© fluent inc. 12/18/2015 d1 fluent software training trn-98-006 modeling turbulent flows
TRANSCRIPT
© Fluent Inc. 04/21/23D1
Fluent Software TrainingTRN-98-006
Modeling Turbulent Flows
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Fluent Software TrainingTRN-98-006
Unsteady, aperiodic motion in which all three velocity components fluctuate mixing matter, momentum, and energy.
Decompose velocity into mean and fluctuating parts:
Ui(t) Ui + ui(t)
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?
Direct numerical simulation of governing equations is only possible for simple low-Re flows.
Instead, we solve Reynolds Averaged Navier-Stokes (RANS) equations:
where (Reynolds stresses)
Time-averaged statistics of turbulent velocity fluctuations are modeled using
functions containing empirical constants and information about the mean flow. Large Eddy Simulation numerically resolves large eddies and models small
eddies.
(steady, incompressible flow w/o body forces)
jiij uuR
j
ij
jj
i
ik
ik x
R
xx
U
x
p
x
UU
2
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Is the Flow Turbulent?
External Flows
Internal Flows
Natural Convection
5105 xRe along a surface
around an obstacle
where
UL
ReL where
Other factors such as free-stream turbulence, surface conditions, and disturbances may cause earlier transition to turbulent flow.
L = x, D, Dh, etc.
108 1010 Ra 3TLg
Ra
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How Complex is the Flow?
Extra strain rates Streamline curvature Lateral divergence Acceleration or deceleration Swirl Recirculation (or separation) Secondary flow
3D perturbations Transpiration (blowing/suction) Free-stream turbulence Interacting shear layers
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Choices to be Made
Turbulence Model&
Near-Wall Treatment
Flow Physics
AccuracyRequired
Computational Resources
Turnaround TimeConstraints
Computational Grid
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Zero-Equation Models
One-Equation Models Spalart-AllmarasTwo-Equation Models Standard k- RNG k-Realizable k-Reynolds-Stress Model
Large-Eddy Simulation
Direct Numerical Simulation
Turbulence Modeling Approaches
IncludeMorePhysics
IncreaseComputationalCostPer Iteration
Availablein FLUENT 5
RANS-basedmodels
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RANS equations require closure for Reynolds stresses.
Turbulent viscosity is indirectly solved for from single transport equation of modified viscosity for One-Equation model.
For Two-Equation models, turbulent viscosity correlated with turbulent kinetic energy (TKE) and the dissipation rate of TKE.
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:
2kCt Turbulent Viscosity:
Boussinesq Hypothesis:(isotropic stresses)
i
j
j
itijjiij x
U
x
UkuuR
3
2
2/iiuuk
i
j
j
i
j
i
x
u
x
u
x
u
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Turbulent viscosity is determined from:
is determined from the modified viscosity transport equation:
The additional variables are functions of the modified turbulent viscosity and velocity gradients.
One Equation Model: Spalart-Allmaras
21
2
2~
1
~~~~1~~~
dfc
xc
xxSc
Dt
Dww
jb
jjb
3
13
3
/~/~
~
ct
~
Generation Diffusion Destruction
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One-Equation Model: Spalart-Allmaras
Designed specifically for aerospace applications involving wall-bounded flows.
Boundary layers with adverse pressure gradients turbomachinery
Can use coarse or fine mesh at wall Designed to be used with fine mesh as a “low-Re” model, i.e., throughout
the viscous-affected region. Sufficiently robust for relatively crude simulations on coarse meshes.
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Two Equation Model: Standard k- Model
Turbulent Kinetic Energy
Dissipation Rate
21, ,, CCk are empirical constants
(equations written for steady, incompressible flow w/o body forces)
Convection Generation DiffusionDestruction
ikt
ii
j
j
i
i
jt
ii x
k
xx
U
x
U
x
U
x
kU )(
DestructionConvection Generation Diffusion
kC
xxx
U
x
U
x
U
kC
xU
it
ii
j
j
i
i
jt
ii
2
21 )(
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Two Equation Model: Standard k- Model
“Baseline model” (Two-equation) Most widely used model in industry Strength and weaknesses well documented
Semi-empirical k equation derived by subtracting the instantaneous mechanical energy
equation from its time-averaged value equation formed from physical reasoning
Valid only for fully turbulent flows Reasonable accuracy for wide range of turbulent flows
industrial flows heat transfer
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Two Equation Model: Realizable k- Distinctions from Standard k- model:
Alternative formulation for turbulent viscosity
where is now variable
(A0, As, and U* are functions of velocity gradients)
Ensures positivity of normal stresses;
Ensures Schwarz’s inequality;
New transport equation for dissipation rate, :
2kCt
kUAA
C
so
*
1
0u2i
2j
2i
2ji u u)uu(
bj
t
j
Gck
ck
cScxxDt
D
31
2
21
GenerationDiffusion Destruction Buoyancy
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Shares the same turbulent kinetic energy equation as Standard k- Superior performance for flows involving:
planar and round jets boundary layers under strong adverse pressure gradients, separation rotation, recirculation strong streamline curvature
Two Equation Model: Realizable k-
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Two Equation Model: RNG k-Turbulent Kinetic Energy
Dissipation Rate
Convection DiffusionDissipation
ik
it
ii x
k
xS
x
kU eff
2
Generation
j
i
i
jijijij x
U
x
USSSS
2
1,2
where
are derived using RNG theory 21, ,, CCk
(equations written for steady, incompressible flow w/o body forces)
Additional termrelated to mean strain& turbulence quantities
Convection Generation Diffusion Destruction
RkC
xxS
kC
xU
iit
ii
2
2eff2
1
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Two Equation Model: RNG k- k- equations are derived from the application of a rigorous statistical
technique (Renormalization Group Method) to the instantaneous Navier-Stokes equations.
Similar in form to the standard k- equations but includes: additional term in equation that improves analysis of rapidly strained flows the effect of swirl on turbulence analytical formula for turbulent Prandtl number differential formula for effective viscosity
Improved predictions for: high streamline curvature and strain rate transitional flows wall heat and mass transfer
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Reynolds Stress Model
k
ijkijijij
k
jik x
JP
x
uuU
Generationk
ikj
k
jkiij x
Uuu
x
UuuP
i
j
j
iij x
u
x
up
k
j
k
iij x
u
x
u
2
Pressure-StrainRedistribution
Dissipation
TurbulentDiffusion
(modeled)
(related to )
(modeled)
(computed)
(equations written for steady, incompressible flow w/o body forces)
Reynolds StressTransport Eqns.
Pressure/velocity fluctuations
Turbulenttransport
)( jikijkkjiijk uupuuuJ
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Reynolds Stress Model RSM closes the Reynolds-Averaged Navier-Stokes equations by
solving additional transport equations for the Reynolds stresses. Transport equations derived by Reynolds averaging the product of the
momentum equations with a fluctuating property Closure also requires one equation for turbulent dissipation Isotropic eddy viscosity assumption is avoided
Resulting equations contain terms that need to be modeled. RSM has high potential for accurately predicting complex flows.
Accounts for streamline curvature, swirl, rotation and high strain rates Cyclone flows, swirling combustor flows Rotating flow passages, secondary flows
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Large Eddy Simulation Large eddies:
Mainly responsible for transport of momentum, energy, and other scalars, directly affecting the mean fields.
Anisotropic, subjected to history effects, and flow-dependent, i.e., strongly dependent on flow configuration, boundary conditions, and flow parameters.
Small eddies: Tend to be more isotropic and less flow-dependent More likely to be easier to model than large eddies.
LES directly computes (resolves) large eddies and models only small eddies (Subgrid-Scale Modeling).
Large computational effort Number of grid points, NLES Unsteady calculation
2Reu
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Comparison of RANS Turbulence Models
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Near-Wall Treatments
Most k- and RSM turbulence models will not predict correct near-wall behavior if integrated down to the wall.
Special near-wall treatment is required.
Standard wall functions Nonequilibrium wall functions Two-layer zonal model
Boundary layer structure
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Standard Wall Functions
/
2/14/1
w
PP kCUU
)(ln1
Pr
)(Pr
**
**
Tt
T
yyPEy
yyyT
PP ykC
y2/14/1
q
kCcTTT PpPw
2/14/1)(*
Mean Velocity
Temperature
where
where and P is a function of the fluid and turbulent Prandtl numbers.
thermal sublayer thickness
EyU ln1
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Nonequilibrium Wall Functions Log-law is sensitized to pressure gradient for better
prediction of adverse pressure gradient flows and separation.
Relaxed local equilibrium assumptions for TKE in wall-neighboring cells.
Thermal law-of-wall unchanged
ykC
EkCU
w
2/14/12/14/1
ln1/
~
y
k
yyyy
k
ydxdp
UU vv
v
v2
2/12/1ln
21~
where
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Two-Layer Zonal Model Used for low-Re flows or
flows with complex near-wall phenomena.
Zones distinguished by a wall-distance-based turbulent Reynolds number
High-Re k- models are used in the turbulent core region. Only k equation is solved in the viscosity-affected region. is computed from the correlation for length scale. Zoning is dynamic and solution adaptive.
yk
Rey
200yRe
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Comparison of Near Wall Treatments
Strengths Weaknesses
Standard wallFunctions
Robust, economical,reasonably accurate
Empirically based on simplehigh-Re flows; poor for low-Reeffects, massive transpiration,p, strong body forces, highly3D flows
Nonequilibriumwall functions
Accounts for p effects,allows nonequilibrium:
-separation-reattachment-impingement
Poor for low-Re effects, massivetranspiration, severe p, strongbody forces, highly 3D flows
Two-layer zonalmodel
Does not rely on empiricallaw-of-the-wall relations,good for complex flows,applicable to low-Re flows
Requires finer mesh resolutionand therefore larger cpu andmemory resources
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Computational Grid GuidelinesWall Function
ApproachTwo-Layer Zonal Model Approach
First grid point in log-law region
At least ten points in the BL.
Better to use stretched quad/hex cells for economy.
First grid point at y+ 1.
At least ten grid points within buffer & sublayers.
Better to use stretched quad/hex cells for economy.
50050 y
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Estimating Placement of First Grid Point
Estimate the skin friction coefficient based on correlations either approximate or empirical:
Flat Plate-
Pipe Flow-
Compute the friction velocity:
Back out required distance from wall: Wall functions • Two-layer model
Use post-processing to confirm near-wall mesh resolution
2.0Re0359.02/ Lfc
2.0Re039.02/ Dfc
2// few cUu
y1 = 250/u y1 = / u
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Setting Boundary Conditions
Characterize turbulence at inlets & outlets (potential backflow) k- models require k and Reynolds stress model requires Rij and
Several options allow input using more familiar parameters Turbulence intensity and length scale
length scale is related to size of large eddies that contain most of energy. For boundary layer flows: l 0.499
For flows downstream of grids /perforated plates: l opening size Turbulence intensity and hydraulic diameter
Ideally suited for duct and pipe flows Turbulence intensity and turbulent viscosity ratio
For external flows: Input of k and explicitly allowed (non-uniform profiles possible).
10/1 t
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GUI for Turbulence Models
Define Models Viscous...
Turbulence Model options
Near Wall Treatments
Inviscid, Laminar, or Turbulent
Additional Turbulence options
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Example: Channel Flow with Conjugate Heat Transfer
adiabatic wall
cold airV = 50 fpmT = 0 °F
constant temperature wall T = 100 °F
insulation
1 ft
1 ft
10 ft
P
Predict the temperature at point P in the solid insulation
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Turbulence Modeling Approach Check if turbulent ReDh
= 5,980
Developing turbulent flow at relatively low Reynolds number and BLs on walls will give pressure gradient use RNG k- with nonequilibrium wall functions.
Develop strategy for the grid Simple geometry quadrilateral cells Expect large gradients in normal direction to horizontal walls fine mesh
near walls with first cell in log-law region. Vary streamwise grid spacing so that BL growth is captured. Use solution-based grid adaption to further resolve temperature gradients.
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Velocitycontours
Temperaturecontours
BLs on upper & lower surfaces accelerate the core flow
Prediction of Momentum & Thermal Boundary Layers
Important that thermal BL was accurately resolved as well
P
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Example: Flow Around a Cylinder
wall
wall
1 ft
2 ft
2 ft
airV = 4 fps
Compute drag coefficient of the cylinder
5 ft 14.5 ft
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Check if turbulent ReD = 24,600
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- with 2-layer zonal model.
Develop strategy for the grid Simple geometry & BLs quadrilateral cells. Large gradients near surface of cylinder & 2-layer model
fine mesh near surface & first cell at y+ = 1. Use solution-based grid adaption to further resolve pressure
gradients.
Turbulence Modeling Approach
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Grid for Flow Over a Cylinder
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Prediction of Turbulent Vortex Shedding
Contours of effective viscosity eff = + t
CD = 0.53 Strouhal Number = 0.297
U
DSt
where
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Summary: Turbulence Modeling Guidelines
Successful turbulence modeling requires engineering judgement of: Flow physics Computer resources available Project requirements
Accuracy Turnaround time
Turbulence models & near-wall treatments that are available Begin with standard k- and change to RNG or Realizable k- if
needed. Use RSM for highly swirling flows. Use wall functions unless low-Re flow and/or complex near-wall
physics are present.