train theory+practical.pdf

8/9/2019 train theory+practical.pdf

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© Fluent Inc. 2/20/01D1

Fluent Software Training

TRN-99-003

Modeling Turbulent Flows


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u Unsteady, aperiodic motion in which all three velocity components

fluctuate Õ mixing matter, momentum, and energy.

u Decompose velocity into mean and fluctuating parts:

U i(t) ≡ U i + ui(t)

u Similar fluctuations for pressure, temperature, and species

concentration values.

What is Turbulence?

Time

U i (t)

U i

ui(t)


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Why Model Turbulence?u Direct numerical simulation of governing equations is only possible for

simple low- Re flows.

u Instead, we solve Reynolds Averaged Navier-Stokes (RANS)

equations:

where (Reynolds stresses)

u Time-averaged statistics of turbulent velocity fluctuations are modeled

using functions containing empirical constants and information about

the mean flow.

u Large Eddy Simulation numerically resolves large eddies and models

small eddies.

(steady, incompressible flow

w/o body forces)

jiij uu R ρ−=

j

ij

j j

i

ik

ik

x

R

x x

U

x

p

x

U U

∂∂+

∂∂∂+

∂∂−=

∂∂ 2

µρ


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Is the Flow Turbulent?External Flows

Internal Flows

Natural Convection

5105×≥ x Re along a surface

around an obstacle

where

µ

ρUL Re L ≡where

Other factors such as free-stream

turbulence, surface conditions, and

disturbances may cause earlier

transition to turbulent flow.

L = x, D, Dh, etc.

,3002≥h D

Re

108 1010 −≥ Raµα

ρβ 3TL g Ra

∆≡

20,000≥ D Re


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How Complex is the Flow?

u Extra strain rates

l Streamline curvature

l Lateral divergence

l Acceleration or decelerationl Swirl

l Recirculation (or separation)

l Secondary flow

u 3D perturbations

u

Transpiration (blowing/suction)u Free-stream turbulence

u Interacting shear layers


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Choices to be Made

Turbulence Model

&

Near-Wall Treatment

Flow

Physics

Accuracy

Required

Computational

Resources

Turnaround

Time

Constraints

Computational

Grid


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Zero-Equation Models

One-Equation Models

Spalart-AllmarasTwo-Equation Models

Standard k-

RNG k-

Realizable k-

Reynolds-Stress Model

Large-Eddy Simulation

Direct Numerical Simulation

Turbulence Modeling Approaches

Include

More

Physics

Increase

Computational

Cost

Per IterationAvailable

in FLUENT 5

RANS-based

models


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u RANS equations require closure for Reynolds stresses.

u Turbulent viscosity is indirectly solved for from single transport

equation of modified viscosity for One-Equation model.

u For Two-Equation models, turbulent viscosity correlated with turbulent

kinetic energy (TKE) and the dissipation rate of TKE.

u Transport equations for turbulent kinetic energy and dissipation rate aresolved so that turbulent viscosity can be computed for RANS equations.

Reynolds Stress Terms in RANS-based Models

Turbulent

Kinetic Energy:Dissipation Rate of

Turbulent Kinetic Energy:

ερµ µ

2k C t ≡Turbulent Viscosity:

Boussinesq Hypothesis:(isotropic viscosity)

∂∂

+∂∂

+−=−=i

j

j

it ij jiij

x

U

x

U k uu R µδρρ

3

2

2/iiuuk ≡

∂

∂+

∂∂

∂∂

≡i

j

j

i

j

i

x

u

x

u

x

u νε


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One-Equation Model: Spalart-Allmaras

u Designed specifically for aerospace applications involving wall-

bounded flows.

l Boundary layers with adverse pressure gradients

l turbomachinery

u Can use coarse or fine mesh at wall

l Designed to be used with fine mesh as a “low-Re” model, i.e., throughout

the viscous-affected region.

l Sufficiently robust for relatively crude simulations on coarse meshes.


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Two Equation Model: Standard k-ε ModelTurbulent Kinetic Energy

Dissipation Rate

εεεσσ 21, ,, C C k are empirical constants

(equations written for steady, incompressible flow w/o body forces)

Convection Generation DiffusionDestruction

{

ρεσµµρ −

∂∂

∂∂

+∂∂

∂∂

+∂

∂=

∂∂

4 4 4 34 4 4 214 4 4 34 4 4 2143421

i

k t

ii

j

j

i

i

j

t

i

i x

k

x x

U

x

U

x

U

x

k U )(

DestructionConvection Generation Diffusion

434214 4 4 34 4 4 214 4 4 4 4 34 4 4 4 4 2143421

−

∂∂

∂∂

+∂

∂

∂∂

+∂

∂

=

∂∂

k C

x x x

U

x

U

x

U

k C

xU

i

t

ii

j

j

i

i

j

t

i

i

2

21 )( ε

ρε

σµµεε

ρ εεε


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Two Equation Model: Standard k-ε Model

u “Baseline model” (Two-equation)

l Most widely used model in industry

l Strength and weaknesses well documented

u Semi-empiricall k equation derived by subtracting the instantaneous mechanical energy

equation from its time-averaged value

l ε equation formed from physical reasoningu Valid only for fully turbulent flows

u Reasonable accuracy for wide range of turbulent flows

l industrial flows

l heat transfer


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Two Equation Model: Realizable k-εu Distinctions from Standard k-ε model:

l Alternative formulation for turbulent viscosity

where is now variable

n (A0, As, and U* are functions of velocity gradients)

n Ensures positivity of normal stresses;

n Ensures Schwarz’s inequality;

l New transport equation for dissipation rate, ε:

ερµ µ

2k

C t ≡

ε

µk U

A A

C

so

*

1

+=

0u2

i ≥2

j

2

i

2

ji uu)uu( ≤

b

j

t

j

Gck

ck

cS c x x Dt

Dεε

ε

ε

νε

ερερ

ε

σ

µµ

ερ 31

2

21 ++−+

∂∂

+

∂∂

=

GenerationDiffusion Destruction Buoyancy


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u Shares the same turbulent kinetic energy equation as Standard k-εu Superior performance for flows involving:

l planar and round jets

l

boundary layers under strong adverse pressure gradients, separationl rotation, recirculation

l strong streamline curvature

Two Equation Model: Realizable k-ε


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Two Equation Model: RNG k-εTurbulent Kinetic Energy

Dissipation Rate

Convection Diffusion

Dissipation

{ {

ρεµαµρ −

∂∂

∂∂

+=∂∂

4 4 34 4 2143421 ik

i

t

i

i x

k

xS

x

k U eff

2

Generation

∂

∂+

∂

∂≡≡

j

i

i

j

ijijij

x

U

x

U S S S S

2

1,2

where

are derived using RNG theoryεεεαα 21, ,, C C k


Additional termrelated to mean strain

& turbulence quantitiesConvection Generation Diffusion Destruction

{ R

k C

x xS

k C

xU

ii

t

i

i −

−

∂∂

∂∂

+

=

∂∂

434214 4 34 4 214 434 421

43421

2

2eff

2

1

ερ

εµαµ

εερ εεε


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Two Equation Model: RNG k-εu k-ε equations are derived from the application of a rigorous statistical

technique (Renormalization Group Method) to the instantaneous Navier-

Stokes equations.

u Similar in form to the standard k-ε equations but includes:l additional term in ε equation that improves analysis of rapidly strained flowsl the effect of swirl on turbulence

l analytical formula for turbulent Prandtl number

l differential formula for effective viscosity

u Improved predictions for:

l high streamline curvature and strain rate

l transitional flows

l wall heat and mass transfer


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Reynolds Stress Model

k

ijk

ijijij

k

ji

k x

J P

x

uuU

∂∂

+−Φ+=∂

∂ερ

Generation

k

ik j

k

j

k iij

x

U uu

x

U uu P

∂

∂+

∂

∂≡

∂∂

+∂∂′−≡Φ

i

j

j

iij

x

u

x

u p

k

j

k

iij

x

u

x

u

∂

∂

∂

∂≡ µε 2

Pressure-Strain

Redistribution

Dissipation

Turbulent

Diffusion

(modeled)

(related to ε)

(modeled)

(computed)


Reynolds Stress

Transport Eqns.

Pressure/velocity

fluctuations

Turbulent

transport

)( jik i jk k jiijk uu puuu J δδ +′+=


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Reynolds Stress Modelu RSM closes the Reynolds-Averaged Navier-Stokes equations by

solving additional transport equations for the Reynolds stresses.

l Transport equations derived by Reynolds averaging the product of the

momentum equations with a fluctuating property

l Closure also requires one equation for turbulent dissipation

l Isotropic eddy viscosity assumption is avoided

u Resulting equations contain terms that need to be modeled.

u RSM has high potential for accurately predicting complex flows.

l Accounts for streamline curvature, swirl, rotation and high strain rates

n Cyclone flows, swirling combustor flowsn Rotating flow passages, secondary flows


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Large Eddy Simulationu Large eddies:

l Mainly responsible for transport of momentum, energy, and other scalars,

directly affecting the mean fields.

l Anisotropic, subjected to history effects, and flow-dependent, i.e., strongly

dependent on flow configuration, boundary conditions, and flow parameters.u Small eddies:

l Tend to be more isotropic and less flow-dependent

l More likely to be easier to model than large eddies.

u LES directly computes (resolves) large eddies and models only small

eddies (Subgrid-Scale Modeling).u Large computational effort

l Number of grid points, NLES ∝l Unsteady calculation

2Re

τu


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Comparison of RANS Turbulence ModelsModel Strengths Weaknesses

Spalart-

Allmaras

Economical (1-eq.); good track record

for mildly complex B.L. type of flows

Not very widely tested yet; lack of

submodels (e.g. combustion, buoyancy)

STD k- Robust, economical, reasonablyaccurate; long accumulated

performance data

Mediocre results for complex flows

involving severe pressure gradients,strong streamline curvature, swirl

and rotation

RNG k-

Good for moderately complex

behavior like jet impingement,

separating flows, swirling flows, andsecondary flows

Subjected to limitations due to

isotropic eddy viscosity

assumption

Realizable

k-

Offers largely the same benefits as

RNG; resolves round-jet anomaly

Subjected to limitations due toisotropic eddy viscosity

assumption

Reynolds

Stress

Model

Physically most complete model

(history, transport, and anisotropy of

turbulent stresses are all accountedfor)

Requires more cpu effort (2-3x);

tightly coupled momentum andturbulence equations


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Near-Wall Treatments

u Most k-ε and RSM turbulencemodels will not predict correct

near-wall behavior if integrated

down to the wall.u Special near-wall treatment is

required.

l Standard wall functions

l Nonequilibrium wall functions

l Two-layer zonal model

Boundary layer structure


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Standard Wall Functions

ρτ

µ

/

2/14/1

w

P P k C U U ≡∗

( )

>

+

<= ∗

∗

∗

)(ln1

Pr

)(Pr

**

**

T t

T

y y P Ey

y y y

T

κ

µ

ρ µ P P yk C y

2/14/1

≡∗

q

k C cT T T

P p P w

′′−

≡&

2/14/1)(*

µρ

Mean Velocity

Temperature

where

where and P is a function of the fluid

and turbulent Prandtl numbers.

thermal sublayer thickness

( )∗∗ = EyU ln1κ


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Nonequilibrium Wall Functionsu Log-law is sensitized to pressure gradient for

better prediction of adverse pressure gradient

flows and separation.

u Relaxed local equilibrium assumptions for

TKE in wall-neighboring cells.u Thermal law-of-wall unchanged

=

µ

ρ

κ ρτ

µµ yk C E

k C U

w

2/14/12/14/1

ln1

/

~

+

−+

−= ∗∗ µρκ ρκ

y

k

y y

y

y

k

y

dx

dpU U vv

v

v

2

2/12/1ln

21~where


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Two-Layer Zonal Modelu Used for low- Re flows or

flows with complex near-wall

phenomena.

u Zones distinguished by a wall-

distance-based turbulent

Reynolds number

u High- Re k-ε models are used in the turbulent core region.u Only k equation is solved in the viscosity-affected region.

u ε is computed from the correlation for length scale.

u Zoning is dynamic and solution adaptive.

µρ yk

Re y ≡

200> y Re

200


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Comparison of Near Wall TreatmentsStrengths Weaknesses

Standard wall

Functions

Robust, economical,

reasonably accurate

Empirically based on simple

high- Re flows; poor for low- Reeffects, massive transpiration,

∇ p, strong body forces, highly3D flows

Nonequilibrium

wall functions

Accounts for ∇ p effects,allows nonequilibrium:

-separation-reattachment

-impingement

Poor for low- Re effects, massive

transpiration, severe ∇ p, strong body forces, highly 3D flows

Two-layer zonal

model

Does not rely on law-of-the-

wall, good for complexflows, especially applicableto low- Re flows

Requires finer mesh resolution

and therefore larger cpu andmemory resources


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Computational Grid GuidelinesWall Function

ApproachTwo-Layer Zonal

Model Approach

l First grid point in log-law region

l At least ten points in the BL.

l Better to use stretched quad/hex

cells for economy.

l First grid point at y+ ≈ 1.

l At least ten grid points within

buffer & sublayers.

l Better to use stretched quad/hex

cells for economy.

50050 ≤≤ + y


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Estimating Placement of First Grid Point

u Estimate the skin friction coefficient based on correlations either

approximate or empirical:

l Flat Plate-

l Pipe Flow-

u Compute the friction velocity:

u Back out required distance from wall:

l Wall functions • Two-layer model

u Use post-processing to confirm near-wall mesh resolution

2.0Re0359.02/

−≈ L f c2.0

Re039.02/ −≈ D f c

2// f ew cU u =≡ ρττ

y1 = 50 ν/uτ y1 = ν/ uτ


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Setting Boundary Conditions

u Characterize turbulence at inlets & outlets (potential backflow)

l k -ε models require k and ε

l Reynolds stress model requires Rij and ε

u Several options allow input using more familiar parametersl Turbulence intensity and length scale

n length scale is related to size of large eddies that contain most of energy.

n For boundary layer flows: l ≈ 0.4δ99n For flows downstream of grids /perforated plates: l ≈ opening size

l Turbulence intensity and hydraulic diameter

n Ideally suited for duct and pipe flows

l Turbulence intensity and turbulent viscosity ratio

n For external flows:

u Input of k and ε explicitly allowed (non-uniform profiles possible).

10/1


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GUI for Turbulence Models

Define Õ Models Õ Viscous...

Turbulence Model options

Near Wall Treatments

Inviscid, Laminar, or Turbulent

Additional Turbulence options


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Example: Channel Flow with Conjugate HeatTransfer

adiabatic wall

cold air V = 50 fpm

T = 0 °F

constant temperature wall T = 100 °F

insulation

1 ft

1 ft

10 ft

P

Predict the temperature at point P in the solid insulation


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Turbulence Modeling Approachu Check if turbulent Õ Re Dh

= 5,980

u Developing turbulent flow at relatively low Reynolds number and

BLs on walls will give pressure gradient Õ use RNG k-ε withnonequilibrium wall functions.

u Develop strategy for the gridl Simple geometryÕ quadrilateral cells

l Expect large gradients in normal direction to horizontal wallsÕ fine

mesh near walls with first cell in log-law region.

l Vary streamwise grid spacing so that BL growth is captured.

l Use solution-based grid adaption to further resolve temperature

gradients.


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Velocity

contours

Temperature

contours

BLs on upper & lower surfaces accelerate the core flow

Prediction of Momentum & Thermal

Boundary Layers

Important that thermal BL was accurately resolved as well

P


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Example: Flow Around a Cylinder

wall

wall

1 ft

2 ft

2 ft

air

V = 4 fps

Compute drag coefficient of the cylinder

5 ft 14.5 ft


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u Check if turbulent Õ Re D = 24,600

u Flow over an object, unsteady vortex shedding is expected,

difficult to predict separation on downstream side, and close proximity of side walls may influence flow around cylinder

Õ use RNG k-ε with 2-layer zonal model.

u Develop strategy for the grid

l Simple geometry & BLs Õ quadrilateral cells.

l Large gradients near surface of cylinder & 2-layer model

Õ fine mesh near surface & first cell at y+ = 1.

Turbulence Modeling Approach


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Grid for Flow Over a Cylinder


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Prediction of Turbulent Vortex Shedding

Contours of effective viscosity µeff = µ + µt

C D = 0.53 Strouhal Number = 0.297

U

DSt

τ≡where


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Summary: Turbulence Modeling Guidelines

u Successful turbulence modeling requires engineering judgement of:

l Flow physics

l Computer resources available

l Project requirementsn Accuracy

n Turnaround time

l Turbulence models & near-wall treatments that are available

u Begin with standard k-ε and change to RNG or Realizable k-ε if needed.

u Use RSM for highly swirling flows.

u Use wall functions unless low- Re flow and/or complex near-wall

physics are present.


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© Fluent Inc. 2/23/01E1


TRN-99-003

Solver Settings


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Outlineu Using the Solver

l Setting Solver Parameters

l Convergence

n Definition

n Monitoring

n Stability

n Accelerating Convergence

l Accuracy

n Grid Independence

n Adaption

u Appendix: Backgroundl Finite Volume Method

l Explicit vs. Implicit

l Segregated vs. Coupled

l Transient Solutions


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Modify solution

parameters or grid

NoYes

No

Set the solution parameters

Initialize the solution

Enable the solution monitors of interest

Calculate a solution

Check for convergence

Check for accuracy

Stop

Yes

Solution Procedure Overviewu Solution Parameters

l Choosing the Solver

l Discretization Schemes

u Initialization

u Convergence

l Monitoring Convergence

l Stability

n Setting Under-relaxation

n Setting Courant number

l Accelerating Convergence

u Accuracy

l Grid Independence

l Adaption


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Choosing a Solver u Choices are Coupled-Implicit, Coupled-Explicit, or Segregated (implicit)

u The Coupled solvers are recommended if a strong inter-dependence exists

between density, energy, momentum, and/or species.

l e.g., high speed compressible flow or finite-rate reaction modeled flows.

l In general, the Coupled-Implicit solver is recommended over the coupled-explicit

solver.

n Time required: Implicit solver runs roughly twice as fast.

n Memory required: Implicit solver requires roughly twice as much memory as coupled-

explicit or segregated-implicit solvers! (Performance varies.)

l The Coupled-Explicit solver should only be used for unsteady flows when the

characteristic time scale of problem is on same order as that of the acoustics.n e.g., tracking transient shock wave

u The Segregated (implicit) solver is preferred in all other cases.

l Lower memory requirements than coupled-implicit solver.

l Segregated approach provides flexibility in solution procedure.


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Discretization (Interpolation Methods)

u Field variables (stored at cell centers) must be interpolated to the faces of

the control volumes in the FVM:

u FLUENT offers a number of interpolation schemes:

l First-Order Upwind Scheme

n easiest to converge, only first order accurate.

l Power Law Scheme

n more accurate than first-order for flows when Recell< 5 (typ. low Re flows).

l Second-Order Upwind Schemen uses larger ‘stencil’ for 2nd order accuracy, essential with tri/tet mesh or

when flow is not aligned with grid; slower convergence

l Quadratic Upwind Interpolation (QUICK)

n applies to quad/hex mesh, useful for rotating/swirling flows, 3rd order

accurate on uniform mesh.

V S A AV V t

f

faces

f f f

faces

f f f

t t t

∆+∇Γ =+∆∆

− ∑∑ ⊥∆+

φφφρρφρφ

,)()()(


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Interpolation Methods for Pressureu Additional interpolation options are available for calculating face pressure when

using the segregated solver.

u FLUENT interpolation schemes for Face Pressure:

l Standard

n

default scheme; reduced accuracy for flows exhibiting large surface-normal pressuregradients near boundaries.

l Linear

n useful only when other options result in convergence difficulties or unphysical

behavior.

l Second-Order

n use for compressible flows or when PRESTO! cannot be applied.

l Body Force Weighted

n use when body forces are large, e.g., high Ra natural convection or highly swirling

flows.

l PRESTO!

n applies to quad/hex cells; use on highly swirling flows, flows involving porous

media, or strongly curved domains.


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Pressure-Velocity Coupling

u Pressure-Velocity Coupling refers to the way mass continuity is

accounted for when using the segregated solver.

u Three methods available:

l SIMPLE

n default scheme, robustl SIMPLEC

n Allows faster convergence for simple problems (e.g., laminar flows with

no physical models employed).

l PISO

n useful for unsteady flow problems or for meshes containing cells with

higher than average skew.


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Initializationu Iterative procedure requires that all solution variables be initialized

before calculating a solution.

SolveÕ InitializeÕ Initialize...

l Realistic ‘guesses’ improves solution stability and accelerates convergence.

l In some cases, correct initial guess is required:n Example: high temperature region to initiate chemical reaction.

u “Patch” values for individual

variables in certain regions.

SolveÕ InitializeÕ Patch...

l Free jet flows

(patch high velocity for jet)

l Combustion problems

(patch high temperature

for ignition)


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Convergence Preliminaries: Residuals

u Transport equation for φ can be presented in simple form:

l Coefficients a p , anb typically depend upon the solution.

l Coefficients updated each iteration.

u At the start of each iteration, the above equality will not hold.

l The imbalance is called the residual, R p, where:

l R p should become negligible as iterations increase.

l The residuals that you monitor are summed over all cells:

n By default, the monitored residuals are scaled.

n

You can also normalize the residuals.u Residuals monitored for the coupled solver are based on the rms value of

the time rate of change of the conserved variable.

l Only for coupled equations; additional scalar equations use segregated

definition.

p

nb

nbnb p p baa =+ ∑ φφ

p

nb

nbnb p p p baa R −+= ∑ φφ

||∑=cells

p R R


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Convergence

u At convergence:

l All discrete conservation equations (momentum, energy, etc.) are

obeyed in all cells to a specified tolerance.

l Solution no longer changes with more iterations.

l Overall mass, momentum, energy, and scalar balances are obtained.

u Monitoring convergence with residuals:

l Generally, a decrease in residuals by 3 orders of magnitude indicates at

least qualitative convergence.

n Major flow features established.

l Scaled energy residual must decrease to 10-6 for segregated solver.

l Scaled species residual may need to decrease to 10-5

to achieve species balance.

u Monitoring quantitative convergence:

l Monitor other variables for changes.

l Ensure that property conservation is satisfied.


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Convergence Monitors: Residualsu Residual plots show when the residual values have reached the

specified tolerance.

SolveÕ MonitorsÕ Residual...

All equations converged.

10-3

10-6


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Convergence Monitors: Forces/Surfacesu In addition to residuals, you can also monitor:

l Lift, drag, or moment

SolveÕ MonitorsÕ Force...

l Variables or functions (e.g., surface integrals)

at a boundary or any defined surface:SolveÕ MonitorsÕ Surface...


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Checking for Property Conservationu In addition to monitoring residual and variable histories, you should

also check for overall heat and mass balances.

l Net imbalance should be less than 0.1% of net flux through domain.

Report Õ Fluxes...


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Decreasing the Convergence Toleranceu If your monitors indicate that the solution is converged, but the

solution is still changing or has a large mass/heat imbalance:

l Reduce Convergence Criterion

or disable Check Convergence.l Then calculate until solution

converges to the new tolerance.


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Convergence Difficulties

u Numerical instabilities can arise with an ill-posed problem, poor

quality mesh, and/or inappropriate solver settings.

l Exhibited as increasing (diverging) or “stuck” residuals.

l Diverging residuals imply increasing imbalance in conservation equations.

l Unconverged results can be misleading!u Troubleshooting:

l Ensure problem is well posed.

l Compute an initial solution with

a first-order discretization scheme.

l Decrease under-relaxation for

equations having convergence

trouble (segregated).

l Reduce Courant number (coupled).

l Re-mesh or refine grid with high

aspect ratio or highly skewed cells.

Continuity equation convergence

trouble affects convergence of

all equations.


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Modifying Under-relaxation Factorsu Under-relaxation factor, α, is

included to stabilize the iterative

process for the segregated solver .

u Use default under-relaxation factors

to start a calculation.

SolveÕ ControlsÕ Solution...

u Decreasing under-relaxation for

momentum often aids convergence.

l Default settings are aggressive but

suitable for wide range of problems.l ‘Appropriate’ settings best learned

from experience.

pold p p φαφφ ∆+= ,

u For coupled solvers, under-relaxation factors for equations outside coupled

set are modified as in segregated solver.


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Modifying the Courant Number u Courant number defines a ‘time

step’ size for steady-state problems.

l A transient term is included in the

coupled solver even for steady state

problems.u For coupled-explicit solver:

l Stability constraints impose a

maximum limit on Courant number.

n Cannot be greater than 2.

s Default value is 1.

n Reduce Courant number whenhaving difficulty converging.

u

xt

∆=∆

)CFL(u For coupled-implicit solver:

l Courant number is not limited by stability constraints.

n Default is set to 5.


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Accelerating Convergenceu Convergence can be accelerated by:

l Supplying good initial conditions

n Starting from a previous solution.

l Increasing under-relaxation factors or Courant number

n Excessively high values can lead to instabilities.

n Recommend saving case and data files before continuing iterations.

l Controlling multigrid solver settings.

n Default settings define robust Multigrid solver and typically do not need

to be changed.


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Starting from a Previous Solutionu Previous solution can be used as an initial condition when changes are

made to problem definition.

l Once initialized, additional iterations uses current data set as starting point.

Actual Problem Initial Condition

flow with heat transfer isothermal solution

natural convection lower Ra solution

combustion cold flow solution

turbulent flow Euler solution


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Multigridu The Multigrid solver accelerates convergence by using solution on

coarse mesh as starting point for solution on finer mesh.

l Influence of boundaries and far-away points are more easily transmitted to

interior of coarse mesh than on fine mesh.

l Coarse mesh defined from original mesh.n Multiple coarse mesh ‘levels’ can be created.

s AMG- ‘coarse mesh’ emulated algebraically.

s FAS- ‘cell coalescing’ defines new grid.

– a coupled-explicit solver option

n Final solution is for original mesh.

l Multigrid operates automatically in the background.

u Accelerates convergence for problems with:

l Large number of cells

l Large cell aspect ratios, e.g., ∆ x/∆ y > 20l Large differences in thermal conductivity

fine (original) mesh

coarse mesh

‘solution

transfer’


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Accuracyu A converged solution is not necessarily an accurate one.

l Solve using 2nd order discretization.

l Ensure that solution is grid-independent.

n Use adaption to modify grid.

u If flow features do not seem reasonable:

l Reconsider physical models and boundary conditions.

l Examine grid and re-mesh.


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Mesh Quality and Solution Accuracy

u Numerical errors are associated with calculation of cell gradients and

cell face interpolations.

u These errors can be contained:

l Use higher order discretization schemes.

l

Attempt to align grid with flow.l Refine the mesh.

n Sufficient mesh density is necessary to resolve salient features of flow.

s Interpolation errors decrease with decreasing cell size.

n Minimize variations in cell size.

s Truncation error is minimized in a uniform mesh.

s Fluent provides capability to adapt mesh based on cell size variation.n Minimize cell skewness and aspect ratio.

s In general, avoid aspect ratios higher than 5:1.

s Optimal quad/hex cells have bounded angles of 90 degrees

s Optimal tri/tet cells are equilateral.


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Determining Grid Independence

u When solution no longer changes with further grid refinement, you

have a “grid-independent” solution.

u Procedure:

l Obtain new grid:

n Adapt

s Save original mesh before adapting.

– If you know where large gradients are expected, concentrate the

original grid in that region, e.g., boundary layer.

s Adapt grid.

– Data from original grid is automatically interpolated to finer grid.

n file → reread-grid and File → Interpolate...s Import new mesh and initialize with old solution.

l Continue calculation to convergence.

l Compare results obtained w/different grids.

l Repeat adaption/calculation procedure if necessary.


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Unsteady Flow Problems

u Transient solutions are possible with both segregated and coupled solvers.

l Solver iterates to convergence at each time level, then advances automatically.

l Solution Initialization provides initial condition, must be realistic.

u For segregated solver:

l

Time step size, ∆t, is input in Iterate panel.n ∆t should be small enough to resolve

time dependent features and to ensure

convergence within 20 iterations.

n May need to start solution with small ∆t.l Number of time steps, N, is also required.

n

N*∆t = total simulated time.l Use TUI command ‘it #’ to iterate without advancing time step.

u For Coupled Solver, Courant number defines in practice:

l global time step size for coupled explicit solver.

l pseudo-time step size for coupled implicit solver.


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Summaryu Solution procedure for the segregated and coupled solvers is the same:

l Calculate until you get a converged solution.

l Obtain second-order solution (recommended).

l Refine grid and recalculate until grid-independent solution is obtained.

u All solvers provide tools for judging and improving convergence and

ensuring stability.

u All solvers provide tools for checking and improving accuracy.

u Solution accuracy will depend on the appropriateness of the physical

models that you choose and the boundary conditions that you specify.


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Appendixu Background

l Finite Volume Method

l Explicit vs. Implicit

l Segregated vs. Coupled

l Transient Solutions


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Background: Finite Volume Method - 1u FLUENT solvers are based on the finite volume method.

l Domain is discretized into a finite set of control volumes or cells.

u General transport equation for mass, momentum, energy, etc. is

applied to each cell and discretized. For cell p,

∫ ∫ ∫ ∫ ∀

+⋅∇Γ =⋅+∂∂

dV S d d dV t

A AV

φφρφρφ AAV

unsteady convection diffusion generation

Eqn.

continuity 1

x-mom. u

y-mom. v

energy h

φ

Fluid region of pipe flow

discretized into finite set of

control volumes (mesh).

control

volume

u All equations are solved to render flow field.


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Background: Finite Volume Method - 2u Each transport equation is discretized into algebraic form. For cell p,

face f

adjacent cells, nb

cell p

u Discretized equations require information at cell centers and faces.

l Field data (material properties, velocities, etc.) are stored at cell centers.

l Face values can be expressed in terms of local and adjacent cell values.

l Discretization accuracy depends upon ‘stencil’ size.

u The discretized equation can be expressed simply as:

l Equation is written out for every control volume in domain resulting in an

equation set .

p

nb


V S A AV V t

f

faces

f f f

faces

f f f

t

p

t t

p ∆+∇Γ =+∆∆

−∑∑ ⊥

∆+

φφφρρφρφ

,)()()(


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u Equation sets are solved iteratively.

l Coefficients a p and anb are typically functions

of solution variables (nonlinear and coupled).

l Coefficients are written to use values of solution variables from previous

iteration.n Linearization: removing coefficients’ dependencies on φ.

n De-coupling: removing coefficients’ dependencies on other solution

variables.

l Coefficients are updated with each iteration.

n For a given iteration, coefficients are constant.

s φ p can either be solved explicitly or implicitly.

Background: Linearization

p

nb



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u Assumptions are made about the knowledge of φnb:

l Explicit linearization - unknown value in each cell computed from relations

that include only existing values (φnb assumed known from previousiteration).

n φ p solved explicitly using Runge-Kutta scheme.

l Implicit linearization - φ p and φnb are assumed unknown and are solvedusing linear equation techniques.

n Equations that are implicitly linearized tend to have less restrictive stability

requirements.

n The equation set is solved simultaneously using a second iterative loop (e.g.,

point Gauss-Seidel).

Background: Explicit vs. Implicit


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Background: Coupled vs. Segregatedu Segregated Solver

l If the only unknowns in a given equation are assumed to be for a single

variable, then the equation set can be solved without regard for the

solution of other variables.

n coefficients a p and anb are scalars.

u Coupled Solver

l If more than one variable is unknown in each equation, and each

variable is defined by its own transport equation, then the equation set is

coupled together.

n coefficients a p

and anb

are N eq

x N eq

matrices

n φ is a vector of the dependent variables, { p, u, v, w, T, Y }T

p

nb



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Background: Segregated Solver

u In the segregated solver , each equation is

solved separately.

u The continuity equation takes the form

of a pressure correction equation as partof SIMPLE algorithm.

u Under-relaxation factors are included in

the discretized equations.

l Included to improve stability of iterative

process.

l Under-relaxation factor, α, in effect,limits change in variable from one

iteration to next:

Update properties.

Solve momentum equations (u, v, w velocity).

Solve pressure-correction (continuity) equation.

Update pressure, face mass flow rate.

Solve energy, species, turbulence, and other

scalar equations.

Converged?

Stop No Yes

pold p p φαφφ ∆+= ,


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Background: Coupled Solver

u Continuity, momentum, energy, and

species are solved simultaneously in the

coupled solver .

u Equations are modified to resolve

compressible and incompressible flow.u Transient term is always included.

l Steady-state solution is formed as time

increases and transients tend to zero.

u For steady-state problem, ‘time step’ is

defined by Courant number.

l Stability issues limit maximum time step

size for explicit solver but not for

implicit solver.

Solve continuity, momentum, energy,

and species equations simultaneously.

Stop

No Yes

Solve turbulence and other scalar equations.

Update properties.

Converged?

u

xt

∆=∆

)CFL( CFL = Courant-Friedrichs-Lewy-number where u = appropriate velocity scale

∆ x = grid spacing


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Background: Segregated/Transient

u Transient solutions are possible with both segregated and coupled solvers.

l 1st- and 2nd-order time implicit discretizations (Euler) available for coupled

and segregated solvers.

n Procedure: Iterate to convergence at each time level, then advance in time.

l 2nd order time-explicit discretization also available for coupled-explicit solver.

u For segregated solver:

l Time step size, ∆t, is input in Iterate panel.n ∆t should be small enough to resolve

time dependent features.

l Number of time steps, N, is also required.

n N*∆t equals total simulated time.l Generally, use ∆t small enough to ensure

convergence within 20 iterations.

l Note: Use TUI command ‘it #’ to iterate

further without advancing time step.


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Background: Coupled/Transientl If implicit scheme is selected, two transient terms are included in discretization.

n Physical-time transient

s Physical-time derivative term is discretized implicitly (1st or 2nd order).

s Time step size, ∆t, defined as with segregated solver.n Pseudo-time transient

s At each physical-time level, a pseudo-time transient is driven to zero through aseries of inner iterations (dual time stepping).

s Pseudo-time derivative term is discretized:

– explicitly in coupled-explicit solver.

– implicitly in coupled-implicit solver.

s Courant number defines pseudo-time step size, ∆τ.

l For explicit time stepping, physical-time derivative isdiscretized explicitly.

n Option only available with coupled-explicit solver

n Physical-time step size is defined by Courant number.

s Same time step size is used throughout domain (global time stepping).


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Heat Transfer and Thermal Boundary

Conditions

Headlamp modeled withDiscrete Ordinates

Radiation Model

l S f i i


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Outlineu Introduction

u Thermal Boundary Conditions

u Fluid Properties

u Conjugate Heat Transfer

u Natural Convection

u Radiation

u Periodic Heat Transfer

Fl S f T i i


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Introduction

u Energy transport equation is solved, subject to a wide range of thermal

boundary conditions.

l Energy source due to chemical reaction is included for reacting flows.

l Energy source due to species diffusion included for multiple species flows.

n Always included in coupled solver.

n Can be disabled in segregated solver.

l Energy source due to viscous heating:

n Describes thermal energy created by viscous shear in the flow.

s Important when shear stress in fluid is large (e.g., lubrication) and/or in

high-velocity, compressible flows.

n Often negligible

s not included by default for segregated solver

s always included for coupled solver.

l In solid regions, simple conduction equation solved.

n Convective term can also be included for moving solids.

Fl t S ft T i i


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User Inputs for Heat Transfer

1. Activate calculation of heat transfer.

l Select the Enable Energy option in the Energy panel.

Define Õ Models Õ Energy...

l Enabling a temperature dependent density model, reacting flow model, or a

radiation model will toggle Enable Energy on without visiting this panel.

2. Enable appropriate options:

l Viscous Heating in Viscous Model panel

l Diffusion Energy Source option in the Species Model panel

3. Define thermal boundary conditions.

Define Õ Boundary Conditions...

4. Define material properties for heat transfer.

Define Õ Materials...

l Heat capacity and thermal conductivity must be defined.

Fl t S ft T i i


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Solution Process for Heat Transfer

u Many simple heat transfer problems can be successfully solved using

default solution parameters.

u However, you may accelerate convergence and/or improve the stability

of the solution process by changing the options below:

l Under-relaxation of energy equation.Solve Õ Controls Õ Solution...

l Disabling species diffusion term.

Define Õ Models Õ Species...

l Compute isothermal flow first, then add calculation of energy equation.

Solve Õ Controls Õ Solution...



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Theoretical Basis of Wall Heat Transfer

u For laminar flows, fluid side heat transfer is approximated as:

n = local coordinate normal to wall

u For turbulent flows:

l Law of the wall is extended to treat wall heat flux.

n The wall-function approach implicitly accounts for viscous sublayer.

l The near-wall treatment is extended to account for viscous dissipation

which occurs in the boundary layer of high-speed flows.

′′ = ≈q k T

nk

T

nwall

∂

∂

∆∆



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Thermal Boundary Conditions at Flow Inletsand Exits

u At flow inlets, must supply

fluid temperature.

u

At flow exits, fluidtemperature extrapolated

from upstream value.

u At pressure outlets, where

flow reversal may occur,

“backflow” temperature is

required.



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Thermal Conditions for Fluids and Solids

u Can specify an energy source

using Source Terms option.



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Thermal Boundary Conditions at Walls

u Use any of following thermal

conditions at walls:

l Specified heat flux

l Specified temperature

l Convective heat transfer

l External radiation

l Combined external radiation

and external convective heat

transfer



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u Fluid properties such as heat capacity, conductivity, and viscosity can

be defined as:

l Constant

l Temperature-dependent

l Composition-dependent

l Computed by kinetic theory

l Computed by user-defined functions

u Density can be computed by ideal gas law.

u Alternately, density can be treated as:

l Constant (with optional Boussinesq modeling)

l Temperature-dependent

l Composition-dependent

l User Defined Function

Fluid Properties



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Conjugate Heat Transfer

u Ability to compute conduction of heat through solids, coupled with

convective heat transfer in fluid.

u Coupled Boundary Condition:

l available to wall zone that

separates two cell zones.Grid

Temperature contours

Velocity vectors

Example: Cooling flow over fuel rods



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Natural Convection - Introduction

u Natural convection occurs

when heat is added to fluid

and fluid density varies

with temperature.

u Flow is induced by force of gravity acting on density

variation.

u When gravity term is

included, pressure gradient

and body force term is written

as:

g x

p g

po )(

'

ρρρ −+∂∂

−⇒+∂∂

−

where gx p p oρ−='

• This format avoids potential roundoff error when gravitational body force term is included.



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Natural Convection - Boussinesq Model

u Makes simplifying assumption that density is uniform.

l Except for body force term in momentum equation, which is replaced by:

l Valid when density variations are small (i.e., small variations in T).

u

Provides faster convergence for many natural-convection flows than by using fluid density as function of temperature.

l Constant density assumptions reduces non-linearity.

l Use when density variations are small.

l Cannot be used with species calculations or reacting flows.

u Natural convection problems inside closed domains:

l For steady-state solver, Boussinesq model must be used.

n Constant density, ρo, allows mass in volume to be defined.l For unsteady solver, Boussinesq model or Ideal gas law can be used.

n Initial conditions define mass in volume.

( ) ( )ρ ρ ρ β− = − −0 0 0 g T T g



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User Inputs for Natural Convection

1. Set gravitational acceleration.

Define Õ Operating Conditions...

2. Define density model.

l If using Boussinesq model:

n Select boussinesq as the Density method

and assign constant value, ρo.Define Õ Materials...

n Set Thermal Expansion Coefficient, β.

n Set Operating Temperature, To.

l If using temperature dependent model,

(e.g., ideal gas or polynomial):n Specify Operating Density or,

n Allow Fluent to calculate ρo from a cellaverage (default, every iteration).

3. Set boundary conditions.



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Radiation

u Radiation intensity transport equations (RTE) are solved.

l Local absorption by fluid and at boundaries links energy equation with RTE.

u Radiation intensity is directionally and spatially dependent.

l Intensity along any direction can be reduced by:

n Local absorption

n Out-scattering (scattering away from the direction)

l Intensity along any direction can be augmented by:

n Local emission

n In-scattering (scattering into the direction)

u Four radiation models are provided in FLUENT:

l Discrete Ordinates Model (DOM)l Discrete Transfer Radiation Model (DTRM)

l P-1 Radiation Model

l Rosseland Model (limited applicability)



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Discrete Ordinates Model

u The radiative transfer equation is solved for a discrete number of finite

solid angles:

u Advantages:

l Conservative method leads to heat balance for coarse discretization.

l Accuracy can be increased by using a finer discretization.

l Accounts for scattering, semi-transparent media, specular surfaces.

l Banded-gray option for wavelength-dependent transmission.

u Limitations:

l Solving a problem with a large number of ordinates is CPU-intensive.

( ) ')'()',(4

),(

4

0

42 Ω⋅Φ+=++

∂

∂

∫ d s s sr I

T an sr I a

x

I s

s

i

i s

π

π

σ

π

σσ

absorption emission scattering



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Discrete Transfer Radiation Model (DTRM)

u Main assumption: radiation leaving surface element in a specific range of

solid angles can be approximated by a single ray.

u Uses ray-tracing technique to integrate radiant intensity along each ray:

u Advantages:

l Relatively simple model.

l Can increase accuracy by increasing number of rays.

l Applies to wide range of optical thicknesses.

u Limitations:

l Assumes all surfaces are diffuse.

l Effect of scattering not included.

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

π

σαα

4T I ds

dI

+−=



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P-1 Model

u Main assumption: radiation intensity can be decomposed into series of

spherical harmonics.

l Only first term in this (rapidly converging) series used in P-1 model.

l Effects of particles, droplets, and soot can be included.

u

Advantages:l Radiative transfer equation easy to solve with little CPU demand.

l Includes effect of scattering.

l Works reasonably well for combustion applications where optical

thickness is large.

l Easily applied to complicated geometries with curvilinear coordinates.

u Limitations:

l Assumes all surfaces are diffuse.

l May result in loss of accuracy, depending on complexity of geometry, if

optical thickness is small.

l Tends to overpredict radiative fluxes from localized heat sources or sinks.



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Choosing a Radiation Model

u For certain problems, one radiation model may be more

appropriate in general.

Define Õ Models Õ Radiation...

l Computational effort: P-1 gives reasonable accuracy with

less effort.

l Accuracy: DTRM and DOM more accurate.

l Optical thickness: DTRM/DOM for optically thin media

(optical thickness


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Periodic Heat Transfer (1)

u Also known as streamwise-periodic or fully-developed flow.

u Used when flow and heat transfer patterns are repeated, e.g.,

l Compact heat exchangers

l Flow across tube banks

u Geometry and boundary conditions repeat in streamwise direction.

Outflow at one periodic boundary

is inflow at the other

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Periodic Heat Transfer (2)

u Temperature (and pressure) vary in streamwise direction.

u Scaled temperature (and periodic pressure) is same at periodic

boundaries.

u For fixed wall temperature problems, scaled temperature defined as:

T b = suitably defined bulk temperature

u Can also model flows with specified wall heat flux.

θ = −

−T T

T T wall

b wall


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Periodic Heat Transfer (3)u Periodic heat transfer is subject to the following constraints:

l Either constant temperature or fixed flux bounds.

l Conducting regions cannot straddle periodic plane.

l Properties cannot be functions of temperature.

l Radiative heat transfer cannot be modeled.

l Viscous heating only available with heat flux wall boundaries.

Contours of Scaled Temperature


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Summary

u Heat transfer modeling is available in all Fluent solvers.

u After activating heat transfer, you must provide:

l Thermal conditions at walls and flow boundaries

l Fluid properties for energy equation

u Available heat transfer modeling options include:

l Species diffusion heat source

l Combustion heat source

l Conjugate heat transfer

l Natural convection

l Radiation

l Periodic heat transfer

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