Chapter 11. Modeling Heat Transfer...Modeling Heat Transfer 11.2 Convective and Conductive Heat...

Chapter 11. Modeling Heat Transfer

This chapter provides details about the heat transfer models available inFLUENT.

Information is presented in the following sections:

• Section 11.1: Overview of Heat Transfer Models in FLUENT

• Section 11.2: Convective and Conductive Heat Transfer

• Section 11.3: Radiative Heat Transfer

• Section 11.4: Periodic Heat Transfer

• Section 11.5: Buoyancy-Driven Flows

11.1 Overview of Heat Transfer Models in FLUENT

The flow of thermal energy from matter occupying one region in spaceto matter occupying a different region in space is known as heat transfer.Heat transfer can occur by three main methods: conduction, convection,and radiation. Physical models involving only conduction and/or convec-tion are the simplest, while buoyancy-driven flow, or natural convection,and radiation models are more complex. Depending on your problem,FLUENT will solve a variation of the energy equation that takes intoaccount the heat transfer methods you have specified. FLUENT is alsoable to predict heat transfer in periodically repeating geometries, thusgreatly reducing the required computational effort in certain cases.

c© Fluent Inc. November 28, 2001 11-1

Modeling Heat Transfer

11.2 Convective and Conductive Heat Transfer

FLUENT allows you to include heat transfer within the fluid and/or solidregions in your model. Problems ranging from thermal mixing within afluid to conduction in composite solids can thus be handled by FLUENTusing the physical models and inputs described in this section. Radiationmodeling is described in Section 11.3 and natural convection is describedin Section 11.5.

Information about heat transfer is presented in the following subsections:

• Section 11.2.1: Theory

• Section 11.2.2: User Inputs for Heat Transfer

• Section 11.2.3: Solution Process for Heat Transfer

• Section 11.2.4: Reporting and Displaying Heat Transfer Quantities

• Section 11.2.5: Exporting Heat Flux Data

11.2.1 Theory

The Energy Equation

FLUENT solves the energy equation in the following form:

∂

∂t(ρE) + ∇ · (~v(ρE + p)) = ∇ ·

keff∇T −

∑j

hj~Jj + (τ eff · ~v)

+ Sh

(11.2-1)

where keff is the effective conductivity (k + kt, where kt is the turbulentthermal conductivity, defined according to the turbulence model beingused), and ~Jj is the diffusion flux of species j. The first three terms onthe right-hand side of Equation 11.2-1 represent energy transfer due toconduction, species diffusion, and viscous dissipation, respectively. Sh

includes the heat of chemical reaction, and any other volumetric heatsources you have defined.

11-2 c© Fluent Inc. November 28, 2001


In Equation 11.2-1,

E = h− p

ρ+v2

2(11.2-2)

where sensible enthalpy h is defined for ideal gases as

h =∑j

Yjhj (11.2-3)

and for incompressible flows as

h =∑j

Yjhj +p

ρ(11.2-4)

In Equations 11.2-3 and 11.2-4, Yj is the mass fraction of species j and

hj =∫ T

Tref

cp,j dT (11.2-5)

where Tref is 298.15 K.

The Energy Equation for the Non-Premixed Combustion Model

When the non-adiabatic non-premixed combustion model is enabled,FLUENT solves the total enthalpy form of the energy equation:

∂

∂t(ρH) + ∇ · (ρ~vH) = ∇ ·

(kt

cp∇H

)+ Sh (11.2-6)

Under the assumption that the Lewis number (Le) = 1, the conductionand species diffusion terms combine to give the first term on the right-hand side of the above equation while the contribution from viscousdissipation appears in the non-conservative form as the second term.The total enthalpy H is defined as



H =∑j

YjHj (11.2-7)

where Yj is the mass fraction of species j and

Hj =∫ T

Tref,j

cp,jdT + h0j (Tref,j) (11.2-8)

h0j (Tref,j) is the formation enthalpy of species j at the reference temper-

ature Tref,j.

Inclusion of Pressure Work and Kinetic Energy Terms

Equation 11.2-1 includes pressure work and kinetic energy terms whichare often negligible in incompressible flows. For this reason, the seg-regated solver by default does not include the pressure work or kineticenergy when you are solving incompressible flow. If you wish to includethese terms, use the define/models/energy? text command to turnthem on.

Pressure work and kinetic energy are always accounted for when you aremodeling compressible flow or using one of the coupled solvers.

Inclusion of the Viscous Dissipation Terms

Equations 11.2-1 and 11.2-6 include viscous dissipation terms, whichdescribe the thermal energy created by viscous shear in the flow.

When the segregated solver is used, FLUENT’s default form of the energyequation does not include them (because viscous heating is often negli-gible). Viscous heating will be important when the Brinkman number,Br, approaches or exceeds unity, where

Br =µU2

e

k∆T(11.2-9)

and ∆T represents the temperature difference in the system.



When your problem requires inclusion of the viscous dissipation termsand you are using the segregated solver, you should activate the terms us-ing the Viscous Heating option in the Viscous Model panel. Compressibleflows typically have Br ≥ 1. Note, however, that when the segregatedsolver is used, FLUENT does not automatically activate the viscous dis-sipation if you have defined a compressible flow model.

When one of the coupled solvers is used, the viscous dissipation termsare always included when the energy equation is solved.

Inclusion of the Species Diffusion Term

Equations 11.2-1 and 11.2-6 both include the effect of enthalpy transportdue to species diffusion.

When the segregated solver is used, the term

∇ ·∑

j

hj~Jj

is included in Equation 11.2-1 by default. If you do not want to includeit, you can turn off the Diffusion Energy Source option in the SpeciesModel panel.

When the non-adiabatic non-premixed combustion model is being used,this term does not explicitly appear in the energy equation, because itis included in the first term on the right-hand side of Equation 11.2-6.

When one of the coupled solvers is used, this term is always included inthe energy equation.

Energy Sources Due to Reaction

Sources of energy, Sh, in Equation 11.2-1 include the source of energydue to chemical reaction:

Sh,rxn = −∑j

(h0

j

Mj+∫ T

Tref,j

cp,j dT

)Rj (11.2-10)



where h0j is the enthalpy of formation of species j and Rj is the volumetric

rate of creation of species j.

In the energy equation used for non-adiabatic non-premixed combustion(Equation 11.2-6), the heat of formation is included in the definitionof enthalpy (see Equation 11.2.1), so reaction sources of energy are notincluded in Sh.

Energy Sources Due To Radiation

When one of the radiation models is being used, Sh in Equation 11.2-1 or11.2-6 also includes radiation source terms. See Section 11.3 for details.

Interphase Energy Sources

It should be noted that the energy sources, Sh, also include heat transferbetween the continuous and the discrete phase. This is discussed furtherin Section 19.5.

Boundary Conditions for Heat Transfer at Walls

Heat transfer boundary conditions at walls are discussed in Section 10.8.2.

Energy Equation in Solid Regions

In solid regions, the energy transport equation used by FLUENT has thefollowing form:

∂

∂t(ρh) + ∇ · (~vρh) = ∇ · (k∇T ) + Sh (11.2-11)

where ρ = densityh = sensible enthalpy,

∫ TTref

cpdT

k = conductivityT = temperatureSh = volumetric heat source

The second term on the left-hand side of Equation 11.2-11 representsconvective energy transfer due to rotational or translational motion ofthe solids. The velocity field ~v is computed from the motion specified



for the solid zone (see Section 6.18). The terms on the right-hand sideof Equation 11.2-11 are the heat flux due to conduction and volumetricheat sources within the solid, respectively.

Anisotropic Conductivity in Solids

When you use the segregated solver, FLUENT allows you to specifyanisotropic conductivity for solid materials. The conduction term foran anisotropic solid has the form

∇ · (kij∇T ) (11.2-12)

where kij is the conductivity matrix. See Section 7.4.5 for details onspecifying anisotropic conductivity for solid materials.

Diffusion at Inlets

The net transport of energy at inlets consists of both the convectionand diffusion components. The convection component is fixed by theinlet temperature specified by you. The diffusion component, however,depends on the gradient of the computed temperature field. Thus thediffusion component (and therefore the net inlet transport) is not speci-fied a priori.

In some cases, you may wish to specify the net inlet transport of energyrather than the inlet temperature. If you are using the segregated solver,you can do this by disabling inlet energy diffusion. By default, FLUENTincludes the diffusion flux of energy at inlets. To turn off inlet diffusion,use the define/models/energy? text command.

Inlet diffusion cannot be turned off if you are using one of the coupledsolvers.



11.2.2 User Inputs for Heat Transfer

When your FLUENT model includes heat transfer you need to activatethe relevant models, supply thermal boundary conditions, and input ma-terial properties that govern heat transfer and/or may vary with tem-perature. These inputs are described in this section.

The procedure for setting up a heat transfer problem is described below.(Note that this procedure includes only those steps necessary for the heattransfer model itself; you will need to set up other models, boundaryconditions, etc. as usual.)

1. To activate the calculation of heat transfer, turn on the EnergyEquation option in the Energy panel (Figure 11.2.1).

Define −→ Models −→Energy...

Figure 11.2.1: The Energy Panel

2. (optional, segregated solver only) If you are modeling viscous flowand you want to include the viscous heating terms in the energyequation, turn on the Viscous Heating option in the Viscous Modelpanel. As noted in Section 11.2.1, the viscous heating terms inthe energy equation are (by default) ignored by FLUENT whenthe segregated solver is used. (They are always included whenone of the coupled solvers is used.) Viscous dissipation should beenabled when the shear stress in the fluid is large (e.g., in lubri-cation problems) and/or in high-velocity, compressible flows (seeEquation 11.2-9).

Define −→ Models −→Viscous...



3. Define thermal boundary conditions at flow inlets, flow outlets, andwalls.

Define −→Boundary Conditions...

At flow inlets and exits you will set the temperature; at walls youmay use any of the following thermal conditions:

• Specified heat flux

• Specified temperature

• Convective heat transfer

• External radiation

• Combined external radiation and external convective heattransfer

Section 6.13.1 provides details on the model inputs that governthese thermal boundary conditions. The default thermal boundarycondition at inlets is a specified temperature of 300 K; at walls thedefault condition is zero heat flux (adiabatic). See Chapter 6 fordetails about boundary condition inputs.

If your heat transfer application involves two separated fluid re-!gions, see the information provided below.

4. Define material properties for heat transfer.

Define −→Materials...

Heat capacity and thermal conductivity must be defined, and youcan specify many properties as functions of temperature, as de-scribed in Chapter 7.

If your heat transfer application involves two separated fluid re-!gions, see the information provided below.

The Temperature Floor and Ceiling

For stability reasons, FLUENT includes a limit on the predicted tem-perature range. The purpose of the temperature ceiling and floor is toimprove the stability of calculations in which the temperature shouldphysically lie within known limits. Sometimes intermediate solutions of



the equations give rise to temperatures beyond these limits for whichproperty definitions, etc. are not well defined. The temperature limitskeep the temperatures within the expected range for your problem. If theFLUENT calculation predicts a temperature above the maximum limit,the stored temperature values are “pegged” at this maximum value. Thedefault for the temperature ceiling is 5000 K. If the FLUENT calculationpredicts a temperature below the minimum limit, the stored tempera-ture values are “pegged” at this minimum value. The default for thetemperature minimum is 1 K.

If you expect the temperature in your domain to exceed 5000 K, youshould use the Solution Limits panel to increase the Maximum Tempera-ture.

Solve −→ Controls −→Limits...

Modeling Heat Transfer in Two Separated Fluid Regions

If your heat transfer application involves two fluid regions separated bya solid zone or a wall, as illustrated in Figure 11.2.2, you will need todefine the problem with some care. Specifically:

• You should not use outflow boundary conditions in either fluid.

• You can establish separate fluid properties by selecting a differentfluid material for each zone. (For species calculations, however, youcan only select a single mixture material for the entire domain.)

fluid 1

fluid 2

Figure 11.2.2: Typical Counterflow Heat Exchanger Involving HeatTransfer Between Two Separated Fluid Streams



11.2.3 Solution Process for Heat Transfer

Although many simple heat transfer problems can be successfully solvedusing the default solution parameters assumed by FLUENT, you mayaccelerate the convergence of your problem and/or improve the stabilityof the solution process using some of the guidelines provided in thissection.

Under-Relaxation of the Energy Equation

When you use the segregated solver, FLUENT under-relaxes the energyequation using the under-relaxation parameter defined by you in theSolution Controls panel, as described in Section 22.9.

Solve −→ Controls −→Solution...

If you are using the non-adiabatic non-premixed combustion model, youwill set the energy under-relaxation factor as usual, but you will alsoset an under-relaxation factor for temperature, which will be used asdescribed below.

FLUENT uses a default under-relaxation factor of 1.0 for the energyequation, regardless of the form in which it is solved (temperature orenthalpy). In problems where the energy field impacts the fluid flow(via temperature-dependent properties or buoyancy) you should use alower value for the under-relaxation factor, in the range of 0.8–1.0. Inproblems where the flow field is decoupled from the temperature field (notemperature-dependent properties or buoyancy forces), you can usuallyretain the default value of 1.0.

Under-Relaxation of Temperature When the Enthalpy Equationis Solved

When the enthalpy form of the energy equation is solved (i.e., when youare using the non-adiabatic non-premixed combustion model), FLUENTalso under-relaxes the temperature, updating the temperature by only afraction of the change that would result from the change in the (under-relaxed) enthalpy values. This second level of under-relaxation can beused to good advantage when you would like to let the enthalpy field



change rapidly, but the temperature response (and its effect on fluidproperties) to lag. FLUENT uses a default setting of 1.0 for the under-relaxation on temperature and you can modify this setting using theSolution Controls panel.

Disabling the Species Diffusion Term

If you are solving for species transport using the segregated solver andyou encounter convergence difficulties, you may want to consider turningoff the Diffusion Energy Source option in the Species Model panel.

Define −→ Models −→Species...

When this option is disabled, FLUENT will neglect the effects of speciesdiffusion on the energy equation.

Note that species diffusion effects are always included when one of thecoupled solvers is used.

Step-by-Step Solutions

Often the most efficient strategy for predicting heat transfer is to com-pute an isothermal flow first and then to add the calculation of the energyequation. The procedure differs slightly, depending on whether or notthe flow and heat transfer are coupled.

Decoupled Flow and Heat Transfer Calculations

If your flow and heat transfer are decoupled (no temperature-dependentproperties or buoyancy forces), you can first solve the isothermal flow(energy equation turned off) to yield a converged flow-field solution andthen solve the energy transport equation alone.

Since the coupled solvers always solve the flow and energy equations!together, the procedure for solving for energy alone applies only to thesegregated solver.

You can temporarily turn off the flow equations or the energy equationby deselecting Energy in the Equations list in the Solution Controls panel.(See also Section 22.19.2.)




Coupled Flow and Heat Transfer Calculations

If your flow and heat transfer are coupled (i.e., your model includestemperature-dependent properties or buoyancy forces), you can firstsolve the flow equations before turning on energy. Once you have aconverged flow-field solution, you can turn on energy and solve the flowand energy equations simultaneously to complete the heat transfer sim-ulation.

11.2.4 Reporting and Displaying Heat Transfer Quantities

FLUENT provides several additional reporting options for simulationsinvolving heat transfer. You can generate graphical plots or reports ofthe following variables/functions:

• Static Temperature

• Total Temperature

• Enthalpy

• Relative Total Temperature

• Rothalpy

• Wall Temperature (Outer Surface)

• Wall Temperature (Inner Surface)

• Total Enthalpy

• Total Enthalpy Deviation

• Entropy

• Total Energy

• Internal Energy

• Total Surface Heat Flux



• Surface Heat Transfer Coef.

• Surface Nusselt Number

• Surface Stanton Number

The first 12 variables listed above are contained in the Temperature...category of the variable selection drop-down list that appears in post-processing panels, and the remaining variables are in the Wall Fluxes...category. See Chapter 27 for their definitions.

Definition of Enthalpy and Energy in Reports and Displays

The definitions of the reported values of enthalpy and energy will be dif-ferent depending on whether the flow is compressible or incompressible.See Section 27.4 for a complete list of definitions.

Reporting Heat Transfer Through Boundaries

You can use the Flux Reports panel to compute the heat transfer througheach boundary of the domain, or to sum the heat transfer through allboundaries to check the heat balance.

Report −→Fluxes...

It is recommended that you perform a heat balance check to ensurethat your solution is truly converged. See Section 26.2 for details aboutgenerating flux reports.

Reporting Heat Transfer Through a Surface

You can use the Surface Integrals panel (described in Section 26.5) tocompute the heat transfer through any boundary or any surface createdusing the methods described in Chapter 24.

Report −→Surface Integrals...

To report the flow rate of enthalpy

Q =∫Hρ~v · d ~A (11.2-13)



choose the Mass Flow Rate option in the Surface Integrals panel, selectEnthalpy (in the Temperature... category) as the Field Variable, and pickthe surface(s) on which to integrate.

Reporting Averaged Heat Transfer Coefficients

The Surface Integrals panel can also be used to generate a report ofaveraged heat transfer coefficient h on a surface ( 1

A

∫h dA).

Report −→Surface Integrals...

In the Surface Integrals panel, choose the Area-Weighted Average option,select Surface Heat Transfer Coef. (in the Wall Fluxes... category) as theField Variable, and pick the surface.

11.2.5 Exporting Heat Flux Data

It is possible to export heat flux data on wall zones (including radiation)to a generic file that you can examine or use in an external program. Tosave a heat flux file, you will use the custom-heat-flux text command.

file −→ export −→custom-heat-flux

Heat transfer data will be exported in the following free format for eachface zone that you select for export:

zone-name nfacesx_f y_f z_f A Q T_w T_c...

Each block of data starts with the name of the face zone (zone-name)and the number of faces in the zone (nfaces). Next there is a line foreach face (i.e., nfaces lines), each containing the components of theface centroid (x f, y f, and, in 3D, z f), the face area (A), the totalheat transfer including radiation heat transfer (Q), the face temperature(T w), and the adjacent cell temperature (T c).



11.3 Radiative Heat Transfer

Information about radiation modeling is presented in the following sec-tions:

• Section 11.3.1: Introduction to Radiative Heat Transfer

• Section 11.3.2: Choosing a Radiation Model

• Section 11.3.3: The Discrete Transfer Radiation Model (DTRM)

• Section 11.3.4: The P-1 Radiation Model

• Section 11.3.5: The Rosseland Radiation Model

• Section 11.3.6: The Discrete Ordinates (DO) Radiation Model

• Section 11.3.7: The Surface-to-Surface (S2S) Radiation Model

• Section 11.3.8: Radiation in Combusting Flows

• Section 11.3.9: Overview of Using the Radiation Models

• Section 11.3.10: Selecting the Radiation Model

• Section 11.3.11: Defining the Ray Tracing for the DTRM

• Section 11.3.12: Computing or Reading the View Factors for theS2S Model

• Section 11.3.13: Defining the Angular Discretization for the DOModel

• Section 11.3.14: Defining Non-Gray Radiation for the DO Model

• Section 11.3.15: Defining Material Properties for Radiation

• Section 11.3.16: Setting Radiation Boundary Conditions

• Section 11.3.17: Setting Solution Parameters for Radiation

• Section 11.3.18: Solving the Problem

• Section 11.3.19: Reporting and Displaying Radiation Quantities

• Section 11.3.20: Displaying Rays and Clusters for the DTRM



11.3.1 Introduction to Radiative Heat Transfer

FLUENT provides five radiation models which allow you to include ra-diation, with or without a participating medium, in your heat transfersimulations:

• Discrete transfer radiation model (DTRM) [30, 208]

• P-1 radiation model [35, 210]

• Rosseland radiation model [210]

• Surface-to-surface (S2S) radiation model [210]

• Discrete ordinates (DO) radiation model [37, 183]

Heating or cooling of surfaces due to radiation and/or heat sources orsinks due to radiation within the fluid phase can be included in yourmodel using one of these radiation models.

Radiative Transfer Equation

The radiative transfer equation (RTE) for an absorbing, emitting, andscattering medium at position ~r in the direction ~s is

dI(~r,~s)ds

+ (a+ σs)I(~r,~s) = an2σT4

π+σs

4π

∫ 4π

0I(~r,~s ′) Φ(~s · ~s ′) dΩ′

(11.3-1)



where ~r = position vector~s = direction vector~s ′ = scattering direction vectors = path lengtha = absorption coefficientn = refractive indexσs = scattering coefficientσ = Stefan-Boltzmann constant (5.672 × 10−8 W/m2-K4)I = radiation intensity, which depends on position (~r)

and direction (~s)T = local temperatureΦ = phase functionΩ′ = solid angle

(a+ σs)s is the optical thickness or opacity of the medium. The refrac-tive index n is important when considering radiation in semi-transparentmedia. Figure 11.3.1 illustrates the process of radiative heat transfer.

ds

Incoming radiation (I)

Outgoing radiation I + (dI/ds)ds

Absorption andscattering loss: I (a+σs) ds

Gas emission:(aσT /π) ds4

Scattering addition

Figure 11.3.1: Radiative Heat Transfer

The DTRM and the P-1, Rosseland, and DO radiation models requirethe absorption coefficient a as input. a and the scattering coefficient σs



can be constants, and a can also be a function of local concentrationsof H2O and CO2, path length, and total pressure. FLUENT providesthe weighted-sum-of-gray-gases model (WSGGM) for computation of avariable absorption coefficient. See Section 11.3.8 for details. The dis-crete ordinates implementation can model radiation in semi-transparentmedia. The refractive index n of the medium must be provided as a partof the calculation for this type of problem.

Applications of Radiative Heat Transfer

Typical applications well suited for simulation using radiative heat trans-fer include the following:

• Radiative heat transfer from flames

• Surface-to-surface radiant heating or cooling

• Coupled radiation, convection, and/or conduction heat transfer

• Radiation through windows in HVAC applications, and cabin heattransfer analysis in automotive applications

• Radiation in glass processing, glass fiber drawing, and ceramic pro-cessing

You should include radiative heat transfer in your simulation when theradiant heat flux, Qrad = σ(T 4

max − T 4min), is large compared to the heat

transfer rate due to convection or conduction. Typically this will occurat high temperatures where the fourth-order dependence of the radiativeheat flux on temperature implies that radiation will dominate.

11.3.2 Choosing a Radiation Model

For certain problems, one radiation model may be more appropriate thanthe others. When deciding which radiation model to use, consider thefollowing:

• Optical thickness: The optical thickness aL is a good indicator ofwhich model to use in your problem. Here, L is an appropriate



length scale for your domain. For flow in a combustor, for exam-ple, L is the diameter of the combustion chamber. If aL 1,your best alternatives are the P-1 and Rosseland models. The P-1model should typically be used for optical thicknesses > 1. Foroptical thickness > 3, the Rosseland model is cheaper and moreefficient. The DTRM and the DO model work across the rangeof optical thicknesses, but are substantially more expensive to use.So you should use the “thick-limit” models, P-1 and Rosseland, ifthe problem allows it. For optically thin problems (aL < 1), onlythe DTRM and the DO model are appropriate.

• Scattering and emissivity: The P-1, Rosseland, and DO modelsaccount for scattering, while the DTRM neglects it. Since theRosseland model uses a temperature slip condition at walls, it isinsensitive to wall emissivity.

• Particulate effects: Only the P-1 and DO models account for ex-change of radiation between gas and particulates (see Equation11.3-15).

• Semi-transparent media and specular boundaries: Only the DOmodel allows specular reflection (e.g., for mirrors) and calculationof radiation in semi-transparent media such as glass.

• Non-gray radiation: Only the DO model allows you to computenon-gray radiation using a gray band model.

• Localized heat sources: In problems with localized sources of heat,the P-1 model may overpredict the radiative fluxes. The DO modelis probably the best suited for computing radiation for this case,although the DTRM, with a sufficiently large number of rays, isalso acceptable.

• Enclosure radiative transfer with non-participating media: Thesurface-to-surface (S2S) model is suitable for this type of prob-lem. The radiation models used with participating media may, inprinciple, be used to compute the surface-to-surface radiation, butthey are not always efficient.



External Radiation

If you need to include radiative heat transfer from the exterior of yourphysical model, you can include an external radiation boundary condi-tion in your model (see Section 6.13.1). If you are not concerned withradiation within the domain, this boundary condition can be used with-out activating one of the radiation models.

Advantages and Limitations of the DTRM

The primary advantages of the DTRM are threefold: it is a relativelysimple model, you can increase the accuracy by increasing the numberof rays, and it applies to a wide range of optical thicknesses.

You should be aware of the following limitations when using the DTRMin FLUENT:

• The DTRM assumes that all surfaces are diffuse. This means thatthe reflection of incident radiation at the surface is isotropic withrespect to solid angle.

• The effect of scattering is not included.

• The implementation assumes gray radiation.

• Solving a problem with a large number of rays is CPU-intensive.

Advantages and Limitations of the P-1 Model

The P-1 model has several advantages over the DTRM. For the P-1model, the RTE (Equation 11.3-1) is a diffusion equation, which is easyto solve with little CPU demand. The model includes the effect of scat-tering. For combustion applications where the optical thickness is large,the P-1 model works reasonably well. In addition, the P-1 model caneasily be applied to complicated geometries with curvilinear coordinates.

You should be aware of the following limitations when using the P-1radiation model:



• The P-1 model assumes that all surfaces are diffuse. This meansthat the reflection of incident radiation at the surface is isotropicwith respect to the solid angle.


• There may be a loss of accuracy, depending on the complexity ofthe geometry, if the optical thickness is small.

• The P-1 model tends to overpredict radiative fluxes from localizedheat sources or sinks.

Advantages and Limitations of the Rosseland Model

The Rosseland model has two advantages over the P-1 model. Since itdoes not solve an extra transport equation for the incident radiation (asthe P-1 model does), the Rosseland model is faster than the P-1 modeland requires less memory.

The Rosseland model can only be used for optically thick media. It isrecommended for use when the optical thickness exceeds 3. Note alsothat the Rosseland model is not available when one of the coupled solversis being used; it is available only with the segregated solver.

Advantages and Limitations of the DO Model

The DO model spans the entire range of optical thicknesses, and allowsyou to solve problems ranging from surface-to-surface radiation to par-ticipating radiation in combustion problems. It also allows the solutionof radiation in semi-transparent media. Computational cost is moderatefor typical angular discretizations, and memory requirements are modest.

The current implementation is restricted to either gray radiation or non-gray radiation using a gray-band model. Solving a problem with a fineangular discretization may be CPU-intensive.

The non-gray implementation in FLUENT is intended for use with par-ticipating media with a spectral absorption coefficient aλ that varies ina stepwise fashion across spectral bands, but varies smoothly within theband. Glass, for example, displays banded behavior of this type. The



current implementation does not model the behavior of gases such ascarbon dioxide or water vapor, which absorb and emit energy at distinctwave numbers [161]. The modeling of non-gray gas radiation is still anevolving field. However, some researchers [66] have used gray-band mod-els to model gas behavior by approximating the absorption coefficientswithin each band as a constant. The implementation in FLUENT can beused in this fashion if desired.

The non-gray implementation in FLUENT is compatible with all the mod-els with which the gray implementation of the DO model can be used.Thus, it is possible to include scattering, anisotropy, semi-transparentmedia, and particulate effects. However, the non-gray implementationassumes a constant absorption coefficient within each wavelength band.The weighted sum of gray gases model (WSGGM) cannot be used tospecify the absorption coefficient in each band. The implementation al-lows the specification of spectral emissivity at walls. The emissivity isassumed to be constant within each band.

Advantages and Limitations of the S2S Model

The surface-to-surface radiation model is good for modeling the enclo-sure radiative transfer without participating media (e.g., spacecraft heatrejection system, solar collector systems, radiative space heaters, andautomotive underhood cooling). In such cases, the methods for partici-pating radiation may not always be efficient. As compared to the DTRMand the DO radiation model, the S2S model has a much faster time periteration, although the view factor calculation itself is CPU-intensive.

You should be aware of the following limitations when using the S2Sradiation model:

• The S2S model assumes that all surfaces are diffuse.


• The storage and memory requirements increase very rapidly as thenumber of surface faces increases. This can be minimized by usinga cluster of surface faces, although the CPU time is independentof the number of clusters that are used.



• The S2S model cannot be used to model participating radiationproblems.

• The surface clustering method does not work with sliding meshesor hanging nodes.

• The S2S model cannot be used if your model contains periodic orsymmetry boundary conditions.

11.3.3 The Discrete Transfer Radiation Model (DTRM)

The main assumption of the DTRM is that the radiation leaving thesurface element in a certain range of solid angles can be approximatedby a single ray. This section provides details about the equations usedin the DTRM.

The DTRM Equations

The equation for the change of radiant intensity, dI, along a path, ds,can be written as

dI

ds+ aI =

aσT 4

π(11.3-2)

where a = gas absorption coefficientI = intensityT = gas local temperatureσ = Stefan-Boltzmann constant (5.672 × 10−8 W/m2-K4)

Here, the refractive index is assumed to be unity. The DTRM integratesEquation 11.3-2 along a series of rays emanating from boundary faces.If a is constant along the ray, then I(s) can be estimated as

I(s) =σT 4

π(1 − e−as) + I0e

−as (11.3-3)

where I0 is the radiant intensity at the start of the incremental path,which is determined by the appropriate boundary condition (see the de-scription of boundary conditions, below). The energy source in the fluid



due to radiation is then computed by summing the change in intensityalong the path of each ray that is traced through the fluid control volume.

The “ray tracing” technique used in the DTRM can provide a predictionof radiative heat transfer between surfaces without explicit view-factorcalculations. The accuracy of the model is limited mainly by the numberof rays traced and the computational grid.

Ray Tracing

The ray paths are calculated and stored prior to the fluid flow calculation.At each radiating face, rays are fired at discrete values of the polar andazimuthal angles (see Figure 11.3.2). To cover the radiating hemisphere,θ is varied from 0 to π

2 and φ from 0 to 2π. Each ray is then traced todetermine the control volumes it intercepts as well as its length withineach control volume. This information is then stored in the radiationfile, which must be read in before the fluid flow calculations begin.

θ

φ t

n

P

Figure 11.3.2: Angles θ and φ Defining the Hemispherical Solid AngleAbout a Point P



Clustering

DTRM is computationally very expensive when there are too many sur-faces to trace rays from and too many volumes crossed by the rays. Toreduce the computational time, the number of radiating surfaces andabsorbing cells is reduced by clustering surfaces and cells into surfaceand volume “clusters”. The volume clusters are formed by starting froma cell and simply adding its neighbors and their neighbors until a spec-ified number of cells per volume cluster is collected. Similarly, surfaceclusters are made by starting from a face and adding its neighbors andtheir neighbors until a specified number of faces per surface cluster iscollected.

The incident radiation flux, qin, and the volume sources are calculatedfor the surface and volume clusters respectively. These values are thendistributed to the faces and cells in the clusters to calculate the wall andcell temperatures. Since the radiation source terms are highly non-linear(proportional to the fourth power of temperature), care must be takento calculate the average temperatures of surface and volume clusters anddistribute the flux and source terms appropriately among the faces andcells forming the clusters.

The surface and volume cluster temperatures are obtained by area andvolume averaging as shown in the following equations:

Tsc =

(∑f AfT

4f∑

Af

)1/4

(11.3-4)

Tvc =

(∑c VcT

4c∑

Vc

)1/4

(11.3-5)

where Tsc and Tvc are the temperatures of the surface and volume clustersrespectively, Af and Tf are the area and temperature of face f , and Vc

and Tc are the volume and temperature of cell c. The summations arecarried over all faces of a surface cluster and all cells of a volume cluster.



Boundary Condition Treatment for the DTRM at Walls

The radiation intensity approaching a point on a wall surface is inte-grated to yield the incident radiative heat flux, qin, as

qin =∫~s·~n>0

Iin~s · ~ndΩ (11.3-6)

where Ω is the hemispherical solid angle, Iin is the intensity of the in-coming ray, ~s is the ray direction vector, and ~n is the normal pointingout of the domain. The net radiative heat flux from the surface, qout, isthen computed as a sum of the reflected portion of qin and the emissivepower of the surface:

qout = (1 − εw)qin + εwσT4w (11.3-7)

where Tw is the surface temperature of the point P on the surface and εwis the wall emissivity which you input as a boundary condition. FLUENTincorporates the radiative heat flux (Equation 11.3-7) in the prediction ofthe wall surface temperature. Equation 11.3-7 also provides the surfaceboundary condition for the radiation intensity I0 of a ray emanating fromthe point P , as

I0 =qout

π(11.3-8)

Boundary Condition Treatment for the DTRM at Flow Inletsand Exits

The net radiative heat flux at flow inlets and outlets is computed in thesame manner as at walls, as described above. FLUENT assumes that theemissivity of all flow inlets and outlets is 1.0 (black body absorption)unless you choose to redefine this boundary treatment.

FLUENT includes an option that allows you to use different temperaturesfor radiation and convection at inlets and outlets. This can be usefulwhen the temperature outside the inlet or outlet differs considerablyfrom the temperature in the enclosure. See Section 11.3.16 for details.



11.3.4 The P-1 Radiation Model

The P-1 radiation model is the simplest case of the more general P-Nmodel, which is based on the expansion of the radiation intensity I intoan orthogonal series of spherical harmonics [35, 210]. This section pro-vides details about the equations used in the P-1 model.

The P-1 Model Equations

As mentioned above, the P-1 radiation model is the simplest case ofthe P-N model. If only four terms in the series are used, the followingequation is obtained for the radiation flux qr:

qr = − 13(a+ σs) −Cσs

∇G (11.3-9)

where a is the absorption coefficient, σs is the scattering coefficient, Gis the incident radiation, and C is the linear-anisotropic phase functioncoefficient, described below. After introducing the parameter

Γ =1

(3(a + σs) − Cσs)(11.3-10)

Equation 11.3-9 simplifies to

qr = −Γ∇G (11.3-11)

The transport equation for G is

∇ (Γ∇G) − aG+ 4aσT 4 = SG (11.3-12)

where σ is the Stefan-Boltzmann constant and SG is a user-defined ra-diation source. FLUENT solves this equation to determine the local ra-diation intensity when the P-1 model is active.

Combining Equations 11.3-11 and 11.3-12, the following equation is ob-tained:



−∇qr = aG− 4aσT 4 (11.3-13)

The expression for −∇qr can be directly substituted into the energyequation to account for heat sources (or sinks) due to radiation.

Anisotropic Scattering

Included in the P-1 radiation model is the capability for modeling aniso-tropic scattering. FLUENT models anisotropic scattering by means of alinear-anisotropic scattering phase function:

Φ(~s ′ · ~s) = 1 + C~s ′ · ~s (11.3-14)

Here, ~s is the unit vector in the direction of scattering, and ~s ′ is theunit vector in the direction of the incident radiation. C is the linear-anisotropic phase function coefficient, which is a property of the fluid.C ranges from −1 to 1. A positive value indicates that more radiantenergy is scattered forward than backward, and a negative value meansthat more radiant energy is scattered backward than forward. A zerovalue defines isotropic scattering (i.e., scattering that is equally likely inall directions), which is the default in FLUENT. You should modify thedefault value only if you are certain of the anisotropic scattering behaviorof the material in your problem.

Particulate Effects in the P-1 Model

When your FLUENT model includes a dispersed second phase of particles,you can include the effect of particles in the P-1 radiation model. Notethat when particles are present, FLUENT ignores scattering in the gasphase. (That is, Equation 11.3-15 assumes that all scattering is due toparticles.)

For a gray, absorbing, emitting, and scattering medium containing ab-sorbing, emitting, and scattering particles, the transport equation forthe incident radiation can be written as



∇ · (Γ∇G) + 4π

(aσT 4

π+ Ep

)− (a+ ap)G = 0 (11.3-15)

where Ep is the equivalent emission of the particles and ap is the equiv-alent absorption coefficient. These are defined as follows:

Ep = limV →0

N∑n=1

εpnApn

σT 4pn

πV(11.3-16)

and

ap = limV →0

N∑n=1

εpnApn

V(11.3-17)

In Equations 11.3-16 and 11.3-17, εpn, Apn, and Tpn are the emissivity,projected area, and temperature of particle n. The summation is overN particles in volume V . These quantities are computed during particletracking in FLUENT.

The projected area Apn of particle n is defined as

Apn =πd2

pn

4(11.3-18)

where dpn is the diameter of the nth particle.

The quantity Γ in Equation 11.3-15 is defined as

Γ =1

3(a+ ap + σp)(11.3-19)

where the equivalent particle scattering factor is defined as

σp = limV →0

N∑n=1

(1 − fpn)(1 − εpn)Apn

V(11.3-20)



and is computed during particle tracking. In Equation 11.3-20 fpn is thescattering factor associated with the nth particle.

Heat sources (sinks) due to particle radiation are included in the energyequation as follows:

−∇qr = −4π

(aσT 4

π+ Ep

)+ (a+ ap)G (11.3-21)

Boundary Condition Treatment for the P-1 Model at Walls

To get the boundary condition for the incident radiation equation, thedot product of the outward normal vector ~n and equation 11.3-11 iscomputed:

qr · ~n = −Γ∇G · ~n (11.3-22)

qr,w = −Γ∂G

∂n(11.3-23)

Thus the flux of the incident radiation, G, at a wall is −qr,w. The wallradiative heat flux is computed using the following boundary condition:

Iw(~r,~s) = fw(~r,~s) (11.3-24)

fw(~r,~s) = εwσT 4

w

π+ ρwI(~r,−~s) (11.3-25)

where ρw is the wall reflectivity. The Marshak boundary condition isthen used to eliminate the angular dependence [171]:

∫ 2π

0Iw(~r,~s) ~n · ~s dΩ =

∫ 2π

0fw(~r,~s) ~n · ~s dΩ (11.3-26)

Substituting Equations 11.3-24 and 11.3-25 into Equation 11.3-26 andperforming the integrations yields



qr,w = − 4πεwσT 4

wπ − (1 − ρw)Gw

2(1 + ρw)(11.3-27)

If it is assumed that the walls are diffuse gray surfaces, then ρw = 1−εw,and Equation 11.3-27 becomes

qr,w = − εw2 (2 − εw)

(4σT 4

w −Gw

)(11.3-28)

Equation 11.3-28 is used to compute qr,w for the energy equation and forthe incident radiation equation boundary conditions.

Boundary Condition Treatment for the P-1 Model at Flow Inletsand Exits

The net radiative heat flux at flow inlets and outlets is computed in thesame manner as at walls, as described above. FLUENT assumes that theemissivity of all flow inlets and outlets is 1.0 (black body absorption)unless you choose to redefine this boundary treatment.


11.3.5 The Rosseland Radiation Model

The Rosseland or diffusion approximation for radiation is valid when themedium is optically thick ((a+σs)L 1), and is recommended for use inproblems where the optical thickness is greater than 3. It can be derivedfrom the P-1 model equations, with some approximations. This sectionprovides details about the equations used in the Rosseland model.

The Rosseland Model Equations

As with the P-1 model, the radiative heat flux vector in a gray mediumcan be approximated by Equation 11.3-11:



qr = −Γ∇G (11.3-29)

where Γ is given by Equation 11.3-10.

The Rosseland radiation model differs from the P-1 model in that theRosseland model assumes that the intensity is the black-body intensityat the gas temperature. (The P-1 model actually calculates a transportequation for G.) Thus G = 4σT 4. Substituting this value for G intoEquation 11.3-29 yields

qr = −16σΓT 3∇T (11.3-30)

Since the radiative heat flux has the same form as the Fourier conductionlaw, it is possible to write

q = qc + qr (11.3-31)= −(k + kr)∇T (11.3-32)

kr = 16σΓT 3 (11.3-33)

where k is the thermal conductivity and kr is the radiative conductiv-ity. Equation 11.3-31 is used in the energy equation to compute thetemperature field.


The Rosseland model allows for anisotropic scattering, using the samephase function (Equation 11.3-14) described for the P-1 model in Sec-tion 11.3.4.

Boundary Condition Treatment for the Rosseland Model at Walls

Since the diffusion approximation is not valid near walls, it is necessaryto use a temperature slip boundary condition. The radiative heat fluxat the wall boundary, qr,w, is defined using the slip coefficient ψ:



qr,w = −σ(T 4

w − T 4g

)ψ

(11.3-34)

where Tw is the wall temperature, Tg is the temperature of the gas atthe wall, and the slip coefficient ψ is approximated by a curve fit to theplot given in [210]:

ψ =

1/2 Nw < 0.012x3+3x2−12x+7

54 0.01 ≤ Nw ≤ 100 Nw > 10

(11.3-35)

where Nw is the conduction to radiation parameter at the wall:

Nw =k(a+ σs)

4σT 3w

(11.3-36)

and x = log10Nw.

Boundary Condition Treatment for the Rosseland Model at FlowInlets and Exits

No special treatment is required at flow inlets and outlets for the Rosse-land model. The radiative heat flux at these boundaries can be deter-mined using Equation 11.3-31.


11.3.6 The Discrete Ordinates (DO) Radiation Model

The discrete ordinates (DO) radiation model solves the radiative trans-fer equation (RTE) for a finite number of discrete solid angles, eachassociated with a vector direction ~s fixed in the global Cartesian system(x, y, z). The fineness of the angular discretization is controlled by you,analogous to choosing the number of rays for the DTRM. Unlike the



DTRM, however, the DO model does not perform ray tracing. Instead,the DO model transforms Equation 11.3-1 into a transport equation forradiation intensity in the spatial coordinates (x, y, z). The DO modelsolves for as many transport equations as there are directions ~s. Thesolution method is identical to that used for the fluid flow and energyequations.

The implementation in FLUENT uses a conservative variant of the dis-crete ordinates model called the finite-volume scheme [37, 183], and itsextension to unstructured meshes [165].

The DO Model Equations

The DO model considers the radiative transfer equation (RTE) in thedirection ~s as a field equation. Thus, Equation 11.3-1 is written as

∇ · (I(~r,~s)~s) + (a+ σs)I(~r,~s) = an2σT4

π+σs

4π

∫ 4π

0I(~r,~s ′) Φ(~s · ~s ′) dΩ′

(11.3-37)

FLUENT also allows the modeling of non-gray radiation using a gray-band model. The RTE for the spectral intensity Iλ(~r,~s) can be writtenas

∇·(Iλ(~r,~s)~s)+(aλ+σs)Iλ(~r,~s) = aλn2Ibλ+

σs

4π

∫ 4π

0Iλ(~r,~s ′) Φ(~s·~s ′) dΩ′

(11.3-38)

Here λ is the wavelength, aλ is the spectral absorption coefficient, and Ibλis the black body intensity given by the Planck function. The scatteringcoefficient, the scattering phase function, and the refractive index n areassumed independent of wavelength.

The non-gray DO implementation divides the radiation spectrum into Nwavelength bands, which need not be contiguous or equal in extent. Thewavelength intervals are supplied by you, and correspond to values invacuum (n = 1). The RTE is integrated over each wavelength interval,



resulting in transport equations for the quantity Iλ∆λ, the radiant en-ergy contained in the wavelength band ∆λ. The behavior in each bandis assumed gray. The black body emission in the wavelength band perunit solid angle is written as

[F (0 → nλ2T ) − F (0 → nλ1T )]n2σT4

π(11.3-39)

where F (0 → nλT ) is the fraction of radiant energy emitted by a blackbody [161] in the wavelength interval from 0 to λ at temperature T in amedium of refractive index n. λ2 and λ1 are the wavelength boundariesof the band.

The total intensity I(~r,~s) in each direction ~s at position ~r is computedusing

I(~r,~s) =∑k

Iλk(~r,~s)∆λk (11.3-40)

where the summation is over the wavelength bands.

Boundary conditions for the non-gray DO model are applied on a bandbasis. The treatment within a band is the same as that for the gray DOmodel.

Angular Discretization and Pixelation

Each octant of the angular space 4π at any spatial location is discretizedinto Nθ×Nφ solid angles of extent ωi, called control angles. The angles θand φ are the polar and azimuthal angles respectively, and are measuredwith respect to the global Cartesian system (x, y, z) as shown in Fig-ure 11.3.3. The θ and φ extents of the control angle, ∆θ and ∆φ, areconstant. In two-dimensional calculations, only four octants are solveddue to symmetry, making a total of 4NθNφ directions in all. In three-dimensional calculations, a total of 8NθNφ directions are solved. In thecase of the non-gray model, 4NθNφ or 8NθNφ equations are solved foreach band.



θ

φ

z

yx

s

Figure 11.3.3: Angular Coordinate System

When Cartesian meshes are used, it is possible to align the global angulardiscretization with the control volume face, as shown in Figure 11.3.4.For generalized unstructured meshes, however, control volume faces donot in general align with the global angular discretization, as shown inFigure 11.3.5, leading to the problem of control angle overhang [165].

Essentially, control angles can straddle the control volume faces, so thatthey are partially incoming and partially outgoing to the face. Fig-ure 11.3.6 shows a 3D example of a face with control angle overhang.

The control volume face cuts the sphere representing the angular spaceat an arbitrary angle. The line of intersection is a great circle. Controlangle overhang may also occur as a result of reflection and refraction.It is important in these cases to correctly account for the overhangingfraction. This is done through the use of pixelation [165].

Each overhanging control angle is divided into Nθp×Nφp pixels, as shownin Figure 11.3.7. The energy contained in each pixel is then treated asincoming or outgoing to the face. The influence of overhang can thus beaccounted for within the pixel resolution. FLUENT allows you to choose



C0 C1 n

face f

incomingdirections

outgoingdirections

Figure 11.3.4: Face with No Control Angle Overhang

C0

C1

n

face f

incomingdirections

outgoingdirections

overhanging control angle

Figure 11.3.5: Face with Control Angle Overhang



x

y

z

outgoingdirections

incomingdirections

overhangingcontrolangle

controlvolumeface

Figure 11.3.6: Face with Control Angle Overhang (3D)

pixel

control volumeface

control angle ω i

s i

Figure 11.3.7: Pixelation of Control Angle



the pixel resolution. For problems involving gray-diffuse radiation, thedefault pixelation of 1 × 1 is usually sufficient. For problems involvingsymmetry, periodic, specular, or semi-transparent boundaries, a pixe-lation of 3 × 3 is recommended. You should be aware, however, thatincreasing the pixelation adds to the cost of computation.


The DO implementation in FLUENT admits a variety of scattering phasefunctions. You can choose an isotropic phase function, a linear anisotropicphase function, a Delta-Eddington phase function, or a user-definedphase function. The linear anisotropic phase function is described inEquation 11.3-14. The Delta-Eddington function takes the followingform:

Φ(~s · ~s ′) = 2fδ(~s · ~s ′) + (1 − f)(1 + C~s · ~s ′) (11.3-41)

Here, f is the forward-scattering factor and δ(~s · ~s ′) is the Dirac deltafunction. The f term essentially cancels a fraction f of the out-scattering;thus, for f = 1, the Delta-Eddington phase function will cause the in-tensity to behave as if there is no scattering at all. C is the asymmetryfactor. When the Delta-Eddington phase function is used, you will spec-ify values for f and C.

When a user-defined function is used to specify the scattering phasefunction, FLUENT assumes the phase function to be of the form

Φ(~s · ~s ′) = 2fδ(~s · ~s ′) + (1 − f)Φ∗(~s · ~s ′) (11.3-42)

The user-defined function will specify Φ∗ and the forward-scattering fac-tor f .

The scattering phase functions available for gray radiation can also beused for non-gray radiation. However, the scattered energy is restrictedto stay within the band.



Particulate Effects in the DO Model

The DO model allows you to include the effect of a discrete second phaseof particulates on radiation. In this case, FLUENT will neglect all othersources of scattering in the gas phase.

The contribution of the particulate phase appears in the RTE as:

∇·(I~s)+(a+ap+σp)I(~r,~s) = an2σT4

π+Ep+

σp

4π

∫ 4π

0I(~r,~s ′) Φ(~s·~s ′) dΩ′

(11.3-43)

where ap is the equivalent absorption coefficient due to the presence ofparticulates, and is given by Equation 11.3-17. The equivalent emissionEp is given by Equation 11.3-16. The equivalent particle scattering factorσp, defined in Equation 11.3-20, is used in the scattering terms.

For non-gray radiation, absorption, emission, and scattering due to theparticulate phase are included in each wavelength band for the radiationcalculation. Particulate emission and absorption terms are also includedin the energy equation.

Boundary Condition Treatment at Gray-Diffuse Walls

For gray radiation, the incident radiative heat flux, qin, at the wall is

qin =∫~s·~n>0

Iin~s · ~ndΩ (11.3-44)

The net radiative flux leaving the surface is given by

qout = (1 − εw)qin + n2εwσT4w (11.3-45)

where n is the refractive index of the medium next to the wall. Theboundary intensity for all outgoing directions ~s at the wall is given by

I0 =qout

π(11.3-46)



For non-gray radiation, the incident radiative heat flux qin,λ in the band∆λ at the wall is

qin,λ = ∆λ∫~s·~n>0

Iin,λ~s · ~ndΩ (11.3-47)

The net radiative flux leaving the surface in the band ∆λ is given by

qout,λ = (1 − εwλ)qin,λ + εwλ[F (0 → nλ2Tw) − F (0 → nλ1Tw)]n2σT 4w

(11.3-48)

where εwλ is the wall emissivity in the band. The boundary intensity forall outgoing directions ~s in the band ∆λ at the wall is given by

I0λ =qout,λ

π∆λ(11.3-49)

Boundary Condition Treatment at Semi-Transparent Walls

FLUENT allows the specification of both diffusely and specularly reflect-ing semi-transparent walls. You can prescribe the fraction of the incom-ing radiation at the semi-transparent wall which is to be reflected andtransmitted diffusely; the rest is treated specularly.

For non-gray radiation, this treatment is applied on a band basis. Theradiant energy within a band ∆λ is transmitted, reflected, and refractedas in the gray case; there is no transmission, reflection, or refraction ofradiant energy from one band to another.

Specular Semi-Transparent Walls

Consider a ray traveling from a semi-transparent medium a with refrac-tive index na to a semi-transparent medium b with a refractive index nb

in the direction ~s, as shown in Figure 11.3.8. Side a of the interface is theside that faces medium a; similarly, side b faces medium b. The interfacenormal ~n is assumed to point into side a. We distinguish between theintensity Ia(~s), the intensity in the direction ~s on side a of the interface,and the corresponding quantity on the side b, Ib(~s).



θa

θ b

s sr

sts’

n

medium b

medium a

n > nb a

Figure 11.3.8: Reflection and Refraction of Radiation at the InterfaceBetween Two Semi-Transparent Media

A part of the energy incident on the interface is reflected, and the rest istransmitted. The reflection is specular, so that the direction of reflectedradiation is given by

~sr = ~s− 2 (~s · ~n)~n (11.3-50)

The radiation transmitted from medium a to medium b undergoes re-fraction. The direction of the transmitted energy, ~st, is given by Snell’slaw:

sin θb =na

nbsin θa (11.3-51)

where θa is the angle of incidence and θb is the angle of transmission, asshown in Figure 11.3.8. We also define the direction

~s ′ = ~st − 2 (~st · ~n)~n (11.3-52)



shown in Figure 11.3.8.

The interface reflectivity on side a [161]

ra(~s) =12

(na cos θb − nb cos θa

na cos θb + nb cos θa

)2

+12

(na cos θa − nb cos θb

na cos θa + nb cos θb

)2

(11.3-53)

represents the fraction of incident energy transferred from ~s to ~sr.

The boundary intensity Iw,a(~sr) in the outgoing direction ~sr on side a ofthe interface is determined from the reflected component of the incomingradiation and the transmission from side b. Thus

Iw,a(~sr) = ra(~s)Iw,a(~s) + τb(~s ′)Iw,b(~s ′) (11.3-54)

where τb(~s ′) is the transmissivity of side b in direction ~s ′. Similarly, theoutgoing intensity in the direction ~st on side b of the interface, Iw,b(~st),is given by

Iw,b(~st) = rb(~s ′)Iw,b(~s ′) + τa(~s)Iw,a(~s) (11.3-55)

For the case na < nb, the energy transmitted from medium a to mediumb in the incoming solid angle 2π must be refracted into a cone of apexangle θc (see Figure 11.3.9) where

θc = sin−1 na

nb(11.3-56)

Similarly, the transmitted component of the radiant energy going frommedium b to medium a in the cone of apex angle θc is refracted into theoutgoing solid angle 2π. For incident angles greater than θc, total internalreflection occurs and all the incoming energy is reflected specularly backinto medium b.

When medium b is external to the domain, Iw,b(~s ′) is given in Equa-tion 11.3-54 as a part of the problem specification. This boundary speci-fication is usually made by providing the incoming radiative flux and the



θa

θ b

medium b

medium a

n > nb a

θ c

n

Figure 11.3.9: Critical Angle θc

solid angle over which the radiative flux is to be applied. The refractiveindex of the external medium is assumed to be unity.

Diffuse Semi-Transparent Walls

In many engineering problems, the semi-transparent interface may bea diffuse reflector. For such a case, the interfacial reflectivity r(~s) isassumed independent of ~s, and equal to the hemispherically averagedvalue rd. For n = na/nb > 1, rd,a and rd,b are given by [211]

rd,a = 1 − (1 − rd,b)n2

(11.3-57)

rd,b =12

+(3n + 1)(n − 1)

6(n+ 1)2+n2(n2 − 1)2

(n2 + 1)3ln(n− 1n+ 1

)−

2n3(n2 + 2n− 1)(n2 + 1)(n4 − 1)

+8n4(n4 + 1)

(n2 + 1)(n4 − 1)2ln(n) (11.3-58)



The boundary intensity for all outgoing directions on side a of the inter-face is given by

Iw,a =rd,aqin,a + τd,bqin,b

π(11.3-59)

Similarly for side b,

Iw,b =rd,bqin,b + τd,aqin,a

π(11.3-60)

where

qin,a = −∫4πIw,a~s · ~ndΩ, ~s · ~n < 0 (11.3-61)

qin,b =∫4πIw,b~s · ~ndΩ, ~s · ~n ≥ 0 (11.3-62)

As before, if medium b is external to the domain, qin,b is given as a partof the boundary specification.

Beam Irradiation

As mentioned above, FLUENT allows the specification of the irradiationat semi-transparent boundaries. The irradiation is specified in termsof an incident radiant heat flux (W/m2). You can specify the solidangle over which the irradiation is distributed, as well as the vector ofthe centroid of the solid angle. To indicate whether the irradiation isreflected specularly or diffusely, you can specify the diffuse fraction.

For non-gray radiation, FLUENT allows you to specify the irradiation atsemi-transparent boundaries on a band basis. The irradiation is specifiedas an incident heat flux (W/m2) for each wavelength band. As in thegray case, you can specify the solid angle over which the irradiation isdistributed, as well as the vector of the centroid of the solid angle.



The Diffuse Fraction

At semi-transparent boundaries, FLUENT allows you to specify the frac-tion of the incoming radiation that is treated as diffuse. The diffusefraction is reflected diffusely, using the treatment described above; thetransmitted portion is also treated diffusely. The remainder of the in-coming energy is treated in a specular fashion.

For non-gray radiation, you can specify the diffuse fraction separatelyfor each band.

Boundary Condition Treatment at Specular Walls and SymmetryBoundaries

At specular walls and symmetry boundaries, the direction of the re-flected ray ~sr corresponding to the incoming direction ~s is given byEquation 11.3-50. Furthermore

Iw(~sr) = Iw(~s) (11.3-63)

Boundary Condition Treatment at Periodic Boundaries

When rotationally periodic boundaries are used, it is important to usepixelation in order to ensure that radiant energy is correctly transferredbetween the periodic and shadow faces. A pixelation between 3 × 3 and10 × 10 is recommended.

Boundary Condition Treatment at Flow Inlets and Exits

The treatment at flow inlets and exits is described in Section 11.3.3.

11.3.7 The Surface-to-Surface (S2S) Radiation Model

The surface-to-surface radiation model can be used to account for theradiation exchange in an enclosure of gray-diffuse surfaces. The energyexchange between two surfaces depends in part on their size, separa-tion distance, and orientation. These parameters are accounted for by ageometric function called a “view factor”.



The main assumption of the S2S model is that any absorption, emission,or scattering of radiation can be ignored; therefore, only “surface-to-surface” radiation need be considered for analysis.

Gray-Diffuse Radiation

FLUENT’s S2S radiation model assumes the surfaces to be gray anddiffuse. Emissivity and absorptivity of a gray surface are independent ofthe wavelength. Also, by Kirchoff’s law [161], the emissivity equals theabsorptivity (ε = α). For a diffuse surface, the reflectivity is independentof the outgoing (or incoming) directions.

The gray-diffuse model is what is used in FLUENT. Also, as stated earlier,for applications of interest, the exchange of radiative energy between sur-faces is virtually unaffected by the medium that separates them. Thus,according to the gray-body model, if a certain amount of radiant energy(E) is incident on a surface, a fraction (ρE) is reflected, a fraction (αE)is absorbed, and a fraction (τE) is transmitted. Since for most appli-cations the surfaces in question are opaque to thermal radiation (in theinfrared spectrum), the surfaces can be considered opaque. The trans-missivity, therefore, can be neglected. It follows, from conservation ofenergy, that α+ ρ = 1, since α = ε (emissivity), and ρ = 1 − ε.

The S2S Model Equations

The energy flux leaving a given surface is composed of directly emittedand reflected energy. The reflected energy flux is dependent on the inci-dent energy flux from the surroundings, which then can be expressed interms of the energy flux leaving all other surfaces. The energy reflectedfrom surface k is

qout,k = εkσT4k + ρkqin,k (11.3-64)

where qout,k is the energy flux leaving the surface, εk is the emissivity,σ is Boltzmann’s constant, and qin,k is the energy flux incident on thesurface from the surroundings.



The amount of incident energy upon a surface from another surface isa direct function of the surface-to-surface “view factor,” Fjk. The viewfactor Fjk is the fraction of energy leaving surface k that is incident onsurface j. The incident energy flux qin,k can be expressed in terms of theenergy flux leaving all other surfaces as

Akqin,k =N∑

j=1

Ajqout,jFjk (11.3-65)

where Ak is the area of surface k and Fjk is the view factor betweensurface k and surface j. For N surfaces, using the view factor reciprocityrelationship gives

AjFjk = AkFkj for j = 1, 2, 3, . . . N (11.3-66)

so that

qin,k =N∑

j=1

Fkjqout,j (11.3-67)

Therefore,

qout,k = εkσT4k + ρk

N∑j=1

Fkjqout,j (11.3-68)

which can be written as

Jk = Ek + ρk

N∑j=1

FkjJj (11.3-69)

where Jk represents the energy that is given off (or radiosity) of surfacek, and Ek represents the emissive power of surface k. This represents Nequations, which can be recast into matrix form as



KJ = E (11.3-70)

where K is an N × N matrix, J is the radiosity vector, and E is theemissive power vector.

Equation 11.3-70 is referred to as the radiosity matrix equation. Theview factor between two finite surfaces i and j is given by

Fij =1Ai

∫Ai

∫Aj

cos θi cos θj

πr2δijdAidAj (11.3-71)

where δij is determined by the visibility of dAj to dAi. δij = 1 if dAj isvisible to dAi and 0 otherwise.

Clustering

The S2S radiation model is computationally very expensive when thereare a large number of radiating surfaces. To reduce the computationaltime as well as the storage requirement, the number of radiating surfacesis reduced by creating surface “clusters”. The surface clusters are madeby starting from a face and adding its neighbors and their neighbors untila specified number of faces per surface cluster is collected.

The radiosity, J , is calculated for the surface clusters. These valuesare then distributed to the faces in the clusters to calculate the walltemperatures. Since the radiation source terms are highly non-linear(proportional to the fourth power of temperature), care must be taken tocalculate the average temperature of the surface clusters and distributethe flux and source terms appropriately among the faces forming theclusters.

The surface cluster temperature is obtained by area averaging as shownin the following equation:

Tsc =

(∑f AfT

4f∑

Af

)1/4

(11.3-72)



where Tsc is the temperature of the surface cluster, and Af and Tf arethe area and temperature of face f . The summation is carried over allfaces of a surface cluster.

Smoothing

Smoothing can be performed on the view factor matrix to enforce thereciprocity relationship and conservation.

The reciprocity relationship is represented by

AiFij = AiFji (11.3-73)

where Ai is the area of surface i, Fij is the view factor between surfacesi and j, and Fji is the view factor between surfaces j and i.

Once the reciprocity relationship has been enforced, a least-squares smooth-ing method [123] can be used to ensure that conservation is satisfied, i.e.,

∑Fij = 1.0 (11.3-74)

11.3.8 Radiation in Combusting Flows

The Weighted-Sum-of-Gray-Gases Model

The weighted-sum-of-gray-gases model (WSGGM) is a reasonable com-promise between the oversimplified gray gas model and a complete modelwhich takes into account particular absorption bands. The basic assump-tion of the WSGGM is that the total emissivity over the distance s canbe presented as

ε =I∑

i=0

aε,i(T )(1 − e−κips) (11.3-75)

where aε,i are the emissivity weighting factors for the ith fictitious graygas, the bracketed quantity is the ith fictitious gray gas emissivity, κi isthe absorption coefficient of the ith gray gas, p is the sum of the partial



pressures of all absorbing gases, and s is the path length. For aε,i and κi

FLUENT uses values obtained from [41] and [219]. These values dependon gas composition, and aε,i also depend on temperature. When thetotal pressure is not equal to 1 atm, scaling rules for κi are used (seeEquation 11.3-81).

The absorption coefficient for i = 0 is assigned a value of zero to accountfor windows in the spectrum between spectral regions of high absorption(∑I

i=1 aε,i < 1) and the weighting factor for i = 0 is evaluated from [219]:

aε,0 = 1 −I∑

i=1

aε,i (11.3-76)

The temperature dependence of aε,i can be approximated by any func-tion, but the most common approximation is

aε,i =J∑

j=1

bε,i,jTj−1 (11.3-77)

where bε,i,j are the emissivity gas temperature polynomial coefficients.The coefficients bε,i,j and κi are estimated by fitting Equation 11.3-75 tothe table of total emissivities, obtained experimentally [41, 49, 219].

The absorptivity α of the radiation from the wall can be approximated ina similar way [219], but, to simplify the problem, it is assumed that ε =α [160]. This assumption is justified unless the medium is optically thinand the wall temperature differs considerably from the gas temperature.

Since the coefficients bε,i,j and κi are slowly varying functions of ps andT , they can be assumed constant for a wide range of these parameters.In [219] these constant coefficients are presented for different relativepressures of the CO2 and H2O vapor, assuming that the total pressure pT

is 1 atm. The values of the coefficients shown in [219] are valid for 0.001 ≤ps ≤ 10.0 atm-m and 600 ≤ T ≤ 2400 K. For T > 2400 K, coefficientvalues suggested by [41] are used. If κips 1 for all i, Equation 11.3-75simplifies to



ε =I∑

i=0

aε,iκips (11.3-78)

Comparing Equation 11.3-78 with the gray gas model with absorptioncoefficient a, it can be seen that the change of the radiation intensityover the distance s in the WSGGM is exactly the same as in the graygas model with the absorption coefficient

a =I∑

i=0

aε,iκip (11.3-79)

which does not depend on s. In the general case, a is estimated as

a = − ln(1 − ε)s

(11.3-80)

where the emissivity ε for the WSGGM is computed using Equation 11.3-75.a as defined by Equation 11.3-80 depends on s, reflecting the non-graynature of the absorption of thermal radiation in molecular gases. In FLU-ENT, Equation 11.3-79 is used when s ≤ 10−4 m and Equation 11.3-80is used for s > 10−4 m. Note that for s ≈ 10−4 m, the values of a pre-dicted by Equations 11.3-79 and 11.3-80 are practically identical (sinceEquation 11.3-80 reduces to Equation 11.3-79 in the limit of small s).

FLUENT allows you to specify s as the characteristic cell size or the meanbeam length. The model based on the mean beam length is appropriateif you have a nearly homogeneous medium and you are interested mainlyin the radiation exchange between the walls of the enclosure. You canspecify the mean beam length or have FLUENT compute it. If you areprimarily interested in the radiation heat exchange between neighboringcells (e.g., the distribution of radiation in the vicinity of a heater), whichis very common for the optically thick media for which the P-1 model isprimarily designed, using the characteristic cell size as s is more appro-priate. Note that the values of a predicted by the WSGGM based onthe characteristic cell size can be somewhat grid-dependent, if s is small.This grid dependence, however, will not necessarily affect the predicted



temperature distribution, since the radiation energy is proportional toT 4. The characteristic-cell-size approach may give a better temperaturedistribution, while the mean-beam-length approach can give more accu-rate fluxes at the boundaries. See Section 7.6.1 for details about settingproperties for the WSGGM.

The WSGGM cannot be used to specify the absorption coefficient in!each band when using the non-gray DO model. If the WSGGM is usedwith the non-gray DO model, the absorption coefficient will be the samein all bands.

When pT 6= 1 atm

The WSGGM, as described above, assumes that pT —the total (static)gas pressure—is equal to 1 atm. In cases where pT is not unity (e.g.,combustion at high temperatures), scaling rules suggested in [59] areused to introduce corrections. When pT < 0.9 atm or pT > 1.1 atm, thevalues for κi in Equations 11.3-75 and 11.3-79 are rescaled:

κi → κipmT (11.3-81)

where m is a nondimensional value obtained from [59], which dependson the partial pressures and temperature T of the absorbing gases, aswell as on pT .

The Effect of Soot on the Absorption Coefficient

When soot formation is computed, FLUENT can include the effect of thesoot concentration on the radiation absorption coefficient. The general-ized soot model estimates the effect of the soot on radiative heat transferby determining an effective absorption coefficient for soot. The absorp-tion coefficient of a mixture of soot and an absorbing (radiating) gas isthen calculated as the sum of the absorption coefficients of pure gas andpure soot:

as+g = ag + as (11.3-82)



where ag is the absorption coefficient of gas without soot (obtained fromthe WSGGM) and

as = b1cm[1 + bT (T − 2000)] (11.3-83)

with

b1 = 1232.4 m2/kg and bT ≈ 4.8 × 10−4 K−1

cm is the soot concentration in kg/m3.

The coefficients b1 and bT were obtained [199] by fitting Equation 11.3-83to data based on the Taylor-Foster approximation [240] and data basedon the Smith et al. approximation [219].

See Sections 7.6 and 17.2.3 for information about including the soot-radiation interaction effects.

The Effect of Particles on the Absorption Coefficient

FLUENT can also include the effect of discrete phase particles on theradiation absorption coefficient, provided that you are using either theP-1 or the DO model. When the P-1 or DO model is active, radiationabsorption by particles can be enabled. The particle emissivity, reflec-tivity, and scattering effects are then included in the calculation of theradiative heat transfer. See Section 19.11 for more details on the inputof radiation properties for the discrete phase.



11.3.9 Overview of Using the Radiation Models

The procedure for setting up and solving a radiation problem is outlinedbelow, and described in detail in Sections 11.3.10–11.3.18. Steps thatare relevant only for a particular radiation model are noted as such.Remember that only the steps that are pertinent to radiation modelingare shown here. For information about inputs related to other modelsthat you are using in conjunction with radiation, see the appropriatesections for those models.

1. Select the radiation model, as described in Section 11.3.10.

2. If you are using the DTRM, define the ray tracing as describedin Section 11.3.11. If you are using the S2S model, compute orread the view factors as described in Section 11.3.12. If you areusing the DO model, define the angular discretization as describedin Section 11.3.13 and, if relevant, define the non-gray radiationparameters as described in Section 11.3.14.

3. Define the material properties, as described in Section 11.3.15.

4. Define the boundary conditions, as described in Section 11.3.16. Ifyour model contains a semi-transparent medium, see the informa-tion below on setting up semi-transparent media.

5. Set the solution parameters (DTRM, DO, S2S, and P-1 only). SeeSection 11.3.17 for details.

6. Solve the problem, as described in Section 11.3.18.

Setup of Semi-Transparent Media

As part of step 4 above, you will take the following steps in order to setup a semi-transparent medium such as glass in your domain.

1. If your semi-transparent material is a solid, enable the calculationof radiation in the solid cell zone, as described in Section 11.3.16.(If your semi-transparent material is a fluid, this step is not neces-sary.)



2. At internal boundaries between the semi-transparent medium andadjacent fluids (or between adjacent semi-transparent media), setthe two-sided wall to be semi-transparent, as described in Sec-tion 11.3.16. This will enable radiation to pass through the in-ternal boundary and will account for effects such as reflection andrefraction.

3. At external semi-transparent boundaries, set the external wall ra-diation boundary condition to be semi-transparent, as described inSection 11.3.16. This will allow the externally specified radiativeflux to enter the domain, and also allow the transmission of radia-tion from the interior to the outside. Both the radiation from theexterior of the domain and the radiation being transmitted fromthe interior to the outside will be reflected and refracted at theboundary appropriately.

4. Specify the degree to which interior and exterior walls reflect dif-fusely or specularly by setting the diffuse fraction, as described inSection 11.3.16.

5. Specify the appropriate refractive index for the material associatedwith the solid cell zone (in the Materials panel).

If you are not interested in the detailed temperature distribution insideyour semi-transparent medium, you can use a thin semi-transparent wallinstead of a semi-transparent solid zone, as described in Section 11.3.16.

11.3.10 Selecting the Radiation Model

You can enable radiative heat transfer by selecting a radiation model inthe Radiation Model panel (Figure 11.3.10).

Define −→ Models −→Radiation...

Select Rosseland, P1, Discrete Transfer (DTRM), Surface to Surface (S2S),or Discrete Ordinates as the Model. To disable radiation, select Off. Notethat when the DTRM or the DO or S2S model is activated, the RadiationModel panel will expand to show additional parameters (described in



Figure 11.3.10: The Radiation Model Panel (DO Model)

Sections 11.3.12, 11.3.13, 11.3.14, and 11.3.17). These parameters willnot appear if you select one of the other radiation models.

The Rosseland model can be used only with the segregated solver.!

When the radiation model is active, the radiation fluxes will be includedin the solution of the energy equation at each iteration. If you set upa problem with the radiation model turned on, and you then decide toturn it off completely, you must select the Off button in the RadiationModel panel.

Note that, when you enable a radiation model, FLUENT will automati-cally turn on solution of the energy equation; you need not turn on theenergy equation first.



11.3.11 Defining the Ray Tracing for the DTRM

When you select the Discrete Transfer model and click OK in the Ra-diation Model panel, the Ray Tracing panel (Figure 11.3.11) will openautomatically. (Should you need to modify the current settings laterin the problem setup or solution procedure, you can open this panelmanually using the Define/Ray Tracing... menu item.)

Figure 11.3.11: The Ray Tracing Panel

In this panel you will set parameters for and create the rays and clustersdiscussed in Section 11.3.3.

The procedure is as follows:

1. To control the number of radiating surfaces and absorbing cells,set the Cells Per Volume Cluster and Faces Per Surface Cluster. (Seethe explanation below.)

2. To control the number of rays being traced, set the number ofTheta Divisions and Phi Divisions. (Guidelines are provided below.)

3. When you click OK in the Ray Tracing panel, a Select File dialogbox will open, prompting you for the name of the “ray file”. Afteryou have specified the file name and written the ray file, FLUENTwill read the file back in again for use in the calculation. See belowfor details.



If you cancel the Ray Tracing panel without writing and reading the ray!file, the DTRM will be disabled.

Controlling the Clusters

Your inputs for Cells Per Volume Cluster and Faces Per Surface Cluster willcontrol the number of radiating surfaces and absorbing cells. By default,each is set to 1, so the number of surface clusters (radiating surfaces)will be the number of boundary faces, and the number of volume clusters(absorbing cells) will be the number of cells in the domain. For small 2Dproblems, these are acceptable numbers, but for larger problems you willwant to reduce the number of surface and/or volume clusters in orderto reduce the ray-tracing expense. (See Section 11.3.3 for details aboutclustering.)

Controlling the Rays

Your inputs for Theta Divisions and Phi Divisions will control the numberof rays being traced from each surface cluster (radiating surface).

Theta Divisions defines the number of discrete divisions in the angle θused to define the solid angle about a point P on a surface. The solidangle is defined as θ varies from 0 to 90 degrees (Figure 11.3.2), and thedefault setting of 2 for the number of discrete settings implies that eachray traced from the surface will be located at a 45 angle from the otherrays.

Phi Divisions defines the number of discrete divisions in the angle φ usedto define the solid angle about a point P on a surface. The solid angleis defined as φ varies from 0 to 180 degrees in 2D and from 0 to 360degrees in 3D (Figure 11.3.2). The default setting of 2 implies that eachray traced from the surface will be located at a 90 angle from the otherrays in 2D calculations, and in combination with the default settingfor Theta Divisions, above, implies that 4 rays will be traced from eachsurface control volume in your 2D model. Note that the Phi Divisionsshould be increased to 4 for equivalent accuracy in 3D models. In manycases, it is recommended that you at least double the number of divisionsin θ and φ.



Writing and Reading the DTRM Ray File

After you have activated the DTRM and defined all of the parameterscontrolling the ray tracing, you must create a ray file which will be readback in and used during the radiation calculation. The ray file containsa description of the ray traces (path lengths, cells traversed by eachray, etc.). This information is stored in the ray file, instead of beingrecomputed, in order to speed up the calculation process.

By default, a binary ray file will be written. You can also create text(formatted) ray files by turning off the Write Binary Files option in theSelect File dialog box.

Do not write or read a compressed ray file, because FLUENT will not be!able to access the ray tracing information properly from a compressedray file.

The ray filename must be specified to FLUENT only once. Thereafter, thefilename is stored in your case file and the ray file will be automaticallyread into FLUENT whenever the case file is read. FLUENT will remindyou that it is reading the ray file after it finishes reading the rest of thecase file by reporting its progress in the text (console) window.

Note that the ray filename stored in your case file may not contain the fullname of the directory in which the ray file exists. The full directory namewill be stored in the case file only if you initially read the ray file throughthe GUI (or if you typed in the directory name along with the filenamewhen using the text interface). In the event that the full directory nameis absent, the automatic reading of the ray file may fail (since FLUENTdoes not know in which directory to look for the file), and you will needto manually specify the ray file, using the File/Read/Rays... menu item.The safest approaches are to use the GUI when you first read the rayfile or to supply the full directory name when using the text interface.

You should recreate the ray file whenever you do anything that changes!the grid, such as:

• change the type of a boundary zone

• adapt or reorder the grid



• scale the grid

• change from 2D space to axisymmetric space or vice versa

You can open the Ray Tracing panel directly with the Define/Ray Trac-ing... menu item.

Displaying the Clusters

Once a ray file has been created or read in manually, you can click on theDisplay Clusters button in the Ray Tracing panel to graphically display theclusters in the domain. See Section 11.3.20 for additional informationabout displaying rays and clusters.

11.3.12 Computing or Reading the View Factors for the S2SModel

When you select the Surface to Surface (S2S) model, the Radiation Modelpanel will expand (see Figure 11.3.12). In this section of the panel,you will compute the view factors for your problem or read previouslycomputed view factors into FLUENT.

The S2S radiation model is computationally very expensive when thereare a large number of radiating surfaces. To reduce the computationaltime as well as the storage requirement, the number of radiating surfacesis reduced by creating surface clusters. The surface cluster information(coordinates and connectivity of the nodes, surface cluster IDs) is usedby FLUENT to compute the view factors for the surface clusters.

You should recreate the surface cluster information whenever you do!anything that changes the grid, such as:

• change the type of a boundary zone

• reorder the grid

• scale the grid

• change from 2D space to axisymmetric space or vice versa



Figure 11.3.12: The Radiation Model Panel (S2S Model)

Note that you do not need to recalculate view factors after shell con-duction at any wall has been enabled or disabled. See Section 6.13.1 formore information about shell conduction.

Computing View Factors

FLUENT can compute the view factors for your problem in the currentsession and save them to a file for use in the current session and futuresessions. Alternatively, you can save the surface cluster information andview factor parameters to a file, calculate the view factors outside FLU-ENT, and then read the view factors into FLUENT. These methods forcomputing view factors are described below.

For large meshes or complex models, it is recommended that you calcu-!late the view factors outside FLUENT and then read them into FLUENTbefore starting your simulation.



Computing View Factors Inside FLUENT

To compute view factors in your current FLUENT session, you must firstset the parameters for the view factor calculation in the View Factorand Cluster Parameters panel (see below for details). When you have setthe view factor and surface cluster parameters, click Compute/Write...under Methods in the Radiation Model panel. A Select File dialog boxwill open, prompting you for the name of the file in which FLUENTshould save the surface cluster information and the view factors. Afteryou have specified the file name, FLUENT will write the surface clusterinformation to the file. FLUENT will use the surface cluster informationto compute the view factors, save the view factors to the same file, andthen automatically read the view factors.

Computing View Factors Outside FLUENT

To compute view factors outside FLUENT, you must save the surfacecluster information and view factor parameters to a file.

File −→ Write −→Surface Clusters...

FLUENT will open the View Factor and Cluster Parameters panel, whereyou will set the view factor and surface cluster parameters (see below fordetails). When you click OK in the View Factor and Cluster Parameterspanel, a Select File dialog box will open, prompting you for the name ofthe file in which FLUENT should save the surface cluster information andview factor parameters. After you have specified the file name, FLUENTwill write the surface cluster information and view factor parametersto the file. If the specified Filename ends in .gz or .Z, appropriate filecompression will be performed. (See Section 3.1.5 for details about filecompression.)

To calculate the view factors outside FLUENT, enter one of the followingcommands:

• For the serial solver:

utility viewfac inputfile



where inputfile is the filename, or the correct path to the filename,for the surface cluster information and view factor parameters filethat you saved from FLUENT. You can then read the view factorsinto FLUENT, as described below.

• For the network parallel solver:

utility viewfac -p -tn -cnf=host1,host2,. . .,hostn inputfile

where n is the number of compute nodes, and host1, host2,. . . arethe names of the machines being used.

Note that host1 must be a host machine.!

• For a dedicated parallel machine with multiple processors:

utility viewfac -tn inputfile

Note that for parallel runs (dedicated or network) using n processors,!the problem is duplicated for each processor. For example, if the viewfactor calculation requires 100 MB of RAM using a single CPU, it willrequire 100n MB of RAM to run the calculation on n processors.

Reading View Factors into FLUENT

If the view factors for your problem have already been computed (eitherinside or outside FLUENT) and saved to a file, you can read them intoFLUENT. To read in the view factors, click Read... under Methods in theRadiation Model panel. A Select File dialog box will open where you canspecify the name of the file containing the view factors. You can alsomanually specify the view factors file, using the File/Read/View Factors...menu item.

Setting View Factor and Surface Cluster Parameters

You will use the View Factor and Cluster Parameters panel (Figure 11.3.13)to set view factor and cluster parameters for the S2S model. To openthis panel, click Set... under Parameters in the Radiation Model panel oruse the File/Write/Surface Clusters... menu item.



Figure 11.3.13: The View Factor and Cluster Parameters Panel



Controlling the Clusters

Your input for Faces Per Surface Cluster will control the number of radi-ating surfaces. By default, it is set to 1, so the number of surface clusters(radiating surfaces) will be the number of boundary faces. For small 2Dproblems, this is an acceptable number. For larger problems, you maywant to reduce the number of surface clusters to reduce both the sizeof the view-factor file and the memory requirement. Such a reductionin the number of clusters, however, comes at the cost of some accuracy.(See Section 11.3.7 for details about clustering.)

In some cases, you may wish to modify the cutoff or “split” angle betweenadjacent face normals for the purpose of controlling surface clustering.The split angle sets the limit for which adjacent surfaces are clustered. Asmaller split angle allows for a better representation of the view factor.By default, no surface cluster will contain any face that has a face normalgreater than 20. To modify the value of this parameter, you can usethe split-angle text command:

define −→ models −→ radiation −→ s2s-parameters −→split-angle

or

file −→ write-surface-clusters −→split-angle

Specifying the Orientation of Surface Pairs

View factor calculations depend on the geometric orientations of surfacepairs with respect to each other. Two situations may be encounteredwhen examining surface pairs:

• If there is no obstruction between the surface pairs under consid-eration, then they are referred to as “non-blocking” surfaces.

• If there is another surface blocking the views between the surfacesunder consideration, then they are referred to as “blocking” sur-faces. Blocking will change the view factors between the surfacepairs and require additional checks to compute the correct value ofthe view factors.



For cases with blocking surfaces, select Blocking under Surfaces in theView Factor and Cluster Parameters panel. For cases with non-blockingsurfaces, you can choose either Blocking or Nonblocking without affectingthe accuracy. However, it is better to choose Nonblocking for such cases,as it takes less time to compute.

Selecting the Method for Smoothing

In order to enforce reciprocity and conservation (see Section 11.3.7),smoothing can be performed on the view factor matrix. To use theleast-squares method for smoothing of the view factor matrix, selectLeast Square under Smoothing in the View Factor and Cluster Parameterspanel. If you do not wish to smooth the view factor matrix, select Noneunder Smoothing.

Selecting the Method for Computing View Factors

FLUENT provides two methods for computing view factors: the hemicubemethod and the adaptive method. The hemicube method is availableonly for 3D cases.

The adaptive method calculates the view factors on a pair-by-pair basisusing a variety of algorithms (analytic or Gauss quadrature) that arechosen adaptively depending on the proximity of the surfaces. To main-tain accuracy, the order of the quadrature increases the closer the facesare together. For surfaces that are very close to each other, the analyticmethod is used. FLUENT determines the method to use by performinga visibility calculation. The Gaussian quadrature method is used if noneof the rays from a surface are blocked by the other surface. If some ofthe rays are blocked by the other surface, then either a Monte Carlointegration method or a quasi-Monte Carlo integration method is used.

To use the adaptive method to compute the view factors, select Adaptivein the View Factor and Cluster Parameters panel. It is recommended thatyou use the adaptive method for simple models, because it is faster thanthe hemicube method for these types of models.

The hemicube method uses a differential area-to-area method and calcu-lates the view factors on a row-by-row basis. The view factors calculated



from the differential areas are summed to provide the view factor for thewhole surface. This method originated from the use of the radiosityapproach in the field of computer graphics [40].

To use the hemicube method to compute the view factors, select Hemicubein the View Factor and Cluster Parameters panel. It is recommended thatyou use the hemicube method for large complex models, because it isfaster than the adaptive method for these types of models.

The hemicube method is based upon three assumptions about the ge-ometry of the surfaces: aliasing, visibility, and proximity. To validatethese assumptions, you can specify three different hemicube parameters,which can help you obtain better accuracy in calculating view factors.In most cases, however, the default settings will be sufficient.

• Aliasing—The true projection of each visible face onto the hemicubecan be accurately accounted for by using a finite-resolution hemicube.As described above, the faces are projected onto a hemicube. Be-cause of the finite resolution of the hemicube, the projected areasand resulting view factors may be over- or under-estimated. Alias-ing effects can be reduced by increasing the value of the Resolutionof the hemicube under Hemicube Parameters.

• Visibility—The visibility between any two faces does not change.In some cases, face i has a complete view of face k from its cen-troid, but some other face j occludes much of face k from face i.In such a case, the hemicube method will overestimate the viewfactor between face i and face k calculated from the centroid offace i. This error can be reduced by subdividing face i into smallersubfaces. You can specify the number of subfaces by entering avalue for Subdivision under Hemicube Parameters.

• Proximity—The distance between faces is great compared to the ef-fective diameter of the faces. The proximity assumption is violatedwhenever faces are close together in comparison to their effectivediameter or are adjacent to one another. In such cases, the dis-tances between the centroid of one face and all points on the otherface vary greatly. Since the view factor dependence on distance isnon-linear, the result is a poor estimate of the view factor.



Under Hemicube Parameters, you can set a limit for the Normal-ized Separation Distance, which is the ratio of the minimum faceseparation to the effective diameter of the face. If the computednormalized separation distance is less than the specified value, theface will then be divided into a number of subfaces until the nor-malized distances of the subfaces are greater than the specifiedvalue. Alternatively, you can specify the number of subfaces tocreate for such faces by entering a value for Subdivision.

11.3.13 Defining the Angular Discretization for the DO Model

When you select the Discrete Ordinates model, the Radiation Model panelwill expand to show inputs for Angular Discretization (see Figure 11.3.10).In this section, you will set parameters for the angular discretization andpixelation described in Section 11.3.6.

Theta Divisions (Nθ) and Phi Divisions (Nφ) will define the number ofcontrol angles used to discretize each octant of the angular space (seeFigure 11.3.3). For a 2D model, FLUENT will solve only 4 octants (due tosymmetry); thus, a total of 4NθNφ directions ~s will be solved. For a 3Dmodel, 8 octants are solved, resulting in 8NθNφ directions ~s. By default,the number of Theta Divisions and the number of Phi Divisions are bothset to 2. For most practical problems, these settings are adequate. Afiner angular discretization can be specified to better resolve the influenceof small geometric features or strong spatial variations in temperature,but larger numbers of Theta Divisions and Phi Divisions will add to thecost of the computation.

Theta Pixels and Phi Pixels are used to control the pixelation that ac-counts for any control volume overhang (see Figure 11.3.7 and the figuresand discussion preceding it). For problems involving gray-diffuse radia-tion, the default pixelation of 1 × 1 is usually sufficient. For problemsinvolving symmetry, periodic, specular, or semi-transparent boundaries,a pixelation of 3 × 3 is recommended. You should be aware, however,that increasing the pixelation adds to the cost of computation.



11.3.14 Defining Non-Gray Radiation for the DO Model

If you want to model non-gray radiation using the DO model, you canspecify the Number Of Bands (N) under Non-Gray Model in the expandedRadiation Model panel (Figure 11.3.14). For a 2D model, FLUENT willsolve 4NθNφN directions. For a 3D model, 8NθNφN directions will besolved. By default, the Number of Bands is set to zero, indicating thatonly gray radiation will be modeled. Because the cost of computationincreases directly with the number of bands, you should try to minimizethe number of bands used. In many cases, the absorption coefficient orthe wall emissivity is effectively constant for the wavelengths of impor-tance in the temperature range of the problem. For such cases, the grayDO model can be used with little loss of accuracy. For other cases, non-gray behavior is important, but relatively few bands are necessary. Fortypical glasses, for example, two or three bands will frequently suffice.

When a non-zero Number Of Bands is specified, the Radiation Model panelwill expand once again to show the Wavelength Intervals (Figure 11.3.14).You can specify a Name for each wavelength band, as well as the Startand End wavelength of the band in µm. Note that the wavelength bandsare specified for vacuum (n = 1). FLUENT will automatically accountfor the refractive index in setting band limits for media with n differentfrom unity.

The frequency of radiation remains constant as radiation travels across asemi-transparent interface. The wavelength, however, changes such thatnλ is constant. Thus, when radiation passes from a medium with refrac-tive index n1 to one with refractive index n2, the following relationshipholds:

n1λ1 = n2λ2 (11.3-84)

Here λ1 and λ2 are the wavelengths associated with the two media. It isconventional to specify the wavelength rather than frequency. FLUENTrequires you to specify wavelength bands for an equivalent medium withn = 1.

For example, consider a typical glass with a step jump in the absorption



Figure 11.3.14: The Radiation Model Panel (Non-Gray DO Model)



coefficient at a cut-off wavelength of λc. The absorption coefficient isa1 for λ ≤ λc µm and a2 for λ > λc µm. The refractive index of theglass is ng. Since nλ is constant across a semi-transparent interface, theequivalent cut-off wavelength for a medium with n = 1 is ngλc usingEquation 11.3-84. You should choose two bands in this case, with thelimits 0 to ngλc and ngλc to 100. Here, the upper wavelength limithas been chosen to be a large number, 100, in order to ensure thatthe entire spectrum is covered by the bands. When multiple materialsexist, you should convert all the cut-off wavelengths to equivalent cut-off wavelengths for an n = 1 medium, and choose the band boundariesaccordingly.

The bands can have different widths and need not be contiguous. Youcan ensure that the entire spectrum is covered by your bands by choosingλmin = 0 and nλmaxTmin ≥ 50, 000. Here λmin and λmax are the minimumand maximum wavelength bounds of your wavelength bands, and Tmin

is the minimum expected temperature in the domain.

11.3.15 Defining Material Properties for Radiation

When you are using the P-1, DO, or Rosseland radiation model in FLU-ENT, you should be sure to define both the absorption and scatteringcoefficients of the fluid in the Materials panel. If you are modeling semi-transparent media using the DO model, you should also define the re-fractive index for the semi-transparent fluid or solid material. For theDTRM, you need to define only the absorption coefficient.


If your model includes gas phase species such as combustion products,absorption and/or scattering in the gas may be significant. The scatter-ing coefficient should be increased from the default of zero if the fluidcontains dispersed particles or droplets which contribute to scattering.FLUENT provides the facility for input of a composition-dependent ab-sorption coefficient for CO2 and H2O mixtures, using the WSGGM. Themethod for computing a variable absorption coefficient is described inSection 11.3.8. Section 7.6 provides a detailed description of the proce-dures used for input of radiation properties.



Absorption Coefficient for a Non-Gray DO Model

If you are using the non-gray DO model, you can specify a differentconstant absorption coefficient for each of the bands used by the gray-band model, as described in Section 7.6. You cannot, however, computea composition-dependent absorption coefficient in each band. If you usethe WSGGM to compute a variable absorption coefficient, the value willbe the same for all bands.

11.3.16 Setting Radiation Boundary Conditions

When you set up a problem that includes radiation, you will set addi-tional boundary conditions at walls, inlets, and exits.


Inlet and Exit Boundary Conditions

Emissivity

When radiation is active, you can define the emissivity at each inletand exit boundary when you are defining boundary conditions in theassociated inlet or exit boundary panel (Pressure Inlet panel, VelocityInlet panel, Pressure Outlet panel, etc.). Enter the appropriate value forInternal Emissivity. The default value for all boundary types is 1.

For non-gray DO models, the specified constant emissivity will be usedfor all wavelength bands.

The Internal Emissivity boundary condition is not available with the!Rosseland model.

Black Body Temperature

FLUENT includes an option that allows you to take into account the in-fluence of the temperature of the gas and the walls beyond the inlet/exitboundaries, and specify different temperatures for radiation and convec-tion at inlets and exits. This is useful when the temperature outsidethe inlet or exit differs considerably from the temperature in the enclo-sure. For example, if the temperature of the walls beyond the inlet is



2000 K and the temperature at the inlet is 1000 K, you can specify theoutside-wall temperature to be used for computing radiative heat flux,while the actual temperature at the inlet is used for calculating convec-tive heat transfer. To do this, you would specify a radiation temperatureof 2000 K as the black body temperature.

Although this option allows you to account for both cooler and hotteroutside walls, you must use caution in the case of cooler walls, sincethe radiation from the immediate vicinity of the hotter inlet or outletalmost always dominates over the radiation from cooler outside walls.If, for example, the temperature of the outside walls is 250 K and theinlet temperature is 1500 K, it might be misleading to use 250 K for theradiation boundary temperature. This temperature might be expectedto be somewhere between 250 K and 1500 K; in most cases it will beclose to 1500 K. (Its value depends on the geometry of the outside wallsand the optical thickness of the gas in the vicinity of the inlet.)

In the flow inlet or exit panel (Pressure Inlet panel, Velocity Inlet panel,etc.), select Specified External Temperature in the External Black BodyTemperature Method drop-down list, and then enter the value of theradiation boundary temperature as the Black Body Temperature.

If you want to use the same temperature for radiation and convection,!retain the default selection of Boundary Temperature as the External BlackBody Temperature Method.

The Black Body Temperature boundary condition is not available with!the Rosseland model.

Wall Boundary Conditions for the DTRM, and the P-1, S2S, andRosseland Models

The DTRM and the P-1, S2S, and Rosseland models assume all walls tobe gray and diffuse. The only radiation boundary condition required inthe Wall panel is the emissivity. For the Rosseland model, the internalemissivity is 1. For the DTRM and the P-1 and S2S models, you canenter the appropriate value for Internal Emissivity in the Radiation sectionof the Wall panel. The default value is 1.



Wall Boundary Conditions for the DO Model

When the DO model is used, you can model diffuse, specular, and semi-transparent walls, as discussed in Section 11.3.6.

You can use a diffuse wall to model wall boundaries in many industrialapplications since, for the most part, surface roughness makes the re-flection of incident radiation diffuse. For highly polished surfaces, suchas reflectors or mirrors, the specular boundary condition is appropriate.The semi-transparent boundary condition is appropriate for modelingglass panes in air, for example.

Diffuse Wall Boundary Conditions for the DO Model

In the Radiation section of the Wall panel, select diffuse in the BC Typedrop-down list to specify a diffuse wall. Diffuse walls are treated as grayif gray radiation is being computed, or non-gray if the non-gray DOmodel is being used. Once you have selected diffuse as the BC Type,the only radiation boundary condition required in the Wall panel is theemissivity.

For gray-radiation DO models, enter the appropriate value for InternalEmissivity. (The default value is 1.) For non-gray DO models, specify aconstant Internal Emissivity for each wavelength band. (The default valuein each band is 1.)

Specular Wall Boundary Conditions for the DO Model

In the Radiation section of the Wall panel, select specular in the BC Typedrop-down list to specify a specular wall. No additional inputs are re-quired.

Semi-Transparent Wall Boundary Conditions for the DO Model

In the Radiation section of the Wall panel, select semi-transparent in theBC Type drop-down list to specify a semi-transparent wall.

For an external semi-transparent wall, you can define an external ir-radiation flux in the Wall panel (see Figure 11.3.15). For an internal



semi-transparent wall, see the discussion on multiple-zone domains, be-low.

Figure 11.3.15: The Wall Panel for a Semi-Transparent Wall

The inputs for an external semi-transparent wall are as follows:

1. Specify the value of the irradiation flux under Irradiation. If thenon-gray DO model is being used, a constant Irradiation can bespecified for each band.

2. Define the Beam Width by specifying the beam Theta and Phi ex-tents.

3. Specify the (X,Y,Z) vector that defines the Beam Direction.

4. Specify the fraction of the irradiation that is to be treated as dif-fuse. By default, the Diffuse Fraction is set to 1, indicating that



all of the irradiation is diffuse. If you specify a value less than 1,the diffuse fraction will be reflected diffusely (as described in Sec-tion 11.3.6), the transmitted portion will also be reflected diffusely,and the remainder will be reflected specularly.

If the non-gray DO model is being used, the Diffuse Fraction canbe specified for each band.

Note that the refractive index of the external medium is assumed to be 1.!

If Heat Flux conditions are specified in the Thermal section of the Wall!panel, the specified heat flux is considered to be only the conduction andconvection portion of the boundary flux. The given irradiation specifiesthe incoming exterior radiative flux; the radiative flux transmitted fromthe domain interior to the outside is computed as a part of the calculationby FLUENT.

Enabling Radiation in Specific Cell Zones (DO Model Only)

With the DO model, you can specify whether or not you want to solve forradiation in each cell zone in the domain. By default, the DO equationsare solved in all fluid zones, but not in any solid zones. If you want tomodel semi-transparent media, for example, you can enable radiation inthe solid zone(s). To do so, turn on the Participates In Radiation optionin the Solid panel (Figure 11.3.16).

In general, you should not turn off the Participates In Radiation option!for any fluid zones.

Two-Sided-Wall Boundary Conditions for the DO Model inMultiple-Zone Domains

For the DO model, you can specify the boundary condition on each sideof a two-sided wall independently to be either diffuse or specular. Notethat the two fluid zones bordering the wall will not be radiatively coupled(although you can choose them to be thermally coupled.)

You can also choose to couple the contiguous fluid or solid zones radia-tively by making the two-sided wall between them semi-transparent. In



Figure 11.3.16: The Solid Panel

this case, radiation will pass through the wall. You can specify the two-sided wall to be semi-transparent only if both the neighboring cell zonesparticipate in radiation; both sides of the wall will be semi-transparentif you define one side to be semi-transparent. You can, however, specifya different diffuse fraction for each side.

It is also possible to associate a thickness with the two-sided wall. In thiscase, the refraction due to the wall thickness is accounted for as radiationtravels through the boundary. You can specify a Wall Thickness and awall Material Name in the Wall panel (as described in Section 6.13.1).The refractive index and the absorption coefficient are those of the spec-ified wall material. Only a constant absorption coefficient is allowedfor a solid material. The effective reflectivity and transmissivity of thewall are computed assuming a planar layer of the given thickness withabsorption but no emission. The refractive indices of the surroundingmedia correspond to those of the surrounding fluid materials. (When an



external wall is specified to be semi-transparent, the refractive index ofthe external medium is assumed to be 1.)

Thermal Boundary Conditions

In general, any well-posed combination of thermal boundary conditionscan be used when any of the radiation models is active. The radiationmodel will be well-posed in combination with fixed temperature walls,conducting walls, and/or walls with set external heat transfer boundaryconditions (Section 6.13.1). You can also use any of the radiation modelswith heat flux boundary conditions defined at walls, in which case theheat flux you define will be treated as the sum of the convective and ra-diative heat fluxes. The exception to this is the case of semi-transparentwalls for the DO model. Here, FLUENT allows you to specify the con-vective and radiative portions of the heat flux separately, as explainedabove. Also, the fixed-temperature boundary condition is not allowed ata semi-transparent wall.

11.3.17 Setting Solution Parameters for Radiation

For the DTRM and the DO, S2S, and P-1 radiation models, there areseveral parameters that control the radiation calculation. You can usethe default solution parameters for most problems, or you can modifythese parameters to control the convergence and accuracy of the solution.There are no solution parameters to be set for the Rosseland model, sinceit impacts the solution only through the energy equation.

DTRM Solution Parameters

When the DTRM is active, FLUENT updates the radiation field dur-ing the calculation and computes the resulting energy sources and heatfluxes via the ray-tracing technique described in Section 11.3.3. FLUENTprovides several solution parameters that control the solver and the so-lution accuracy. These parameters appear in the expanded portion ofthe Radiation Model panel (Figure 11.3.17).

You can control the maximum number of sweeps of the radiation cal-culation during each global iteration by changing the Number of DTRM



Figure 11.3.17: The Radiation Model Panel (DTRM)

Sweeps. The default setting of 1 sweep implies that the radiant intensitywill be updated just once. If you increase this number, the radiant in-tensity at the surfaces will be updated multiple times, until the tolerancecriterion is met or the number of radiation sweeps is exceeded.

The Tolerance parameter (0.001 by default) determines when the radi-ation intensity update is converged. It is defined as the maximum nor-malized change in the surface intensity from one DTRM sweep to thenext (see Equation 11.3-85).

You can also control the frequency with which the radiation field is up-dated as the continuous phase solution proceeds. The Flow Iterations PerRadiation Iteration parameter is set to 10 by default. This implies thatthe radiation calculation is performed once every 10 iterations of the so-lution process. Increasing the number can speed the calculation process,but may slow overall convergence.

S2S Solution Parameters

For the S2S model, as for the DTRM, you can control the frequencywith which the radiosity is updated as the continuous-phase solutionproceeds. See the description of Flow Iterations Per Radiation Iterationfor the DTRM, above.



If you are using the segregated solver and you first solve the flow equa-tions with the energy equation turned off, you should reduce the FlowIterations Per Radiation Iteration from 10 to 1 or 2. This will ensurethe convergence of the radiosity. If the default value of 10 is kept inthis case, it is possible that the flow and energy residuals may convergeand the solution will terminate before the radiosity is converged. SeeSection 11.3.18 for more information about residuals for the S2S model.

You can control the maximum number of sweeps of the radiation calcula-tion during each global iteration by changing the Number of S2S Sweeps.The default setting of 1 sweep implies that the radiosity will be updatedjust once. If you increase this number, the radiosity at the surfaces willbe updated multiple times, until the tolerance criterion is met or thenumber of radiation sweeps is exceeded.

The Tolerance parameter (0.001 by default) determines when the radios-ity update is converged. It is defined as the maximum normalized changein the radiosity from one S2S sweep to the next (see Equation 11.3-86).

DO Solution Parameters

For the discrete ordinates model, as for the DTRM, you can control thefrequency with which the surface intensity is updated as the continu-ous phase solution proceeds. See the description of Flow Iterations PerRadiation Iteration for the DTRM, above.

For most problems, the default under-relaxation of 1.0 for the DO equa-tions is adequate. For problems with large optical thicknesses (aL > 10),you may experience slow convergence or solution oscillation. For suchcases, under-relaxing the energy and DO equations is useful. Under-relaxation factors between 0.9 and 1.0 are recommended for both equa-tions.

P-1 Solution Parameters

For the P-1 radiation model, you can control the convergence criterionand under-relaxation factor. You should also pay attention to the opticalthickness, as described below.



The default convergence criterion for the P-1 model is 10−6, the sameas that for the energy equation, since the two are closely linked. SeeSection 22.16.1 for details about convergence criteria. You can set theConvergence Criterion for p1 in the Residual Monitors panel.

Solve −→ Monitors −→Residual...

The under-relaxation factor for the P-1 model is set with those for othervariables, as described in Section 22.9. Note that since the equationfor the radiation temperature (Equation 11.3-12) is a relatively stablescalar transport equation, in most cases you can safely use large valuesof under-relaxation (0.9–1.0).

For optimal convergence with the P-1 model, the optical thickness(a + σs)L must be between 0.01 and 10 (preferably not larger than 5).Smaller optical thicknesses are typical for very small enclosures (charac-teristic size of the order of 1 cm), but for such problems you can safelyincrease the absorption coefficient to a value for which (a+σs)L = 0.01.Increasing the absorption coefficient will not change the physics of theproblem because the difference in the level of transparency of a mediumwith optical thickness = 0.01 and one with optical thickness < 0.01 isindistinguishable within the accuracy level of the computation.

11.3.18 Solving the Problem

Once the radiation problem has been set up, you can proceed as usualwith the calculation. Note that while the P-1 and DO models will solveadditional transport equations and report residuals, the DTRM and theRosseland and S2S models will not (since they impact the solution onlythrough the energy equation). Residuals for the DTRM and S2S modelsweeps are reported by FLUENT every time a DTRM or S2S model iter-ation is performed, as described below.

Residual Reporting for the P-1 Model

The residual for radiation as calculated by the P-1 model is updatedafter each iteration and reported with the residuals for all other vari-ables. FLUENT reports the normalized P-1 radiation residual as definedin Section 22.16.1 for the other transport equations.



Residual Reporting for the DO Model

After each DO iteration, the DO model reports a composite normal-ized residual for all the DO transport equations. The definition of theresiduals is similar to that for the other transport equations (see Sec-tion 22.16.1).

Residual Reporting for the DTRM

FLUENT does not include a DTRM residual in its usual residual reportthat is issued after each iteration. The effect of radiation on the solutioncan be gathered, instead, via its impact on the energy field and theenergy residual. However, each time a DTRM iteration is performed,FLUENT will print out the normalized radiation error for each DTRMsweep. The normalized radiation error is defined as

E =

∑all radiating surfaces

(Inew − Iold)

N (σT 4/π)(11.3-85)

where the error E is the maximum change in the intensity (I) at the cur-rent sweep, normalized by the maximum surface emissive power, and Nis the total number of radiating surfaces. Note that the default radiationconvergence criterion, as noted in Section 11.3.17, defines the radiationcalculation to be converged when E decreases to 10−3 or less.

Residual Reporting for the S2S Model

FLUENT does not include an S2S residual in its usual residual report thatis issued after each iteration. The effect of radiation on the solution canbe gathered, instead, via its impact on the energy field and the energyresidual. However, each time an S2S iteration is performed, FLUENTwill print out the normalized radiation error for each S2S sweep. Thenormalized radiation error is defined as

E =

∑all radiating surface clusters

(Jnew − Jold)

NσT 4(11.3-86)



where the error E is the maximum change in the radiosity (J) at thecurrent sweep, normalized by the maximum surface emissive power, andN is the total number of radiating surface clusters. Note that the defaultradiation convergence criterion, as noted in Section 11.3.17, defines theradiation calculation to be converged when E decreases to 10−3 or less.

Disabling the Update of the Radiation Fluxes

Sometimes, you may wish to set up your FLUENT model with the radi-ation model active and then disable the radiation calculation during theinitial calculation phase. For the P-1 and DO models, you can turn offthe radiation calculation temporarily by deselecting P1 or Discrete Ordi-nates in the Equations list in the Solution Controls panel. For the DTRMand the S2S model, there is no item in the Equations list. You can in-stead set a very large number for Flow Iterations Per Radiation Iterationin the expanded portion of the Radiation Model panel.

If you turn off the radiation calculation, FLUENT will skip the update ofthe radiation field during subsequent iterations, but will leave in placethe influence of the current radiation field on energy sources due to ab-sorption, wall heat fluxes, etc. Turning the radiation calculation off inthis way can thus be used to initiate your modeling work with the ra-diation model inactive and/or to focus the computational effort on theother equations if the radiation model is relatively well converged.

11.3.19 Reporting and Displaying Radiation Quantities

FLUENT provides several additional reporting options when your modelincludes solution of radiative heat transfer. You can generate graphicalplots or alphanumeric reports of the following variables/functions:

• Absorption Coefficient (DTRM, P-1, DO, and Rosseland modelsonly)

• Scattering Coefficient (P-1, DO, and Rosseland models only)

• Refractive Index (DO model only)

• Radiation Temperature (P-1 and DO models only)



• Incident Radiation (P-1 and DO models only)

• Incident Radiation (Band n) (non-gray DO model only)

• Surface Cluster ID (S2S model only)

• Radiation Heat Flux

The first seven variables are contained in the Radiation... category ofthe variable selection drop-down list that appears in postprocessing pan-els, and the last one is contained in the Wall Fluxes... category. SeeChapter 27 for their definitions.

Note the sign convention on the radiative heat flux: heat flux from the!wall surface is a positive quantity.

Note that it is possible to export heat flux data on wall zones (includingradiation) to a generic file that you can examine or use in an externalprogram. See Section 11.2.5 for details.

Reporting Radiative Heat Transfer Through Boundaries

You can use the Flux Reports panel to compute the radiative heat transferthrough each boundary of the domain, or to sum the radiative heattransfer through all boundaries.

Report −→Fluxes...

See Section 26.2 for details about generating flux reports.

Overall Heat Balances When Using the DTRM

The DTRM yields a global heat balance and a balance of radiant heatfluxes only in the limit of a sufficient number of rays. In any givencalculation, therefore, if the number of rays is insufficient you may findthat the radiant fluxes do not obey a strict balance. Such imbalancesare the inevitable consequence of the discrete ray tracing procedure andcan be minimized by selecting a larger number of rays from each wallboundary.



11.3.20 Displaying Rays and Clusters for the DTRM

When you use the DTRM, FLUENT allows you to display surface or vol-ume clusters, as well as the rays that emanate from a particular surfacecluster. You will use the DTRM Graphics panel (Figure 11.3.18) for allof these displays.

Display −→DTRM Graphics...

Figure 11.3.18: The DTRM Graphics Panel

Displaying Clusters

To view clusters, select Cluster under Display Type and then select eitherSurface or Volume under Cluster Type.

To display all of the surface or volume clusters, select the Display AllClusters option under Cluster Selection and click the Display button.



To display only the cluster (surface or volume) nearest to a specifiedpoint, deselect the Display All Clusters option and specify the coordinatesunder Nearest Point. You may also use the mouse to choose the nearestpoint. Click on the Select Point With Mouse button and then right-clickon a point in the graphics window.

Displaying Rays

To display the rays emanating from the surface cluster nearest to thespecified point, select Ray under Display Type. Set the appropriate valuesfor Theta and Phi Divisions under Ray Parameters (see Section 11.3.11 fordetails), and then click on the Display button. Figure 11.3.19 shows aray plot for a simple 2D geometry.

DTRM Rays

Figure 11.3.19: Ray Display

Including the Grid in the Display

For some problems, especially complex 3D geometries, you may wantto include portions of the grid in your ray or cluster display as spatialreference points. For example, you may want to show the location of an


11.4 Periodic Heat Transfer

inlet and an outlet along with displaying the rays. This is accomplishedby turning on the Draw Grid option in the DTRM Graphics panel. TheGrid Display panel will appear automatically when you turn on the DrawGrid option, and you can set the grid display parameters there. Whenyou click on Display in the DTRM Graphics panel, the grid display, asdefined in the Grid Display panel, will be included in the ray or clusterdisplay.


FLUENT is able to predict heat transfer in periodically repeating ge-ometries, such as compact heat exchangers, by including only a singleperiodic module for analysis.

This section discusses streamwise-periodic heat transfer. The treatmentof streamwise-periodic flows is discussed in Section 8.3, and a descriptionof no-pressure-drop periodic flow is provided in Section 6.15.

Information about streamwise-periodic heat transfer is presented in thefollowing sections:

• Section 11.4.1: Overview and Limitations

• Section 11.4.2: Theory

• Section 11.4.3: Modeling Periodic Heat Transfer

• Section 11.4.4: Solution Strategies for Periodic Heat Transfer

• Section 11.4.5: Monitoring Convergence

• Section 11.4.6: Postprocessing for Periodic Heat Transfer

11.4.1 Overview and Limitations

Overview

As discussed in Section 8.3.1, streamwise-periodic flow conditions existwhen the flow pattern repeats over some length L, with a constant pres-sure drop across each repeating module along the streamwise direction.



Periodic thermal conditions may be established when the thermal bound-ary conditions are of the constant wall temperature or wall heat flux type.In such problems, the temperature field (when scaled in an appropriatemanner) is periodically fully-developed [173]. As for periodic flows, suchproblems can be analyzed by restricting the numerical model to a singlemodule or periodic length.

Constraints for Periodic Heat Transfer Predictions

In addition to the constraints for streamwise-periodic flow discussed inSection 8.3.1, the following constraints must be met when periodic heattransfer is to be considered:

• The segregated solver must be used.

• The thermal boundary conditions must be of the specified heat fluxor constant wall temperature type. Furthermore, in a given prob-lem, these thermal boundary types cannot be combined: all bound-aries must be either constant temperature or specified heat flux.(You can, however, include constant-temperature walls and zero-heat-flux walls in the same problem.) For the constant-temperaturecase, all walls must be at the same temperature (profiles are notallowed) or zero heat flux. For the heat flux case, profiles and/ordifferent values of heat flux may be specified at different walls.

• When constant-temperature wall boundaries are used, you cannotinclude viscous heating effects or any volumetric heat sources.

• In cases that involve solid regions, the regions cannot straddle theperiodic plane.

• The thermodynamic and transport properties of the fluid (heatcapacity, thermal conductivity, viscosity, and density) cannot befunctions of temperature. (You cannot, therefore, model reactingflows.) Transport properties may, however, vary spatially in a peri-odic manner, and this allows you to model periodic turbulent flowsin which the effective turbulent transport properties (effective con-ductivity, effective viscosity) vary with the (periodic) turbulencefield.



Sections 11.4.2 and 11.4.3 provide more detailed descriptions of the inputrequirements for periodic heat transfer.

11.4.2 Theory

Streamwise-periodic flow with heat transfer from constant-temperaturewalls is one of two classes of periodic heat transfer that can be modeledby FLUENT. A periodic fully-developed temperature field can also beobtained when heat flux conditions are specified. In such cases, thetemperature change between periodic boundaries becomes constant andcan be related to the net heat addition from the boundaries as describedin this section.

Periodic heat transfer can be modeled only if you are using the segregated!solver.

Definition of the Periodic Temperature for Constant-Temperature Wall Conditions

For the case of constant wall temperature, as the fluid flows through theperiodic domain, its temperature approaches that of the wall boundaries.However, the temperature can be scaled in such a way that it behavesin a periodic manner. A suitable scaling of the temperature for periodicflows with constant-temperature walls is [173]

θ =T (~r) − Twall

Tbulk,inlet − Twall(11.4-1)

The bulk temperature, Tbulk,inlet, is defined by

Tbulk,inlet =∫A T |ρ~v · d ~A|∫A |ρ~v · d ~A| (11.4-2)

where the integral is taken over the inlet periodic boundary (A). It isthe scaled temperature, θ, which obeys a periodic condition across thedomain of length L.



Definition of the Periodic Temperature Change σ for SpecifiedHeat Flux Conditions

When periodic heat transfer with heat flux conditions is considered, theform of the unscaled temperature field becomes analogous to that of thepressure field in a periodic flow:

T (~r + ~L) − T (~r)L

=T (~r + 2~L) − T (~r + ~L)

L= σ. (11.4-3)

where ~L is the periodic length vector of the domain. This temperaturegradient, σ, can be written in terms of the total heat addition within thedomain, Q, as

σ =Q

mcpL=Tbulk,exit − Tbulk,inlet

L(11.4-4)

where m is the specified or calculated mass flow rate.

11.4.3 Modeling Periodic Heat Transfer

Overview of Streamwise-Periodic Flow and Heat TransferModeling Procedures

A typical calculation involving both streamwise-periodic flow and peri-odic heat transfer is performed in two parts. First, the periodic velocityfield is calculated (to convergence) without consideration of the tempera-ture field. Next, the velocity field is frozen and the resulting temperaturefield is calculated. These periodic flow calculations are accomplished us-ing the following procedure:

1. Set up a grid with translationally periodic boundary conditions.

2. Input constant thermodynamic and molecular transport proper-ties.

3. Specify either the periodic pressure gradient or the net mass flowrate through the periodic boundaries.



4. Compute the periodic flow field, solving momentum, continuity,and (optionally) turbulence equations.

5. Specify the thermal boundary conditions at walls as either heatflux or constant temperature.

6. Define an inlet bulk temperature.

7. Solve the energy equation (only) to predict the periodic tempera-ture field.

These steps are detailed below.

User Inputs for Periodic Heat Transfer

In order to model the periodic heat transfer, you will need to set up yourperiodic model in the manner described in Section 8.3.3 for periodicflow models with the segregated solver, noting the restrictions discussedin Sections 8.3.1 and 11.4.1. In addition, you will need to provide thefollowing inputs related to the heat transfer model:

1. Activate solution of the energy equation in the Energy panel.


2. Define the thermal boundary conditions according to one of thefollowing procedures:


• If you are modeling periodic heat transfer with specified-temp-erature boundary conditions, set the wall temperature Twall

for all wall boundaries in their respective Wall panels. Notethat all wall boundaries must be assigned the same tempera-ture and that the entire domain (except the periodic bound-aries) must be “enclosed” by this fixed-temperature condition,or by symmetry or adiabatic (q=0) boundaries.

• If you are modeling periodic heat transfer with specified-heat-flux boundary conditions, set the wall heat flux in the Wallpanel for each wall boundary. You can define different values



of heat flux on different wall boundaries, but you should haveno other types of thermal boundary conditions active in thedomain.

3. Define solid regions, if appropriate, according to one of the follow-ing procedures:


• If you are modeling periodic heat transfer with specified-temp-erature conditions, conducting solid regions can be used withinthe domain, provided that on the perimeter of the domainthey are enclosed by the fixed-temperature condition. Heatgeneration within the solid regions is not allowed when youare solving periodic heat transfer with fixed-temperature con-ditions.

• If you are modeling periodic heat transfer with specified-heat-flux conditions, you can define conducting solid regions atany location within the domain, including volumetric heataddition within the solid, if desired.

4. Set constant material properties (density, heat capacity, viscosity,thermal conductivity), not temperature-dependent properties, us-ing the Materials panel.


5. Specify the Upstream Bulk Temperature in the Periodicity Conditionspanel.

Define −→Periodic Conditions...

If you are modeling periodic heat transfer with specified-temperature!conditions, the bulk temperature should not be equal to the walltemperature, since this will give you the trivial solution of constanttemperature everywhere.

11.4.4 Solution Strategies for Periodic Heat Transfer

After completing the inputs described in Section 11.4.3, you can solvethe flow and heat transfer problem to convergence. The most efficient



approach to the solution, however, is a sequential one in which the peri-odic flow is first solved without heat transfer and then the heat transferis solved leaving the flow field unaltered. This sequential approach isaccomplished as follows:

1. Disable solution of the energy equation under Equations in the So-lution Controls panel.


2. Solve the remaining equations (continuity, momentum, and, op-tionally, turbulence parameters) to convergence to obtain the pe-riodic flow field.

When you initialize the flow field before beginning the calculation,!use the mean value between the inlet bulk temperature and thewall temperature for the initialization of the temperature field.

3. Return to the Solution Controls panel and turn off solution of theflow equations and turn on the energy solution.

4. Solve the energy equation to convergence to obtain the periodictemperature field of interest.

While you can solve your periodic flow and heat transfer problems byconsidering both the flow and heat transfer simultaneously, you will findthat the procedure outlined above is more efficient.

11.4.5 Monitoring Convergence

If you are modeling periodic heat transfer with specified-temperatureconditions, you can monitor the value of the bulk temperature ratio

θ =Twall − Tbulk,inlet

Twall − Tbulk,exit(11.4-5)

during the calculation using the Statistic Monitors panel to ensure thatyou reach a converged solution. Select per/bulk-temp-ratio as the variableto be monitored. See Section 22.16.2 for details about using this feature.



11.4.6 Postprocessing for Periodic Heat Transfer

The actual temperature field predicted by FLUENT in periodic mod-els will not be periodic, and viewing the temperature results duringpostprocessing will display this actual temperature field (T (~r) of Equa-tion 11.4-1). The displayed temperature may exhibit values outside therange defined by the inlet bulk temperature and the wall temperature.This is permissible since the actual temperature profile at the inlet peri-odic face will have temperatures that are higher or lower than the inletbulk temperature.

Static Temperature is found in the Temperature... category of the variableselection drop-down list that appears in postprocessing panels.

Figure 11.4.1 shows the temperature field in a periodic heat exchangergeometry.

Contours of Static Temperature (k)

4.00e+02

3.87e+02

3.74e+02

3.61e+02

3.48e+02

3.35e+02

3.22e+02

3.09e+02

2.96e+02

2.83e+02

2.70e+02

Figure 11.4.1: Temperature Field in a 2D Heat Exchanger GeometryWith Fixed Temperature Boundary Conditions


11.5 Buoyancy-Driven Flows


When heat is added to a fluid and the fluid density varies with tem-perature, a flow can be induced due to the force of gravity acting onthe density variations. Such buoyancy-driven flows are termed natural-convection (or mixed-convection) flows and can be modeled by FLUENT.

11.5.1 Theory

The importance of buoyancy forces in a mixed convection flow can bemeasured by the ratio of the Grashof and Reynolds numbers:

GrRe2 =

∆ρghρv2

(11.5-1)

When this number approaches or exceeds unity, you should expect strongbuoyancy contributions to the flow. Conversely, if it is very small, buoy-ancy forces may be ignored in your simulation. In pure natural con-vection, the strength of the buoyancy-induced flow is measured by theRayleigh number:

Ra =gβ∆TL3ρ

µα(11.5-2)

where β is the thermal expansion coefficient:

β = −1ρ

(∂ρ

∂T

)p

(11.5-3)

and α is the thermal diffusivity:

α =k

ρcp(11.5-4)

Rayleigh numbers less than 108 indicate a buoyancy-induced laminarflow, with transition to turbulence occurring over the range of108 < Ra < 1010.



11.5.2 Modeling Natural Convection in a Closed Domain

When you model natural convection inside a closed domain, the solutionwill depend on the mass inside the domain. Since this mass will not beknown unless the density is known, you must model the flow in one ofthe following ways:

• Perform a transient calculation. In this approach, the initial den-sity will be computed from the initial pressure and temperature, sothe initial mass is known. As the solution progresses over time, thismass will be properly conserved. If the temperature differences inyour domain are large, you must follow this approach.

• Perform a steady-state calculation using the Boussinesq model (de-scribed in Section 11.5.3). In this approach, you will specify a con-stant density, so the mass is properly specified. This approach isvalid only if the temperature differences in the domain are small;if not, you must use the transient approach.

For a closed domain, you cannot use the incompressible ideal gas lawwith a fixed operating pressure. You can use the compressible ideal gaslaw with a fixed operating pressure, but the incompressible ideal gas lawcan be used only with a floating operating pressure. See Section 8.5.4for information about the floating operating pressure option.

11.5.3 The Boussinesq Model

For many natural-convection flows, you can get faster convergence withthe Boussinesq model than you can get by setting up the problem withfluid density as a function of temperature. This model treats density asa constant value in all solved equations, except for the buoyancy term inthe momentum equation:

(ρ− ρ0)g ≈ −ρ0β(T − T0)g (11.5-5)

where ρ0 is the (constant) density of the flow, T0 is the operating tem-perature, and β is the thermal expansion coefficient. Equation 11.5-5 is



obtained by using the Boussinesq approximation ρ = ρ0(1 − β∆T ) toeliminate ρ from the buoyancy term. This approximation is accurate aslong as changes in actual density are small; specifically, the Boussinesqapproximation is valid when β(T − T0) 1.

Limitations of the Boussinesq Model

The Boussinesq model should not be used if the temperature differencesin the domain are large. In addition, it cannot be used with speciescalculations, combustion, or reacting flows.

11.5.4 User Inputs for Buoyancy-Driven Flows

You must provide the following inputs to include buoyancy forces in thesimulation of mixed or natural convection flows:

1. Turn on solution of the energy equation in the Energy panel.


2. Turn on Gravity in the Operating Conditions panel (Figure 11.5.1)and set the Gravitational Acceleration in each Cartesian coordinatedirection by entering the appropriate values in the X, Y, and (for3D) Z fields.

Define −→Operating Conditions

Note that the default gravitational acceleration in FLUENT is zero.

3. If you are using the incompressible ideal gas law, check that theOperating Pressure is set to an appropriate (non-zero) value in theOperating Conditions panel.

4. Depending on whether or not you use the Boussinesq approxima-tion, specify the appropriate parameters described below:

• If you are not using the Boussinesq model, the inputs are asfollows:

(a) If necessary, enable the Specified Operating Density op-tion in the Operating Conditions panel, and specify theOperating Density. See below for details.



Figure 11.5.1: The Operating Conditions Panel

(b) Define the fluid density as a function of temperature, asdescribed in Sections 7.1.3 and 7.2.


• If you are using the Boussinesq model (described in Section 11.5.3),the inputs are as follows:

(a) Specify the Operating Temperature (T0 in Equation 11.5-5)in the Operating Conditions panel.

(b) Select boussinesq as the method for Density in the Mate-rials panel, as described in Sections 7.1.3 and 7.2.



(c) Also in the Materials panel, set the Thermal ExpansionCoefficient (β in Equation 11.5-5) for the fluid materialand specify a constant density.

Note that, if your model involves multiple fluid materials, you canchoose whether or not to use the Boussinesq model for each mate-rial. As a result, you may have some materials using the Boussinesqmodel and others not. In such cases, you will need to set all theparameters described above in this step.

5. The boundary pressures that you input at pressure inlet and outletboundaries are the redefined pressures as given by Equation 11.5-6.In general you should input equal pressures, p′, at the inlet andexit boundaries of your FLUENT model if there are no externally-imposed pressure gradients.


6. Select Body Force Weighted or Second Order as the Discretizationmethod for Pressure in the Solution Controls panel.


You may also want to add cells near the walls to resolve boundarylayers.

If you are using the segregated solver, selecting PRESTO! as theDiscretization method for Pressure is another recommended ap-proach.

See also Section 11.2.2 for information on setting up heat transfer calcu-lations.

Definition of the Operating Density

When the Boussinesq approximation is not used, the operating den-sity, ρ0, appears in the body-force term in the momentum equations as(ρ− ρ0)g.

This form of the body-force term follows from the redefinition of pressurein FLUENT as



p′s = ps − ρ0gx (11.5-6)

The hydrostatic pressure in a fluid at rest is then

p′s = 0 (11.5-7)

The definition of the operating density is thus important in all buoyancy-driven flows.

Setting the Operating Density

By default, FLUENT will compute the operating density by averagingover all cells. In some cases, you may obtain better results if you explic-itly specify the operating density instead of having the solver computeit for you. For example, if you are solving a natural-convection problemwith a pressure boundary, it is important to understand that the pres-sure you are specifying is p′s in Equation 11.5-6. Although you will knowthe actual pressure ps, you will need to know the operating density ρ0

in order to determine p′s from ps. Therefore, you should explicitly spec-ify the operating density rather than use the computed average. Thespecified value should, however, be representative of the average value.

In some cases, the specification of an operating density will improveconvergence behavior, rather than the actual results. For such cases,use the approximate bulk density value as the operating density, andbe sure that the value you choose is appropriate for the characteristictemperature in the domain.

Note that, if you are using the Boussinesq approximation for all fluidmaterials, the operating density is not used, so you need not specify it.

11.5.5 Solution Strategies for Buoyancy-Driven Flows

For high-Rayleigh-number flows, you may want to consider the solu-tion guidelines below. In addition, the guidelines presented in Sec-tion 11.2.3 for solving other heat transfer problems can also be appliedto buoyancy-driven flows. Note, however, that for some laminar, high-Rayleigh-number flows, no steady-state solution exists.



Guidelines for Solving High-Rayleigh-Number Flows

When you are solving a high-Rayleigh-number flow (Ra > 108), youshould follow one of the procedures outlined below for best results.

The first procedure uses a steady-state approach:

1. Start the solution with a lower value of Rayleigh number (e.g., 107)and run to convergence using the first-order scheme.

2. To change the effective Rayleigh number, change the value of gravi-tational acceleration (e.g., from 9.8 to 0.098 to reduce the Rayleighnumber by two orders of magnitude).

3. Use the resulting data file as an initial guess for the higher Rayleighnumber, and start the higher-Rayleigh-number solution using thefirst-order scheme.

4. After you obtain a solution with the first-order scheme, you maycontinue the calculation with a higher-order scheme.

The second procedure uses a time-dependent approach to obtain a steady-state solution [89]:

1. Start the solution from a steady-state solution obtained for thesame or a lower Rayleigh number.

2. Estimate the time constant as [16]

τ =L

U∼ L2

α(PrRa)−1/2 =

L√gβ∆TL

(11.5-8)

where L and U are the length and velocity scales, respectively. Usea time step ∆t such that

∆t ≈ τ

4(11.5-9)

Using a larger time step ∆t may lead to divergence.



3. After oscillations with a typical frequency of fτ = 0.05–0.09 havedecayed, the solution reaches steady state. Note that τ is thetime constant estimated in Equation 11.5-8 and f is the oscillationfrequency in Hz. In general, this solution process may take as manyas 5000 time steps to reach steady state.

11.5.6 Postprocessing for Buoyancy-Driven Flows

The postprocessing reports of interest for buoyancy-driven flows are thesame as for other heat transfer calculations. See Section 11.2.4 for details.


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