introduction to statistical turbulence modelling overview webseite...introduction to statistical...

Folie 1Introduction to statistical turbulence modelling. Overview, RWTH Aachen, 08./09.03.2010

Introduction to statistical turbulence modelling

Overview

Bernhard Eisfeld 08./09.03.2010


Overview: Introduction to statistical turbulence modelling

Guiding principles

• Presentation of ideas and concepts• Derivation of equations

• Tough due to short time frame• Slides should carry all necessary information

Turbulence modelling is rational, not miraculous• Focus on boundary layers (not free shear layers)• Field of application: Aerodynamics (aeronautics)



Part I: Fundamental equations• Averaging• Flow equations• Turbulence equations

Part II: Characteristics of turbulent flows• Realisability• Homogeneous isotropic turbulence• Wall-bounded flows:

• Near wall asymptotics• Profiles (velocity, temperature, turbulent quantities)

Part III: Turbulence models• Boussinesq hypothesis• Eddy viscosity models

• Algebraic models• 2-equation models (k-, k-)• 1-equation models (Spalart-Allmaras)



(Optional supplement)Part IV: Differential Reynolds stress models• Problems of Boussinesq hypothesis• Differential Reynolds stress models

• Redistribution• Diffusion• Dissipation• Length scale• Engineering approach

Folie 1Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010

Introduction

to statistical

turbulence

modelling

Fundamental equations

Bernhard Eisfeld 08.03.2010


Overview: Fundamentals

• Introduction• Averaging• Flow

equations

• Exact• Averaged

• Turbulence

equations• Fluctuations• Correlations

• Summary


Overview: Fundamentals


equations


• Turbulence


• Summary


Reynolds’

experiment:Inject dye into pipe flowObserve filament at different flow speed

Occurrence of turbulence

Beyond critical Reynolds number: Irregular flow pattern turbulent

Introduction

Below critical Reynolds number: Regular flow pattern laminar


IntroductionCharacteristics of turbulent flow

No clear-cut definition, but common features of turbulent flow:•

Irregular (random) fluctuations of all quantities –

in time

unsteady–

in space three-dimensional•

Vortical

structures (“eddies”)–

Continuous spectrum of

scales–

Energy cascade

from large to small scales•

Small scale motion Enhanced transfer of

–

Momentum Turbulent (Reynolds) stresses–

Heat Turbulent heat flux–

Mass Turbulent diffusion flux


Major approaches

DNS LES Statistical

• No modelling• Extremelyexpensive

• No technicalapplication

• Resolve

largescales

• Model smallscales

• Very

expensive• Massiveseparation

• Low Re

• Model averageeffect

on mean

flow• Cheap

• Attached

flow

• Technical

Re

Introduction


Direct

Numerical

Simulation (DNS)Accept

instability

of Navier-Stokes

solution

Resolve

3D time-dependent

fluctuations

Problem:Smallest

scales

become

smaller

with

increasing

Reynolds

numberExtremely

expensive

for

technical

problems

(~Re3.2)

Dependence

on boundary/initial

conditions

Technical

applicabilityNot yet

achieved

Introduction


Large Eddy Simulation (LES)Resolve

only

largest

fluctuations

(80% of kinetic

energy)Subgrid

scale

model

for

influence

of smaller

fluctuations

by

assumption

of local

isotropy

Problem:Isotropic

cells

required

(near

wall region)

Reynolds number

dependence

remainsStill very

expensive

for

technical

problems

(~Re2.8)

Technical

applicability:Flows

with

massive separation

(bluff bodies)

Flows

at low

Reynolds numberFlows

at high Reynolds number

(wall modelling): ~2040

Introduction


Statistical

turbulence

modellingTechnical

applications:

Information on average

flow

often

suffices

(forces)Model average

effect

of fluctuations

on average

flow

Moments (mean

values) of the

statistical

distributions

Problem:All fluctuations

in one

model

Requires

influence

of fluctuations

= f (mean

flow)Inherent

assumption

of steady

mean

flow

Lack of accuracy

in complex

flows

(separation)

Technical

applicability:Backbone

of (steady) technical

applications

Introduction


Restrictions

of statistical

turbulence

modellingMost often

assumption

of incompressible

fluid

density

= const.

viscosity

= const.

Transfer to compressible

flow

via variable density„Compressibility

corrections“

HereWherever

possible:

Generalisation to simple compressible

fluidIdeal gasPerfect

gas (specific

heats

Cp

, Cv

= const.)

Introduction


Overview


equations


• Turbulence


• Summary


TheoryEnsemble average: N realisations

of the same flow

avTurb ttt

Spectral gap(existence not guaranteed)

• Implies steady mean flow• Extension to unsteady flows requires

Averaging

EngineeringTime average: Monitor flow over time t

tt

ttt dttx

tx ),(1)( lim

N

n nN

E txN

tx1

,1, lim


Simple or Reynolds averaging

where simple average fluctuation

Averaging

• Decomposition

• Average of average

• Average of fluctuations

0

General features


Averaging rules

Averaging

• Average of a sum

2121

• Average of a product with a constant

• Average of a product '22

'1121

'2

'12

'1

'2121

'2

'12

'1

'2121

00

2121

Averaging of non-linear termsCorrelations of fluctuations:

new unknownsModelling:

Correlations = functions of averages


Averages of differentials

Averaging

tt

ii xx

(spectral gap)

Linear operation

Differentials in transport equations remain unchanged under averaging


Example: Averaging of the continuity equation

• Constant density

00

k

k

k

k

xU

xU

inconvenient

Averaging

• Compressible

00

k

k

k

k

xU

txU

t

0

k

k

k

k

xU

xU

t


• Compressible


Mass weighted or Favre averaging

~

~

where mass weighted average

fluctuation

• Decomposition

Relation between averages

Averaging

~


• Average of fluctuations

• Average of average

~~,~~~

Averaging rules

Averaging

• Average of a sum

21

2121 ~~

• Average of a product with a constant

~

0~0


Average of triple products with the density

Averaging

''22''1121~~

2121~~

''2

''1

''12

''2121~~~~

''2

''1

''12

''2121

0

~

0

~~~

New unknown requiring modeling as function of average


Example: Mass weighted averaging of the continuity equation


convenient

Averaging

• Compressible

0~

0

k

k

k

k

k

k

xU

txU

txU

t


• Compressible

0~

0

k

k

k

k

k

k

xU

xU

xU

Note − Mass weighted averages simplify the notation, not the physics− For constant density both mass weighted and simple averages are

equal

no change


Overview


equations


• Turbulence


• Summary


Navier-Stokes equations for compressible flow

Flow

equations: Exact

form

• Momentum

0

k

k

xU

t

• Continuity

k

ik

ik

kii

xxp

xUU

tU

• Total energy

k

k

k

iik

k

k

xq

xU

xHU

tE

wherepEH

Specific

total enthalpy


Flow

equations: Exact

form

*2 ijij S

• Viscous stress tensor (Newtonian fluid)

ijkkijij SSS 31*

• Heat flux vector (Fourier’s law)

ii x

Tq

where Tracelessstrain

rate tensor

i

j

j

iij x

UxUS

21 (Simple)

strain

rate tensor

• Molecular viscosity (Sutherland’s law)

STST

TT ref

refref

2/3

Material laws

• Heat conductivity

PrpC

S Sutherland constant Pr Prandtl

number


Flow

equations: Exact

form

RTp

Caloric equations (perfect gas)

where

Thermal equation of state (ideal gas)

vp CCR Specific

gas constant

• Specific total energy

• Specific internal energy • Specific enthalpy

• Specific total enthalpy

2kkUUeE

pEUUhH kk

2

TCe vpeTCh p

., constCC vp Specific heatswhere


Alternative energy equations (1)

Flow

equations: Exact

form

• Step 1: Multiply momentum equation by Ui

and re-arrange terms

k

ik

ii

k

kkinkin

xxpU

xUE

tE )()(

• Step 2: Subtract Kinetic energy equation from total energy equation

kk

iikik

k

k

xq

xUp

xeU

te

where iikin UUE

21)( Specific

kinetic

energy

where )(kinEEe Specific

internal

energy


Alternative energy equations (2)

Flow

equations: Exact

form

• Step 3: Exploit definition of internal energy

kk

iikik

k

k

xq

xUp

xCTU

tCT

vCC

Perfect

gas

Specific

heat

at constant

volume

)( flpCC

Liquid (incompressible

fluid)

Specific

heat

of fluid


Overview


equations


• Turbulence


• Summary


Reynolds averaged Navier-Stokes (RANS) equations for compressible flow (1)

Flow

equations: Averaged

form

0

k

k

xU

t

• Continuity (repetition)

0~

kk

Uxt


Reynolds averaged Navier-Stokes equations for compressible flow (2)

Flow

equations: Averaged

form

• Momentum

k

ik

ik

kii

xxp

xUU

tU

k

ik

ik

kii

xxp

xUU

tU

k

ik

iik

kk

kii

xxpR

xxUU

tU

~~~~

where kiik uuR ~ Reynolds stress tensor

Requires modelling



Flow

equations: Averaged

form

• Total energy

k

k

k

iik

k

k

xq

xU

xHU

tE

where

k

k

i

iik

k

k

k

k

xq

xU

xuH

xUH

tE

~~~

kiiiikk

kiiii

kk

uuuURuh

uuUuUhh

uHuHHHH

21~~

~~21~

~

iikiikiiikiik uUuUU ~~Requiresmodelling



Flow

equations: Averaged

form

• Averaged total energy transport

where

)~()(~~~~~~

k

k

tk

k

k

i

kik

k

iik

k

k Dxq

xq

xU

xUR

xUH

tE

kkikiki

k uuuux

D 21)~(

kt

k uhq )(

Diffusion fluxof specific

kinetic

turbulence

energy

Turbulent heat

flux

Terms require modelling


Reynolds averaged Navier-Stokes equations for compressible flow (supplement)

Flow

equations: Averaged

form

• Averaged internal energy transport

k

k

k

iik

kk

k

iik

kk

k

k

k

k

xq

xu

xpu

xU

xpU

xuh

xUh

te

~~~~~

• Averaged temperature transport

kkk

iikik

k

iikik

k

k

k

k

xT

xxup

xUp

xuTC

xUTC

tTC ~~~~


Flow

equations: Averaged

form

Viscous stress tensor (Newtonian fluid)

**

** ~22~22 ikik

ikikik SSSS

where Averaged molecular viscosity

ikmmikik SSS ~31~~* Averaged traceless

strain rate tensor

Heat flux vector (Fourier’s law)

kkkkk x

TxT

xT

xTq

~1~

where averaged heat conductivity

Exact for

= const.

Exact for

= const.

i

k

k

iik x

UxUS

~~

21~ Averaged (simple)

strain rate tensor


Averaged Sutherland law

STST

TT

STST

TT ref

refref

ref

refref

~

~ 2/32/3

where KS 4.110 Sutherland constant (air)

Thermal conductivity (constant Prandtl

number)

PrpC

Flow

equations: Averaged

form


Thermal equation of state (ideal gas)

RTp

vp CCR specific gas constantwhere

TRp ~

Flow

equations: Averaged

form


Caloric equations of state (perfect gas)

• Specific total energy

• Specific internal energy

• Specific enthalpy

TCeTCe vv~~

TChTCh pp~~

Specific turbulentkinetic energy

iiii Ruuk ~21

21~

., constCC vp

Constantspecific heats

where

kUUeEUUeE kkkk

~2

~~~~

21

where

• Specific total enthalpy

kUUhHUUhH kkkk

~2

~~~~21

Flow

equations: Averaged

form


Special case: Incompressible Newtonian fluid

• Continuity

Kinematic viscosity.constwhere

• Momentum

0

k

k

xU

k

ik

ik

ik

k

kii

xxp

xR

xUU

tU

kk

i

ik

ik

k

ik

i

xxU

xp

xR

xUU

tU

21

kiik uuR Specific Reynolds stress tensor

Flow

equations: Averaged

form


Overview


equations


• Turbulence


• Summary


Turbulence

equations: Fluctuations

Continuity (consider incompressible fluid only)

0

k

k

xU

0

k

k

xU

• exact

• averaged

00

k

kkk

k xuUU

x• difference Fluctuating

velocity

field

divergence

free


Momentum (consider incompressible fluid only)

kk

i

ik

ik

i

xxU

xp

xUU

tU

21

i

ik

kk

i

ik

ik

i

xR

xxU

xp

xUU

tU

21

• exact

• averaged

k

ik

kk

i

ik

ik

k

ik

k

ik

i

xR

xxu

xp

xuu

xUu

xuU

tu

21

• difference

Starting

point for

dissipation

equation

Turbulence

equations: Fluctuations


Overview


equations


• Turbulence


• Summary


Reynolds stress transport equation (1)

Turbulence

equations: Correlations

1. Write exact momentum equation as

0)(

k

ik

iki

k

ii xx

pUUxt

UUN

2. Take following average

0 ij UNu

3. Re-arrange terms by splitting quantities into averages and fluctuations

0 jiij UNuUNu 0)()( jiij UNuUNu→ →



Turbulence


• Time derivatives

t

uutR

tUu

tU

u jiiji

jj

i

~

• Convective terms

k

kji

k

jik

k

ijk

k

kji

k

kij

k

kij

k

kji

xUuu

xU

RxUR

xuuu

xUR

xUUu

xUU

u

~

~~~~~

=0



Turbulence


• Pressure terms/viscous terms

j

ij

i

j

i

j

i

j

i

j

i

j

i

j

i

j

i

ji

xu

xu

xu

xu

xu

xu

xu

xu

xu

xu

where ijp ,

Compressible

only


Reynolds stress transport equation

ijijijijijkijk

ij MDPURxt

R

~~~

k

ijk

k

jikij x

URxU

RP

~~~

~

ij

ij

ijD

Production (exact)

Re-distribution

Dissipation

Diffusion

ijM Mass flux (compressibility)

Turbulence



Re-distribution (pressure-strain correlation)

i

j

j

iij x

uxup

− Excitation of fluctuation ui

’’

by fluctuation uj

’’

(and vice versa)

iU~ iu

p

ji xu /

ji uu ,

Physical significance

Turbulence


Incompressible: traceless 02

i

iii x

up


Diffusion

321ikjjkijikijkkji

kij upupuuuuu

xD

turbulent transport viscous

diffusion pressure

diffusion

contribution

to mean

energy

equation in general

neglected

Turbulence



Why diffusion?

dSndV

xdVD kijk

k

ijkij

Surface

integral

fluxes

diffusion

Volume

integral

Turbulence



Dissipation

k

ijk

k

jikij x

uxu

• Split into

deviatoric

(traceless) and isotropic

part

ijtotD

ijij )()(

32

where ijtot

k

ijk

k

jik

Dij x

uxu

)()(

32

l

kkl

tot

xu )(

deviatoric

(traceless)

total dissipation rate

Common procedureAssume

local

isotropy

neglect

deviatoric

part

(„lumped together with re-distrbution term“)

Turbulence



Incompressible treatment

• Consider difference in viscous diffusion and dissipation Common terms cancel Reconsider remaining terms

ij

xu

xu

xxRD

k

j

k

i

kk

ijij

vij

ˆ

''22

)(

.const

kinematic viscosity

Common procedure

Assumek

i

k

iiitot

xu

xu

2ˆ)(

(= neglect cancelled term)

Turbulence


isotropic dissipation rate


Mass flux

ikikk

jjkjkk

iij px

upx

uM

• Purely

compressible

since

for

constant

density

• Only

important

at high Mach numbers

(> 3…5)

• Neglected

in transonic

flows

0iu

Turbulence



Transport equation of the specific turbulent kinetic energy

• Take trace of the Reynolds stress transport equation

• Consider that kRii~2~

)~()~()()~()~(~~~

kktotkkk

k

MDPUkxt

k

• Result

Compressible

only

Turbulence



Terms of the k-equation (1)

• Production

k

iik

k

xURP

~~)~(

− Production by gradients of the mean velocity field− Reynolds stresses no longer provided need modelling

Turbulence



Effects of curvature and rotation

• Velocity gradient tensor

ijijj

i SxU

~~~

i

j

j

iij

i

j

j

iij

xU

xU

xU

xUS

~~

21~

~~

21~

where Strain

rate tensor(symmetric)

Rotation tensor(anti-symmetric)

• Reynolds stress production term

ikikjkjkjkikij SRSRP ~~~~~~

• k-production term

ikikiik SRPP ~~2

)~(

Rotation tensor

drops

out

• Reynolds stress tensor is symmetric

Turbulence




• Dissipation

− Dissipation by viscous stresses

l

kkl

tot

xu )( Total dissipation rate (≈

isotropic dissipation rate )

• Pressure dilatation (1/2 trace of redistribution term)

k

kk

xup )~(

− Zero for constant density no contribution to k ij

=

redistribution− Usually neglected for transonic flow

Turbulence




• Diffusion),~(),~()~()~( pkvkkk DDTD

− Turbulent transport

kiik

k uuux

T

21)~(

− Viscous diffusion

iikk

vk ux

D

),~(

− Pressure diffusion

kk

pk upx

D

),~( usually neglected

Note )~(),~()~( kvkk DDT

Contribution to total energytransport equation

Turbulence



Turbulence

equations


• Fluctuating mass flux contribution

k

ik

ii

k

xxpuM )~(

− Contribution due to fluctuating density− Only important at high Mach numbers (> 3…5)− Usually neglected


Transport equation for the isotropic dissipation rate

Consider incompressible fluid with ., const

1. Momentum equation of the fluctuations

k

ik

kk

i

ik

ik

k

ik

k

ik

i

xR

xxu

xp

xuu

xUu

xuU

tu

21

2. Multiply byl

i

xu2 and average

3. Result

kkk

kxx

DPx

Ut

2)()()(

Transport equation for the isotropic dissipation rateor -equation

Turbulence



Terms of the -equation

• Production of dissipation

lk

i

l

ik

k

i

l

k

l

i

l

k

l

i

k

i

l

i

k

i

l

k

xxU

xuu

xu

xu

xu

xu

xu

xU

xu

xu

xUP

2)( 2

• Dissipation of dissipation

lk

i

lk

i

xxu

xxu

22

2)( 2

• Turbulent diffusion of dissipation

l

i

l

ik

kll

i

i xu

xuu

xxp

xu

xD

2)(

Huge number of terms

Turbulence



Transport equation for the turbulent heat flux

1. Write exact transport equation for the specific enthalpy

0)(

k

k

hUxt

hhN

2. Take following average

0 hNui

3. Re-arrange terms by splitting quantities into averages and fluctuations

0)( hNui→

ki

k

i Uuhxt

uh ~

Terms on right hand side to be modelled(rarely done)

Turbulence



Overview


equations


• Turbulence


• Summary


Summary

Motiviation

of statistical

modelling

• Applicability

to engineering

problems

Averaging

• Reynolds averages• Favre averages

Mean

flow

equations

• Continuity• Momentum• Total energy• Material laws• Thermodynamics

Turbulence

equations

• Fluctuations• Reynolds stresses• Specific turbulent kinetic energy• Isotropic dissipation rate

Folie 1Introduction to statistical turbulence modelling. Part II, RWTH Aachen, 08.03.2010

Introduction

to statistical

turbulence

modelling

Characteristics

of turbulent flows



Overview: Characteristics

of turbulent flows

• Repetition• Realisability

• Conditions• Invariants

of the

anisotropy

tensor

• Homogeneous

isotropic

turbulence• Wall-bounded

flows

• Near

wall asymptotics• Profiles

• Velocity• Temperature• Turbulent quantities

• Summary



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near



• Summary


RepetitionCharacteristics of turbulent flow

No clear-cut definition, but common features of turbulent flow:• Irregular (random) fluctuations of all quantities

– in time

unsteady– in space three-dimensional

• Vortical

structures (“eddies”)– Continuous spectrum of

scales

– Energy cascade

from large to small scales• Small scale motion Enhanced transfer of

– Momentum Turbulent (Reynolds) stresses– Heat Turbulent heat flux– Mass Turbulent diffusion flux


RepetitionTheoretical description of turbulent flow

Equations derived for • Mean flow (RANS)• Fluctuations• Correlations (Reynolds stresses, kinetic turbulence energy, dissipation

rate)

Questions

Which additional characteristics can be found by• Theoretical conclusion?• Experimental observation?



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near



• Summary


RealisabilityQuestions

• Are there restrictions for the turbulence?• Is any state of turbulent flow possible?

Concentrate on Reynolds stresses• Are there conditions the components have to obey? • What are the bounds of possible states?



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near



• Summary


Realisability: ConditionsReynolds stress tensor

Transformation to principal axes Coordinate system, where off-diagonals = zero

333231

322221

312111

uuuuuuuuuuuuuuuuuu

R symmetric

3

2

1

000000

R where 3,2,1, uu (no summation)

Eigenvalues

0 R is

a positive semi-definite matrix


Realisability: ConditionsFeatures of symmetric positive semi-definite 3x3 matrices

Reynolds stress tensor:

Squares

of fluctuating

velocities

0A

AAA 2Schwarz inequality

6 conditions

for

the

6 components

• Main diagonal elements

• Off-diagonal elements


Realisability: ConditionsComments

• Realisability

not

necessarily

guaranteed

by

turbulence

models

• Enforcing

realisability

component

wise

may

depend

on coordinate

system

• Importance

of realisability

unclear(popular

non-realisable

models)



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near



• Summary


Realisability: Invariants

of the

anisotropy

tensorEigenvalues

of a 3x3 matrix

• Characteristic

equation

0det 23 IIIIII AAAA

where

ii

I

AAAAA

332211

jiijjjii

II

AAAA

AAAAAAAAAAAAA

21

322333223113331121122211

kijkijkjjkiikkjjii

III

AAAAAAAAA

AA

2361

det

(trace)

Eigenvalues

independent of coordinate

system

Coefficients


system

invariants


Reynolds stress anisotropy tensor

• Definition

ijij

ij kR

b 31

~2

~~

• Characteristics

• parallel to Reynolds stresses

(identical

principal

axes)

• non-dimensional

• symmetric

• traceless


of the

anisotropy

tensor


Anisotropy tensor in principal axes

3,2,1,31

~2

~~

kRb

• Minimum

,3,2,1,,0~b

31~min0~min

bR

• Maximum

32~min~min~max0~~~

bbbbbb

• One component

on minimum

bbbbbb ~

31~min~~

31~min~

Straight

line


of the

anisotropy

tensor


Bounds of the anisotropy tensor

Tracelessness

allows

consider

two

components

only


of the

anisotropy

tensor


Invariants of the anisotropy tensor

0~~~~

bbbbI

• First invariant: Trace

• Second invariant

222 ~~~21~

bbbbII

• Third

invariant

333 ~~~31~

bbbbIII

Symmetry

with

respect

to all indices

in principal

axes


of the

anisotropy

tensor


Invariants of the anisotropy tensor

• Exploit

tracelessness

• plot

over

b

-b

-plane


of the

anisotropy

tensor


Exploit symmetry of the invariants

• Symmetry

axes

in the

b

-b

-plane

(tracelessness)

bb ~~

bbbb ~

21~~~

bbbb ~2~~~


of the

anisotropy

tensor

bbb ~~~

• Tracelessness



3 intersecting

symmetry

lines

6 equivalent

triangles

B-A-C

A: zero

b-components

equal

R-components

isotropic

turbulence

B: two

minimumb-components

2 zero

R-components

1-C turbulence


of the

anisotropy

tensor



B-C: one

minimumb-component

One zero

R-component

2-C turbulence

B-A-C: symmetry

lines two

identical

b/R-components

Axisymmetricturbulence C: axisymmetric

2-C turbulence


of the

anisotropy

tensor

3 intersecting

symmetry

lines

6 equivalent

triangles

B-A-C


of the

anisotropy

tensor


Invariants along B-A-C

A: Isotropic

turbulence

Introduce

b-relations

into

invariant definitions

0~IIb

B: 1-C turbulence 31~

IIb272~

IIIb

0~IIIb

C: axisymmetric

2-C turbulence 121~

IIb1081~

IIIb

B-C: 2-C turbulence

91

31~~~

bbbII

31~~

31~

bbbIII

B-A-C: axisymmetric

turbulence

2~3~ bbII 3~2~

bbIII

2~43~

bbII 3~41~

bbIII

B-A

A-C


of the

anisotropy

tensor


Invariants along B-A-C and B-C


of the

anisotropy

tensor


Invariant map: Range of possible turbulent states

Obtain

bII

= f(bIII

) by

eliminating

b

B-C: 2-C turbulence

IIIII bb ~391~

B-A-C: axisym. turbulence2/3

3

~2~

II

IIIbb


system

isotropic


of the

anisotropy

tensor

2-Caxisym.

axisym.

axisym.

2-C

1-C



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near



• Summary


Homogeneous

isotropic

turbulence

Defintions

Homogeneous

Statistics

independent of position

no gradients

of averages

Isotropic

Invariant against

rotation and reflection

of coordinate

system

Identical

principal

Reynolds stresses

k

k

k

R

~3200

0~320

00~32

~ (in any

coordinate

system)


Homogeneous

isotropic

turbulence

K-equation

• Incompressible

fluid,

= const.

• No spatial

gradients

tk

0

k

i

k

i

xu

xu

• Isotropic

dissipation

rate

k decays

in time

• Experiments for

large t

06.025.1, ntk n



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near



• Summary


Wall-bounded

flows: Near-wall

asymptotics

Incompressible

fluid

• Taylor series

expansion

around

wall point

33

32

2

2

61

21

ww

iw

w

iw

w

iwii yy

yuyy

yuyy

yuuu

where

0

0

w

wi

y

u

33

32

2

2

61

21 y

yuy

yuy

yuu

w

i

w

i

w

ii

so that

any

velocity

component


Wall-bounded

flows: Near-wall

asymptotics

Continuity

00

00

wwww yv

zw

yv

xu

where

,!

1

wn

n

n yu

na

,!

1

wn

n

n yv

nb

w

n

n

n yw

nc

!

1

Thus

33

221

33

22

33

221

ycycycw

ybybv

yayayauv‘

negligible


Wall-bounded

flows: Near-wall

asymptotics

Specific

Reynolds stresses

consistent

with

boundary

conditions

0

0

w

ij

wij

yR

R

43

12

3211

4321

3211

5422

3211

yOycbwvR

yOycawuR

yOybavuR

yOyccwwR

yOybbvvR

yOyaauuR

yz

xz

xy

zz

yy

xx

Specific

kinetic

turbulence

energy

3211112

121 yOyccaawwvvuuk


Wall-bounded

flows: Near-wall

asymptotics

Repetition: Specific

kinetic

turbulence

energy

3211112

1 yOyccaak

Isotropic

dissipation

rate

2221111 4 ybbccaa

xu

xu

j

i

j

i 21111 yOccaa

w

Non-zero

wall value

Relations

20

2limyk

yw or

ww y

k2

2

(near-wall

extensionsof k-

models)



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near

wall asymptotics• Profiles:


• Summary


Wall-bounded

flows: Profiles: Velocity

Incompressible

boundary

layers

w

ref uU

Self

similarnear

wall profile

Frictionvelocity

• Inner scaling

u

Lref lengthscale

uyy

uUu

/,

yfu

where


Wall-bounded


Incompressible

turbulent boundary

layers

xyxy R

yxp

yUV

xUU

1

• x-Momentum

where2

21yU

yxy

0

yyRpy

• y-Momentum

wherepR yy (approx.) constant

pressure

along

y


Wall-bounded


Channel

0,0 V

xU

xyxy R

yxp

10

• x-Momentum

Flat

plate

0

xp

• x-Momentum

xyxy R

yyUV

xUU

Zero convection Zero pressure

gradient


Wall-bounded


Viscous

sublayer

(y+

≤

3…5) (region

of near

wall asymptotics)

xyxy R

yxp

10

• Immediately

at the

wall convection

negligible

• Integrate

over

y with

b.c.

2

0

uw

y

xy

whereyUxy

21

uR

yUy

xp

xy

• Near-wall

limit, y→0:2 u

yU

• Integrate

over

y with

no-slip

b.c. 00

yU yu

0lim1lim00

xy

yyRy

xp

linear


Wall-bounded


Log-layer

(y+

≥

30…60)

• Universally

observed

relation

Cyu ln1

where 41.0 von Karman

constant

5.55C

• Fairly

independent of pressure

gradient

(except

separation)

• Important

relation

for

turbulence

model

calibration

• y+

range

increases

with

Reynolds number

Ongoing

debate

• Universality

of coefficients

• Power law

vs. log-law


Wall-bounded


Defect

layer

(y ≥

0.2 )

• Similarity

of velocity

defect

in outer

scaling

Fu

UU e where

eU velocity

at boundary

layer

edge

y

boundary

layer

thickness( displacement

thickness)

Defect

layer

is

largest

portion

of the

boundary

layer


Wall-bounded


Intermittency

• Varying

periods

of turbulent and laminar

flow

for

0.4 ≤

y/

≤

1.2

16

5.51

y

• Intermittency

factor

:

turbulent time fraction

• Approximation


Wall-bounded


Interpretation of log-law

as overlap

region

(1)

• Viscous

sublayer: yfuU sub

• Postulate overlap

region

where

both

are

valid

• Defect

layer: FuUU edef

FuUyfu e

n

n

nn

nn

n

n

dFdu

dyfduu

yU

Identical

velocity

Identical

derivatives

region

• Self-similarity

only

variables y+

and

occur

multiply

with

yn/u n

nn

n

nn

dFd

dyfdy


Wall-bounded


Interpretation of log-law

as overlap

region

(2)

• Condition must

hold in more

than

one

point

).(nconstd

Fddy

fdy n

nn

n

nn

• const.(n) independent of y+

and 1,1 y

• Define

const.(n=1) = 1/

• Check higher

derivatives

Cyyfdydfy

ln11

).(!11 1 nconstndy

fdy nn

nn


Wall-bounded


Determining

cf

from

the

log-law

• Re-write

velocity

and wall-normal coordinate

• log-linear with

A,B = f(u

/Ue

)

• Re-write

log-lawe

e

e

e

UuUyy

uU

UUu

B

CUu

UuUy

AUu

UU

ee

e

ee

ln1ln1

2,

2locf

e

w

e

c

UUu

U(y), Ue

measured

known


Wall-bounded


Clauser

plot

• Plot experimental data

asU/Ue

vs. ln(yUe

/)

• Best agreement

with

exp.

cf

• Plot log-law

as

BUyAUU e

e

ln

for

various

cf

Result

depends

on chosen

values

for

and C



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near



• Summary


Wall-bounded

flows: Profiles: Temperature

Temperature

profile

• Inner scaling

uCqT

p

wref Friction

temperature

u

Lref Length

scale

wq Wall heat

flux

pC Specific

heat

with

• Self-similar

normalised

temperature

difference

near

the

wall

yfT

TT w


Wall-bounded


Wall layer

• Immediately

near-wall wall normal molecular

heat

conduction

dominating

0

yqy

• Energy equation

• Integrate

with

b.c.

wyy qq 0

whereyTqy

wqyT

• Integrate

with

b.c.

wyTT

0

yyqTT ww Pr

analogous

to

viscous

sublayer


Wall-bounded


Log layer

• Experimental observation

where

5.7Pr7.13Pr

47.03/2

fC

Cy ln1

• Turbulent Prandtl number

87.0Pr

t

Wall layer

decreasing

with

increasing

Pr

(strictly

valid

only

forboundary

layers)

Exp. Data collected

by

Kays



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near



• Summary


Wall-bounded

flows: Profiles: Total stress

Constant

total stress layer

• Integrated

near-wall

x-momentum

equation

(convection

neglected)

21

uRyxp

xyxy

whereyUxy

• Pressure

term

decreases

faster

than

Reynolds stress towards

wall

2

uR

yU

xytot

Constant

specific

total stress

• Inner scaling

1

xytot R

yu

where 2

uRR xy

xy


Wall-bounded

flows: Profiles: Total stress

Channel

flow

DNS by

Moser et al.

Constant

total stress layer

• Only

very

close

to the

wall

• Extent

increases

with

Reynolds number


Wall-bounded

flows: Profiles: Reynolds stress

Constant

Reynolds stress layer

• Outer

part

of constant

stress layer

assume

viscous

effects

negligible

1

xyxy RRyU Constant

specific

Reynolds stress

• Underlying

assumptions:

• Close enough

to the

wall convection

and pressure

term

negligible

• Far enough

away

of the

wall viscous

effects

negligible

„overlap

region“

identified

with

log layer


Wall-bounded

flows: Profiles: Reynolds stress

Collection

of experimental flat

plate

data

by

Fernholz&Finley

Constant

Reynolds stress layer:

• existence

not

evident

• probably

requiresvery

high Re

• probably

fairly

small(re-call

total stress layer)


Wall-bounded

flows: Profiles: Bradshaw relation

Observation

• Large part

of flat

plate

boundary

layer

shows

12a

kRxy

where 15.01 a Structure

parameter

• Constant


1

2

1 22 au

aRk xy

Constant

specific

kinetic

turbulence

energy


Wall-bounded

flows: Profiles: Bradshaw relation

Collection

of experimental flat

plate

data

by

Fernholz&Finley

Structure

parameter

• range: 0.1 ≤

a1 ≤

0.16

• indication

of plateau

with

0.14 ≤

a1

≤

0.16


Wall-bounded

flows: Profiles: Turbulent equilibrium

Zero pressure

gradient

boundary

layers, pipe/channel

flow

• Assumption

for

k-equation kP equilibrium

• Boundary

layer

yURP xy

k

• log law

yu

yU

• constant

Reynolds stress

2uRxy

• isotropic

dissipation

rate

yu

3


Wall-bounded

flows: Profiles: Turbulent equilibrium

Channel

flow

DNS data

by

del Alamo

et al.

Equilibrium

layer

• reasonable

approximation

• Increasing

Re

better

agreement


Wall-bounded

flows: Profiles: Maximum k-production

Constant

total stress (near

wall region)

• integrated

x-momentum

yUuRxy

2

• k-production

yU

yUu

yURP xy

k

2

• Maximum at

41

/maxmax

4max

2 4,

42

max

uPPuPu

yU k

kk

P

• Reynolds stress at production

maximum

21

2/max

2max

max

2

maxmax

u

RRu

yUPR P

xy

Pxy

P

k

Pxy


Wall-bounded

flows: Profiles: Maximum k-production

Channel

flow

DNS data

by

del Alamo

et al.

Maximum k-production

• max(P(k),+) = 0.25 asymptotically

reached

for

high Re

• -Rxy+|max(P)

= 0.5 confirmed

• at constant

positionin buffer

layer,

approximately

at intersection

of viscous

sublayer

and log layer


Wall-bounded

flows: Profiles: Heat

flux

Constant

total heat

flux

layer

• Near-wall

energy

equation

(convection/mechanical

forces

neglected)

.0 constqqqqqx w

tt

k

• Outer

part

of constant

total heat

flux

layer: viscous effects negligible

w

t qq Constant

turbulent heat

flux

Analogous

to constant

total/Reynolds stress layer

Associated

with

temperature

log law

Constant

turbulent heat

flux

layer



of turbulent flows



of the

anisotropy

tensor

• Homogeneous

isotropic


flows

• Near



• Summary


Summary

Realisability

• Reynolds stress tensor

is

positive semi-definite

• Anisotropy

tensor

is

bounded: -1/3 ≤

b()

≤

2/3

• Invariants

bII

, bIII

are

bounded

• Limiting

states:

• 2-C turbulence

• axisymmetric

turbulence

• Additional limiting

states

• isotropic

turbulence

• 1-C turbulence


Summary

Homogeneous

isotropic

turbulence

• specific

kinetic

turbulence

energy

decays

• for

long

t: k ~ t-n, where

n ≈

1.25


Summary

Wall bounded

flows

(boundary

layers, channels)

• Velocity profile:

• Viscous

sublayer

(y+

≤

3…5)

• defect

layer

(y ≥

0.2)

• log layer

(overlap

region, y+

≥

30…60)

• Temperature

profile: similar

structure

• Turbulence:

• Constant

total stress layer

(near

wall layer)

• Constant


(outer

part

of near

wall layer

log law)

• Bradshaw relation

(-Rxy

/k

= const.)

• Turbulent equilibrium

(P(k)

= )

• Maximum k-production

Folie 1Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010


Turbulence models



Overview

• Repetition: Characteristics of turbulent flows• Eddy viscosity models

– Boussinesq hypothesis– Algebraic models– 2-equation models

• K-

model• K-

models

– Wilcox model– Menter BSL model– Menter SST model

– 1-equation models• Eddy viscosity transport• Spalart-Allmaras model

• Summary


Overview



• K-

model• K-

models



• Summary


Repetition: Characteristics of turbulent flows

Homogeneous isotropic turbulence

• No spatial gradients of means• Experiments:

ntk

where 06.025.1 n

for long times t



Viscous sublayer• Turbulent effects negligibleLog layer• overlap region• viscous effects negligible• convection/pressure gradientnegligible

with inner scaling

uUu

yuy

Friction velocity

wu

Cyu ln1



Log layer

2uRxy

• Constant Reynolds stress

y

uP k

3

• Equilibrium

1

2

1 215.0

2 auka

kRxy


Near-wall layer

• Constant total stress

2 uR

yU

xy


Overview



• K-

model• K-

models



• Summary


Boussinesq hypothesisPhenomenological consideration

1. Turbulence increases the drag increase the viscosity *)( ~2~ij

tij SR

2. Trace of the Reynolds stress tensor

where

kRii~2~

i

j

j

iij

ijkkijij

xU

xUS

SSS

~~

21~

~31~~* Traceless strain rate tensor

(Simple) strain rate tensor

t Eddy viscosity• to be provided byturbulence model

• depends on flow (not on fluid)

3. Extension necessary

ijijt

ij kSR ~32~2~ *)(

Boussinesq hypothesis

where


Boussinesq hypothesisTurbulent heat flux

Turbulence increases the heat flux increase the heat conductivity

i

tti x

Tq

~

)( t Eddy heat conductivity• to be provided by turbulence model• depends on flow (not on fluid)

Equivalent Boussinesq hypothesisfor turbulent heat flux

where


Boussinesq hypothesisTurbulent Scales

tU

• Dimensional analysis

Turbulent velocity scale, standard choice

tttt LUlengthvelocitytime

length

2

kU t ~

tL Turbulent length scale („characteristic size of eddies“)

t

tt

ULT Turbulent time scale

• Re-write Boussinesq hypothesis

ijijt

ij kSR ~32~2~ *)(

tt Kinematic eddy viscositywhere

Turbulent Reynolds number

tt

tLUR High turb. Reynolds number

Viscous effects negligible


Boussinesq hypothesisEddy viscosity in log layer

• Constant Reynolds shear stress

yuuyUR tt

xy

2)(

• Consider incompressible fluid

.., constconst

• Log layer

yu

yUCyu

ln1

Linear increase

uU t

yL t

Velocity scale

Length scale

where


Boussinesq hypothesisEddy viscosity in viscous sublayer layer

• Herring and Mellor (1968)

t

yFut

• Higher damping towards wall

• Van Driest (1956)

yl

F mixt

2141 2

t

F Damping function (empirical)where

where

26

exp1

0

0

A

Ayylmix

33

3

HMyyF

t

where 9.6HM


Boussinesq hypothesisEddy viscosity in inner part of boundary layer

• Van DriestDamping extends intolog layer (y+ > 100)

• Mellor and HerringDamping restrictedto buffer layer (y+ < 60)


Boussinesq hypothesisEddy viscosity in defect layer

• Clauser (1956)Turbulent scales in defect layer

e

t UU Velocity at boundary layer edge

dyU

UULe

et

0

* Displacement thickness

• Eddy viscosity by formal procedure * et U

• Calibration to experiments

0180.0

0168.0

Zero pressure gradient

Average of all experiments


Boussinesq hypothesisEddy heat conductivity in log layer

• Constant turbulent heat flux

yuCqyTq p

tw

tty

• Log layer

yT

yTCy

ln1

uC

qTp

w

Derivation for log layer, but used generally

where

• Compare with eddy viscosity

t

tpt

t

pt

t CC

PrPr/1

87.0Pr twhere

T

TT wand


Overview



• K-

model• K-

models



• Summary


Algebraic modelsMixing length concept (Prandtl, 1925)

• Consider eddy in shear layerat velocity U(y)

• Lateral fluctuation velocity Vmix lateral shift lmix lateral momentum exchange

• Assumption

yUClV mixmix

• Turbulent scales

mix

t

mixmixt

lL

yUl

CVU

Eddy viscosity

yUlLU mix

ttt

2


Algebraic modelsMixing length in boundary layers

• Log layer

• Viscous sublayer: Damping required mixl

mix yFl

• Van Driest (1956)

0

exp1AyF mixl

yLlyL tmix

t

where 260 A


Algebraic modelsMixing length and eddy viscosity damping

• Consider constant total stress layer

mix

mix

l

ul

yU

2

4112

2

1412

2

ul

yUl

mix

mixt

2

2

2 u

R

yUl

yU

xy

mix

• Comparison with general formula

t

Fyt

yl

F mixt

2141 2


Algebraic modelsBaldwin-Lomax model (1978) (1)

• Inner layer: Generalized log law with Van Driest damping

2mix

ti l

• Generalized velocity gradient

U

0

exp1Ayylmix

Vorticity



• Outer layer: Clauser relation

*

ett

Klebwakecpt

o

UUL

FFC

• Wake function (defect layer)

max

2max

maxmax ;minF

UyCFyF dif

wakewake

where maxmax

max max1

Fyy

lF mixy

along wall normal direction y

U

U

UUy

e

ydif

minmax

(crossover at i(t) = o

(t))



• Intermittency: Klebanoff function

• Model coefficients

16

max

5.51

yyCF Kleb

Kleb

von Karman constant40.0

260 A Van Driest damping

0168.0 Clauser (defect layer)

3.01

6.1

Kleb

wake

cp

CC

CBy calibration


Algebraic modelsDegani-Schiff modification (1986)

• Free vortices (delta wings) Strong 2nd maximumWrong predictions

• RemedyTake wall-nearest maximuminstead of absolute maximum


Algebraic modelsRemark on algebraic models

• Mixing length depends on specific flow, e.g.• boundary layers• free shear layers

models lack generality

• In the past popular for attached boundary layer flows (airfoils)• Bad predictions in case of separation or strong shocks• Requires wall-normal search implies structured solution method (grid line = search direction)

• Search conflicts in case of opposite walls not suitable for general 3D flows

Remarks on Baldwin-Lomax model


Overview



• K-

model• K-

models



• Summary


Consider incompressible fluid

.

~.

const

const

2-equation modelsIdea

• Independent turbulent scales L(t), U(t)

Provide individual transport equations• Velocity scale U(t)

provided by k• Length scale L(t)

provided by some suitable variable


Overview



• K-

model• K-

models



• Summary


k-equation

)(2

k

kk

k

k

k DxxkP

xkU

tk

k

iik

k

xURP

j

i

j

i

xu

xu

jjiij

k upuuux

D21

Production

Isotropic dissipation rate

Turbulent diffusion

2-equation models: K-

model



model

Modelling the k-equation

jk

t

j

ijijt

k

kxk

xSS

xkU

tk

2

• Production: Boussinesq hypothesis

ijij

t

j

iij

k SSxURP 2

where

i

j

j

iij

xU

xUS

21

• Turbulent diffusion: Gradient hypothesis

kk

t

k

pkkk

xk

xDTD

)(

),()()(

• High turbulent Reynolds number neglect viscous diffusion

Result


-equation

)(2

Dxx

Px

Ut kkk

k

Production

Dissipation

Turbulent diffusion

Huge number of terms to be modelled !


model

lk

i

l

ik

k

i

l

k

l

i

l

k

l

i

k

i

l

i

k

i

l

k

xxU

xuu

xu

xu

xu

xu

xu

xU

xu

xu

xUP

2)( 2

lk

i

lk

i

xxu

xxu

22

2)( 2

l

i

l

ik

kll

i

i xu

xuu

xxp

xu

xD

2)(



model

Modelling the -equation

• Production: Scale k-production

kPk

CP

1


j

t

j xxD

)(

)(

• High turbulent Reynolds number neglect viscous diffusion

• Result

• Destruction: Scale k-dissipation

kC

kC

2

22

j

t

j

k

k

kxxk

CPk

Cx

Ut

2

21



model

Eddy viscosity

• Velocity scale

kU t

• Length scale

2/3kL t

• Result

2kct


Calibration for homogeneous isotropic turbulence

kC

dtddtkd

2

2

Comparison with experiment for large t

84.1176.1 2

n

nC

11

0

02

0

211

Ct

kC

kk

where 06.025.1 nntk


model

00 ,kwhere initial values


Calibration for boundary layer (log law)

cuk

2

yukct

2

• Equilibrium

yuP k

3

• Constant Reynolds stress


09.0222

21

1

2

1

aca

uaRk xy

Equilibrium parameter

• -equation, convection neglected

cCC

xxkCP

kC

j

t

j

k2

21

2

21 0

• Diffusion coefficients from numerical experiments


model


Standard values of closure coefficents (Launder-Sharma, 1974)

• Diffusion

• -destruction

92.12 C

Rule of thumb: Order one

• -production

44.11 C

• Equilibrium parameter

3.10.1

k

slightly high

implies

= 0.433 slightly high

09.0c

(fine-tuned for free shear layers)


model


Remarks on the k-

model

• Most popular model in numerical codes• Wide range of applications• Not preferred in aerodynamics• Known problems

•„Low-Re“ extensions for near-wall region required:- Numerically stiff (non-linearities)- Missing physical background- Alternative: Bridging by wall-functions

• Bad predictions for positive pressure gradient (separation)

• Reason of problemsModeled -equation


model


Overview



• K-

model• K-

models


– 1-equation models• Eddy viscosity transport• Spalart-Almaras model

• Summary



models: Wilcox model

Wilcox‘ reasoning

• Drastic simplification of exact -equation physical accuracy is fictitious

• Replace -equation by emprical length scale equation• Enforce favourable near-wall characteristics

Wilcox‘ choice

k

*

1

„Specific dissipation rate“

where c*




Modelling the k-equation

j

t

j

ijijt

k

kxk

xkSS

xkU

tk **2

• Production: Boussinesq hypothesis

ijij

t

j

iij

k SSxURP 2

where

i

j

j

iij

xU

xUS

21


j

t

j

k

xk

xD )(*)(

• Low turbulent Reynolds number keep viscous diffusion

Result

• Destruction: convert

into

k* where c*



model: Wilcox model

Modelling the -equation

• Production: Scale k-production

kPk

P


j

t

j xxD )()(

• Low turbulent Reynolds number add viscous diffusion

Result

• Destruction: Scale k-dissipation

2*ˆ kk

j

t

j

k

k

kxx

Pkx

Ut

2

DPx

Ut k

k

• Transport equation

same structure as -equation




Eddy viscosity

• Velocity scale

kU t

• Length scale

*

kL t

• Result

kt




Calibration

• Same procedure as for k-

model

/0

0

*

1 tkk

• Homogeneous isotropic turbulence

31.119.1*

n

*

2

uk

• Equilibrium • Bradshaw relation

09.0* 0756.00687.0

*

2

*

• -equation, log law


Values of closure coefficents (Wilcox, 1988)

• Diffusion

• -destruction

075.0

• -production

5556.09/5

• Equilibrium parameter

5.05.0*

within experimental range

Implies

= 0.408

09.0*




Near wall behaviour

• Convection and production ((t)) negligible only viscous diffusion and destruction



22

2

*2

2

y

kyk

• Solution

202

*

1lim66

23.341121,

yy

nyk

yw

n

Near-wall asymptotics: n = 2

Singularity at the wall


Wall boundary conditions

• k-equation:Natural boundary condition



0w

k

• -equation: Approximation of infinite value

21

16y

Fw

where

Menter (1994): Extrapolate

from near wall point

10F


Features of the Wilcox model• Improved near-wall behaviour (viscous sublayer)• Improved prediction of flows with positive pressure gradient• Free stream sensitivity (not observed with k-

model)

- Solution depends on

at boundary layer edge- Low -value spoils solution- -value cannot be controlled (decay in free stream)

Note: Free stream sensitivity mainly important in free shear layers




Overview



• K-

model• K-

models



• Summary



models: Menter BSL model

Menter idea• Combine k-

near the wall with k-

further away

Method• Write -equation in terms of

= *k

• Insert k-equation• Neglect minor terms

Result

j

t

j

ijijt

k

kxk

xkSS

xkU

tk **2

D

jj

d

j

t

j

k

k

k

C

xxk

xxP

kxU

t

2

identical

Cross-diffusion term suppresses free stream sensitivity


Model coefficients• Blending between k-

and k-

11 1 FF kk

• Blending function

41 tanh F

where 321 ,,maxmin

and

06

500lim5002022

yy

yL

yk t

*1

2103

10,

2

yxx

kk

kk




Remarks on the Menter BSL model• Not very common• Intermediate step towards SST model• Similar behaviour as Wilcox model, except freestream sensitivity




Overview



• K-

model• K-

models



• Summary



models: Menter SST model

Menter idea• Improve sensitivity to separation

Method• Write Reynolds stress –Rxy in terms of P(k)/

yUk

yUR t

xy

Reynolds stress

k-production

xy

k

xyk

RP

yU

yURP

Isotropic dissipation ratek

k **

Result

k

xyPkR * Bradshaw relation kaRxy

*

2 1

Increases with P(k)/(pressure rise) Independent of P(k)/




Limit eddy viscosity• Standard definition

kt

• From Bradshaw relation

yUka

kayUR

t

txy

/2

2

1

1

• Limitation

yUaka

yUkakt

/,2max2

/2,min

1

11

where

yU vorticity

(general 3D flows)




Restrict limitation• Bradshaw relation holds only for boundary layers Restrict limitation to near wall region

21

1

,2max2

Fakat

Notes• F2 reaches further than F1• Re-write -production term

kt

k PPPkP

• k-diffusion coefficient re-calibratedfor proper flat plate cf

where 2212 ,2maxtanh F




Closure coefficients

* * d

K- 0.09 0.0750 0.85 0.500 0

K- 0.09 0.0828 1.00 0.856 2*

Coefficient of -production from calibration for log law

*

2*

*




Results for RAE 2822

Improvement at shock (pressure rise) by SST limitiation

Ma = 0.73 Ma = 0.75


Remarks on the Menter SST model• Standard model in aerodynamics• Enhanced sensitivity to separation Convergence problems possible in case of separation

• Often improved prediction of shock position• Sometimes separation predicted to early

Model variant (Menter 2003)• Replace vorticity by strain rate in SST limitation may reduce sensitivity to separation




Overview



• K-

model• K-

models



• Summary


Consider k-

model (Menter, 1997)

1-equation models: Eddy viscosity transport

• Eddy viscosity

2kct

• Convection of eddy viscosity

k

k

k

k

k

k

k

t

k

t

xU

tkc

xkU

tkkc

kx

Uckt

cx

Ut

2

22

2

k-equation -equation


Eddy viscosity transport equation


ttt

DPx

Ut k

t

k

t

where

kPkcCPt

12

kcCt

22

j

t

jj

t

jk xxk

xk

xkcD

t

12

with ijij

tk SSP 2

Problem• Missing relation between (t), k and


Idea (Menter, 1997)


yU

ccRk

txy


where

• Generalization for 3D

Sc

kt

ijij SSS 2

• Consequence

2

2

Skc tt




(Neglect minor contributions to diffusion term)

• Production ScP tt

1

ttt

DPx

Ut k

t

k

t ~

2

2

vK

t

Lc

t

jj

tt

j

t

j

t

j

tt

j xxxxxxD

t

21~

• Dissipation

• Diffusion

Von Karman length scale

jj

vK

xS

xS

SL

where

Log layer yLvK


Comments


• Explains relation between models• Not the basis for common model development


Overview



• K-

model• K-

models



• Summary


Modelling ideas

1-equation models: Spalart-Allmaras model

• Empirical transport equation for modified eddy viscosity• Step-by-step extension for various flow types

• Free shear flows• Boundary layers at high turbulent Reynolds numbers• Boundary layers at low turbulent Reynolds numbers

• Results of previous steps always maintained


Step 1: Free shear flows (1)


• Define generic transport equation for eddy viscosity

tt

DPx

Ut k

t

k

t

• Production ScP t

bt

1

Almost identical to Menter derivation

S vorticity

• Diffusion

j

t

j

t

bj

tt

j xxc

xxD

t

2

1

where

• No dissipation


Step 1: Free shear flows (2)


• Solution for constant velocity

• Calibration by consideration of various shear flowsOptimum values:

• No decay in isotropic homogeneous turbulence• Accepted, because focus is on boundary layers

.constt

32

622.01355.0

2

1

b

b

cc Menter derivation: 144.0121 CCccb


Step 2: Extension to boundary layers, high Rt (1)


• Boundary layer at high Rt = log layer

Velocity gradient:

yut Eddy viscosity:

yuS

Convection negligible


ttt

uccDP bb

2221 1

Balance requires dissipation term in boundary layers




• Dissipation modelling

Log layer

Far away the wall

2

1

221 1

y

c

ccft

w

bbw

t

uy

yuy

t

0lim

y

t

y

• Too slow decay observed in defect layer Damping function fw < 1 needed for correct cf




• Damping function fw :Consider mixing length

• Mixing length models

SlSl

tDef

mixmixt 2

• Log layer yl Logmix

Ratio is suitable argument

22

2

ySllr

t

Logmix

Defmix

Log layer: r = 1

Boundary layer edge: r = 0




• Modelling the damping function fw :

where rrcrrg w 62

ghrgfw

Intended damping function

6/1

63

6

631

w

w

cgcgh

Prevent singularity

grSS 00limlimFree stream Singular

6/16300

1limlim wSwScghrgf

Free stream Regular




Calibration: cw2 = 0.3, cw3 = 2


Step 3: Extension to boundary layers, low Rt (1)


Idea:• Maintain linear behaviour in sublayer new transport variable

~

• Wall damping of eddy viscosity

1~

vt f

• Damping function: According to Herring & Mellor

3

13

3

1 /~/~

vv c

f

where cv1 = 7.1 (instead of 6.9)




Production term modification

yuS

2uS

Log layer

Viscous sublayer

Log layer behaviour requires

ScP b~~

1~

where

yuSSS

~

Behaviour changes !

and 22~~

ySr

(dissipation)




Determining S

tvu

yuSSS

2~

2

uSR txy

xy

• Constant total stress layer = log layer + viscous sublayer

by definition

• Exploit definition

1

~v

t f

• Result

22

~vfy

S

where

yu

~ Log layer behaviour

where 12 /~1

/~1

vv f

f


Complete model


Transport equation

• Dissipation

• Production

~~~~~DP

xU

t k

k

ScP b~~

1~

2

1~ ~

ycf ww

• Diffusion

jj

bjj xx

cxx

D

~~~~1

2~

• Eddy viscosity 1

~v

t f Low Rt extension


Remarks on the Spalart-Allmaras model• Standard model in aerodynamics• Numerically very robust, in particular on unstructured grids• Not particularly sensitive to separation

Model variant: Edwards modificationClaim: Enhance numerical robustness• Replace vorticity by strain rate• Approximate r differently• No particular improvement noticed• 2% lower cf



Overview



• K-

model• K-

models



• Summary


Boussinesq hypothesis• Eddy viscosity concept• Near wall behaviour (viscous sublayer, log layer, defect layer)

Algebraic models• Mixing length concept• Baldwin-Lomax model

2-equation models• Relation to fundamental transport equations• Calibration methodology• Development history: k-

Wilcox k-

Menter BSL & SST

1-equation models• Relation of eddy viscosity transport to k-

models

• Development steps of Spalart-Allmaras model

Summary

Folie 1Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010


Differential Reynolds stress models



Overview: Differential Reynolds stress models

• Repetition: Boussinesq hypothesis• Differential Reynolds stress models

• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach

• Summary





• Summary


Repetition: Boussinesq hypothesis

Reynolds stress tensor

ijijt

ij kSR ~32~2~ * Parallel to viscous stresses

not generally valid

Normal stress anisotropy

Channel flow: 0~~),(~ WVyfU

k

kyU

yUk

R t

t

ij

~3200

0~32/~

0/~~32

~

00000/~0/~0

21~* yU

yUSij

Identical normal stresses Contradicts experimental observation


Repetition: Boussinesq hypothesis

K-production

ijij

ijijijij

ijijij

j

iij

k

SR

RSR

SR

xURP

~~0

~~~~

~~~

~~

where

i

j

j

iij x

UxUS

~~

21~

i

j

j

iij x

UxU

~~

21~

symmetric

antisymmetric

Contribution of rotation drops out Excessive dissipation of vortices





• Summary


Differential Reynolds stress models

Reynolds stress transport equation

k

ijk

k

jikij x

URxU

RP

~~~

~

ij

ij

ijD

Production (exact)

Re-distribution

Dissipation

Diffusion

ijM Mass flux (compressibility)

ijijijijijkijk

ij MDPURxt

R

~~~





• Summary


Analytical solution for homogeneous turbulence (Rotta, 1951)

“Slow term” Aij

m

lmjli

miljijij x

UaaA

(far away from walls)

• By physical consideration: Return to isotropy (Rotta, 1951)

Constraints on (by theoretical considerations)milja

1u2u

3u

1u2u

3u

anisotropic isotropic

Differential Reynolds stress models: Redistribution


Launder-Reece-Rodi model (LRR)

Consider most general tensor that• is linear in Reynolds stresses• fulfills the constraints by Rotta on

• Result: Only one free parameter for calibration

milja

“Rapid term”

Extension to wall bounded flows

Wall-reflexion terms• Enhance anisotropy • According to “slow”and “rapid” terms

• Dependence on wall normals ni nj



Speziale-Sarkar-Gatski model (SSG)

• Extension of LRR by non-linear terms• Coefficients calibrated individually• No wall-reflexion terms needed• Extension to wall-bounded flows by Chen

Hanjalic-Jakirlic model (HJ)

• Simplified LRR with non-constant coefficients (functions of complex turbulence parameters, e.g. anisotropy invariants) Enhance correct near-wall anisotropy Introduces non-linearity

• Wall-reflexion terms included

Model development: Suad Jakirlic, TU DarmstadtExtensions/Technical applications: Axel Probst, TU Braunschweig






• Summary


Approach

Simple gradient diffusion hypothesis (SGDH)

• Scalar diffusion coefficient

k

ijequivt

kij x

Rx

D ),(* where

kkcequivt

2/3),( equivalent

eddy viscosity

• Tensorial diffusion coefficient

l

ijklkl

kij x

RRx

D

*

Generalized gradient diffusion hypothesis (GGDH)

• Gradient diffusion as with k-/k-

models

Differential Reynolds stress models: Diffusion





• Summary


Popular assumtion

ijij 32

• Consider only isotropic contribution

Differential Reynolds stress models: Dissipation

• Isotropic dissipation rate from length scale equation

Anisotropic model: Hanjalic-Jakirlic

ijh

s

h

ijshij f

kRf

321

where ijijhij D

21

„homogeneous dissipation rate“

fs function of invariants of bij and ij





• Summary


Possibilities

• Any length scale known from 2-equation models possible

• Calibration of coefficients depends on length scale

Differential Reynolds stress models: Length scale

• -equationStandard choice

• -equationException: Wilcox stress-

model (= LRR without wall reflexion)





• Summary


Background (EU project FLOMANIA)

• SSG model considered better suited than LRR model• -equation preferred in aerodynamics

Differential Reynolds stress models: Engineering approach

Idea:• Combination of SSG in far field with LRR close to walls• Continuous change of coefficients Transfer of Menter‘s ideas for k-

models to RSM

SSG/LRR-

model• Far field: SSG + • Near wall: LRR + • Coefficients: Blending function F1 by Menter

• BSL--equation by Menter





• Summary


Boussinesq hypothesis• Reynolds stresses || viscous stresses• Anisotropy of normal stresses not captured• Effects of rotation drop out

Summary

Differential Reynolds stress models• Modelled Reynolds stress transport equation

• Production exact• Redistribution modelling:Theory of homogeneous flows

• Diffusion:Gradient diffusion as k-/k- models

• Dissipation:Isotropic (most often)

• Length scaleas with 2-equation models

• Engineering approach:Transfer Menter‘s ideas SSG/LRR- model

introduction to statistical turbulence modelling overview webseite...introduction to statistical...

Documents