Turbulence and Fluent

Turbulence and Fluent

Turbulence and Fluent

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.

Turbulent Flows

Larger Structures Smaller

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

Computational Approaches(2)

LES, DNS

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 j i k i k i x R x U x x p x U U t U ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ∂ ∂ − = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂

µ

ρ

j i ij

u

u

R

=

−

ρ

′

′

( )

x

t

U

( ) ( )

x

t

u

x

t

u

_i

r

,

=

_i

r

,

+

_i

′

r

,

Turbulent fluctuation Mean u'_i U_i u_i time u

The Closure Problem

Reynolds 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.

ij ij k k i j j i j i ij

k

x

U

x

U

x

U

u

u

R

ρ

µ

µ

δ

ρ

δ

3

2

3

2

t t

−

∂

∂

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

=

′

′

−

=

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-Allmaras

A 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 1 3 3 1 1

/

~

/

~

,

~

v v v t

C

f

f

+

≡

=

ν

ν

ν

ν

ν

ρ

µ

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

C

G

k

C

x

x

Dt

D

G

x

k

x

k

Dt

D

k e j t j k j k t j 2 2 1

ε

ρ

ε

ε

σ

µ

µ

ε

ρ

ρε

σ

µ

µ

ρ

ε ε

−

+

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

∂

∂

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

∂

∂

=

−

+

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

∂

∂

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

∂

∂

=

3

.

1

,

0

.

1

,

92

.

1

,

44

.

1

,

09

.

0

₁

=

₂

=

=

=

=

_{ε ε ε} µ

C

C

σ

k

σ

C

where

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 j j i ij j k t j j i ij t x x f x U k Dt D x k x k f x U Dt Dk k ω σ µ µ ω β ρ τ ω α ω ρ σ µ µ ω β ρ τ ρ ω ρ α µ ω β β 2 * * *

τ

ε

ω

≈

∝

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)

(

)

(

)

ij ij T ij ij ij j i k k j i

U

u

u

P

F

D

x

u

u

t

ρ

∂

ρ

′

′

=

+

+

+

Φ

−

ε

∂

+

′

′

∂

∂

Turbulent diffusion Stress-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.

The Structure of Near-Wall

Flows

Near-Wall Modeling

Wall Functions Wall Integration

Accurate near-wall modeling is

important to correctly predict

frictional drag, 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

/

/

_{e f} w

U

c

u

_τ

≡

τ

ρ

=

300

30

−

≈

+ p

y

1

≈

+ p

y

τ τ

ν

y

y

ν

u

u

y

y

+_p

≡

_p

/

⇒

_p

≡

+_p

/

300 30− ≈ + p y

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

Enhanced Wall Treatment

‹ Fully-Developed Channel Flow (Re_t = 590)

z For fixed pressure drop cross periodic boundaries, different

near-wall mesh resolutions yielded different volume flux as follows

z 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. W all fn. 12.68 13.77 16.77 19.08

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

≈

0.4d

₉₉

– 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 < m

_t

/m < 10

j iu u j iu u

Is the Flow Turbulent?

External Flows

Internal Flows

5

10

5

×

≥

x

Re

along a surface around an obstacle

,300

2

≥

h D

Re

µ

ρ

UL

Re

_L

≡

where L = x, D, D_h, etc.

20,000

≥

D

Re

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

Natural Convection

µα

ρ

β

3

TL

g

Ra

≡

∆

where 10 8 10 10 − ≥ Ra

Turbulence Models in Fluent

Zero-Equation Models

One-Equation Models

Spalart-Allmaras

Two-Equation Models

Standard k-ε RNG k-ε Realizable k-ε Standard k-ω SST k-ω

V2F Model

Reynolds-Stress Model

Detached Eddy Simulation

Large-Eddy Simulation

Direct Numerical Simulation

Increase in Computational

Cost

Per Iteration Available_{in FLUENT}

RANS models

Near-wall options

Customization

Auxiliary Models

z Standard wall functions z Non-equilibrium wall

functions

z Enhanced wall treatment z Buoyancy effects

z Compressibility effects z Low Re effects

z Pressure gradient effects

z Turbulent viscosity z Source terms

z Turbulence transport equations

GUI for Turbulence Models

Define → Models → Viscous...

Turbulence Model options

Near Wall Treatments

Inviscid, Laminar, or Turbulent

RANS Turbulence Model

Behavior and Usage

Model

Behavior and Usage

Spalart-Allmaras Standard 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)

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

Impact of Boundary Conditions

Run

X-Velocity

B.C.

Thermal

B.C.

Turbulence B.C.

1

Profile

Uniform

Uniform

Uniform

Profile

2

Uniform

Intensity & Hydraulic

Diameter

Impact of Grid Quality

Structured

Tri w b/l

Quad Pave

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

Rotating Flow in a Cyclone

0.2 m 0.97 m 0.1 m • U_in = 20 m/s 0.12 m

• Highly swirling flows (

W

_max

= 1.8

U

_in

)

• 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