Modeling Turb. Flow with Fluent

© Fluent Inc. 2/20/01D1

Fluent Software TrainingTRN-99-003

Modeling Turbulent Flows



u Unsteady, aperiodic motion in which all three velocity componentsfluctuate Õ mixing matter, momentum, and energy.

u Decompose velocity into mean and fluctuating parts:Ui(t) ≡ Ui + ui(t)

u Similar fluctuations for pressure, temperature, and speciesconcentration values.

What is Turbulence?

Time

U i (t)

Ui

ui(t)



Why Model Turbulence?

u Direct numerical simulation of governing equations is only possible forsimple 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 modeledusing functions containing empirical constants and information aboutthe mean flow.

u Large Eddy Simulation numerically resolves large eddies and modelssmall eddies.

(steady, incompressible floww/o body forces)

jiij uuR ρ−=

j

ij

jj

i

ik

ik x

Rxx

Uxp

xU

U∂∂

+∂∂

∂+

∂∂

−=∂∂ 2

µρ



Is the Flow Turbulent?

External Flows

Internal Flows

Natural Convection

5105×≥xRe along a surface

around an obstacle

where

µρUL

ReL ≡where

Other factors such as free-streamturbulence, surface conditions, anddisturbances may cause earliertransition to turbulent flow.

L = x, D, Dh, etc.

,3002 ≥hD Re

108 1010 −≥Raµα

ρβ 3TLgRa

∆≡

20,000≥DRe



How Complex is the Flow?

u Extra strain ratesl Streamline curvaturel Lateral divergencel Acceleration or decelerationl Swirll Recirculation (or separation)l Secondary flow

u 3D perturbationsu Transpiration (blowing/suction)u Free-stream turbulenceu Interacting shear layers



Choices to be Made

Turbulence Model&

Near-Wall Treatment

Flow Physics

AccuracyRequired

Computational Resources

Turnaround TimeConstraints

Computational Grid



Zero-Equation Models

One-Equation Models Spalart-AllmarasTwo-Equation Models Standard k-ε RNG k-ε Realizable k-ε Reynolds-Stress Model

Large-Eddy Simulation

Direct Numerical Simulation

Turbulence Modeling Approaches

IncludeMorePhysics

IncreaseComputationalCostPer Iteration

Availablein FLUENT 5

RANS-basedmodels



u RANS equations require closure for Reynolds stresses.

u Turbulent viscosity is indirectly solved for from single transportequation of modified viscosity for One-Equation model.

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

2kCt ≡Turbulent Viscosity:

Boussinesq Hypothesis:(isotropic viscosity)

∂∂

+∂∂+−=−=

i

j

j

itijjiij x

UxU

kuuR µδρρ32

2/iiuuk ≡

∂

∂+

∂∂

∂∂

≡i

j

j

i

j

i

x

u

xu

xu

νε



u Turbulent viscosity is determined from:

u is determined from the modified viscosity transport equation:

u The additional variables are functions of the modified turbulentviscosity and velocity gradients.

One Equation Model: Spalart-Allmaras

( ) 21

2

2~

1

~~~~1~~~

dfc

xc

xxSc

DtD

wwj

bjj

b

νρ

νρ

ννρµ

σνρ

νρ

ν

−

∂∂

+

∂∂

+∂∂

+=

( )( )

+= 3

13

3

/~/~

~ννν

νννρµ

ct

ν~

Generation Diffusion Destruction



One-Equation Model: Spalart-Allmaras

u Designed specifically for aerospace applications involving wall-bounded flows.l Boundary layers with adverse pressure gradientsl turbomachinery

u Can use coarse or fine mesh at walll 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.



Two Equation Model: Standard k-ε Model

Turbulent Kinetic Energy

Dissipation Rate

εεεσσ 21, ,, CCk are empirical constants

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

Convection Generation DiffusionDestructionρεσµµρ −

∂∂

∂∂

+∂

∂

∂∂

+∂

∂=

∂∂

444 3444 21444 3444 2143421 ikt

ii

j

j

i

i

jt

ii x

kxx

U

xU

x

U

xk

U )(

DestructionConvection Generation Diffusion

43421444 3444 2144444 344444 2143421

−

∂∂

∂∂

+∂

∂

∂∂

+∂

∂

=

∂∂

kC

xxx

U

xU

x

U

kC

xU

it

ii

j

j

i

i

jt

ii

2

21 )(ε

ρε

σµµεε

ρ εεε



Two Equation Model: Standard k-ε Model

u “Baseline model” (Two-equation)l Most widely used model in industryl Strength and weaknesses well documented

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

equation from its time-averaged valuel ε equation formed from physical reasoning

u Valid only for fully turbulent flowsu Reasonable accuracy for wide range of turbulent flows

l industrial flowsl heat transfer



Two Equation Model: Realizable k-ε

u 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

2kCt ≡

ε

µ kUAA

C

so

*

1

+=

0u2i ≥

2j

2i

2ji u u)uu( ≤

bj

t

j

Gck

ck

cScxxDt

Dεε

ε

ενε

ερερ

εσµ

µε

ρ 31

2

21 ++

−+

∂∂

+

∂∂

=

GenerationDiffusion Destruction Buoyancy



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

l planar and round jetsl boundary layers under strong adverse pressure gradients, separationl rotation, recirculationl strong streamline curvature

Two Equation Model: Realizable k-ε



Two Equation Model: RNG k-ε

Turbulent Kinetic Energy

Dissipation Rate

Convection DiffusionDissipation

ρεµαµρ −

∂∂

∂∂

+=∂∂

44 344 2143421 ik

it

ii x

kx

Sxk

U eff2

Generation

∂∂

+∂∂

≡≡j

i

i

jijijij x

UxU

SSSS21

,2where

are derived using RNG theoryεεεαα 21, ,, CCk


Additional termrelated to mean strain& turbulence quantities

Convection Generation Diffusion Destruction

RkC

xxS

kC

xU

iit

ii −

−

∂∂

∂∂

+

=

∂∂

4342144 344 21443442143421

2

2eff2

1

ερ

εµαµ

εερ εεε



Two Equation Model: RNG k-ε

u k-ε equations are derived from the application of a rigorous statisticaltechnique (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 turbulencel analytical formula for turbulent Prandtl numberl differential formula for effective viscosity

u Improved predictions for:l high streamline curvature and strain ratel transitional flowsl wall heat and mass transfer



Reynolds Stress Model

k

ijkijijij

k

jik x

JP

xuu

U∂∂

+−Φ+=∂

∂ερ

Generationk

ikj

k

jkiij x

UuuxU

uuP∂∂+

∂∂

≡

∂∂

+∂∂′−≡Φ

i

j

j

iij x

u

xu

p

k

j

k

iij x

uxu

∂∂

∂∂

≡ µε 2

Pressure-StrainRedistribution

Dissipation

TurbulentDiffusion

(modeled)

(related to ε)

(modeled)

(computed)


Reynolds StressTransport Eqns.

Pressure/velocity fluctuations

Turbulenttransport

)( jikijkkjiijk uupuuuJ δδ +′+=



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 propertyl Closure also requires one equation for turbulent dissipationl 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 ratesn Cyclone flows, swirling combustor flowsn Rotating flow passages, secondary flows



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., stronglydependent on flow configuration, boundary conditions, and flow parameters.

u Small eddies:l Tend to be more isotropic and less flow-dependentl More likely to be easier to model than large eddies.

u LES directly computes (resolves) large eddies and models only smalleddies (Subgrid-Scale Modeling).

u Large computational effortl Number of grid points, NLES ∝l Unsteady calculation

2Reτu



Comparison of RANS Turbulence Models

Model Strengths WeaknessesSpalart-Allmaras

Economical (1-eq.); good track recordfor mildly complex B.L. type of flows

Not very widely tested yet; lack ofsubmodels (e.g. combustion,buoyancy)

STD k-εRobust, economical, reasonablyaccurate; long accumulatedperformance data

Mediocre results for complex flowsinvolving severe pressure gradients,strong streamline curvature, swirland rotation

RNG k-εGood for moderately complexbehavior like jet impingement,separating flows, swirling flows, andsecondary flows

Subjected to limitations due toisotropic eddy viscosityassumption

Realizablek-ε

Offers largely the same benefits asRNG; resolves round-jet anomaly

Subjected to limitations due toisotropic eddy viscosityassumption

ReynoldsStressModel

Physically most complete model(history, transport, and anisotropy ofturbulent stresses are all accountedfor)

Requires more cpu effort (2-3x);tightly coupled momentum andturbulence equations



Near-Wall Treatments

u Most k-ε and RSM turbulencemodels will not predict correctnear-wall behavior if integrateddown to the wall.

u Special near-wall treatment isrequired.l Standard wall functionsl Nonequilibrium wall functionsl Two-layer zonal model

Boundary layer structure



Standard Wall Functions

ρτµ

/

2/14/1

w

PP kCUU ≡∗

( )

>

+

<= ∗

∗

∗

)(ln1

Pr

)(Pr**

**

Tt

T

yyPEy

yyyT

κ

µ

ρ µ PP ykCy

2/14/1

≡∗

q

kCcTTT PpPw

′′−

≡&

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



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

better prediction of adverse pressure gradientflows and separation.

u Relaxed local equilibrium assumptions forTKE in wall-neighboring cells.

u Thermal law-of-wall unchanged

=

µρ

κρτµµ ykCEkCU

w

2/14/12/14/1

ln1/

~

+

−+

−= ∗∗ µρκρκ

y

k

yyyy

ky

dxdpUU vv

v

v2

2/12/1 ln21~

where



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

flows with complex near-wallphenomena.

u Zones distinguished by a wall-distance-based turbulentReynolds 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 ykRey ≡

200>yRe

200<yRe



Comparison of Near Wall TreatmentsStrengths Weaknesses

Standard wallFunctions

Robust, economical,reasonably accurate

Empirically based on simplehigh-Re flows; poor for low-Reeffects, massive transpiration,∇p, strong body forces, highly3D flows

Nonequilibriumwall functions

Accounts for ∇p effects,allows nonequilibrium:

-separation-reattachment-impingement

Poor for low-Re effects, massivetranspiration, severe ∇p, strongbody forces, highly 3D flows

Two-layer zonalmodel

Does not rely on law-of-the-wall, good for complexflows, especially applicableto low-Re flows

Requires finer mesh resolutionand therefore larger cpu andmemory resources



Computational Grid GuidelinesWall Function

ApproachTwo-Layer ZonalModel Approach

l First grid point in log-law region

l At least ten points in the BL.

l Better to use stretched quad/hexcells for economy.

l First grid point at y+ ≈ 1.

l At least ten grid points withinbuffer & sublayers.

l Better to use stretched quad/hexcells for economy.

50050 ≤≤ +y



Estimating Placement of First Grid Point

u Estimate the skin friction coefficient based on correlations eitherapproximate 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/ −≈ Lfc2.0Re039.02/ −≈ Dfc

2// few cUu =≡ ρττ

y1 = 50ν/uτ y1 = ν/ uτ



Setting Boundary Conditions

u Characterize turbulence at inlets & outlets (potential backflow)l k-ε models require k and εl 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 ≈ 0.4δ99

n For flows downstream of grids /perforated plates: l ≈ opening sizel 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 << µµ

t



GUI for Turbulence Models

Define Õ Models Õ Viscous...

Turbulence Model options

Near Wall Treatments

Inviscid, Laminar, or Turbulent

Additional Turbulence options



Example: Channel Flow with Conjugate HeatTransfer

adiabatic wall

cold airV = 50 fpmT = 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



Turbulence Modeling Approachu Check if turbulent Õ ReDh

= 5,980u 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 geometry Õ quadrilateral cellsl Expect large gradients in normal direction to horizontal walls Õ 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.



Velocitycontours

Temperaturecontours

BLs on upper & lower surfaces accelerate the core flow

Prediction of Momentum & ThermalBoundary Layers

Important that thermal BL was accurately resolved as well

P



Example: Flow Around a Cylinder

wall

wall

1 ft

2 ft

2 ft

airV = 4 fps

Compute drag coefficient of the cylinder

5 ft 14.5 ft



u Check if turbulent Õ ReD = 24,600

u Flow over an object, unsteady vortex shedding is expected,difficult to predict separation on downstream side, and closeproximity of side walls may influence flow around cylinderÕ use RNG k-ε with 2-layer zonal model.

u Develop strategy for the gridl 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



Grid for Flow Over a Cylinder



Prediction of Turbulent Vortex Shedding

Contours of effective viscosity µeff = µ + µt

CD = 0.53 Strouhal Number = 0.297

UD

Stτ

≡where



Summary: Turbulence Modeling Guidelines

u Successful turbulence modeling requires engineering judgement of:l Flow physicsl Computer resources availablel Project requirements

n Accuracyn 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 ifneeded.

u Use RSM for highly swirling flows.u Use wall functions unless low-Re flow and/or complex near-wall

physics are present.

© Fluent Inc. 2/23/01E1


Solver Settings



Outlineu Using the Solver

l Setting Solver Parametersl Convergence

n Definitionn Monitoringn Stabilityn Accelerating Convergence

l Accuracyn Grid Independencen Adaption

u Appendix: Backgroundl Finite Volume Methodl Explicit vs. Implicitl Segregated vs. Coupledl Transient Solutions



Modify solutionparameters 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 Overview

u Solution Parametersl Choosing the Solverl Discretization Schemes

u Initializationu Convergence

l Monitoring Convergencel Stability

n Setting Under-relaxationn Setting Courant number

l Accelerating Convergence

u Accuracyl Grid Independencel Adaption



Choosing a Solveru 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.



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.

VSAAVVt f

facesfff

facesfff

ttt

∆+∇Γ=+∆∆

− ∑∑ ⊥

∆+

φφφρρφρφ

,)()()(



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 Standardn default scheme; reduced accuracy for flows exhibiting large surface-normal pressure

gradients near boundaries.l Linear

n useful only when other options result in convergence difficulties or unphysicalbehavior.

l Second-Ordern use for compressible flows or when PRESTO! cannot be applied.

l Body Force Weightedn 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 porousmedia, or strongly curved domains.



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

accounted for when using the segregated solver.u Three methods available:

l SIMPLEn default scheme, robust

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

no physical models employed).

l PISOn useful for unsteady flow problems or for meshes containing cells with

higher than average skew.



Initializationu Iterative procedure requires that all solution variables be initialized

before calculating a solution.Solve Õ Initialize Õ 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 individualvariables in certain regions.Solve Õ Initialize Õ Patch...l Free jet flows

(patch high velocity for jet)l Combustion problems

(patch high temperaturefor ignition)



Convergence Preliminaries: Residuals

u Transport equation for φ can be presented in simple form:l Coefficients ap, 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, Rp, where:

l Rp 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 ofthe time rate of change of the conserved variable.l Only for coupled equations; additional scalar equations use segregated

definition.

pnb

nbnbpp baa =+ ∑ φφ

pnb

nbnbppp baaR −+= ∑ φφ

||∑=cells

pRR



Convergenceu At convergence:

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



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

specified tolerance.Solve Õ Monitors Õ Residual...

All equations converged.

10-3

10-6



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

l Lift, drag, or momentSolve Õ Monitors Õ Force...

l Variables or functions (e.g., surface integrals)at a boundary or any defined surface:Solve Õ Monitors Õ Surface...



Checking for Property Conservation

u In addition to monitoring residual and variable histories, you shouldalso check for overall heat and mass balances.l Net imbalance should be less than 0.1% of net flux through domain.

Report Õ Fluxes...



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 Criterionor disable Check Convergence.

l Then calculate until solutionconverges to the new tolerance.



Convergence Difficultiesu 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 convergencetrouble (segregated).

l Reduce Courant number (coupled).l Re-mesh or refine grid with high

aspect ratio or highly skewed cells.

Continuity equation convergencetrouble affects convergence ofall equations.



Modifying Under-relaxation Factors

u Under-relaxation factor, α, isincluded to stabilize the iterativeprocess for the segregated solver.

u Use default under-relaxation factorsto start a calculation.

Solve Õ Controls Õ Solution...

u Decreasing under-relaxation formomentum often aids convergence.l Default settings are aggressive but

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

from experience.

poldpp φαφφ ∆+= ,

u For coupled solvers, under-relaxation factors for equations outside coupledset are modified as in segregated solver.



Modifying the Courant Numberu Courant number defines a ‘time

step’ size for steady-state problems.l A transient term is included in the

coupled solver even for steady stateproblems.

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 when

having difficulty converging.

ux

t∆

=∆)CFL(

u For coupled-implicit solver:l Courant number is not limited by stability constraints.

n Default is set to 5.



Accelerating Convergence

u Convergence can be accelerated by:l Supplying good initial conditions

n Starting from a previous solution.

l Increasing under-relaxation factors or Courant numbern 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.



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



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 optionn Final solution is for original mesh.

l Multigrid operates automatically in the background.

u Accelerates convergence for problems with:l Large number of cellsl Large cell aspect ratios, e.g., ∆x/∆y > 20l Large differences in thermal conductivity

fine (original) mesh

coarse mesh

‘solutiontransfer’



Accuracy

u 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.



Mesh Quality and Solution Accuracyu 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 degreess Optimal tri/tet cells are equilateral.



Determining Grid Independenceu 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.



Unsteady Flow Problemsu 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 resolvetime dependent features and to ensureconvergence 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.



Summary

u 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 andensuring 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.



Appendix

u Backgroundl Finite Volume Methodl Explicit vs. Implicitl Segregated vs. Coupledl Transient Solutions



Background: Finite Volume Method - 1

u 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. isapplied to each cell and discretized. For cell p,

∫∫∫∫∀

+⋅∇Γ=⋅+∂∂

dVSdddVt AAV

φφρφρφ AAV

unsteady convection diffusion generation

Eqn.continuity 1

x-mom. uy-mom. venergy h

φ

Fluid region of pipe flowdiscretized into finite set ofcontrol volumes (mesh).

controlvolume

u All equations are solved to render flow field.



Background: Finite Volume Method - 2

u 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 anequation set.

pnb


VSAAVVt f

facesfff

facesfff

tp

ttp ∆+∇Γ=+∆

∆−

∑∑ ⊥

∆+

φφφρρφρφ

,)()()(



u Equation sets are solved iteratively.l Coefficients ap 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 φp can either be solved explicitly or implicitly.

Background: Linearization

pnb




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 φp solved explicitly using Runge-Kutta scheme.

l Implicit linearization - φp 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



Background: Coupled vs. Segregated

u Segregated Solverl 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 thesolution of other variables.n coefficients ap and anb are scalars.

u Coupled Solverl If more than one variable is unknown in each equation, and each

variable is defined by its own transport equation, then the equation set iscoupled together.n coefficients ap and anb are Neqx Neq matricesn φ is a vector of the dependent variables, p, u, v, w, T, YT

pnb




Background: Segregated Solver

u In the segregated solver, each equation issolved separately.

u The continuity equation takes the formof a pressure correction equation as partof SIMPLE algorithm.

u Under-relaxation factors are included inthe discretized equations.l Included to improve stability of iterative

process.l Under-relaxation factor, α, in effect,

limits change in variable from oneiteration 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 otherscalar equations.

Converged?

StopNo Yes

poldpp φαφφ ∆+= ,



Background: Coupled Solveru Continuity, momentum, energy, and

species are solved simultaneously in thecoupled solver.

u Equations are modified to resolvecompressible 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’ isdefined by Courant number.l Stability issues limit maximum time step

size for explicit solver but not forimplicit solver.

Solve continuity, momentum, energy,and species equations simultaneously.

Stop

No Yes

Solve turbulence and other scalar equations.

Update properties.

Converged?

ux

t∆

=∆)CFL( CFL = Courant-Friedrichs-Lewy-number

where u = appropriate velocity scale∆x = grid spacing



Background: Segregated/Transientu Transient solutions are possible with both segregated and coupled solvers.

l 1st- and 2nd-order time implicit discretizations (Euler) available for coupledand 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 resolvetime 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 ensureconvergence within 20 iterations.

l Note: Use TUI command ‘it #’ to iteratefurther without advancing time step.



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

n Physical-time transients Physical-time derivative term is discretized implicitly (1st or 2nd order).s Time step size, ∆t, defined as with segregated solver.

n Pseudo-time transients At each physical-time level, a pseudo-time transient is driven to zero through a

series 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 solvern Physical-time step size is defined by Courant number.

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

© Fluent Inc. 2/20/01F1


Heat Transfer and Thermal BoundaryConditions

Headlamp modeled withDiscrete OrdinatesRadiation Model



Outlineu Introductionu Thermal Boundary Conditionsu Fluid Propertiesu Conjugate Heat Transferu Natural Convectionu Radiationu Periodic Heat Transfer



Introductionu 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 inhigh-velocity, compressible flows.

n Often negligibles not included by default for segregated solvers always included for coupled solver.

l In solid regions, simple conduction equation solved.n Convective term can also be included for moving solids.



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

2. Enable appropriate options:l Viscous Heating in Viscous Model panell 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.



Solution Process for Heat Transferu 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...



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

′′ = ≈q kTn

kTnwall

∂∂

∆∆



Thermal Boundary Conditions at Flow Inletsand Exits

u At flow inlets, must supplyfluid temperature.

u At flow exits, fluidtemperature extrapolatedfrom upstream value.

u At pressure outlets, whereflow reversal may occur,“backflow” temperature isrequired.



Thermal Conditions for Fluids and Solids

u Can specify an energy sourceusing Source Terms option.



Thermal Boundary Conditions at Walls

u Use any of following thermalconditions at walls:l Specified heat fluxl Specified temperaturel Convective heat transferl External radiationl Combined external radiation

and external convective heattransfer



u Fluid properties such as heat capacity, conductivity, and viscosity canbe defined as:l Constantl Temperature-dependentl Composition-dependentl Computed by kinetic theoryl 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-dependentl Composition-dependentl User Defined Function

Fluid Properties



Conjugate Heat Transfer

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



Natural Convection - Introduction

u Natural convection occurswhen heat is added to fluidand fluid density varieswith temperature.

u Flow is induced by force ofgravity acting on densityvariation.

u When gravity term isincluded, pressure gradientand body force term is writtenas:

gxp

gxp

o )('

ρρρ −+∂∂

−⇒+∂∂

−

where gxpp oρ−='

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



Natural Convection - Boussinesq Modelu 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 thanby 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 0g T T g



User Inputs for Natural Convection1. Set gravitational acceleration.

Define Õ Operating Conditions...

2. Define density model.l If using Boussinesq model:

n Select boussinesq as the Density methodand 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 cell

average (default, every iteration).

3. Set boundary conditions.



Radiationu 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 absorptionn Out-scattering (scattering away from the direction)

l Intensity along any direction can be augmented by:n Local emissionn 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 Modell Rosseland Model (limited applicability)



Discrete Ordinates Model

u The radiative transfer equation is solved for a discrete number of finitesolid 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+=++

∂∂

∫ dsssrIT

ansrIaxI s

si

isπ

πσ

πσ

σ

absorption emission scattering



Discrete Transfer Radiation Model (DTRM)

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

πσ

αα4T

IdsdI

+−=



P-1 Modelu 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.



Choosing a Radiation Modelu 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 withless effort.

l Accuracy: DTRM and DOM more accurate.l Optical thickness: DTRM/DOM for optically thin media

(optical thickness << 1); P-1 better for optically thick media.l Scattering: P-1 and DOM account for scattering.l Particulate effects: P-1 and DOM account for radiation exchange between gas

and particulates.l Localized heat sources: DTRM/DOM with sufficiently large number of rays/

ordinates is more appropriate.



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 exchangersl Flow across tube banks

u Geometry and boundary conditions repeat in streamwise direction.

Outflow at one periodic boundaryis inflow at the other

inflow outflow



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:

Tb = suitably defined bulk temperature

u Can also model flows with specified wall heat flux.

θ =−−

T TT T

wall

b wall



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



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 boundariesl Fluid properties for energy equation

u Available heat transfer modeling options include:l Species diffusion heat sourcel Combustion heat sourcel Conjugate heat transferl Natural convectionl Radiationl Periodic heat transfer

Documents

Modeling Turb. Flow with Fluent