Turbulence and FluentTurbulence and Fluent

Turbulence Modeling

What is Turbulence?

• We do not really know

• 3D, unsteady, irregular motion in which transported quantities fluctuate in time and space.– Turbulent eddies (spatial structures).– Diffusive (mixing).– Self-sustaining if a mean shear exist.– Entrainment.

• Energy cascade.– Energy is added at the large eddies.– Energy is dissipated at the small eddies.

Turbulent Flows

Larger Structures

Smaller Structures

Computational Approaches

• DNS (Direct Numerical Simulation)– Solves the Navier-Stokes (N-S) equations. No turbulence modeling required.– Not practical for industrial flows (requires Low Re and simple geometries).

• LES (Large Eddy Simulation)– Solves a filtered version of the N-S equations.– Less expensive than DNS, but still too expensive for most applications.

• RANS (Reynolds-Averaged N-S)– Solve the ensemble-averaged N-S equations. All turbulence is modeled.– The most widely used approach for calculating industrial flows.

• There is not yet a single turbulence model that can reliably predict all turbulent flows found in industrial applications with sufficient accuracy.

Computational Approaches(2)

LES, DNS

RANS

RANS Modeling

• Reynolds decomposition:

• The Reynolds-averaged momentum equations are as follows:

where is called the Reynolds stresses. The Reynolds stresses must be modeled to close the equations.

j

ij

j

i

jik

ik

i

xR

xU

xxp

xUU

tU

∂

∂+⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂

∂ µρ

jiij uuR ′′−= ρ

( ) ( ) ( )txutxUtxu iii ,,, rrr ′+=

Turbulentfluctuation

Mean

u'i

Ui ui

time

u

The Closure ProblemReynolds equations does not contain enough equations to solve for all the

uknown variables. Thus, the Reynolds stresses must be modeled.

Modeling approaches• Eddy-Viscosity Models (EVM):

– Boussinesq hypothesis: Reynolds stresses are modeled using an eddy (or turbulent) viscosity µt . Assumes Isotropic turbulence.

• Reynolds-Stress Models (RSM): – solves transport equations for all individual Reynolds stresses.– Require modeling for many terms in the Reynolds stress equations.– Does NOT assume isotropic turbulence.

ijijk

k

i

j

j

ijiij k

xU

xU

xUuuR δρδµµρ

32

32

tt −∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=′′−=

Modeling the Eddy Viscosity

• Basic approach made through dimensional arguments– Units of νt = µt/ρ are [m2/s]– Typically one needs 2 out of the 3 scales:

• velocity - length - time

• Commonly used scales– is the turbulent kinetic energy [L2/T2]– is the turbulence dissipation rate [L2/T3]– is the specific dissipation rate [1/T]

• Models classified in terms of number of transport equations solved,– zero-equation models– one-equation models– two-equation models

Spalart-AllmarasA one-equation RANS model

A low-cost model solving an equation for the modified eddy viscosity

• Eddy-viscosity is obtained from

• Mainly for aerodynamic/turbo-machinery applications with mild separation(supersonic/transonic flows over airfoils, boundary-layer flows, etc).

( )( ) 3

13

3

11 /~/~

,~v

vvt Cff

+≡=

νννννρµ

ν~

Standard k-ε (SKE)A two-equation RANS model

• Transport equations for k and ε:

• The most widely-used engineering turbulence model for industrial applications

• Robust • Performs poorly for flows with strong separation, large streamline

curvature, and large pressure gradient.

( )

( )k

CGk

CxxDt

D

Gxk

xk

DtD

kej

t

j

kjk

t

j

2

21ερεε

σµµερ

ρεσµµρ

εε

−+⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=

−+⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=

3.1,0.1,92.1,44.1,09.0 21 ===== εεεµ σσ kCCCwhere

Realizable k-ε (RKE)• Realizable k-ε (RKE)

– Positivity of normal stresses– Schwarz’ inequality for Reynolds shear-stresses

• Good performance for flows with axisymmetric jets.

RNG k-ε (RNG)• Constants in the k-ε equations are derived using the Renormalization

Group theory.

• RNG’s sub-models include:

– Differential viscosity model to account for low-Re effects

– Analytically derived algebraic formula for turbulent Prandtl/Schmidt number

– Swirl modification

• Performs better than SKE for more complex shear flows, and flows with high strain rates, swirl, and separation.

k-ω models

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

+−∂∂

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

+−∂∂

=

=

j

t

jj

iij

jk

t

jj

iij

t

xxf

xU

kDtD

xk

xkf

xU

DtDk

k

ωσµµωβρτωαωρ

σµµωβρτρ

ωραµ

ωβ

β

2

*

*

*

τεω 1

∝≈k

specific dissipation rate: ω

Two-equation RANS models

• Fluent supports the standard k-ω model by Wilcox (1998), and Menter’s SST k-ω model (1994).

• k-ω models are inherently low-Re models: – Can be integrated to the wall without using any damping functions– Accurate and robust for a wide range of boundary layer flows with pressure

gradient• Most widely adopted in the aerospace and turbo-machinery communities.• Several sub-models/options of k-ω : compressibility effects, transitional flows

and shear-flow corrections.

Reynolds-Stress Model (RSM)

( ) ( ) ijijTijijijjik

kji DFPuuU

xuu

tερρ −Φ+++=′′

∂∂

+′′∂∂

Turbulent diffusionStress-production

Rotation-production Pressure strain

Dissipation

Modeling required for these terms

• Attempts to address the deficiencies of the EVM.• Anisotropy, history effects of Reynolds stresses. • RSM requires more modeling (the pressure-strain is most critical and difficult

one among them).• More expensive and harder to converge.• Most suitable for complex 3-D flows with strong streamline curvature, swirl and

rotation.

Near Wall Modeling

The Structure of Near-Wall Flows

• The structure of turbulent boundary layers in the near-wall region:

Near-Wall Modeling

Wall Functions Wall Integration

Accurate near-wall modeling is important to correctly predict frictionaldrag, pressure drop, separation, heat transfer etc.

Near-Wall Modeling Options • Wall functions provide boundary conditions for momentum, energy, species

and turbulent quantities.

• The Standard and Non-equilibrium Wall Functions(SWF and NWF) use the law of the wall.

• Enhanced Wall Treatment– Combines the use of blended law-of-the wall and a

two-layer zonal model.– Suitable for low-Re flows or flows with complex

near-wall phenomena.– Turbulence models are modified for the inner layer.– Generally requires a fine near-wall mesh capable of

resolving the viscous sub-layer (more than 10 cells within the inner layer)

inner layer

outer layer

Placement of The First Grid Point

• For standard or non-equilibrium wall functions, each wall-adjacent cell’s centroid should be located within:

• For the enhanced wall treatment (EWT), each wall-adjacent cell’s centroid should be located:

– Within the viscous sublayer, , for the two-layer zonal model:– Preferably within for the blended wall function

• How to estimate the size of wall-adjacent cells before creating the grid:– ,– The skin friction coefficient can be estimated from empirical

correlations:

2// few cUu =≡ ρττ

30030 −≈+py

1≈+py

ττ νν uyyuyy pppp // ++ ≡⇒≡

30030 −≈+py

Near-Wall Modeling: Recommended Strategy

• Use SWF or NWF in high Re applications (Re > 106) where you cannot afford to resolve the viscous sub-layer.– Use NWF for mildly separating, reattaching, or impinging flows.

• You may consider using EWT if:– Near wall characteristics are important.– The physics and near-wall mesh of the case is such that y+ is

likely to vary significantly over a wide portion of the wall region.

• Try to make the mesh either coarse or fine enough to avoid placing the wall-adjacent cells in the buffer layer (y+ = 5 ~ 30).

Enhanced Wall Treatment

Fully-Developed Channel Flow (Ret = 590)

For fixed pressure drop cross periodic boundaries, different near-wall mesh resolutions yielded different volume flux as follows

The enhanced near-wall treatment gives a much smaller variation for different near-wall mesh resolutions compared to the variations found using standard wall functions.

y+ = 1 y+ = 4 y+ = 8 y+ = 16

Std. Wall fn. 12.68 13.77 16.77 19.08EWT 18.31 17.58 17.70 18.48

Inlet/Outlet Conditions

• Boundary conditions for k, ε, w and/or must be specified.

• Direct or indirect specification of turbulence parameters:– Explicitly input k, ε, w, or

• This method allows for profile definition.– Turbulence intensity and length scale

– For boundary layer flows: l ≈ 0.4d99

– For flows downstream of grid: l ≈ opening size– Turbulence intensity and hydraulic diameter

• Internal flows– Turbulence intensity and turbulent viscosity ratio

• For external flows: 1 < mt/m < 10

jiuu

jiuu

Choosing Models

Is the Flow Turbulent?

External Flows

Internal Flows

5105×≥xRe along a surface

around an obstacle

,3002 ≥hD Re

µρULReL ≡where

L = x, D, Dh, etc.

20,000≥DReOther factors such as free-stream turbulence, surface conditions, and disturbances may cause earlier transition to turbulent flow.

Natural Convection

µαρβ 3TLgRa ∆

≡where108 1010 −≥Ra

Turbulence Models in Fluent

Zero-Equation ModelsOne-Equation Models

Spalart-AllmarasTwo-Equation Models

Standard k-εRNG k-εRealizable k-εStandard k-ωSST k-ω

V2F ModelReynolds-Stress ModelDetached Eddy SimulationLarge-Eddy Simulation

Direct Numerical Simulation

Increase inComputational

CostPer Iteration

Availablein FLUENT

RANSmodels

Near-wall options

Customization

Auxiliary Models

Standard wall functionsNon-equilibrium wall functions Enhanced wall treatment

Buoyancy effectsCompressibility effectsLow Re effectsPressure gradient effects

Turbulent viscositySource terms Turbulence transport equations

GUI for Turbulence ModelsDefine → Models → Viscous...

Turbulence Model options

Near Wall Treatments

Inviscid, Laminar, or Turbulent

Additional Turbulence options

RANS Turbulence Model Behavior and Usage

Model Behavior and Usage

Spalart-AllmarasStandard k-ε

RNG k-ε

Realizable k-ε

Standard k-ω

SST k-ω

RSM

• Economical for large meshes• Performs poorly for 3D flows, free shear flows, flows with strong separation• Suitable for mildy complex (quasi-2D) flows (turbo, wings, fuselages, missilies)

• Robust, but performs poorly for complex flows• Suitable for initial conditions, fast design screening and parametric studies

• Suitable for complex shear flows involving rapid strain, moderate swirl, vortices, locally transitional flows (e.g. b.l. Separation, massive separation, vortex shedding)

• Similar benefits and applications as the RNG model• Possibly more accurate and easier to converge

• Superior for wall-bounded, free shear, and low-Re flows• Suitable for complex b.l flows (e.g. external aero, turbomachinery, vortex shedding)• Can predict transition (usually predict to early transition, though)

• Similar benefits as SKO, less sensitive to outer disturbances• Suitable for wall bounded flows, less suited for free shear flows

• The most physically sound RANS model (handels anisotrophy)• Computationally expensive and harder to converge• Suitable for complex 3D flows with strong streamline curvature, strong swirl(e.g. Curved duct, swirl combustors, cyclones)

Examples

Heat Transfer Behind a 2D Backstep

• Heat transfer predictions along the bottom• Measured by Vogel and Eaton (1980)• SKE, RNG, and RKE models are employed with standard wall

functions.

Factors affecting accuracy

• The accuracy of turbulent flow predictions can be affected by user decisions involving– Turbulence model– Boundary conditions– Grid resolution and near wall modeling– Grid quality

Impact of Turbulence Model

• k-ε Results

Impact of Boundary ConditionsRun X-Velocity

B.C.Thermal

B.C. Turbulence B.C.

1 Profile Uniform

Uniform

Uniform

Profile

2 Uniform Intensity & Hydraulic Diameter

3 Profile k=1, ε=1

Impact of Grid Quality

Structured

Tri w b/l

Quad Pave

Tri

Impact of Near Wall Modeling• y+ values must be appropriate for selected near wall treatment• Realizable k-ε with SWF

Stream Function Contours for 180 Degree Bend

Spalart-Allmaras Standard k-ε

RNG k-ε RSM

Rotating Flow in a Cyclone

0.2 m

0.97 m

0.1 m•

Uin = 20 m/s

0.12 m

• Highly swirling flows (Wmax= 1.8 Uin)

• High-order discretization on 40,000 cell hexahedral mesh

• Computed using a family of k-ε models (SKE, RNG, RKE), k-ω models (Wilcox’, SST) and RSM models

Cyclone Velocity Profiles