Equations

The conservation of momentum for a fluid phase q is

∂

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

αqρq(~Fq + ~Flift,q + ~Fvm,q) +n∑

p=1

(Kpq(~vp − ~vq) + mpq~vpq)

(20.4-14)


20.4 Eulerian Model

Here ~g is the acceleration due to gravity and ~Fq, ~Flift,q, and ~Fvm,q are asdefined for Equation 20.4-7.

Fluid-Solid Momentum Equations

Following the work of [2, 32, 50, 76, 131, 145, 167, 235], FLUENT uses amulti-fluid granular model to describe the flow behavior of a fluid-solidmixture. The solid-phase stresses are derived by making an analogy be-tween the random particle motion arising from particle-particle collisionsand the thermal motion of molecules in a gas, taking into account theinelasticity of the granular phase. As is the case for a gas, the intensity ofthe particle velocity fluctuations determines the stresses, viscosity, andpressure of the solid phase. The kinetic energy associated with the parti-cle velocity fluctuations is represented by a “pseudothermal” or granulartemperature which is proportional to the mean square of the randommotion of particles.

The conservation of momentum for the fluid phases is similar to Equa-tion 20.4-14, and that for the sth solid phase is

∂

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

αsρs(~Fs + ~Flift,s + ~Fvm,s) +N∑

l=1

(Kls(~vl − ~vs) + mls~vls) (20.4-15)

where ps is the sth solids pressure, Kls = Ksl is the momentum exchangecoefficient between fluid or solid phase l and solid phase s, N is thetotal number of phases, and ~Fq, ~Flift,q, and ~Fvm,q are as defined forEquation 20.4-7.

20.4.3 Interphase Exchange Coefficients

It can be seen in Equations 20.4-14 and 20.4-15 that momentum exchangebetween the phases is based on the value of the fluid-fluid exchangecoefficient Kpq and, for granular flows, the fluid-solid and solid-solidexchange coefficients Kls.



Fluid-Fluid Exchange Coefficient

For fluid-fluid flows, each secondary phase is assumed to form dropletsor bubbles. This has an impact on how each of the fluids is assignedto a particular phase. For example, in flows where there are unequalamounts of two fluids, the predominant fluid should be modeled as theprimary fluid, since the sparser fluid is more likely to form droplets orbubbles. The exchange coefficient for these types of bubbly, liquid-liquidor gas-liquid mixtures can be written in the following general form:

Kpq =αpρpf

τp(20.4-16)

where f , the drag function, is defined differently for the different exchange-coefficient models (as described below) and τp, the “particulate relax-ation time”, is defined as

τp =ρpd

2p

18µq(20.4-17)

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

Nearly all definitions of f include a drag coefficient (CD) that is based onthe relative Reynolds number (Re). It is this drag function that differsamong the exchange-coefficient models.

• For the model of Schiller and Naumann [202]

f =CDRe

24(20.4-18)

where

CD =

{24(1 + 0.15Re0.687)/Re Re ≤ 10000.44 Re > 1000

(20.4-19)

and Re is the relative Reynolds number. The relative Reynoldsnumber for the primary phase q and secondary phase p is obtainedfrom


20.4 Eulerian Model

Re =ρq|~vp − ~vq|dp

µq(20.4-20)

The relative Reynolds number for secondary phases p and r isobtained from

Re =ρrp|~vr − ~vp|drp

µrp(20.4-21)

where µrp = αpµp + αrµr is the mixture viscosity of the phases pand r.

The Schiller and Naumann model is the default method, and it isacceptable for general use for all fluid-fluid pairs of phases.

• For the Morsi and Alexander model [163]

f =CDRe

24(20.4-22)

where

CD = a1 +a2

Re+

a3

Re2(20.4-23)

and Re is defined by Equation 20.4-20 or 20.4-21. The a’s aredefined as follows:

a1, a2, a3 =

0, 18, 0 0 < Re < 0.13.690, 22.73, 0.0903 0.1 < Re < 11.222, 29.1667, −3.8889 1 < Re < 100.6167, 46.50, −116.67 10 < Re < 1000.3644, 98.33, −2778 100 < Re < 10000.357, 148.62, −47500 1000 < Re < 50000.46, −490.546, 578700 5000 < Re < 100000.5191, −1662.5, 5416700 Re ≥ 10000

(20.4-24)

The Morsi and Alexander model is the most complete, adjustingthe function definition frequently over a large range of Reynolds



numbers, but calculations with this model may be less stable thanwith the other models.

• For the symmetric model

Kpq =αp(αpρp + αqρq)f

τpq(20.4-25)

where

τpq =(αpρp + αqρq)(

dp+dq

2 )2

18(αpµp + αqµq)(20.4-26)

and

f =CDRe

24(20.4-27)

where

CD =

{24(1 + 0.15Re0.687)/Re Re ≤ 10000.44 Re > 1000

(20.4-28)

and Re is defined by Equation 20.4-20 or 20.4-21.

The symmetric model is recommended for flows in which the sec-ondary (dispersed) phase in one region of the domain becomes theprimary (continuous) phase in another. For example, if air is in-jected 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 secondaryphases.

You can specify different exchange coefficients for each pair of phases.It is also possible to use user-defined functions to define exchange coeffi-cients 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 fluidswill be computed independently, with the only “interaction” being theircomplementary volume fractions within each computational cell.


20.4 Eulerian Model

Fluid-Solid Exchange Coefficient

The fluid-solid exchange coefficient Ksl can be written in the followinggeneral form:

Ksl =αsρsf

τs(20.4-29)

where f is defined differently for the different exchange-coefficient models(as described below), and τs, the “particulate relaxation time”, is definedas

τs =ρsd

2s

18µl(20.4-30)

where ds is the diameter of particles of phase s.

All definitions of f include a drag function (CD) that is based on therelative Reynolds number (Res). It is this drag function that differsamong the exchange-coefficient models.

• For the Syamlal-O’Brien model [234]

f =CDResαl

24v2r,s

(20.4-31)

where the drag function has a form derived by Dalla Valle [47]

CD =

0.63 +

4.8√Res/vr,s

2

(20.4-32)

This model is based on measurements of the terminal velocities ofparticles in fluidized or settling beds, with correlations that are afunction of the volume fraction and relative Reynolds number [193]:

Res =ρlds|~vs − ~vl|

µl(20.4-33)



where the subscript l is for the lth fluid phase, s is for the sth solidphase, and ds is the diameter of the sth solid phase particles.

The fluid-solid exchange coefficient has the form

Ksl =3αsαlρl

4v2r,sds

CD

(Resvr,s

)|~vs − ~vl| (20.4-34)

where vr,s is the terminal velocity correlation for the solid phase [73]:

vr,s = 0.5(A− 0.06Res +

√(0.06Res)

2 + 0.12Res (2B −A) +A2

)(20.4-35)

with

A = α4.14l (20.4-36)

and

B = 0.8α1.28l (20.4-37)

for αl ≤ 0.85, and

B = α2.65l (20.4-38)

for αl > 0.85.

This model is appropriate when the solids shear stresses are definedaccording to Syamlal et al. [235] (Equation 20.4-52).

• For the model of Wen and Yu [262], the fluid-solid exchange coef-ficient is of the following form:

Ksl =34CD

αsαlρl|~vs − ~vl|ds

α−2.65l (20.4-39)

where


20.4 Eulerian Model

CD =24

αlRes

[1 + 0.15(αlRes)0.687

](20.4-40)

and Res is defined by Equation 20.4-33.

This model is appropriate for dilute systems.

• The Gidaspow model [76] is a combination of the Wen and Yumodel [262] and the Ergun equation [62].

When αl > 0.8, the fluid-solid exchange coefficient Ksl is of thefollowing form:

Ksl =34CD

αsαlρl|~vs − ~vl|ds

α−2.65l (20.4-41)

where

CD =24

αlRes

[1 + 0.15(αlRes)0.687

](20.4-42)

When αl ≤ 0.8,

Ksl = 150αs(1 − αl)µl

αld2s

+ 1.75ρlαs|~vs − ~vl|

ds(20.4-43)

This model is recommended for dense fluidized beds.

Solid-Solid Exchange Coefficient

The solid-solid exchange coefficient Kls has the following form [233]:

Kls =3 (1 + els)

(π2 + Cfr,ls

π2

8

)αsρsαlρl (dl + ds)

2 g0,ls

2π(ρld

3l + ρsd3

s

) |~vl − ~vs|(20.4-44)



whereels = the coefficient of restitution (described in

Section 20.4.4)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 lg0,ls = the radial distribution coefficient (described in

Section 20.4.4)

20.4.4 Solids Pressure

For granular flows in the compressible regime (i.e., where the solids vol-ume fraction is less than its maximum allowed value), a solids pressure iscalculated independently and used for the pressure gradient term, ∇ps,in the granular-phase momentum equation. Because a Maxwellian ve-locity distribution is used for the particles, a granular temperature isintroduced into the model, and appears in the expression for the solidspressure and viscosities. The solids pressure is composed of a kineticterm and a second term due to particle collisions:

ps = αsρsΘs + 2ρs(1 + ess)α2sg0,ssΘs (20.4-45)

where ess is the coefficient of restitution for particle collisions, g0,ss is theradial distribution function, and Θs is the granular temperature. FLU-ENT uses a default value of 0.9 for ess, but the value can be adjustedto suit the particle type. The granular temperature Θs is proportionalto the kinetic energy of the fluctuating particle motion, and will be de-scribed later in this section. The function g0,ss (described below in moredetail) is a distribution function that governs the transition from the“compressible” condition with α < αs,max, where the spacing betweenthe solid particles can continue to decrease, to the “incompressible” con-dition with α = αs,max, where no further decrease in the spacing canoccur. A value of 0.63 is the default for αs,max, but you can modify itduring the problem setup.


20.4 Eulerian Model

Radial Distribution Function

The radial distribution function, g0, is a correction factor that modifiesthe probability of collisions between grains when the solid granular phasebecomes dense. This function may also be interpreted as the nondimen-sional distance between spheres:

g0 =s+ dp

s(20.4-46)

where s is the distance between grains. From Equation 20.4-46 it can beobserved that for a dilute solid phase s → ∞, and therefore g0 → 1. Inthe limit when the solid phase compacts, s→ 0 and g0 → ∞. The radialdistribution function is closely connected to the factor χ of Chapmanand Cowling’s [32] theory of non-uniform gases. χ is equal to one for arare gas, and increases and tends to infinity when the molecules are soclose together that motion is not possible.

In the literature there is no unique formulation for the radial distributionfunction. FLUENT employs that proposed in [167]:

g0 =

1 −

(αs

αs,max

) 13

−1

(20.4-47)

When the number of solid phases is greater than 1, Equation 20.4-47 isextended to

g0,ll =

1 −

(αl

αl,max

) 13

−1

(20.4-48)

where αl,max is specified by you during the problem setup, and

g0,lm =dmg0,ll + dlg0,mm

dm + dl(20.4-49)



20.4.5 Solids Shear Stresses

The solids stress tensor contains shear and bulk viscosities arising fromparticle momentum exchange due to translation and collision. A fric-tional component of viscosity can also be included to account for theviscous-plastic transition that occurs when particles of a solid phase reachthe maximum solid volume fraction.

The collisional and kinetic parts, and the optional frictional part, areadded to give the solids shear viscosity:

µs = µs,col + µs,kin + µs,fr (20.4-50)

Collisional Viscosity

The collisional part of the shear viscosity is modeled as [76, 235]

µs,col =45αsρsdsg0,ss(1 + ess)

(Θs

π

)1/2

(20.4-51)

Kinetic Viscosity

FLUENT provides two expressions for the kinetic part.

The default expression is from Syamlal et al. [235]:

µs,kin =αsdsρs

√Θsπ

6 (3 − ess)

[1 +

25

(1 + ess) (3ess − 1)αsg0,ss

](20.4-52)

The following optional expression from Gidaspow et al. [76] is also avail-able:

µs,kin =10ρsds

√Θsπ

96αs (1 + ess) g0,ss

[1 +

45g0,ssαs (1 + ess)

]2(20.4-53)


20.4 Eulerian Model

Bulk Viscosity

The solids bulk viscosity accounts for the resistance of the granular par-ticles to compression and expansion. It has the following form from Lunet al. [145]:

λs =43αsρsdsg0,ss(1 + ess)

(Θs

π

)1/2

(20.4-54)

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-definedfunction.

Frictional Viscosity

In dense flow at low shear, where the secondary volume fraction fora solid phase nears the packing limit, the generation of stress is mainlydue to friction between particles. The solids shear viscosity computed byFLUENT does not, by default, account for the friction between particles.

If the frictional viscosity is included in the calculation, FLUENT usesSchaeffer’s [200] expression:

µs,fr =ps sinφ2√I2D

(20.4-55)

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 possibleto specify a constant or user-defined frictional viscosity.

20.4.6 Granular Temperature

The granular temperature for the sth solids phase is proportional to thekinetic energy of the random motion of the particles. The transportequation derived from kinetic theory takes the form [50]

32

[∂

∂t(ρsαsΘs) + ∇ · (ρsαs~vsΘs)

]= (−psI + τ s) : ∇~vs



∇ · (kΘs∇Θs) − γΘs +φls (20.4-56)

where(−psI + τ s) : ∇~vs = the generation of energy by the

solid stress tensorkΘs∇Θs = the diffusion of energy (kΘs is the

diffusion coefficient)γΘs = the collisional dissipation of energyφls = the energy exchange between the lth

fluid or solid phase and the sth solidphase

Equation 20.4-56 contains the term kΘs∇Θs describing the diffusive fluxof granular energy.

The collisional dissipation of energy, γΘs , represents the rate of energydissipation within the sth solids phase due to collisions between particles.This term is represented by the expression derived by Lun et al. [145]

γΘm =12(1 − e2ss)g0,ss

ds√π

ρsα2sΘ

3/2s (20.4-57)

The transfer of the kinetic energy of random fluctuations in particlevelocity from the sth solids phase to the lth fluid or solid phase is repre-sented by φls [76]:

φls = −3KlsΘs (20.4-58)

FLUENT currently uses an algebraic relation for the granular tempera-ture. This has been obtained by neglecting convection and diffusion inthe transport equation, Equation 20.4-56 [235].

20.4.7 Turbulence Models

To describe the effects of turbulent fluctuations of velocities and scalarquantities in a single phase, FLUENT uses various types of closure mod-els, as described in Chapter 10. In comparison to single-phase flows,


20.4 Eulerian Model

the number of terms to be modeled in the momentum equations in mul-tiphase flows is large, and this makes the modeling of turbulence inmultiphase simulations extremely complex.

FLUENT provides three methods for modeling turbulence in multiphaseflows within the context of the k-ε models:

• mixture turbulence model (default)

• dispersed turbulence model

• turbulence model for each phase

The choice of model depends on the importance of the secondary-phaseturbulence in your application.

Note that the descriptions of each method below are presented based!on the standard k-ε model. The multiphase modifications to the RNGand realizable k-ε models are similar, and are therefore not presentedexplicitly.

Mixture Turbulence Model

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

The k and ε equations describing this model are as follows:

∂

∂t(ρmk) + ∇ · (ρm~vmk) = ∇ ·

(µt,m

σk∇k)

+Gk,m − ρmε (20.4-59)

and



∂

∂t(ρmε)+∇·(ρm~vmε) = ∇·

(µt,m

σε∇ε)

+ε

k(C1εGk,m−C2ερmε) (20.4-60)

where the mixture density and velocity, ρm and ~vm, are computed from

ρm =N∑

i=1

αiρi (20.4-61)

and

~vm =

N∑i=1

αiρi~vi

N∑i=1

αiρi

(20.4-62)

the turbulent viscosity, µt,m, is computed from

µt,m = ρmCµk2

ε(20.4-63)

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

Gk,m = µt,m(∇~vm + (∇~vm)T ) : ∇~vm (20.4-64)

The constants in these equations are the same as those described inSection 10.4.1 for the single-phase k-ε model.

Dispersed Turbulence Model

The dispersed turbulence model is the appropriate model when the con-centrations of the secondary phases are dilute. In this case, interparticlecollisions are negligible and the dominant process in the random motionof the secondary phases is the influence of the primary-phase turbulence.Fluctuating quantities of the secondary phases can therefore be given in


20.4 Eulerian Model

terms of the mean characteristics of the primary phase and the ratio ofthe particle relaxation time and eddy-particle interaction time.

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

Assumptions

The dispersed method for modeling turbulence in FLUENT involves thefollowing assumptions:

• A modified k-ε model for the continuous phase: Turbulent predic-tions for the continuous phase are obtained using the standard k-εmodel supplemented with extra terms that include the interphaseturbulent momentum transfer.

• Tchen-theory correlations for the dispersed phases: Predictions forturbulence quantities for the dispersed phases are obtained usingthe Tchen theory of dispersion of discrete particles by homogeneousturbulence [91].

• Interphase turbulent momentum transfer: In turbulent multiphaseflows, the momentum exchange terms contain the correlation be-tween the instantaneous distribution of the dispersed phases andthe turbulent fluid motion. It is possible to take into account thedispersion of the dispersed phases transported by the turbulentfluid motion.

• A phase-weighted averaging process: The choice of averaging pro-cess has an impact on the modeling of dispersion in turbulent multi-phase flows. A two-step averaging process leads to the appearanceof fluctuations in the phase volume fractions. When the two-stepaveraging process is used with a phase-weighted average for the tur-bulence, however, turbulent fluctuations in the volume fractions donot appear. FLUENT uses phase-weighted averaging, so no volumefraction fluctuations are introduced into the continuity equations.



Turbulence in the Continuous Phase

The eddy viscosity model is used to calculate averaged fluctuating quan-tities. The Reynolds stress tensor for continuous phase q takes the fol-lowing form:

τ′′q = −2

3(ρqkq + ρqµt,q∇ · ~Uq)I + ρqµt,q(∇~Uq + ∇~Uq

T) (20.4-65)

where ~Uq is the phase-weighted velocity.

The turbulent viscosity µt,q is written in terms of the turbulent kineticenergy of phase q:

µt,q = ρqCµ

k2q

εq(20.4-66)

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

τt,q =32Cµ

kq

εq(20.4-67)

where εq is the dissipation rate and Cµ = 0.09.

The length scale of the turbulent eddies is

Lt,q =√

32Cµ

k32q

εq(20.4-68)

Turbulent predictions are obtained from the modified k-ε model:

∂

∂t(αqρqkq) + ∇ · (αqρq

~Uqkq) = ∇ · (αqµt,q

σk∇kq) + αqGk,q − αqρqεq +

αqρqΠkq (20.4-69)

and


20.4 Eulerian Model

∂

∂t(αqρqεq) + ∇ · (αqρq

~Uqεq) = ∇ · (αqµt,q

σε∇εq) +

αqεqkq

(C1εGk,q − C2ερqεq) +

αqρqΠεq (20.4-70)

Here Πkq and Πεq represent the influence of the dispersed phases onthe continuous phase q, and Gk,q is the production of turbulent kineticenergy, as defined in Section 10.4.4. All other terms have the samemeaning as in the single-phase k-ε model.

The term Πkq can be derived from the instantaneous equation of thecontinuous phase and takes the following form, where M represents thenumber of secondary phases:

Πkq =M∑

p=1

Kpq

αqρq(< ~v′′q · ~v′′p > +(~Up − ~Uq) · ~vdr) (20.4-71)

which can be simplified to

Πkq =M∑

p=1

Kpq

αqρq(kpq − 2kq + ~vpq · ~vdr) (20.4-72)

where klq is the covariance of the velocities of the continuous phase q andthe dispersed phase l (calculated from Equation 20.4-80 below), ~vpq is therelative velocity, and ~vdr is the drift velocity (defined by Equation 20.4-85below).

Πεq is modeled according to Elgobashi et al. [61]:

Πεq = C3εεqkq

Πkq (20.4-73)

where C3ε = 1.2.



Turbulence in the Dispersed Phase

Time and length scales that characterize the motion are used to evaluatedispersion coefficients, correlation functions, and the turbulent kineticenergy of each dispersed phase.

The characteristic particle relaxation time connected with inertial effectsacting on a dispersed phase p is defined as

τF,pq = αpρpK−1pq

(ρp

ρq+ CV

)(20.4-74)

The Lagrangian integral time scale calculated along particle trajectories,mainly affected by the crossing-trajectory effect [43], is defined as

τt,pq =τt,q√

(1 + Cβξ2)(20.4-75)

where

ξ =|~vpq|τt,qLt,q

(20.4-76)

and

Cβ = 1.8 − 1.35 cos2 θ (20.4-77)

where θ is the angle between the mean particle velocity and the meanrelative velocity. The ratio between these two characteristic times iswritten as

ηpq =τt,pq

τF,pq(20.4-78)

Following Simonin [212], FLUENT writes the turbulence quantities fordispersed phase p as follows:


20.4 Eulerian Model

kp = kq

(b2 + ηpq

1 + ηpq

)(20.4-79)

kpq = 2kq

(b+ ηpq

1 + ηpq

)(20.4-80)

Dt,pq =13kpqτt,pq (20.4-81)

Dp = Dt,pq +(

23kp − b

13kpq

)τF,pq (20.4-82)

b = (1 + CV )

(ρp

ρq+ CV

)−1

(20.4-83)

and CV = 0.5 is the added-mass coefficient.

Interphase Turbulent Momentum Transfer

The turbulent drag term for multiphase flows (Kpq(~vp − ~vq) in Equa-tion 20.4-9) is modeled as follows, for dispersed phase p and continuousphase q:

Kpq(~vp − ~vq) = Kpq(~Up − ~Uq) −Kpq~vdr (20.4-84)

The second term on the right-hand side of Equation 20.4-84 contains thedrift velocity:

~vdr = −(

Dp

σpqαp∇αp − Dq

σpqαq∇αq

)(20.4-85)

Here Dp and Dq are diffusivities, and σpq is a turbulent Schmidt number.When using Tchen theory in multiphase flows, FLUENT assumes Dp =Dq = Dt,pq and the default value for σpq is 0.67.

The drift velocity results from turbulent fluctuations in the volume frac-tion. When multiplied by the exchange coefficient Kpq, it serves as acorrection to the momentum exchange term for turbulent flows. This



correction is not included, by default, but you can enable it during theproblem setup.

Turbulence Model for Each Phase

The most general multiphase turbulence model solves a set of k and εtransport equations for each phase. This turbulence model is the ap-propriate choice when the turbulence transfer among the phases plays adominant role.

Note that, since FLUENT is solving two additional transport equationsfor each secondary phase, the per-phase turbulence model is more com-putationally intensive than the dispersed turbulence model.

Transport Equations

The Reynolds stress tensor and turbulent viscosity are computed usingEquations 20.4-65 and 20.4-66. Turbulence predictions are obtained from

∂

∂t(αqρqkq) + ∇ · (αqρq

~Uqkq) = ∇ · (αqµt,q

σk∇kq) + (αqGk,q − αqρqεq) +

N∑l=1

Klq(Clqkl − Cqlkq) −

N∑l=1

Klq(~Ul − ~Uq) · µt,l

αlσl∇αl +

N∑l=1

Klq(~Ul − ~Uq) · µt,q

αqσq∇αq (20.4-86)

and

∂

∂t(αqρqεq) + ∇ · (αqρq

~Uqεq) = ∇ · (αqµt,q

σε∇εq) +

εqkq

(C1εαqGk,q − C2εαqρqεq +


20.4 Eulerian Model

C3ε(N∑

l=1

Klq(Clqkl − Cqlkq) −

N∑l=1

Klq(~Ul − ~Uq) · µt,l

αlσl∇αl +

N∑l=1

Klq(~Ul − ~Uq) · µt,q

αqσq∇αq))

(20.4-87)

The terms Clq and Cql can be approximated as

Clq = 2

Cql = 2

(ηlq

1 + ηlq

)(20.4-88)

where ηlq is defined by Equation 20.4-78.

Interphase Turbulent Momentum Transfer

The turbulent drag term (Kpq(~vp−~vq) in Equation 20.4-9) is modeled asfollows, where l is the dispersed phase (replacing p in Equation 20.4-9)and q is the continuous phase:

N∑l=1

Klq(~vl − ~vq) =N∑

l=1

Klq(~Ul − ~Uq) −N∑

l=1

Klq~vdr,lq (20.4-89)

Here ~Ul and ~Uq are phase-weighted velocities, and ~vdr,lq is the drift ve-locity for phase l (computed using Equation 20.4-85, substituting l forp). Note that FLUENT will compute the diffusivities Dl and Dq directlyfrom the transport equations, rather than using Tchen theory (as it doesfor the dispersed turbulence model).

As noted above, the drift velocity results from turbulent fluctuations inthe volume fraction. When multiplied by the exchange coefficient Klq,it serves as a correction to the momentum exchange term for turbulent



flows. This correction is not included, by default, but you can enable itduring the problem setup.

The turbulence model for each phase in FLUENT accounts for the effectof the turbulence field of one phase on the other(s). If you want tomodify or enhance the interaction of the multiple turbulence fields andinterphase turbulent momentum transfer, you can supply these termsusing user-defined functions.

20.4.8 Solution Method in FLUENT

For Eulerian multiphase calculations, FLUENT uses the Phase CoupledSIMPLE (PC-SIMPLE) algorithm [244] for the pressure-velocity cou-pling. PC-SIMPLE is an extension of the SIMPLE algorithm [172] tomultiphase flows. The velocities are solved coupled by phases, but in asegregated fashion. The block algebraic multigrid scheme used by thecoupled solver described in [260] is used to solve a vector equation formedby the velocity components of all phases simultaneously. Then, a pres-sure correction equation is built based on total volume continuity ratherthan mass continuity. Pressure and velocities are then corrected so as tosatisfy the continuity constraint.

The Pressure-Correction Equation

For incompressible multiphase flow, the pressure-correction equation takesthe form

n∑k=1

{∂

∂tαk + ∇ · αk~v

′k + ∇ · αk~v

∗k − 1

ρk

n∑l=1

mlk

}= 0 (20.4-90)

where ~v′k is the velocity correction for the kth phase and ~v∗k is the valueof ~vk at the current iteration. The velocity corrections are themselvesexpressed as functions of the pressure corrections.

Volume Fractions

The volume fractions are obtained from the phase continuity equations.In discretized form, the equation of the kth volume fraction is


20.5 Cavitation Effects

ap,kαk =∑nb

(anb,kαnb,k) + bk = Rk (20.4-91)

In order to satisfy the condition that all the volume fractions sum toone,

n∑k=1

αk = 1 (20.4-92)


Cavitation effects can be included in two-phase flows when the mixturemodel or Eulerian model is used. This section provides informationabout the cavitation model used in FLUENT.

20.5.1 Overview and Limitations of the Cavitation Model

A liquid at constant temperature can be subjected to a decreasing pres-sure, which may fall below the saturated vapor pressure. The process ofrupturing the liquid by a decrease of pressure at constant temperatureis called cavitation. The liquid contains micro-bubbles of air or nuclei,and under decreasing pressure these may grow and form cavities.

The cavitation model, designed for two interpenetrating fluids, modelsthe formation of bubbles when the local pressure becomes less than thevaporization pressure. The cavitation model can be used with the mix-ture model (with or without slip velocities) or the Eulerian multiphasemodel.

The following assumptions are made in the cavitation model:

• The system under investigation involves only two phases.

• Bubbles are neither created nor destroyed.

• The population or number of bubbles per unit volume is known inadvance.



The following limitations apply to the cavitation model in FLUENT:

• The cavitation model cannot be used with the VOF model, becausethe surface tracking schemes for the VOF model are incompatiblewith the interpenetrating continua assumption of the cavitationmodel.

• The cavitation model can be used only for multiphase simula-tions that use the mixture or Eulerian model and involve onlytwo phases. It is always preferable to solve for cavitation using themixture model without slip velocity; Eulerian multiphase or slipvelocities can be turned on if the problem suggests that there issignificant slip between phases.

• With the cavitation model, only the secondary phase can be com-pressible; the primary phase must be incompressible.

Volume and Number of Bubbles

The volume of the individual bubbles is changing with respect to spaceand time and denoted by

φ(~r, t) =43πR3 (20.5-1)

where R is the bubble radius.

The volume fraction of vapor is defined as

αv =φη

1 + φη(20.5-2)

where η is the population or number of bubbles per unit volume of liquid.

The Volume Fraction Equation

The volume fraction equation is derived from the continuity equation forthe mixture (m). After some manipulation, assuming an incompressibleliquid (l), the following expression can be obtained:



∂

∂t(αp) + ∇(αp~vm) =

ρl

ρm

η

(1 + ηφ)2dφ

dt+αρv

ρm

dρv

dt(20.5-3)

Bubble Dynamics

Since cavitation bubbles will form in a liquid at low temperatures, FLU-ENT models the cavitating flow as isothermal, neglecting the latent heatof vaporization. The Rayleigh-Plesset equation relates the pressure andthe bubble volume φ:

Rd2R

dt2+

32

(dR

dt

)2

=pB − p

ρl− 2σρlR

− 4µl

ρlR

dR

dt(20.5-4)

Here pB denotes the pressure within the bubble, represented by the sumof a partial pressure (pv) of the vapor and a partial pressure of non-condensable gas (p), and σ is the surface tension coefficient.

To simplify the calculation, FLUENT assumes that the process of bubblegrowth and collapse is given by

dR

dt=

√2(pB−p)

3ρl, pv > p

−√

2(pB−p)3ρl

, pv < p(20.5-5)



20.6 Setting Up a General Multiphase Problem

This section provides instructions and guidelines for using the VOF,mixture, and Eulerian multiphase models. Information is presented inthe following subsections:

• Section 20.6.1: Steps for Using the General Multiphase Models

• Section 20.6.2: Additional Guidelines for Eulerian Multiphase Sim-ulations

• Section 20.6.3: Enabling the Multiphase Model and Specifying theNumber of Phases

• Section 20.6.4: Selecting the VOF Formulation

• Section 20.6.5: Defining a Homogeneous Multiphase Flow

• Section 20.6.6: Including Cavitation Effects

• Section 20.6.7: Overview of Defining the Phases

• Section 20.6.8: Defining Phases for the VOF Model

• Section 20.6.9: Defining Phases for the Mixture Model

• Section 20.6.10: Defining Phases for the Eulerian Model

• Section 20.6.11: Including Body Forces

• Section 20.6.12: Setting Time-Dependent Parameters for the VOFModel

• Section 20.6.13: Selecting a Turbulence Model for an Eulerian Mul-tiphase Calculation

• Section 20.6.14: Setting Boundary Conditions

• Section 20.6.15: Setting Initial Volume Fractions

• Section 20.6.16: Inputs for Compressible VOF and Mixture ModelCalculations

• Section 20.6.17: Inputs for Solidification/Melting VOF Calcula-tions



20.6.1 Steps for Using the General Multiphase Models

The procedure for setting up and solving a general multiphase problemis outlined below, and described in detail in the subsections that fol-low. Remember that only the steps that are pertinent to general multi-phase calculations are shown here. For information about inputs relatedto other models that you are using in conjunction with the multiphasemodel, see the appropriate sections for those models.

See also Section 20.6.2 for guidelines on simplifying Eulerian multiphasesimulations.

1. Enable the multiphase model you want to use (VOF, mixture, orEulerian) and specify the number of phases. For the VOF model,specify the VOF formulation as well.

Define −→ Models −→Multiphase...

See Sections 20.6.3 and 20.6.4 for details.

2. Copy the material representing each phase from the materials data-base.

Define −→Materials...

If the material you want to use is not in the database, create anew material. See Section 7.1.2 for details about copying fromthe database and creating new materials. See Section 20.6.16 foradditional information about specifying material properties for acompressible phase (VOF and mixture models only).

If your model includes a particulate (granular) phase, you will need!to create a new material for it in the fluid materials category (notthe solid materials category).

3. Define the phases, and specify any interaction between them (e.g.,surface tension if you are using the VOF model, slip velocity func-tions if you are using the mixture model, or drag functions if youare using the Eulerian model).

Define −→Phases...

See Sections 20.6.7–20.6.10 for details.



4. (Eulerian model only) If the flow is turbulent, define the multiphaseturbulence model.

Define −→ Models −→Viscous...

See Section 20.6.13 for details.

5. If body forces are present, turn on gravity and specify the gravita-tional acceleration.

Define −→Operating Conditions...


6. Specify the boundary conditions, including the secondary-phasevolume fractions at flow boundaries and (if you are modeling walladhesion in a VOF simulation) the contact angles at walls.

Define −→Boundary Conditions...


7. Set any model-specific solution parameters.

Solve −→ Controls −→Solution...

See Sections 20.6.12 and 20.7 for details.

8. Initialize the solution and set the initial volume fractions for thesecondary phases.

Solve −→ Initialize −→Patch...


9. Calculate a solution and examine the results.

See Sections 20.7 and 20.8 for details.

20.6.2 Additional Guidelines for Eulerian MultiphaseSimulations

Once you have determined that the Eulerian multiphase model is ap-propriate for your problem (as described in Sections 18.4 and 20.1), youshould consider the computational effort required to solve your multi-phase problem. The required computational effort depends strongly on



the number of transport equations being solved and the degree of cou-pling. For the Eulerian multiphase model, which has a large number ofhighly coupled transport equations, computational expense will be high.Before setting up your problem, try to reduce the problem statement tothe simplest form possible.

Instead of trying to solve your multiphase flow in all of its complexityon your first solution attempt, you can start with simple approximationsand work your way up to the final form of the problem definition. Somesuggestions for simplifying a multiphase flow problem are listed below:

• Use a hexahedral or quadrilateral mesh (instead of a tetrahedralor triangular mesh).

• Reduce the number of phases.

You may find that even a very simple approximation will provide youwith useful information about your problem.

See Section 20.7.3 for more solution strategies for Eulerian multiphasecalculations.

20.6.3 Enabling the Multiphase Model and Specifying theNumber of Phases

To enable the VOF, mixture, or Eulerian multiphase model, select Vol-ume of Fluid, Mixture, or Eulerian as the Model in the Multiphase Modelpanel (Figure 20.6.1).

Define −→ Models −→Multiphase...

The panel will expand to show the relevant inputs for the selected mul-tiphase model.

If you selected the VOF model, the inputs are as follows:

• number of phases

• VOF formulation (see Section 20.6.4)



Figure 20.6.1: The Multiphase Model Panel

• (optional) implicit body force formulation (see Section 20.6.11)

If you selected the mixture model, the inputs are as follows:


• whether or not to compute the slip velocities (see Section 20.6.5)

• (optional) implicit body force formulation (see Section 20.6.11)

• (optional) cavitation effects (see Section 20.6.6)



If you selected the Eulerian model, the inputs are as follows:


• (optional) cavitation effects (see Section 20.6.6)

To specify the number of phases for the multiphase calculation, enter theappropriate value in the Number of Phases field. You can specify up to20 phases.

20.6.4 Selecting the VOF Formulation

To specify the VOF formulation to be used, select the appropriate VOFScheme under VOF Parameters in the Multiphase Model panel.

The VOF formulations that are available in FLUENT are as follows:

• Time-dependent with the geometric reconstruction interpolationscheme: This formulation should be used whenever you are inter-ested in the time-accurate transient behavior of the VOF solution.

To use this formulation, select Geo-Reconstruct (the default) as theVOF Scheme. FLUENT will automatically turn on the unsteadyformulation with first-order discretization for time in the Solverpanel.

• Time-dependent with the donor-acceptor interpolation scheme:This formulation should be used instead of the time-dependentformulation with the geometric reconstruction scheme if your meshcontains highly twisted hexahedral cells. For such cases, the donor-acceptor scheme may provide more accurate results.

To use this formulation, select Donor-Acceptor as the VOF Scheme.FLUENT will automatically turn on the unsteady formulation withfirst-order discretization for time in the Solver panel.

• Time-dependent with the Euler explicit interpolation scheme: Sincethe donor-acceptor scheme is available only for quadrilateral andhexahedral meshes, it cannot be used for a hybrid mesh containing



twisted hexahedral cells. For such cases, you should use the time-dependent Euler explicit scheme. This formulation can also beused for other cases in which the geometric reconstruction schemedoes not give satisfactory results, or the flow calculation becomesunstable.

To use this formulation, select Euler Explicit as the VOF Scheme.FLUENT will automatically turn on the unsteady formulation withfirst-order discretization for time in the Solver panel.

While the Euler explicit time-dependent formulation is less com-putationally expensive than the geometric reconstruction scheme,the interface between phases will not be as sharp as that predictedwith the geometric reconstruction scheme. To reduce this diffusiv-ity, it is recommended that you use the second-order discretizationscheme for the volume fraction equations. In addition, you maywant to consider turning the geometric reconstruction scheme backon after calculating a solution with the implicit scheme, in orderto obtain a sharper interface.

• Time-dependent with the implicit interpolation scheme: This for-mulation can be used if you are looking for a steady-state solutionand you are not interested in the intermediate transient flow behav-ior, but the final steady-state solution is dependent on the initialflow conditions and/or you do not have a distinct inflow boundaryfor each phase.

To use this formulation, select Implicit as the VOF Scheme, andenable an Unsteady calculation in the Solver panel (opened withthe Define/Models/Solver... menu item).

The issues discussed above for the Euler explicit time-dependent!formulation also apply to the implicit time-dependent formulation.You should take the precautions described above to improve thesharpness of the interface.

• Steady-state with the implicit interpolation scheme: This formu-lation can be used if you are looking for a steady-state solution,you are not interested in the intermediate transient flow behavior,and the final steady-state solution is not affected by the initial flowconditions and there is a distinct inflow boundary for each phase.



To use this formulation, select Implicit as the VOF Scheme.

The issues discussed above for the Euler explicit time-dependent!formulation also apply to the implicit steady-state formulation.You should take the precautions described above to improve thesharpness of the interface.

For the geometric reconstruction and donor-acceptor schemes, if you are!using a conformal grid (i.e., if the grid node locations are identical at theboundaries where two subdomains meet), you must ensure that there areno two-sided (zero-thickness) walls within the domain. If there are, youwill need to slit them, as described in Section 5.7.8.

Examples

To help you determine the best formulation to use for your problem,examples that use different formulations are listed below:

• jet breakup: time-dependent with the geometric reconstructionscheme (or the donor-acceptor or Euler explicit scheme if prob-lems occur with the geometric reconstruction scheme)

• shape of the liquid interface in a centrifuge: time-dependent withthe implicit interpolation scheme

• flow around a ship’s hull: steady-state with the implicit interpola-tion scheme

20.6.5 Defining a Homogeneous Multiphase Flow

If you are using the mixture model, you have the option to disable thecalculation 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 de-scribed in Section 20.3.4. If you want to solve a homogeneous multiphaseflow, turn off Slip Velocity under Mixture Parameters.



20.6.6 Including Cavitation Effects

For mixture and Eulerian calculations, it is possible to include the effectsof cavitation. To enable the cavitation model, turn on Cavitation underInterphase Mass Transfer in the Multiphase Model panel.

Next you will specify two parameters to be used in the calculation of masstransfer due to cavitation. Specification of these parameters should be inagreement with the characteristic parameters of the flow under investiga-tion: Reynolds number and cavitation number. Under Cavitation Param-eters in the Multiphase Model panel, set the Vaporization Pressure (pv inEquation 20.5-5) and the Bubble Number Density (η in Equation 20.5-2).The default value of η is 10000, as recommended by Kubota et al. [119].The default value of pv is 2367.8, the vaporization pressure for water atambient temperature.

See Section 20.5 for details about modeling cavitation.

20.6.7 Overview of Defining the Phases

To define the phases (including their material properties) and any inter-phase interaction (e.g., surface tension and wall adhesion for the VOFmodel, slip velocity function for the mixture model, drag functions forthe Eulerian model), you will use the Phases panel (Figure 20.6.2).

Define −→Phases...

Each item in the Phase list in this panel is one of two types, as indicatedin the Type list: primary-phase indicates that the selected item is theprimary phase, and secondary-phase indicates that the selected item is asecondary phase. To specify any interaction between the phases, clickthe Interaction... button.

Instructions for defining the phases and interaction are provided in Sec-tions 20.6.8, 20.6.9, and 20.6.10 for the VOF, mixture, and Eulerianmodels, respectively.



Figure 20.6.2: The Phases Panel

20.6.8 Defining Phases for the VOF Model

Instructions for specifying the necessary information for the primary andsecondary phases and their interaction in a VOF calculation are providedbelow.

In general, you can specify the primary and secondary phases whichever!way you prefer. It is a good idea, especially in more complicated prob-lems, 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 1for one phase in a portion of the domain, it may be more convenient tomake that phase a secondary phase. Also, if one of the phases is com-pressible, it is recommended that you specify it as the primary phase toimprove solution stability.

Recall that only one of the phases can be compressible. Be sure that!you do not select a compressible material (i.e., a material that uses thecompressible ideal gas law for density) for more than one of the phases.See Section 20.6.16 for details.



Defining the Primary Phase

To define the primary phase in a VOF calculation, follow these steps:

1. Select phase-1 in the Phase list.

2. Click Set..., and the Primary Phase panel (Figure 20.6.3) will open.

Figure 20.6.3: 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 appro-priate 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 themif necessary. (See Chapter 7 for general information aboutsetting material properties, Section 20.6.16 for specific infor-mation related to compressible VOF calculations, and Sec-tion 20.6.17 for specific information related to melting/solidi-fication VOF calculations.)

If you make changes to the properties, remember to click!Change before closing the Material panel.

6. Click OK in the Primary Phase panel.



Defining a Secondary Phase

To define a secondary phase in a VOF calculation, follow these steps:

1. Select the phase (e.g., phase-2) in the Phase list.

2. Click Set..., and the Secondary Phase panel (Figure 20.6.4) willopen.

Figure 20.6.4: 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 theprocedure outlined above for setting the material properties for theprimary phase.

6. Click OK in the Secondary Phase panel.

Including Surface Tension and Wall Adhesion Effects

As discussed in Section 20.2.8, the importance of surface tension effectsdepends on the value of the capillary number, Ca (defined by Equa-tion 20.2-16), or the Weber number, We (defined by Equation 20.2-17).Surface tension effects can be neglected if Ca � 1 or We � 1.

Note that the calculation of surface tension effects will be more accu-!



rate if you use a quadrilateral or hexahedral mesh in the area(s) of thecomputational domain where surface tension is significant. If you can-not use a quadrilateral or hexahedral mesh for the entire domain, thenyou should use a hybrid mesh, with quadrilaterals or hexahedra in theaffected areas.

If you want to include the effects of surface tension along the interfacebetween one or more pairs of phases, as described in Section 20.2.8, clickInteraction... to open the Phase Interaction panel (Figure 20.6.5).

Figure 20.6.5: The Phase Interaction Panel for the VOF Model

Follow the steps below to include surface tension (and, if appropriate,wall adhesion) effects along the interface between one or more pairs ofphases:

1. Turn on the Surface Tension option.



2. If you want to include wall adhesion, turn on the Wall Adhesion op-tion. (When Wall Adhesion is enabled, you will need to specify thecontact angle at each wall as a boundary condition (as describedin Section 20.6.14.)

3. For each pair of phases between which you want to include theeffects of surface tension, specify a constant surface tension coef-ficient. All surface tension coefficients are equal to 0 by default,representing no surface tension effects along the interface betweenthe two phases.

For calculations involving surface tension, it is recommended that you!also turn on the Implicit Body Force treatment for the Body Force Formu-lation in the Multiphase Model panel. This treatment improves solutionconvergence by accounting for the partial equilibrium of the pressuregradient and surface tension forces in the momentum equations. SeeSection 22.3.3 for details.

20.6.9 Defining Phases for the Mixture Model

Instructions for specifying the necessary information for the primary andsecondary phases and their interaction for a mixture model calculationare provided below.

Recall that only one of the phases can be compressible. Be sure that!you do not select a compressible material (i.e., a material that uses thecompressible ideal gas law for density) for more than one of the phases.See Section 20.6.16 for details.


The procedure for defining the primary phase in a mixture model cal-culation is the same as for a VOF calculation. See Section 20.6.8 fordetails.

Defining a Secondary Phase

To define a secondary phase in a mixture model calculation, follow thesesteps:





Figure 20.6.6: The Secondary Phase Panel for the Mixture Model



5. Define the material properties for the Phase Material, following thesame procedure you used to set the material properties for theprimary phase (see Section 20.6.8). For a particulate phase (whichmust be placed in the fluid materials category, as mentioned inSection 20.6.1), you need to specify only the density; you can ignorethe 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 (dp in Equation 20.3-12). Youcan specify a constant value, or use a user-defined function. Seethe separate UDF Manual for details about user-defined functions.


Defining the Slip Velocity

If you are solving for slip velocities during the mixture calculation, andyou want to modify the slip velocity definition, click Interaction... toopen the Phase Interaction panel (Figure 20.6.7).

Figure 20.6.7: The Phase Interaction Panel for the Mixture Model

Under Slip Velocity, you can specify the slip velocity function for eachsecondary phase with respect to the primary phase by choosing the ap-propriate item in the adjacent drop-down list.

• Select maninnen-et-al (the default) to use the algebraic slip methodof Manninen et al. [150], described in Section 20.3.4.



• Select none if the secondary phase has the same velocity as theprimary phase (i.e., no slip velocity).

• Select user-defined to use a user-defined function for the slip veloc-ity. See the separate UDF Manual for details.

20.6.10 Defining Phases for the Eulerian Model

Instructions for specifying the necessary information for the primaryand secondary phases and their interaction for an Eulerian multiphasecalculation are provided below.


The procedure for defining the primary phase in an Eulerian multiphasecalculation is the same as for a VOF calculation. See Section 20.6.8 fordetails.

Defining a Non-Granular Secondary Phase

To define a non-granular (i.e., liquid or vapor) secondary phase in anEulerian multiphase calculation, follow these steps:





5. Define the material properties for the Phase Material, following thesame procedure you used to set the material properties for theprimary phase (see Section 20.6.8).

6. In the Secondary Phase panel, specify the Diameter of the bubblesor droplets of this phase. You can specify a constant value, or usea user-defined function. See the separate UDF Manual for detailsabout user-defined functions.



Figure 20.6.8: The Secondary Phase Panel for a Non-Granular Phase


Defining a Granular Secondary Phase

To define a granular (i.e., particulate) secondary phase in an Eulerianmultiphase calculation, follow these steps:






Figure 20.6.9: The Secondary Phase Panel for a Granular Phase


5. Define the material properties for the Phase Material, following thesame procedure you used to set the material properties for theprimary phase (see Section 20.6.8). For a granular phase (whichmust be placed in the fluid materials category, as mentioned inSection 20.6.1), you need to specify only the density; you can ignorethe values for the other properties, since they will not be used.

6. In the Secondary Phase panel, specify the following properties ofthe 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, orselect user-defined to use a user-defined function. See the sep-arate UDF Manual for details about user-defined functions.

Granular Viscosity specifies the kinetic part of the granular viscos-ity of the particles (µs,kin in Equation 20.4-50). You can selectconstant (the default) in the drop-down list and specify a con-stant value, select syamlal-obrien to compute the value usingEquation 20.4-52, select gidaspow to compute the value usingEquation 20.4-53, or select user-defined to use a user-definedfunction.

Granular Bulk Viscosity specifies the solids bulk viscosity (λq in Equa-tion 20.4-8). You can select constant (the default) in the drop-down list and specify a constant value, select lun-et-al to com-pute the value using Equation 20.4-54, or select user-definedto use a user-defined function.

Frictional Viscosity specifies a shear viscosity based on the viscous-plastic flow (µs,fr in Equation 20.4-50). By default, the fric-tional viscosity is neglected, as indicated by the default se-lection of none in the drop-down list. If you want to includethe frictional viscosity, you can select constant and specifya constant value, select schaeffer to compute the value usingEquation 20.4-55, or select user-defined to use a user-definedfunction.

Angle of Internal Friction specifies a constant value for the angle φused in Schaeffer’s expression for frictional viscosity (Equa-tion 20.4-55). This parameter is relevant only if you haveselected schaeffer or user-defined for the Frictional Viscosity.

Packing Limit specifies the maximum volume fraction for the gran-ular phase (αs,max). For monodispersed spheres the packinglimit is about 0.63, which is the default value in FLUENT. Inpolydispersed cases, however, smaller spheres can fill the smallgaps between larger spheres, so you may need to increase themaximum packing limit.




Defining the Interaction Between Phases

For both granular and non-granular flows, you will need to specify thedrag function to be used in the calculation of the momentum exchangecoefficients. For granular flows, you will also need to specify the restitu-tion coefficient(s) for particle collisions. It is also possible to include anoptional lift force and/or virtual mass force (described below) for bothgranular and non-granular flows.

To specify these parameters, click Interaction... to open the Phase Inter-action panel (Figure 20.6.10).

Figure 20.6.10: The Phase Interaction Panel for the Eulerian Model

Specifying the Drag Function

FLUENT allows you to specify a drag function for each pair of phases.Follow the steps below:



1. Click the Drag tab to display the Drag Function inputs.

2. For each pair of phases, select the appropriate drag function fromthe corresponding drop-down list.

• Select schiller-naumann to use the fluid-fluid drag functiondescribed by Equation 20.4-19. The Schiller and Naumannmodel is the default method, and it is acceptable for generaluse in all fluid-fluid multiphase calculations.

• Select morsi-alexander to use the fluid-fluid drag function de-scribed by Equation 20.4-23. The Morsi and Alexander modelis the most complete, adjusting the function definition fre-quently over a large range of Reynolds numbers, but calcula-tions with this model may be less stable than with the othermodels.

• Select symmetric to use the fluid-fluid drag function describedby Equation 20.4-28. The symmetric model is recommendedfor flows in which the secondary (dispersed) phase in one re-gion of the domain becomes the primary (continuous) phasein another. For example, if air is injected into the bottom ofa container filled halfway with water, the air is the dispersedphase in the bottom half of the container; in the top half ofthe container, the air is the continuous phase.

• Select wen-yu to use the fluid-solid drag function described byEquation 20.4-40. The Wen and Yu model is applicable fordilute phase flows, in which the total secondary phase volumefraction is significantly lower than that of the primary phase.

• Select gidaspow to use the fluid-solid drag function describedby Equation 20.4-42. The Gidaspow model is recommendedfor dense fluidized beds.

• Select syamlal-obrien to use the fluid-solid drag function de-scribed by Equation 20.4-32. The Syamlal-O’Brien model isrecommended for use in conjunction with the Syamlal-O’Brienmodel for granular viscosity.

• Select syamlal-obrien-symmetric to use the solid-solid drag func-tion described by Equation 20.4-44. The symmetric Syamlal-



O’Brien model is appropriate for a pair of solid phases.

• Select constant to specify a constant value for the drag func-tion, and then specify the value in the text field.

• Select user-defined to use a user-defined function for the dragfunction (see the separate UDF Manual for details).

• If you want to temporarily ignore the interaction between twophases, select none.

Specifying the Restitution Coefficients (Granular Flow Only)

For granular flows, you need to specify the coefficients of restitution forcollisions between particles (els in Equation 20.4-44 and ess in Equa-tion 20.4-45). In addition to specifying the restitution coefficient forcollisions between each pair of granular phases, you will also specify therestitution coefficient for collisions between particles of the same phase.

Follow the steps below:

1. Click the Collisions tab 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.

Including the Lift Force

For both granular and non-granular flows, it is possible to include theeffect of lift forces (Flift in Equation 20.4-10) on the secondary phaseparticles, droplets, or bubbles. These lift forces act on a particle, droplet,or bubble mainly due to velocity gradients in the primary-phase flowfield. In most cases, the lift force is insignificant compared to the dragforce, 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.

Note that the lift force will be more significant for larger particles, but!the FLUENT model assumes that the particle diameter is much smallerthan the interparticle spacing. Thus, the inclusion of lift forces is notappropriate for closely packed particles or for very small particles.

To include the effect of lift forces, follow these steps:



1. Click the Lift tab to display the Lift Coefficient inputs.

2. For each pair of phases, select the appropriate specification methodfrom the corresponding drop-down list. Note that, since the liftforces for a particle, droplet, or bubble are due mainly to velocitygradients in the primary-phase flow field, you will not specify liftcoefficients for pairs consisting of two secondary phases; lift coeffi-cients are specified only for pairs consisting of a secondary phaseand the primary phase.

• Select none (the default) to ignore the effect of lift forces.

• Select constant to specify a constant lift coefficient, and thenspecify the value in the text field.

• Select user-defined to use a user-defined function for the liftcoefficient (see the separate UDF Manual for details).

Including the Virtual Mass Force

For both granular and non-granular flows, it is possible to include the“virtual mass force” (Fvm in Equation 20.4-11) that is present whena secondary phase accelerates relative to the primary phase. The vir-tual mass effect is significant when the secondary phase density is muchsmaller than the primary phase density (e.g., for a transient bubble col-umn).

To include the effect of the virtual mass force, turn on the Virtual Massoption in the Phase Interaction panel. The virtual mass effect will beincluded for all secondary phases; it is not possible to enable it just fora particular phase.

20.6.11 Including Body Forces

In many cases, the motion of the phases is due, in part, to gravitationaleffects. To include this body force, turn on Gravity in the OperatingConditions panel and specify the Gravitational Acceleration.




For VOF calculations, you should also turn on the Specified OperatingDensity option in the Operating Conditions panel, and set the Operat-ing Density to be the density of the lightest phase. (This excludes thebuildup of hydrostatic pressure within the lightest phase, improving theround-off accuracy for the momentum balance.) If any of the phases iscompressible, set the Operating Density to zero.

For VOF and mixture calculations involving body forces, it is recom-!mended that you also turn on the Implicit Body Force treatment for theBody Force Formulation in the Multiphase Model panel. This treatmentimproves solution convergence by accounting for the partial equilibriumof the pressure gradient and body forces in the momentum equations.See Section 22.3.3 for details.

20.6.12 Setting Time-Dependent Parameters for the VOFModel

If you are using the time-dependent VOF formulation in FLUENT, anexplicit solution for the volume fraction is obtained either once each timestep or once each iteration, depending upon your inputs to the model.You also have control over the time step used for the volume fractioncalculation.

To compute a time-dependent VOF solution, you will need to enablethe Unsteady option in the Solver panel (and choose the appropriateUnsteady Formulation, as discussed in Section 22.15.1). If you choose theGeo-Reconstruct, Donor-Acceptor, or Euler Explicit scheme, FLUENT willturn on the first-order unsteady formulation for you automatically, soyou need not visit the Solver panel yourself.

Define −→ Models −→Solver...

There are two inputs for the time-dependent calculation in the MultiphaseModel panel:

• By default, FLUENT will solve the volume fraction equation(s) oncefor each time step. This means that the convective flux coefficientsappearing in the other transport equations will not be completely



updated each iteration, since the volume fraction fields will notchange from iteration to iteration.

If you want FLUENT to solve the volume fraction equation(s) atevery iteration within a time step, turn on the Solve VOF EveryIteration option under VOF Parameters. When FLUENT solves theseequations every iteration, the convective flux coefficients in theother transport equations will be updated based on the updatedvolume fractions at each iteration.

In general, if you anticipate that the location of the interface willchange as the other flow variables converge during the time step,you should enable the Solve VOF Every Iteration option. This situa-tion arises when large time steps are being used in hopes of reachinga steady-state solution, for example. If small time steps are beingused, however, it is not necessary to perform the additional workof solving for the volume fraction every iteration, so you can leavethis option turned off. This choice is the more stable of the two,and requires less computational effort per time step than the firstchoice.

If you are using sliding meshes, using the Solve VOF Every Itera-!tion option may yield more accurate results, although at a greatercomputational cost.

• When FLUENT performs a time-dependent VOF calculation, thetime step used for the volume fraction calculation will not be thesame as the time step used for the rest of the transport equations.FLUENT will refine the time step for VOF automatically, basedon your input for the maximum Courant Number allowed near thefree surface. The Courant number is a dimensionless number thatcompares the time step in a calculation to the characteristic timeof transit of a fluid element across a control volume:

∆t∆xcell/vfluid

(20.6-1)

In the region near the fluid interface, FLUENT divides the volumeof each cell by the sum of the outgoing fluxes. The resulting timerepresents the time it would take for the fluid to empty out of the



cell. The smallest such time is used as the characteristic time oftransit for a fluid element across a control volume, as describedabove. Based upon this time and your input for the maximumallowed Courant Number, a time step is computed for use in theVOF calculation. For example, if the maximum allowed Courantnumber is 0.25 (the default), the time step will be chosen to beat most one-fourth the minimum transit time for any cell near theinterface.

Note that these inputs are not required when the implicit scheme is used.

20.6.13 Selecting a Turbulence Model for an EulerianMultiphase Calculation

If you are using the Eulerian model to solve a turbulent flow, you willneed to choose one of the three turbulence models described in Sec-tion 20.4.7 in the Viscous Model panel (Figure 20.6.11).

The procedure is as follows:

1. Select k-epsilon under Model.

2. Select the desired k-epsilon Model and any other related parameters,as described for single-phase calculations in Section 10.10.

3. Under k-epsilon Multiphase Model, indicate the desired multiphaseturbulence model (see Section 20.4.7 for details about each):

• Select Mixture to use the mixture turbulence model. This isthe default model.

• Select Dispersed to use the dispersed turbulence model. Thismodel is applicable when there is clearly one primary continu-ous phase and the rest are dispersed dilute secondary phases.

• Select Per Phase to use a k-ε turbulence model for each phase.This model is appropriate when the turbulence transfer amongthe phases plays a dominant role.



Figure 20.6.11: The Viscous Model Panel for an Eulerian MultiphaseCalculation

Including Source Terms

By default, the interphase momentum, k, and ε sources are not includedin the calculation. If you want to include any of these source terms,you can enable them using the multiphase-options command in thedefine/models/viscous/multiphase-turbulence/ text menu. Notethat the inclusion of these terms can slow down convergence noticeably.If you are looking for additional accuracy, you may want to compute asolution first without these sources, and then continue the calculationwith these terms included. In most cases these terms can be neglected.



20.6.14 Setting Boundary Conditions

Multiphase boundary conditions are set in the Boundary Conditions panel(Figure 20.6.12), but the procedure for setting multiphase boundary con-ditions is slightly different than for single-phase models. You will needto set some conditions separately for individual phases, while other con-ditions are shared by all phases (i.e., the mixture), as described in detailbelow.

Define −→Boundary Conditions...

Figure 20.6.12: The Boundary Conditions Panel

Boundary Conditions for the Mixture and the Individual Phases

The conditions you need to specify for the mixture and those you needto specify for the individual phases will depend on which of the threemultiphase models you are using. Details for each model are providedbelow.



VOF Model

If you are using the VOF model, the conditions you need to specify foreach type of zone are listed below and summarized in Table 20.6.1.

• For an exhaust fan, inlet vent, intake fan, mass flow inlet, outletvent, pressure inlet, pressure outlet, or velocity inlet, there are noconditions to be specified for the primary phase. For each sec-ondary phase, you will need to set the volume fraction as a con-stant, a profile (see Section 6.25), or a user-defined function (seethe separate UDF Manual). All other conditions are specified forthe 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.

• 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 forthe mixture.

If the fluid zone is porous, you will enable the Porous Zone optionin the Fluid panel for the mixture. The porosity inputs (if rele-vant) are also specified for the mixture. The resistance coefficientsand direction vectors, however, are specified separately for eachphase. See Section 6.19.6 for details about these inputs. All otherconditions are specified for the mixture.

See Chapter 6 for details about the relevant conditions for each type ofboundary. Note that the pressure far-field boundary is not available withthe VOF model.



Table 20.6.1: Phase-Specific and Mixture Conditions for the VOF Model

Type Primary Phase Secondary Phase Mixtureexhaust fan nothing volume fraction all othersinlet ventintake fan

mass flow inletoutlet vent

pressure inletpressure outletvelocity inlet

axis nothing nothing all othersfan

outflowperiodic

porous jumpradiator

solidsymmetry

wallpressure far-field not available not available not available

fluid mass source; mass source; porous zone;other porous other porous porosity;

inputs inputs all others



Mixture Model

If you are using the mixture model, the conditions you need to specifyfor each type of zone are listed below and summarized in Table 20.6.2.

• For an exhaust fan, outlet vent, or pressure outlet, there are no con-ditions to be specified for the primary phase. For each secondaryphase, you will need to set the volume fraction as a constant, a pro-file (see Section 6.25), or a user-defined function (see the separateUDF Manual). All other conditions are specified for the mixture.

• For an inlet vent, intake fan, mass flow inlet, or pressure inlet, youwill specify for the mixture which direction specification methodwill be used at this boundary (Normal to Boundary or DirectionVector). If you select the Direction Vector specification method,you will specify the coordinate system (3D only) and flow-directioncomponents for the individual phases. For each secondary phase,you will need to set the volume fraction (as described above). Allother conditions are specified for the mixture.

• For a velocity inlet, you will specify the velocity for the individualphases. For each secondary phase, you will need to set the volumefraction (as described above). All other conditions are specified forthe 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.

• 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 optionin the Fluid panel for the mixture. The porosity inputs (if rele-vant) are also specified for the mixture. The resistance coefficientsand direction vectors, however, are specified separately for eachphase. See Section 6.19.6 for details about these inputs. All otherconditions are specified for the mixture.



See Chapter 6 for details about the relevant conditions for each type ofboundary. Note that the pressure far-field boundary is not available withthe mixture model.

Table 20.6.2: Phase-Specific and Mixture Conditions for the Mixture Model

Type Primary Phase Secondary Phase Mixtureexhaust fan nothing volume fraction all othersoutlet vent

pressure outletinlet vent coord. system; coord. system; dir. spec.intake fan flow direction flow direction; method;

mass flow inlet volume fraction all otherspressure inletvelocity inlet velocity velocity; all others

volume fractionaxis nothing nothing all othersfan

outflowperiodic

porous jumpradiator

solidsymmetry

wallpressure far-field not available not available not available

fluid mass source; mass source; porous zone;other porous other porous porosity;

inputs inputs all others



Eulerian Model

If you are using the Eulerian model, the conditions you need to specifyfor each type of zone are listed below and summarized in Tables 20.6.3,20.6.4, 20.6.5, and 20.6.6. Note that the specification of turbulence pa-rameters will depend on which of the three multiphase turbulence modelsyou are using, as indicated in Tables 20.6.4–20.6.6. See Sections 20.4.7and 20.6.13 for more information about multiphase turbulence models.

• For an exhaust fan, outlet vent, or pressure outlet, there are noconditions to be specified for the primary phase if you are modelinglaminar flow or using the mixture turbulence model (the defaultmultiphase turbulence model).

For each secondary phase, you will need to set the volume fractionas a constant, a profile (see Section 6.25), or a user-defined function(see the separate UDF Manual). If the phase is granular, you willalso need to set its granular temperature.

If you are using the mixture turbulence model, you will need tospecify the turbulence boundary conditions for the mixture; if youare using the dispersed turbulence model, you will need to specifythem for the primary phase; if you are using the per-phase turbu-lence 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 inlet vent, intake fan, mass flow inlet, or pressure inlet, youwill specify for the mixture which direction specification methodwill be used at this boundary (Normal to Boundary or DirectionVector). If you select the Direction Vector specification method,you will specify the coordinate system (3D only) and flow-directioncomponents 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 needto set its granular temperature.

If you are using the mixture turbulence model, you will need tospecify the turbulence boundary conditions for the mixture; if you



are using the dispersed turbulence model, you will need to specifythem for the primary phase; if you are using the per-phase turbu-lence model, you will need to specify them for the primary phaseand for each secondary phase.


• For a velocity inlet, you will specify the velocity for the individualphases.

For each secondary phase, you will need to set the volume fraction(as described above). If the phase is granular, you will also needto set its granular temperature.

If you are using the mixture turbulence model, you will need tospecify the turbulence boundary conditions for the mixture; if youare using the dispersed turbulence model, you will need to specifythem for the primary phase; if you are using the per-phase turbu-lence model, you will need to specify them for the primary phaseand for each secondary phase.


• For an axis, outflow, periodic, solid, or symmetry zone, all condi-tions are specified for the mixture; there are no conditions to beset for the individual phases.

• For a wall zone, shear conditions are specified for the individualphases; all other conditions are specified for the mixture.

• For a fluid zone, all source terms and fixed values are specified forthe individual phases, unless you are using the mixture turbulencemodel or the dispersed turbulence model. If you are using the mix-ture turbulence model, source terms and fixed values for turbulenceare specified instead for the mixture; if you are using the dispersedturbulence model, they are specified only for the primary phase.


If the fluid zone is porous, you will enable the Porous Zone optionin the Fluid panel for the mixture. The porosity inputs (if rele-vant) are also specified for the mixture. The resistance coefficients



and direction vectors, however, are specified separately for eachphase. See Section 6.19.6 for details about these inputs. All otherconditions are specified for the mixture.

See Chapter 6 for details about the relevant conditions for each typeof boundary. Note that the pressure far-field, fan, porous jump, andradiator boundaries are not available with the Eulerian model.



Table 20.6.3: Phase-Specific and Mixture Conditions for the Eulerian Model(for Laminar Flow)

Type Primary Phase Secondary Phase Mixtureexhaust fan nothing volume fraction; all othersoutlet vent gran. temperature


mass flow inlet volume fraction; all otherspressure inlet gran. temperaturevelocity inlet velocity velocity; all others

volume fraction;gran. temperature

axis nothing nothing all othersoutflowperiodic

solidsymmetry

wall shear condition shear condition all otherspressure far-field not available not available not available

fanporous jump

radiatorfluid all source terms; all source terms; porous zone;

all fixed values; all fixed values; porosity;other porous other porous all others

inputs inputs



Table 20.6.4: Phase-Specific and Mixture Conditions for the Eulerian Model(with the Mixture Turbulence Model)

Type Primary Phase Secondary Phase Mixtureexhaust fan nothing volume fraction; all othersoutlet vent gran. temperature


mass flow inlet volume fraction; all otherspressure inlet gran. temperaturevelocity inlet velocity velocity; all others

volume fraction;gran. temperature


solidsymmetry


fanporous jump

radiatorfluid other source other source source terms

terms; terms; for turbulence;other fixed other fixed fixed values

values; values; for turbulence;other porous other porous porous zone;

inputs inputs porosity;all others



Table 20.6.5: Phase-Specific and Mixture Conditions for the Eulerian Model(with the Dispersed Turbulence Model)

Type Primary Phase Secondary Phase Mixtureexhaust fan turb. parameters volume fraction; all othersoutlet vent gran. temperature

pressure outletinlet vent coord. system; coord. system; dir. spec.intake fan flow direction; flow direction; method

mass flow inlet turb. parameters; volume fraction; all otherspressure inlet gran. temperaturevelocity inlet velocity; velocity; all others

turb. parameters volume fraction;gran. temperature


solidsymmetry


fanporous jump

radiatorfluid momentum, mass, momentum and porous zone;

turb. sources; mass sources; porosity;momentum, mass, momentum and all othersturb. fixed values; mass fixed values;

other porous other porousinputs inputs



Table 20.6.6: Phase-Specific and Mixture Conditions for the Eulerian Model(with the Per-Phase Turbulence Model)

Type Primary Phase Secondary Phase Mixtureexhaust fan turb. parameters volume fraction; all othersoutlet vent turb. parameters;

pressure outlet gran. temperatureinlet vent coord. system; coord. system; dir. spec.intake fan flow direction; flow direction; method;

mass flow inlet turb. parameters volume fraction; all otherspressure inlet turb. parameters;

gran. temperaturevelocity inlet velocity; velocity; all others

turb. parameters volume fraction;turb. parameters;gran. temperature


solidsymmetry


fanporous jump

radiatorfluid momentum, mass, momentum, mass, porous zone;

turb. sources; turb. sources; porosity;momentum, mass, momentum, mass, all othersturb. fixed values; turb. fixed values;

other porous other porousinputs inputs



Steps for Setting Boundary Conditions

The steps you need to perform for each boundary are as follows:

1. Select the boundary in the Zone list in the Boundary Conditionspanel.

2. Set the conditions for the mixture at this boundary, if necessary.(See above for information about which conditions need to be setfor the mixture.)

(a) In the Phase drop-down list, select mixture.

(b) If the current Type for this zone is correct, click Set... to openthe corresponding panel (e.g., the Pressure Inlet panel); oth-erwise, choose the correct zone type in the Type list, confirmthe change (when prompted), and the corresponding panelwill open automatically.

(c) In the corresponding panel for the zone type you have selected(e.g., the Pressure Inlet panel, shown in Figure 20.6.13), specifythe mixture boundary conditions.

Figure 20.6.13: The Pressure Inlet Panel for a Mixture

Note that only those conditions that apply to all phases, asdescribed above, will appear in this panel.

For a VOF calculation, if you enabled the Wall Adhesion op-!



tion in the Phase Interaction panel, you can specify the con-tact angle at the wall for each pair of phases (as shown inFigure 20.6.14).

The contact angle (θw in Figure 20.2.2) is the angle betweenthe wall and the tangent to the interface at the wall, measuredinside the first phase of the pair listed in the Wall panel. Forexample, if you are setting the contact angle between the oiland air phases in the Wall panel shown in Figure 20.6.14, θw

is measured inside the oil phase.

The default value for all pairs is 90 degrees, which is equiv-alent to no wall adhesion effects (i.e., the interface is normalto the adjacent wall). A contact angle of 45◦, for example,corresponds to water creeping up the side of a container, asis common with water in a glass.

(d) Click OK when you are done setting the mixture boundaryconditions.

3. Set the conditions for each phase at this boundary, if necessary.(See above for information about which conditions need to be setfor the individual phases.)

(a) In the Phase drop-down list, select the phase (e.g., water).

Note that, when you select one of the individual phases (rather!than the mixture), only one type of zone appears in the Typelist. It is not possible to assign phase-specific zone types ata 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 20.6.15).

(c) Specify the conditions for the phase. Note that only thoseconditions that apply to the individual phase, as describedabove, will appear in this panel.

(d) Click OK when you are done setting the phase-specific bound-ary conditions.



Figure 20.6.14: The Wall Panel for a Mixture in a VOF Calculation withWall Adhesion



Figure 20.6.15: The Pressure Inlet Panel for a Phase

Steps for Copying Boundary Conditions

The steps for copying boundary conditions for a multiphase flow areslightly different from those described in Section 6.1.5 for a single-phaseflow. The modified steps are listed below:

1. In the Boundary Conditions panel, click the Copy... button. Thiswill open the Copy BCs panel.

2. In the From Zone list, select the zone that has the conditions youwant to copy.

3. In the To Zones list, select the zone or zones to which you want tocopy the conditions.

4. In the Phase drop-down list, select the phase for which you want tocopy the conditions (either mixture or one of the individual phases).

Note that copying the boundary conditions for one phase does!not automatically result in the boundary conditions for the otherphases and the mixture being copied as well. You need to copy theconditions for each phase on each boundary of interest.

5. Click Copy. FLUENT will set all of the selected phase’s (or mix-ture’s) boundary conditions on the zones selected in the To Zones



list to be the same as that phase’s conditions on the zone selectedin the From Zone list. (You cannot copy a subset of the conditions,such as only the thermal conditions.)

See Section 6.1.5 for additional information about copying boundaryconditions, including limitations.

20.6.15 Setting Initial Volume Fractions

Once you have initialized the flow (as described in Section 22.13), youcan define the initial distribution of the phases. For a transient simula-tion, this distribution will serve as the initial condition at t = 0; for asteady-state simulation, setting an initial distribution can provide addedstability in the early stages of the calculation.

You can patch an initial volume fraction for each secondary phase usingthe Patch panel.

Solve −→ Initialize −→Patch...

If the region in which you want to patch the volume fraction is definedas a separate cell zone, you can simply patch the value there. Otherwise,you can create a cell “register” that contains the appropriate cells andpatch the value in the register. See Section 22.13.2 for details.

20.6.16 Inputs for Compressible VOF and Mixture ModelCalculations

If you are using the VOF or mixture model for a compressible flow, notethe following:

• Only one of the phases can be compressible (i.e., you can select theideal gas law for the density of only one phase’s material).

• If you are using the VOF model, for stability reasons it is better(although not required) if the primary phase is compressible.

• If you specify the total pressure at a boundary (e.g., for a pressureinlet or intake fan) the specified value for temperature at that



boundary will be used as total temperature for the compressiblephase, and as static temperature for the other phases (which areincompressible).

See Section 8.5 for more information about compressible flows.

20.6.17 Inputs for Solidification/Melting VOF Calculations

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 orin multiple phases.

• For phases that are not melting or solidifying, you must set thelatent heat (L), liquidus temperature (Tliquidus), and solidus tem-perature (Tsolidus) to zero.

See Chapter 21 for more information about melting and solidification.



20.7 Solution Strategies for General Multiphase Problems

Solution strategies for the VOF, mixture, and Eulerian models are pro-vided in Sections 20.7.1, 20.7.2, and 20.7.3, respectively.

20.7.1 Solution Strategies for the VOF Model

Several recommendations for improving the accuracy and convergence ofthe VOF solution are presented here.

Setting the Reference Pressure Location

The site of the reference pressure can be moved to a location that willresult in less round-off in the pressure calculation. By default, the refer-ence 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 Pres-sure Location in the Operating Conditions panel.


The position that you choose should be in a region that will alwayscontain the least dense of the fluids (e.g., the gas phase, if you have agas phase and one or more liquid phases). This is because variations inthe static pressure are larger in a more dense fluid than in a less densefluid, given the same velocity distribution. If the zero of the relativepressure field is in a region where the pressure variations are small, lessround-off will occur than if the variations occur in a field of large non-zero values. Thus in systems containing air and water, for example, it isimportant that the reference pressure location be in the portion of thedomain filled with air rather than that filled with water.

Pressure Interpolation Scheme

For all VOF calculations, you should use the body-force-weighted pres-sure interpolation scheme or the PRESTO! scheme.



20.7 Solution Strategies for General Multiphase Problems

Discretization Scheme Selection for the Implicit and EulerExplicit Formulations

When the implicit or Euler explicit scheme is used you should use thesecond-order or QUICK discretization scheme for the volume fractionequations in order to improve the sharpness of the interface betweenphases.


Pressure-Velocity Coupling and Under-Relaxation for theTime-Dependent Formulations

Another change that you should make to the solver settings is in thepressure-velocity coupling scheme and under-relaxation factors that youuse. The PISO scheme is recommended for transient calculations ingeneral. Using PISO allows for increased values on all under-relaxationfactors, without a loss of solution stability. You can generally increasethe under-relaxation factors for all variables to 1 and expect stabilityand a rapid rate of convergence (in the form of few iterations requiredper time step). For calculations on tetrahedral or triangular meshes,an under-relaxation factor of 0.7–0.8 for pressure is recommended forimproved stability with the PISO scheme.


As with any FLUENT simulation, the under-relaxation factors will needto be decreased if the solution exhibits unstable, divergent behavior withthe under-relaxation factors set to 1. Reducing the time step is anotherway to improve the stability.

Under-Relaxation for the Steady-State Formulation

If you are using the steady-state implicit VOF scheme, the under-relaxationfactors for all variables should be set to values between 0.2 and 0.5 forimproved stability.



20.7.2 Solution Strategies for the Mixture Model

Setting the Under-Relaxation Factor for the Slip Velocity

You should begin the mixture calculation with a low under-relaxationfactor for the slip velocity. A value of 0.2 or less is recommended. If thesolution shows good convergence behavior, you can increase this valuegradually.

Calculating an Initial Solution

For some cases (e.g., cyclone separation), you may be able to obtain asolution more quickly if you compute an initial solution without solvingthe volume fraction and slip velocity equations. Once you have set upthe mixture model, you can temporarily disable these equations andcompute an initial solution.


In the Solution Controls panel, deselect Volume Fraction and Slip Velocityin the Equations list. You can then compute the initial flow field. Oncea converged flow field is obtained, turn the Volume Fraction and SlipVelocity equations back on again, and compute the mixture solution.

20.7.3 Solution Strategies for the Eulerian Model

Calculating an Initial Solution

To improve convergence behavior, you may want to compute an initialsolution before solving the complete Eulerian multiphase model. Thereare two methods you can use to obtain an initial solution for an Eulerianmultiphase calculation:

• Set up and solve the problem using the mixture model (with orwithout slip velocities) instead of the Eulerian model. You can thenenable the Eulerian model, complete the setup, and continue thecalculation using the mixture-model solution as a starting point.

• Set up the Eulerian multiphase calculation as usual, but computethe flow for only the primary phase. To do this, deselect Volume


20.8 Postprocessing for General Multiphase Problems

Fraction in the Equations list in the Solution Controls panel. Onceyou have obtained an initial solution for the primary phase, turnthe volume fraction equations back on and continue the calculationfor all phases.

You should not try to use a single-phase solution obtained without the!mixture or Eulerian model as a starting point for an Eulerian multiphasecalculation. Doing so will not improve convergence, and may make iteven more difficult for the flow to converge.

Temporarily Ignoring Lift and Virtual Mass Forces

If you are planning to include the effects of lift and/or virtual massforces in a steady-state Eulerian multiphase simulation, you can oftenreduce stability problems that sometimes occur in the early stages of thecalculation by temporarily ignoring the action of the lift and the virtualmass forces. Once the solution without these forces starts to converge,you can interrupt the calculation, define these forces appropriately, andcontinue the calculation.


Each of the three general multiphase models provides a number of ad-ditional field functions that you can plot or report. You can also reportflow rates for individual phases for all three models, and display velocityvectors for the individual phases in a mixture or Eulerian calculation.

Information about these postprocessing topics is provided in the follow-ing subsections:

• Section 20.8.1: Available Postprocessing Variables

• Section 20.8.2: Displaying Velocity Vectors for Individual Phases

• Section 20.8.3: Reporting Fluxes for Individual Phases

• Section 20.8.4: Reporting Forces on Walls for Individual Phases

• Section 20.8.5: Reporting Flow Rates for Individual Phases



20.8.1 Available Postprocessing Variables

When you use one of the general multiphase models, several additionalfield functions will be available for postprocessing, as listed in this sec-tion. See Chapter 27 for a complete list of field functions and their def-initions. Chapters 25 and 26 explain how to generate graphics displaysand reports of data.

VOF Model

For VOF calculations you can generate graphical plots or alphanumericreports of the following additional items:

• Volume fraction of phase-n (in the Phases... category)

• Density of phase-n (in the Density... category)

• Molecular Viscosity of phase-n (in the Properties... category)

• Thermal Conductivity of phase-n (in the Properties... category)

• Specific Heat of phase-n (in the Properties... category)

• Enthalpy of phase-n (in the Temperature... category)

• Total Enthalpy of phase-n (in the Temperature... category)

• Total Energy of phase-n (in the Temperature... category)

• Internal Energy of phase-n (in the Temperature... category)

The non-phase-specific variables that are available (e.g., Molecular Viscos-ity and Thermal Conductivity) represent mixture quantities. The thermalquantities listed above will be available only for calculations that includethe energy equation.



Mixture Model

For calculations with the mixture model, you can generate graphicalplots or alphanumeric reports of the following additional items:



• phase-n Velocity Magnitude (in the Velocity... category)

• phase-n Relative Velocity Magnitude (in the Velocity... category)

• phase-n X, Y, Z, etc. Velocity (in the Velocity... category)

• phase-n Relative X, Y, Z, etc. Velocity (in the Velocity... category)

• phase-n Stream Function (in the Velocity... category)


• Diameter of phase-n (in the Properties... category)

• Thermal Conductivity of phase-n (in the Properties... category)

• Specific Heat of phase-n (in the Properties... category)

• Enthalpy of phase-n (in the Temperature... category)

• Total Enthalpy of phase-n (in the Temperature... category)

• Total Energy of phase-n (in the Temperature... category)

• Internal Energy of phase-n (in the Temperature... category)

The non-phase-specific variables that are available (e.g., Velocity Magni-tude and X Velocity) represent mixture quantities. The thermal quanti-ties listed above will be available only for calculations that include theenergy equation.

Note that if you read a mixture-model data file into FLUENT, you will!need to run the solver for one iteration before plotting or reporting theitems listed above. (This is not necessary if you plot or report thesevariables for a data set that you just computed in the current FLUENTsession.)



Eulerian Model

For Eulerian multiphase calculations you can generate graphical plots oralphanumeric reports of the following additional items:



• phase-n Velocity Magnitude (in the Velocity... category)

• phase-n Relative Velocity Magnitude (in the Velocity... category)

• phase-n X, Y, Z, etc. Velocity (in the Velocity... category)

• phase-n Relative X, Y, Z, etc. Velocity (in the Velocity... category)

• phase-n Stream Function (in the Velocity... category)

• phase-n Turbulent Viscosity (in the Turbulence... category)

• phase-n Wall Yplus (in the Turbulence... category)

• phase-n Turbulent Kinetic Energy (in the Turbulence... category)

• phase-n Turbulent Dissipation Rate (in the Turbulence... category)

• phase-n Production of k (in the Turbulence... category)


• Diameter of phase-n (in the Properties... category)

• phase-n Wall Shear Stress (in the Wall Fluxes... category)

• phase-n X, Y, Z Wall Shear Stress (in the Wall Fluxes... category)

• phase-n Skin Friction Coefficient (in the Wall Fluxes... category)

The availability of the turbulence quantities listed above will depend onwhich multiphase turbulence model you used in the calculation.

Note that if you read an Eulerian multiphase data file into FLUENT, you!



will need to run the solver for one iteration before plotting or reportingthe items listed above. (This is not necessary if you plot or report thesevariables for a data set that you just computed in the current FLUENTsession.)

20.8.2 Displaying Velocity Vectors for Individual Phases

For mixture and Eulerian calculations, it is possible to display velocityvectors for the individual phases using the Vectors panel.

Display −→Vectors...

To display the velocity of a particular phase, select phase-n Velocity(where phase-n is replaced by the name of the phase of interest, e.g., air-bubbles Velocity) in the Vectors Of drop-down list. You can also chooseRelative phase-n Velocity to display the phase velocity relative to a mov-ing reference frame. To display the mixture velocity ~vm (relevant formixture model calculations only), select Velocity (or Relative Velocity forthe mixture velocity relative to a moving reference frame).

20.8.3 Reporting Fluxes for Individual Phases

When you use the Flux Reports panel to compute fluxes through bound-aries, you will be able to specify whether the report is for the mixtureor for an individual phase.

Report −→Fluxes...

Select mixture in the Phase drop-down list at the bottom of the panelto report fluxes for the mixture, or select the name of a phase to reportfluxes just for that phase.

20.8.4 Reporting Forces on Walls for Individual Phases

For Eulerian calculations, when you use the Force Reports panel to com-pute forces or moments on wall boundaries, you will be able to specifythe 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 theleft side of the panel.

20.8.5 Reporting Flow Rates for Individual Phases

You can obtain a report of mass flow rate for each phase (and the mix-ture) through each flow boundary using the report/mass-flow textcommand:

report −→mass-flow

When you specify the phase of interest (the mixture or an individualphase), FLUENT will list each zone, followed by the mass flow ratethrough 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): 0zone 4 (velocity-inlet): 4.9132133

net mass-flow: -5.4608185


chapter 20. general multiphase models - · pdf filechapter 20. general multiphase models ......

Documents