? Fluent Inc,2/20/01D1
Fluent Software Training
TRN-99-003
Modeling Turbulent Flows
Fluent Inc,2/20/01D2
Fluent Software Training
TRN-99-003
u Unsteady,aperiodic motion in which all three velocity components
fluctuate? 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 species
concentration values.
What is Turbulence?
Time
U i (t)
Ui
ui(t)
Fluent Inc,2/20/01D3
Fluent Software Training
TRN-99-003
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 uuR r?=
j
ij
jj
i
ik
i
k x
R
xx
U
x
p
x
UU
+
+
=
2mr
Fluent Inc,2/20/01D4
Fluent Software Training
TRN-99-003
Is the Flow Turbulent?
External Flows
Internal Flows
Natural Convection
5105×≥
xRe along a surface
around an obstacle
where
m
rULRe
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 ≥
hD
Re
108 1010?≥Ra
ma
rb 3TLgRa?≡
20,000≥DRe
Fluent Inc,2/20/01D5
Fluent Software Training
TRN-99-003
How Complex is the Flow?
u Extra strain rates
l Streamline curvature
l Lateral divergence
l Acceleration or deceleration
l Swirl
l Recirculation (or separation)
l Secondary flow
u 3D perturbations
u Transpiration (blowing/suction)
u Free-stream turbulence
u Interacting shear layers
Fluent Inc,2/20/01D6
Fluent Software Training
TRN-99-003
Choices to be Made
Turbulence Model
&
Near-Wall Treatment
Flow
Physics
Accuracy
Required
Computational
Resources
Turnaround
Time
Constraints
Computational
Grid
Fluent Inc,2/20/01D7
Fluent Software Training
TRN-99-003
Zero-Equation Models
One-Equation Models
Spalart-Allmaras
Two-Equation Models
Standard k-e
RNG k-e
Realizable k-e
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
Fluent Inc,2/20/01D8
Fluent Software Training
TRN-99-003
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 are
solved 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:
erm m
2k
Ct ≡Turbulent Viscosity:
Boussinesq Hypothesis:
(isotropic viscosity)?
+
+?=?=
i
j
j
i
tijjiij x
U
x
UkuuR mdrr
3
2
2/iiuuk ≡
+
≡
i
j
j
i
j
i
x
u
x
u
x
une
Fluent Inc,2/20/01D9
Fluent Software Training
TRN-99-003
u Turbulent viscosity is determined from:
u is determined from the modified viscosity transport equation:
u The additional variables are functions of the modified turbulent
viscosity and velocity gradients.
One Equation Model,Spalart-Allmaras
( ) 21
2
2
~
1
~~~~1~~~
dfcxcxxScDt
D
ww
j
b
jj
b
nrnrnnrm
snr
nr
n
+
+
+=
( )
( )
+= 313
3
/~
/~~
nnn
nnnrm
ct
n~
Generation Diffusion Destruction
Fluent Inc,2/20/01D10
Fluent Software Training
TRN-99-003
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.
Fluent Inc,2/20/01D11
Fluent Software Training
TRN-99-003
Two Equation Model,Standard k-e Model
Turbulent Kinetic Energy
Dissipation Rate
eeess 21,,,CCk are empirical constants
(equations written for steady,incompressible flow w/o body forces)
Convection Generation Diffusion
Destruction
{resmmr
+
+
=
444 3444 21444 3444 2143421 i
kt
ii
j
j
i
i
j
t
i
i x
k
xx
U
x
U
x
U
x
kU )(
DestructionConvection Generation Diffusion
43421444 3444 2144444 344444 2143421
+
+
=
kCxxx
U
x
U
x
U
kCxU itii
j
j
i
i
j
t
i
i
2
21 )(
eresmmeer
eee
Fluent Inc,2/20/01D12
Fluent Software Training
TRN-99-003
Two Equation Model,Standard k-e Model
u,Baseline model” (Two-equation)
l Most widely used model in industry
l Strength and weaknesses well documented
u Semi-empirical
l k equation derived by subtracting the instantaneous mechanical energy
equation from its time-averaged value
l ε equation formed from physical reasoning
u Valid only for fully turbulent flows
u Reasonable accuracy for wide range of turbulent flows
l industrial flows
l heat transfer
Fluent Inc,2/20/01D13
Fluent Software Training
TRN-99-003
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:
erm m
2k
Ct ≡
e
m kU
AA
C
so
*
1
+
=
0u2i ≥
2
j
2
i
2
ji u u)uu( ≤
b
j
t
j
GckckcScxxDtD ee
e
e
ne
erere
s
mmer
31
2
21 ++?+
+
=
GenerationDiffusion Destruction Buoyancy
Fluent Inc,2/20/01D14
Fluent Software Training
TRN-99-003
u Shares the same turbulent kinetic energy equation as Standard k-ε
u Superior performance for flows involving:
l planar and round jets
l boundary layers under strong adverse pressure gradients,separation
l rotation,recirculation
l strong streamline curvature
Two Equation Model,Realizable k-ε
Fluent Inc,2/20/01D15
Fluent Software Training
TRN-99-003
Two Equation Model,RNG k-e
Turbulent Kinetic Energy
Dissipation Rate
Convection Diffusion
Dissipation
{ {remamr
+=
44 344 2143421 i
k
i
t
i
i x
k
xSx
kU
eff
2
Generation
+
≡≡
j
i
i
j
ijijij x
U
x
USSSS
2
1,2where
are derived using RNG theoryeeeaa 21,,,CCk
(equations written for steady,incompressible flow w/o body forces)
Additional term
related to mean strain
& turbulence quantities
Convection Generation Diffusion Destruction
{RkCxxSkCxU
ii
t
i
i
+?
=
4342144 344 2144342143421
2
2eff
2
1
eremameer
eee
Fluent Inc,2/20/01D16
Fluent Software Training
TRN-99-003
Two Equation Model,RNG k-e
u 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 flows
l 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
Fluent Inc,2/20/01D17
Fluent Software Training
TRN-99-003
Reynolds Stress Model
k
ijk
ijijij
k
ji
k x
JP
x
uuU
+?Φ+=
er
Generation
k
i
kj
k
j
kiij x
Uuu
x
UuuP
+
≡
+
′?≡Φ
i
j
j
i
ij x
u
x
up
k
j
k
i
ij x
u
x
u
≡ me 2
Pressure-Strain
Redistribution
Dissipation
Turbulent
Diffusion
(modeled)
(related to e)
(modeled)
(computed)
(equations written for steady,incompressible flow w/o body forces)
Reynolds Stress
Transport Eqns.
Pressure/velocity
fluctuations
Turbulent
transport
)( jikijkkjiijk uupuuuJ dd +′+=
Fluent Inc,2/20/01D18
Fluent Software Training
TRN-99-003
Reynolds Stress Model
u 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 flows
n Rotating flow passages,secondary flows
Fluent Inc,2/20/01D19
Fluent Software Training
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Large Eddy Simulation
u 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
tu
Fluent Inc,2/20/01D20
Fluent Software Training
TRN-99-003
Comparison of RANS Turbulence Models
Model 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-e
Robust,economical,reasonably
accurate; long accumulated
performance data
Mediocre results for complex flows
involving severe pressure gradients,
strong streamline curvature,swirl
and rotation
RNG k-e
Good for moderately complex
behavior like jet impingement,
separating flows,swirling flows,and
secondary flows
Subjected to limitations due to
isotropic eddy viscosity
assumption
Realizable
k-e
Offers largely the same benefits as
RNG; resolves round-jet anomaly
Subjected to limitations due to
isotropic eddy viscosity
assumption
Reynolds
Stress
Model
Physically most complete model
(history,transport,and anisotropy of
turbulent stresses are all accounted
for)
Requires more cpu effort (2-3x);
tightly coupled momentum and
turbulence equations
Fluent Inc,2/20/01D21
Fluent Software Training
TRN-99-003
Near-Wall Treatments
u Most k-e and RSM turbulence
models 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
Fluent Inc,2/20/01D22
Fluent Software Training
TRN-99-003
Standard Wall Functions
rt
m
/
2/14/1
w
PP kCUU ≡?
( )?
>?
+
<
=?
)(ln1Pr
)(Pr
**
**
Tt
T
yyPEy
yyy
T
k
m
r m PP ykCy 2/14/1≡?
q
kCcTTT PpPw
′′
≡
&
2/14/1)(
* mr
Mean Velocity
Temperature
where
where and P is a function of the fluid
and turbulent Prandtl numbers.
thermal sublayer thickness
( ) = EyU ln1k
Fluent Inc,2/20/01D23
Fluent Software Training
TRN-99-003
Nonequilibrium Wall Functions
u 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
=
m
r
krt
mm ykCEkCU
w
2/14/12/14/1
ln1/
~
+?+?
=
mrkrk
y
k
yy
y
y
k
y
dx
dpUU vv
v
v
2
2/12/1 ln2
1~where
Fluent Inc,2/20/01D24
Fluent Software Training
TRN-99-003
Two-Layer Zonal Model
u 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-e models are used in the turbulent core region.
u Only k equation is solved in the viscosity-affected region.
u e is computed from the correlation for length scale.
u Zoning is dynamic and solution adaptive.
m
r ykRe
y ≡
200>yRe
200<yRe
Fluent Inc,2/20/01D25
Fluent Software Training
TRN-99-003
Comparison of Near Wall Treatments
Strengths Weaknesses
Standard wall
Functions
Robust,economical,
reasonably accurate
Empirically based on simple
high-Re flows; poor for low-Re
effects,massive transpiration,
p,strong body forces,highly
3D 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 complex
flows,especially applicable
to low-Re flows
Requires finer mesh resolution
and therefore larger cpu and
memory resources
Fluent Inc,2/20/01D26
Fluent Software Training
TRN-99-003
Computational Grid Guidelines
Wall Function
Approach
Two-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
Fluent Inc,2/20/01D27
Fluent Software Training
TRN-99-003
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/?≈
Lfc
2.0Re039.02/?≈
Dfc
2// few cUu =≡ rtt
y1 = 50ν/uτ y1 = ν/ uτ
Fluent Inc,2/20/01D28
Fluent Software Training
TRN-99-003
Setting Boundary Conditions
u Characterize turbulence at inlets & outlets (potential backflow)
l k-e models require k and e
l Reynolds stress model requires Rij and e
u Several options allow input using more familiar parameters
l 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? 0.4d99
n 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 e explicitly allowed (non-uniform profiles possible).
10/1 << mm
t
Fluent Inc,2/20/01D29
Fluent Software Training
TRN-99-003
GUI for Turbulence Models
Define? Models? Viscous...
Turbulence Model options
Near Wall Treatments
Inviscid,Laminar,or Turbulent
Additional Turbulence options
Fluent Inc,2/20/01D30
Fluent Software Training
TRN-99-003
Example,Channel Flow with Conjugate Heat
Transfer
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
Fluent Inc,2/20/01D31
Fluent Software Training
TRN-99-003
Turbulence Modeling Approach
u Check if turbulent? ReDh= 5,980
u Developing turbulent flow at relatively low Reynolds number and
BLs on walls will give pressure gradient? use RNG k-e with
nonequilibrium wall functions.
u Develop strategy for the grid
l Simple geometry? quadrilateral cells
l Expect large gradients in normal direction to horizontal walls? 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.
Fluent Inc,2/20/01D32
Fluent Software Training
TRN-99-003
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
Fluent Inc,2/20/01D33
Fluent Software Training
TRN-99-003
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
Fluent Inc,2/20/01D34
Fluent Software Training
TRN-99-003
u Check if turbulent? ReD = 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
use RNG k-e with 2-layer zonal model.
u Develop strategy for the grid
l Simple geometry & BLs? quadrilateral cells.
l Large gradients near surface of cylinder & 2-layer model
fine mesh near surface & first cell at y+ = 1.
Turbulence Modeling Approach
Fluent Inc,2/20/01D35
Fluent Software Training
TRN-99-003
Grid for Flow Over a Cylinder
Fluent Inc,2/20/01D36
Fluent Software Training
TRN-99-003
Prediction of Turbulent Vortex Shedding
Contours of effective viscosity meff = m + mt
CD = 0.53 Strouhal Number = 0.297
U
DSt
t≡where
Fluent Inc,2/20/01D37
Fluent Software Training
TRN-99-003
Summary,Turbulence Modeling Guidelines
u Successful turbulence modeling requires engineering judgement of:
l Flow physics
l Computer resources available
l Project requirements
n Accuracy
n Turnaround time
l Turbulence models & near-wall treatments that are available
u Begin with standard k-e and change to RNG or Realizable k-e 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.
Fluent Inc,2/23/01E1
Fluent Software Training
TRN-99-003
Solver Settings
Fluent Inc,2/23/01E2
Fluent Software Training
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Outline
u 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,Background
l Finite Volume Method
l Explicit vs,Implicit
l Segregated vs,Coupled
l Transient Solutions
Fluent Inc,2/23/01E3
Fluent Software Training
TRN-99-003
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 Overview
u 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
Fluent Inc,2/23/01E4
Fluent Software Training
TRN-99-003
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.
Fluent Inc,2/23/01E5
Fluent Software Training
TRN-99-003
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 Scheme
n 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
faces
fff
faces
fff
ttt
+?Γ=+ ∑∑ ⊥
+
fffr
rfrf
,)(
)()(
Fluent Inc,2/23/01E6
Fluent Software Training
TRN-99-003
Interpolation Methods for Pressure
u 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 pressure
gradients 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.
Fluent Inc,2/23/01E7
Fluent Software Training
TRN-99-003
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,robust
l 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.
Fluent Inc,2/23/01E8
Fluent Software Training
TRN-99-003
Initialization
u Iterative procedure requires that all solution variables be initialized
before calculating a solution.
Solve? Initialize? 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.
Solve? Initialize? Patch...
l Free jet flows
(patch high velocity for jet)
l Combustion problems
(patch high temperature
for ignition)
Fluent Inc,2/23/01E9
Fluent Software Training
TRN-99-003
Convergence Preliminaries,Residuals
u Transport equation for f 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 of
the time rate of change of the conserved variable.
l Only for coupled equations; additional scalar equations use segregated
definition.
p
nb
nbnbpp baa =+ ∑ ff
p
nb
nbnbppp baaR?+= ∑ ff
||∑=
cells
pRR
Fluent Inc,2/23/01E10
Fluent Software Training
TRN-99-003
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.
Fluent Inc,2/23/01E11
Fluent Software Training
TRN-99-003
Convergence Monitors,Residuals
u Residual plots show when the residual values have reached the
specified tolerance.
Solve? Monitors? Residual...
All equations converged.
10-3
10-6
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Convergence Monitors,Forces/Surfaces
u In addition to residuals,you can also monitor:
l Lift,drag,or moment
Solve? Monitors? Force...
l Variables or functions (e.g.,surface integrals)
at a boundary or any defined surface:
Solve? Monitors? Surface...
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Checking for Property Conservation
u 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? Fluxes...
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Decreasing the Convergence Tolerance
u 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 Factors
u Under-relaxation factor,a,is
included to stabilize the iterative
process for the segregated solver.
u Use default under-relaxation factors
to start a calculation.
Solve? Controls? 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.
poldpp faff?+=,
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 when
having 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 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 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 Solution
u 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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Multigrid
u 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 > 20
l Large differences in thermal conductivity
fine (original) mesh
coarse mesh
‘solution
transfer’
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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.
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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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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 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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Appendix
u Background
l Finite Volume Method
l Explicit vs,Implicit
l Segregated vs,Coupled
l Transient Solutions
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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,is
applied to each cell and discretized,For cell p,
∫∫∫∫
+Γ=?+ dVSdddVt
AAV
ffrfrf AAV
unsteady convection diffusion generation
Eqn.
continuity 1
x-mom,u
y-mom,v
energy h
f
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 - 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 an
equation set.
p
nb
nbnbpp baa =+ ∑ ff
VSAAVVt f
faces
fff
faces
fff
t
p
tt
p?+?Γ=+?
∑∑
⊥
+
fffr
rfrf
,)(
)()(
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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 f.
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 fp can either be solved explicitly or implicitly.
Background,Linearization
p
nb
nbnbpp baa =+ ∑ ff
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u Assumptions are made about the knowledge of fnb:
l Explicit linearization - unknown value in each cell computed from relations
that include only existing values (fnb assumed known from previous
iteration).
n fp solved explicitly using Runge-Kutta scheme.
l Implicit linearization - fp and fnb are assumed unknown and are solved
using 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,Segregated
u 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 ap 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 ap and anb are Neqx Neq matrices
n f is a vector of the dependent variables,{p,u,v,w,T,Y}T
p
nb
nbnbpp baa =+ ∑ ff
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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 part
of 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
poldpp faff?+=,
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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
Dx = 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/Transient
l 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 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 is
discretized 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 with
Discrete Ordinates
Radiation Model
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Outline
u Introduction
u Thermal Boundary Conditions
u Fluid Properties
u Conjugate Heat Transfer
u Natural Convection
u Radiation
u Periodic Heat Transfer
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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.
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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? Models? 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? Boundary Conditions...
4,Define material properties for heat transfer.
Define? Materials...
l Heat capacity and thermal conductivity must be defined.
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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? Controls? Solution...
l Disabling species diffusion term.
Define? Models? Species...
l Compute isothermal flow first,then add calculation of energy equation.
Solve? Controls? 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 Tn k Tn
wall
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Thermal Boundary Conditions at Flow Inlets
and Exits
u At flow inlets,must supply
fluid temperature.
u At flow exits,fluid
temperature 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:
gxpgxp o )(
'
rrr?++
where gxpp or?='
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.
( ) ( )r r r b? =0 0 0g T T g
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User Inputs for Natural Convection
1,Set gravitational acceleration.
Define? Operating Conditions...
2,Define density model.
l If using Boussinesq model:
n Select boussinesq as the Density method
and assign constant value,ρo.
Define? 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.
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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
4
2Φ+=++
∫ dsssrITansrIa
x
I s
s
i
is
p
p
s
p
ss
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.
p
saa 4TI
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? Models? 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 << 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.
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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
inflow outflow
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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:
Tb = suitably defined bulk temperature
u Can also model flows with specified wall heat flux.
q =T TT Twall
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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Modeling Turbulent Flows
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u Unsteady,aperiodic motion in which all three velocity components
fluctuate? 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 species
concentration values.
What is Turbulence?
Time
U i (t)
Ui
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 uuR r?=
j
ij
jj
i
ik
i
k x
R
xx
U
x
p
x
UU
+
+
=
2mr
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Is the Flow Turbulent?
External Flows
Internal Flows
Natural Convection
5105×≥
xRe along a surface
around an obstacle
where
m
rULRe
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 ≥
hD
Re
108 1010?≥Ra
ma
rb 3TLgRa?≡
20,000≥DRe
Fluent Inc,2/20/01D5
Fluent Software Training
TRN-99-003
How Complex is the Flow?
u Extra strain rates
l Streamline curvature
l Lateral divergence
l Acceleration or deceleration
l Swirl
l Recirculation (or separation)
l Secondary flow
u 3D perturbations
u Transpiration (blowing/suction)
u Free-stream turbulence
u Interacting shear layers
Fluent Inc,2/20/01D6
Fluent Software Training
TRN-99-003
Choices to be Made
Turbulence Model
&
Near-Wall Treatment
Flow
Physics
Accuracy
Required
Computational
Resources
Turnaround
Time
Constraints
Computational
Grid
Fluent Inc,2/20/01D7
Fluent Software Training
TRN-99-003
Zero-Equation Models
One-Equation Models
Spalart-Allmaras
Two-Equation Models
Standard k-e
RNG k-e
Realizable k-e
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
Fluent Inc,2/20/01D8
Fluent Software Training
TRN-99-003
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 are
solved 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:
erm m
2k
Ct ≡Turbulent Viscosity:
Boussinesq Hypothesis:
(isotropic viscosity)?
+
+?=?=
i
j
j
i
tijjiij x
U
x
UkuuR mdrr
3
2
2/iiuuk ≡
+
≡
i
j
j
i
j
i
x
u
x
u
x
une
Fluent Inc,2/20/01D9
Fluent Software Training
TRN-99-003
u Turbulent viscosity is determined from:
u is determined from the modified viscosity transport equation:
u The additional variables are functions of the modified turbulent
viscosity and velocity gradients.
One Equation Model,Spalart-Allmaras
( ) 21
2
2
~
1
~~~~1~~~
dfcxcxxScDt
D
ww
j
b
jj
b
nrnrnnrm
snr
nr
n
+
+
+=
( )
( )
+= 313
3
/~
/~~
nnn
nnnrm
ct
n~
Generation Diffusion Destruction
Fluent Inc,2/20/01D10
Fluent Software Training
TRN-99-003
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.
Fluent Inc,2/20/01D11
Fluent Software Training
TRN-99-003
Two Equation Model,Standard k-e Model
Turbulent Kinetic Energy
Dissipation Rate
eeess 21,,,CCk are empirical constants
(equations written for steady,incompressible flow w/o body forces)
Convection Generation Diffusion
Destruction
{resmmr
+
+
=
444 3444 21444 3444 2143421 i
kt
ii
j
j
i
i
j
t
i
i x
k
xx
U
x
U
x
U
x
kU )(
DestructionConvection Generation Diffusion
43421444 3444 2144444 344444 2143421
+
+
=
kCxxx
U
x
U
x
U
kCxU itii
j
j
i
i
j
t
i
i
2
21 )(
eresmmeer
eee
Fluent Inc,2/20/01D12
Fluent Software Training
TRN-99-003
Two Equation Model,Standard k-e Model
u,Baseline model” (Two-equation)
l Most widely used model in industry
l Strength and weaknesses well documented
u Semi-empirical
l k equation derived by subtracting the instantaneous mechanical energy
equation from its time-averaged value
l ε equation formed from physical reasoning
u Valid only for fully turbulent flows
u Reasonable accuracy for wide range of turbulent flows
l industrial flows
l heat transfer
Fluent Inc,2/20/01D13
Fluent Software Training
TRN-99-003
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:
erm m
2k
Ct ≡
e
m kU
AA
C
so
*
1
+
=
0u2i ≥
2
j
2
i
2
ji u u)uu( ≤
b
j
t
j
GckckcScxxDtD ee
e
e
ne
erere
s
mmer
31
2
21 ++?+
+
=
GenerationDiffusion Destruction Buoyancy
Fluent Inc,2/20/01D14
Fluent Software Training
TRN-99-003
u Shares the same turbulent kinetic energy equation as Standard k-ε
u Superior performance for flows involving:
l planar and round jets
l boundary layers under strong adverse pressure gradients,separation
l rotation,recirculation
l strong streamline curvature
Two Equation Model,Realizable k-ε
Fluent Inc,2/20/01D15
Fluent Software Training
TRN-99-003
Two Equation Model,RNG k-e
Turbulent Kinetic Energy
Dissipation Rate
Convection Diffusion
Dissipation
{ {remamr
+=
44 344 2143421 i
k
i
t
i
i x
k
xSx
kU
eff
2
Generation
+
≡≡
j
i
i
j
ijijij x
U
x
USSSS
2
1,2where
are derived using RNG theoryeeeaa 21,,,CCk
(equations written for steady,incompressible flow w/o body forces)
Additional term
related to mean strain
& turbulence quantities
Convection Generation Diffusion Destruction
{RkCxxSkCxU
ii
t
i
i
+?
=
4342144 344 2144342143421
2
2eff
2
1
eremameer
eee
Fluent Inc,2/20/01D16
Fluent Software Training
TRN-99-003
Two Equation Model,RNG k-e
u 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 flows
l 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
Fluent Inc,2/20/01D17
Fluent Software Training
TRN-99-003
Reynolds Stress Model
k
ijk
ijijij
k
ji
k x
JP
x
uuU
+?Φ+=
er
Generation
k
i
kj
k
j
kiij x
Uuu
x
UuuP
+
≡
+
′?≡Φ
i
j
j
i
ij x
u
x
up
k
j
k
i
ij x
u
x
u
≡ me 2
Pressure-Strain
Redistribution
Dissipation
Turbulent
Diffusion
(modeled)
(related to e)
(modeled)
(computed)
(equations written for steady,incompressible flow w/o body forces)
Reynolds Stress
Transport Eqns.
Pressure/velocity
fluctuations
Turbulent
transport
)( jikijkkjiijk uupuuuJ dd +′+=
Fluent Inc,2/20/01D18
Fluent Software Training
TRN-99-003
Reynolds Stress Model
u 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 flows
n Rotating flow passages,secondary flows
Fluent Inc,2/20/01D19
Fluent Software Training
TRN-99-003
Large Eddy Simulation
u 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
tu
Fluent Inc,2/20/01D20
Fluent Software Training
TRN-99-003
Comparison of RANS Turbulence Models
Model 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-e
Robust,economical,reasonably
accurate; long accumulated
performance data
Mediocre results for complex flows
involving severe pressure gradients,
strong streamline curvature,swirl
and rotation
RNG k-e
Good for moderately complex
behavior like jet impingement,
separating flows,swirling flows,and
secondary flows
Subjected to limitations due to
isotropic eddy viscosity
assumption
Realizable
k-e
Offers largely the same benefits as
RNG; resolves round-jet anomaly
Subjected to limitations due to
isotropic eddy viscosity
assumption
Reynolds
Stress
Model
Physically most complete model
(history,transport,and anisotropy of
turbulent stresses are all accounted
for)
Requires more cpu effort (2-3x);
tightly coupled momentum and
turbulence equations
Fluent Inc,2/20/01D21
Fluent Software Training
TRN-99-003
Near-Wall Treatments
u Most k-e and RSM turbulence
models 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
Fluent Inc,2/20/01D22
Fluent Software Training
TRN-99-003
Standard Wall Functions
rt
m
/
2/14/1
w
PP kCUU ≡?
( )?
>?
+
<
=?
)(ln1Pr
)(Pr
**
**
Tt
T
yyPEy
yyy
T
k
m
r m PP ykCy 2/14/1≡?
q
kCcTTT PpPw
′′
≡
&
2/14/1)(
* mr
Mean Velocity
Temperature
where
where and P is a function of the fluid
and turbulent Prandtl numbers.
thermal sublayer thickness
( ) = EyU ln1k
Fluent Inc,2/20/01D23
Fluent Software Training
TRN-99-003
Nonequilibrium Wall Functions
u 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
=
m
r
krt
mm ykCEkCU
w
2/14/12/14/1
ln1/
~
+?+?
=
mrkrk
y
k
yy
y
y
k
y
dx
dpUU vv
v
v
2
2/12/1 ln2
1~where
Fluent Inc,2/20/01D24
Fluent Software Training
TRN-99-003
Two-Layer Zonal Model
u 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-e models are used in the turbulent core region.
u Only k equation is solved in the viscosity-affected region.
u e is computed from the correlation for length scale.
u Zoning is dynamic and solution adaptive.
m
r ykRe
y ≡
200>yRe
200<yRe
Fluent Inc,2/20/01D25
Fluent Software Training
TRN-99-003
Comparison of Near Wall Treatments
Strengths Weaknesses
Standard wall
Functions
Robust,economical,
reasonably accurate
Empirically based on simple
high-Re flows; poor for low-Re
effects,massive transpiration,
p,strong body forces,highly
3D 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 complex
flows,especially applicable
to low-Re flows
Requires finer mesh resolution
and therefore larger cpu and
memory resources
Fluent Inc,2/20/01D26
Fluent Software Training
TRN-99-003
Computational Grid Guidelines
Wall Function
Approach
Two-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
Fluent Inc,2/20/01D27
Fluent Software Training
TRN-99-003
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/?≈
Lfc
2.0Re039.02/?≈
Dfc
2// few cUu =≡ rtt
y1 = 50ν/uτ y1 = ν/ uτ
Fluent Inc,2/20/01D28
Fluent Software Training
TRN-99-003
Setting Boundary Conditions
u Characterize turbulence at inlets & outlets (potential backflow)
l k-e models require k and e
l Reynolds stress model requires Rij and e
u Several options allow input using more familiar parameters
l 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? 0.4d99
n 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 e explicitly allowed (non-uniform profiles possible).
10/1 << mm
t
Fluent Inc,2/20/01D29
Fluent Software Training
TRN-99-003
GUI for Turbulence Models
Define? Models? Viscous...
Turbulence Model options
Near Wall Treatments
Inviscid,Laminar,or Turbulent
Additional Turbulence options
Fluent Inc,2/20/01D30
Fluent Software Training
TRN-99-003
Example,Channel Flow with Conjugate Heat
Transfer
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
Fluent Inc,2/20/01D31
Fluent Software Training
TRN-99-003
Turbulence Modeling Approach
u Check if turbulent? ReDh= 5,980
u Developing turbulent flow at relatively low Reynolds number and
BLs on walls will give pressure gradient? use RNG k-e with
nonequilibrium wall functions.
u Develop strategy for the grid
l Simple geometry? quadrilateral cells
l Expect large gradients in normal direction to horizontal walls? 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.
Fluent Inc,2/20/01D32
Fluent Software Training
TRN-99-003
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
Fluent Inc,2/20/01D33
Fluent Software Training
TRN-99-003
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
Fluent Inc,2/20/01D34
Fluent Software Training
TRN-99-003
u Check if turbulent? ReD = 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
use RNG k-e with 2-layer zonal model.
u Develop strategy for the grid
l Simple geometry & BLs? quadrilateral cells.
l Large gradients near surface of cylinder & 2-layer model
fine mesh near surface & first cell at y+ = 1.
Turbulence Modeling Approach
Fluent Inc,2/20/01D35
Fluent Software Training
TRN-99-003
Grid for Flow Over a Cylinder
Fluent Inc,2/20/01D36
Fluent Software Training
TRN-99-003
Prediction of Turbulent Vortex Shedding
Contours of effective viscosity meff = m + mt
CD = 0.53 Strouhal Number = 0.297
U
DSt
t≡where
Fluent Inc,2/20/01D37
Fluent Software Training
TRN-99-003
Summary,Turbulence Modeling Guidelines
u Successful turbulence modeling requires engineering judgement of:
l Flow physics
l Computer resources available
l Project requirements
n Accuracy
n Turnaround time
l Turbulence models & near-wall treatments that are available
u Begin with standard k-e and change to RNG or Realizable k-e 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.
Fluent Inc,2/23/01E1
Fluent Software Training
TRN-99-003
Solver Settings
Fluent Inc,2/23/01E2
Fluent Software Training
TRN-99-003
Outline
u 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,Background
l Finite Volume Method
l Explicit vs,Implicit
l Segregated vs,Coupled
l Transient Solutions
Fluent Inc,2/23/01E3
Fluent Software Training
TRN-99-003
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 Overview
u 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
Fluent Inc,2/23/01E4
Fluent Software Training
TRN-99-003
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.
Fluent Inc,2/23/01E5
Fluent Software Training
TRN-99-003
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 Scheme
n 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
faces
fff
faces
fff
ttt
+?Γ=+ ∑∑ ⊥
+
fffr
rfrf
,)(
)()(
Fluent Inc,2/23/01E6
Fluent Software Training
TRN-99-003
Interpolation Methods for Pressure
u 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 pressure
gradients 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.
Fluent Inc,2/23/01E7
Fluent Software Training
TRN-99-003
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,robust
l 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.
Fluent Inc,2/23/01E8
Fluent Software Training
TRN-99-003
Initialization
u Iterative procedure requires that all solution variables be initialized
before calculating a solution.
Solve? Initialize? 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.
Solve? Initialize? Patch...
l Free jet flows
(patch high velocity for jet)
l Combustion problems
(patch high temperature
for ignition)
Fluent Inc,2/23/01E9
Fluent Software Training
TRN-99-003
Convergence Preliminaries,Residuals
u Transport equation for f 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 of
the time rate of change of the conserved variable.
l Only for coupled equations; additional scalar equations use segregated
definition.
p
nb
nbnbpp baa =+ ∑ ff
p
nb
nbnbppp baaR?+= ∑ ff
||∑=
cells
pRR
Fluent Inc,2/23/01E10
Fluent Software Training
TRN-99-003
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.
Fluent Inc,2/23/01E11
Fluent Software Training
TRN-99-003
Convergence Monitors,Residuals
u Residual plots show when the residual values have reached the
specified tolerance.
Solve? Monitors? Residual...
All equations converged.
10-3
10-6
Fluent Inc,2/23/01E12
Fluent Software Training
TRN-99-003
Convergence Monitors,Forces/Surfaces
u In addition to residuals,you can also monitor:
l Lift,drag,or moment
Solve? Monitors? Force...
l Variables or functions (e.g.,surface integrals)
at a boundary or any defined surface:
Solve? Monitors? Surface...
Fluent Inc,2/23/01E13
Fluent Software Training
TRN-99-003
Checking for Property Conservation
u 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? Fluxes...
Fluent Inc,2/23/01E14
Fluent Software Training
TRN-99-003
Decreasing the Convergence Tolerance
u 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.
Fluent Inc,2/23/01E15
Fluent Software Training
TRN-99-003
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 Factors
u Under-relaxation factor,a,is
included to stabilize the iterative
process for the segregated solver.
u Use default under-relaxation factors
to start a calculation.
Solve? Controls? 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.
poldpp faff?+=,
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 when
having 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 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 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 Solution
u 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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Multigrid
u 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 > 20
l Large differences in thermal conductivity
fine (original) mesh
coarse mesh
‘solution
transfer’
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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.
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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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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 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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Appendix
u Background
l Finite Volume Method
l Explicit vs,Implicit
l Segregated vs,Coupled
l Transient Solutions
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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,is
applied to each cell and discretized,For cell p,
∫∫∫∫
+Γ=?+ dVSdddVt
AAV
ffrfrf AAV
unsteady convection diffusion generation
Eqn.
continuity 1
x-mom,u
y-mom,v
energy h
f
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 - 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 an
equation set.
p
nb
nbnbpp baa =+ ∑ ff
VSAAVVt f
faces
fff
faces
fff
t
p
tt
p?+?Γ=+?
∑∑
⊥
+
fffr
rfrf
,)(
)()(
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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 f.
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 fp can either be solved explicitly or implicitly.
Background,Linearization
p
nb
nbnbpp baa =+ ∑ ff
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u Assumptions are made about the knowledge of fnb:
l Explicit linearization - unknown value in each cell computed from relations
that include only existing values (fnb assumed known from previous
iteration).
n fp solved explicitly using Runge-Kutta scheme.
l Implicit linearization - fp and fnb are assumed unknown and are solved
using 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,Segregated
u 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 ap 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 ap and anb are Neqx Neq matrices
n f is a vector of the dependent variables,{p,u,v,w,T,Y}T
p
nb
nbnbpp baa =+ ∑ ff
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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 part
of 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
poldpp faff?+=,
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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
Dx = 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/Transient
l 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 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 is
discretized 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 with
Discrete Ordinates
Radiation Model
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Outline
u Introduction
u Thermal Boundary Conditions
u Fluid Properties
u Conjugate Heat Transfer
u Natural Convection
u Radiation
u Periodic Heat Transfer
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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.
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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? Models? 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? Boundary Conditions...
4,Define material properties for heat transfer.
Define? Materials...
l Heat capacity and thermal conductivity must be defined.
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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? Controls? Solution...
l Disabling species diffusion term.
Define? Models? Species...
l Compute isothermal flow first,then add calculation of energy equation.
Solve? Controls? 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 Tn k Tn
wall
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Thermal Boundary Conditions at Flow Inlets
and Exits
u At flow inlets,must supply
fluid temperature.
u At flow exits,fluid
temperature 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:
gxpgxp o )(
'
rrr?++
where gxpp or?='
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.
( ) ( )r r r b? =0 0 0g T T g
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User Inputs for Natural Convection
1,Set gravitational acceleration.
Define? Operating Conditions...
2,Define density model.
l If using Boussinesq model:
n Select boussinesq as the Density method
and assign constant value,ρo.
Define? 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.
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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
4
2Φ+=++
∫ dsssrITansrIa
x
I s
s
i
is
p
p
s
p
ss
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.
p
saa 4TI
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? Models? 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 << 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.
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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
inflow outflow
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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:
Tb = suitably defined bulk temperature
u Can also model flows with specified wall heat flux.
q =T TT Twall
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
