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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
Folie 2Introduction to statistical turbulence modelling. Overview, RWTH Aachen, 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)
Folie 3Introduction to statistical turbulence modelling. Overview, RWTH Aachen, 08./09.03.2010
Overview: Introduction to statistical turbulence modelling
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)
Folie 4Introduction to statistical turbulence modelling. Overview, RWTH Aachen, 08./09.03.2010
Overview: Introduction to statistical turbulence modelling
(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
Folie 2Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Overview: Fundamentals
• Introduction• Averaging• Flow
equations
• Exact• Averaged
• Turbulence
equations• Fluctuations• Correlations
• Summary
Folie 3Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Overview: Fundamentals
• Introduction• Averaging• Flow
equations
• Exact• Averaged
• Turbulence
equations• Fluctuations• Correlations
• Summary
Folie 4Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 5Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 6Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 7Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 8Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 9Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 10Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 11Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Overview
• Introduction• Averaging• Flow
equations
• Exact• Averaged
• Turbulence
equations• Fluctuations• Correlations
• Summary
Folie 12Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 13Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Simple or Reynolds averaging
where simple average fluctuation
Averaging
• Decomposition
• Average of average
• Average of fluctuations
0
General features
Folie 14Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 15Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Averages of differentials
Averaging
tt
ii xx
(spectral gap)
Linear operation
Differentials in transport equations remain unchanged under averaging
Folie 16Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
• Constant density
• Compressible
Folie 17Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Mass weighted or Favre averaging
~
~
where mass weighted average
fluctuation
• Decomposition
Relation between averages
Averaging
~
Folie 18Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
• Average of fluctuations
• Average of average
~~,~~~
Averaging rules
Averaging
• Average of a sum
21
2121 ~~
• Average of a product with a constant
~
0~0
Folie 19Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 20Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Example: Mass weighted averaging of the continuity equation
• Constant density
convenient
Averaging
• Compressible
0~
0
k
k
k
k
k
k
xU
txU
txU
t
• Constant density
• 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
Folie 21Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Overview
• Introduction• Averaging• Flow
equations
• Exact• Averaged
• Turbulence
equations• Fluctuations• Correlations
• Summary
Folie 22Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 23Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 24Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 25Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 26Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 27Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Overview
• Introduction• Averaging• Flow
equations
• Exact• Averaged
• Turbulence
equations• Fluctuations• Correlations
• Summary
Folie 28Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Reynolds averaged Navier-Stokes (RANS) equations for compressible flow (1)
Flow
equations: Averaged
form
0
k
k
xU
t
• Continuity (repetition)
0~
kk
Uxt
Folie 29Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 30Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Reynolds averaged Navier-Stokes equations for compressible flow (3)
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
Folie 31Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Reynolds averaged Navier-Stokes equations for compressible flow (4)
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
Folie 32Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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 ~~~~
Folie 33Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 34Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 35Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Thermal equation of state (ideal gas)
RTp
vp CCR specific gas constantwhere
TRp ~
Flow
equations: Averaged
form
Folie 36Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 37Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 38Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Overview
• Introduction• Averaging• Flow
equations
• Exact• Averaged
• Turbulence
equations• Fluctuations• Correlations
• Summary
Folie 39Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 40Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 41Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Overview
• Introduction• Averaging• Flow
equations
• Exact• Averaged
• Turbulence
equations• Fluctuations• Correlations
• Summary
Folie 42Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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→ →
Folie 43Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Reynolds stress transport equation (2)
Turbulence
equations: Correlations
• 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
Folie 44Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Reynolds stress transport equation (3)
Turbulence
equations: Correlations
• 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
Folie 45Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Folie 46Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Incompressible: traceless 02
i
iii x
up
Folie 47Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Diffusion
321ikjjkijikijkkji
kij upupuuuuu
xD
turbulent transport viscous
diffusion pressure
diffusion
contribution
to mean
energy
equation in general
neglected
Turbulence
equations: Correlations
Folie 48Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Why diffusion?
dSndV
xdVD kijk
k
ijkij
Surface
integral
fluxes
diffusion
Volume
integral
Turbulence
equations: Correlations
Folie 49Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Folie 50Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
isotropic dissipation rate
Folie 51Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Folie 52Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Folie 53Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Folie 54Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Folie 55Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Terms of the k-equation (2)
• 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
equations: Correlations
Folie 56Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Terms of the k-equation (3)
• 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
equations: Correlations
Folie 57Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Turbulence
equations
Terms of the k-equation (4)
• Fluctuating mass flux contribution
k
ik
ii
k
xxpuM )~(
− Contribution due to fluctuating density− Only important at high Mach numbers (> 3…5)− Usually neglected
Folie 58Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Folie 59Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Folie 60Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
equations: Correlations
Folie 61Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Overview
• Introduction• Averaging• Flow
equations
• Exact• Averaged
• Turbulence
equations• Fluctuations• Correlations
• Summary
Folie 62Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Bernhard Eisfeld 08.03.2010
Folie 2Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 3Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 4Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 5Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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?
Folie 6Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 7Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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?
Folie 8Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 9Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 10Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 11Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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)
Folie 12Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 13Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
independent of coordinate
system
invariants
Folie 14Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Reynolds stress anisotropy tensor
• Definition
ijij
ij kR
b 31
~2
~~
• Characteristics
• parallel to Reynolds stresses
(identical
principal
axes)
• non-dimensional
• symmetric
• traceless
Realisability: Invariants
of the
anisotropy
tensor
Folie 15Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Realisability: Invariants
of the
anisotropy
tensor
Folie 16Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Bounds of the anisotropy tensor
Tracelessness
allows
consider
two
components
only
Realisability: Invariants
of the
anisotropy
tensor
Folie 17Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Realisability: Invariants
of the
anisotropy
tensor
Folie 18Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Invariants of the anisotropy tensor
• Exploit
tracelessness
• plot
over
b
-b
-plane
Realisability: Invariants
of the
anisotropy
tensor
Folie 19Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Exploit symmetry of the invariants
• Symmetry
axes
in the
b
-b
-plane
(tracelessness)
bb ~~
bbbb ~
21~~~
bbbb ~2~~~
Realisability: Invariants
of the
anisotropy
tensor
bbb ~~~
• Tracelessness
Folie 20Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Exploit symmetry of the invariants
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
Realisability: Invariants
of the
anisotropy
tensor
Folie 21Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Exploit symmetry of the invariants
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
Realisability: Invariants
of the
anisotropy
tensor
3 intersecting
symmetry
lines
6 equivalent
triangles
B-A-C
Realisability: Invariants
of the
anisotropy
tensor
Folie 22Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Realisability: Invariants
of the
anisotropy
tensor
Folie 23Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Invariants along B-A-C and B-C
Realisability: Invariants
of the
anisotropy
tensor
Folie 24Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
independent of coordinate
system
isotropic
Realisability: Invariants
of the
anisotropy
tensor
2-Caxisym.
axisym.
axisym.
2-C
1-C
Folie 25Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 26Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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)
Folie 27Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 28Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 29Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 30Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 31Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 32Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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)
Folie 33Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 34Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 35Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
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
Folie 36Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
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
Folie 37Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
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
Folie 38Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
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
Folie 39Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
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
Folie 40Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
Intermittency
• Varying
periods
of turbulent and laminar
flow
for
0.4 ≤
y/
≤
1.2
16
5.51
y
• Intermittency
factor
:
turbulent time fraction
• Approximation
Folie 41Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
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
Folie 42Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
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
Folie 43Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
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
Folie 44Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Velocity
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
Folie 45Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 46Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 47Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Temperature
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
Folie 48Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Temperature
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
Folie 49Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 50Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 51Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 52Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 53Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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)
Folie 54Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Bradshaw relation
Observation
• Large part
of flat
plate
boundary
layer
shows
12a
kRxy
where 15.01 a Structure
parameter
• Constant
Reynolds stress layer
1
2
1 22 au
aRk xy
Constant
specific
kinetic
turbulence
energy
Folie 55Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 56Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 57Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Wall-bounded
flows: Profiles: Turbulent equilibrium
Channel
flow
DNS data
by
del Alamo
et al.
Equilibrium
layer
• reasonable
approximation
• Increasing
Re
better
agreement
Folie 58Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 59Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 60Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 61Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 62Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Folie 63Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
Summary
Homogeneous
isotropic
turbulence
• specific
kinetic
turbulence
energy
decays
• for
long
t: k ~ t-n, where
n ≈
1.25
Folie 64Introduction to statistical turbulence modelling. Part I, RWTH Aachen, 08.03.2010
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
Reynolds stress layer
(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
Introduction to statistical turbulence modelling
Turbulence models
Bernhard Eisfeld 09.03.2010
Folie 2Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 3Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 4Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Repetition: Characteristics of turbulent flows
Homogeneous isotropic turbulence
• No spatial gradients of means• Experiments:
ntk
where 06.025.1 n
for long times t
Folie 5Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Repetition: Characteristics of turbulent flows
Viscous sublayer• Turbulent effects negligibleLog layer• overlap region• viscous effects negligible• convection/pressure gradientnegligible
with inner scaling
uUu
yuy
Friction velocity
wu
Cyu ln1
Folie 6Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Repetition: Characteristics of turbulent flows
Log layer
2uRxy
• Constant Reynolds stress
y
uP k
3
• Equilibrium
1
2
1 215.0
2 auka
kRxy
• Bradshaw relation
Near-wall layer
• Constant total stress
2 uR
yU
xy
Folie 7Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 8Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 9Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 10Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 11Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 12Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 13Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Boussinesq hypothesisEddy viscosity in inner part of boundary layer
• Van DriestDamping extends intolog layer (y+ > 100)
• Mellor and HerringDamping restrictedto buffer layer (y+ < 60)
Folie 14Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 15Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 16Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 17Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 18Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 19Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 20Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 21Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Algebraic modelsBaldwin-Lomax model (1978) (2)
• 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))
Folie 22Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Algebraic modelsBaldwin-Lomax model (1978) (3)
• 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
Folie 23Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Algebraic modelsDegani-Schiff modification (1986)
• Free vortices (delta wings) Strong 2nd maximumWrong predictions
• RemedyTake wall-nearest maximuminstead of absolute maximum
Folie 24Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 25Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 26Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 27Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 28Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 29Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
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
Folie 30Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
-equation
)(2
Dxx
Px
Ut kkk
k
Production
Dissipation
Turbulent diffusion
Huge number of terms to be modelled !
2-equation models: K-
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)(
Folie 31Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
model
Modelling the -equation
• Production: Scale k-production
kPk
CP
1
• Turbulent diffusion: Gradient hypothesis
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
Folie 32Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
model
Eddy viscosity
• Velocity scale
kU t
• Length scale
2/3kL t
• Result
2kct
Folie 33Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
2-equation models: K-
model
00 ,kwhere initial values
Folie 34Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Calibration for boundary layer (log law)
cuk
2
yukct
2
• Equilibrium
yuP k
3
• Constant Reynolds stress
• Bradshaw relation
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
2-equation models: K-
model
Folie 35Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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)
2-equation models: K-
model
Folie 36Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
2-equation models: K-
model
Folie 37Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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-Almaras model
• Summary
Folie 38Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
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*
Folie 39Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
models: Wilcox model
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
• Turbulent diffusion: Gradient hypothesis
j
t
j
k
xk
xD )(*)(
• Low turbulent Reynolds number keep viscous diffusion
Result
• Destruction: convert
into
k* where c*
Folie 40Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
model: Wilcox model
Modelling the -equation
• Production: Scale k-production
kPk
P
• Turbulent diffusion: Gradient hypothesis
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
Folie 41Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
models: Wilcox model
Eddy viscosity
• Velocity scale
kU t
• Length scale
*
kL t
• Result
kt
Folie 42Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
models: Wilcox model
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
Folie 43Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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*
2-equation models: K-
models: Wilcox model
Folie 44Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Near wall behaviour
• Convection and production ((t)) negligible only viscous diffusion and destruction
2-equation models: K-
models: Wilcox model
22
2
*2
2
y
kyk
• Solution
202
*
1lim66
23.341121,
yy
nyk
yw
n
Near-wall asymptotics: n = 2
Singularity at the wall
Folie 45Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Wall boundary conditions
• k-equation:Natural boundary condition
2-equation models: K-
models: Wilcox model
0w
k
• -equation: Approximation of infinite value
21
16y
Fw
where
Menter (1994): Extrapolate
from near wall point
10F
Folie 46Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
2-equation models: K-
models: Wilcox model
Folie 47Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 48Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
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
Folie 49Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
2-equation models: K-
models: Menter BSL model
Folie 50Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Remarks on the Menter BSL model• Not very common• Intermediate step towards SST model• Similar behaviour as Wilcox model, except freestream sensitivity
2-equation models: K-
models: Menter BSL model
Folie 51Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 52Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
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)/
Folie 53Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
models: Menter SST model
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)
Folie 54Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
models: Menter SST model
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
Folie 55Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
models: Menter SST model
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*
*
Folie 56Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
2-equation models: K-
models: Menter SST model
Results for RAE 2822
Improvement at shock (pressure rise) by SST limitiation
Ma = 0.73 Ma = 0.75
Folie 57Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
2-equation models: K-
models: Menter SST model
Folie 58Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 59Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 60Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Eddy viscosity transport equation
1-equation models: Eddy viscosity transport
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
Folie 61Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Idea (Menter, 1997)
1-equation models: Eddy viscosity transport
yU
ccRk
txy
• Bradshaw relation
where
• Generalization for 3D
Sc
kt
ijij SSS 2
• Consequence
2
2
Skc tt
Folie 62Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Eddy viscosity transport equation
1-equation models: Eddy viscosity transport
(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
Folie 63Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Comments
1-equation models: Eddy viscosity transport
• Explains relation between models• Not the basis for common model development
Folie 64Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 65Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 66Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 1: Free shear flows (1)
1-equation models: Spalart-Allmaras model
• 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
Folie 67Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 1: Free shear flows (2)
1-equation models: Spalart-Allmaras model
• 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
Folie 68Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 2: Extension to boundary layers, high Rt (1)
1-equation models: Spalart-Allmaras model
• Boundary layer at high Rt = log layer
Velocity gradient:
yut Eddy viscosity:
yuS
Convection negligible
Eddy viscosity transport equation
ttt
uccDP bb
2221 1
Balance requires dissipation term in boundary layers
Folie 69Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 2: Extension to boundary layers, high Rt (2)
1-equation models: Spalart-Allmaras model
• 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
Folie 70Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 2: Extension to boundary layers, high Rt (3)
1-equation models: Spalart-Allmaras model
• 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
Folie 71Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 2: Extension to boundary layers, high Rt (4)
1-equation models: Spalart-Allmaras model
• 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
Folie 72Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 2: Extension to boundary layers, high Rt (5)
1-equation models: Spalart-Allmaras model
Calibration: cw2 = 0.3, cw3 = 2
Folie 73Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 3: Extension to boundary layers, low Rt (1)
1-equation models: Spalart-Allmaras model
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)
Folie 74Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 3: Extension to boundary layers, low Rt (2)
1-equation models: Spalart-Allmaras model
Production term modification
yuS
2uS
Log layer
Viscous sublayer
Log layer behaviour requires
ScP b~~
1~
where
yuSSS
~
Behaviour changes !
and 22~~
ySr
(dissipation)
Folie 75Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Step 3: Extension to boundary layers, low Rt (3)
1-equation models: Spalart-Allmaras model
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
Folie 76Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
Complete model
1-equation models: Spalart-Allmaras 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
Folie 77Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
1-equation models: Spalart-Allmaras model
Folie 78Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Folie 79Introduction to statistical turbulence modelling. Part III, RWTH Aachen, 09.03.2010
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
Introduction to statistical turbulence modelling
Differential Reynolds stress models
Bernhard Eisfeld 09.03.2010
Folie 2Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
Overview: Differential Reynolds stress models
• Repetition: Boussinesq hypothesis• Differential Reynolds stress models
• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach
• Summary
Folie 3Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
Overview: Differential Reynolds stress models
• Repetition: Boussinesq hypothesis• Differential Reynolds stress models
• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach
• Summary
Folie 4Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
Folie 5Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
Folie 6Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
Overview: Differential Reynolds stress models
• Repetition: Boussinesq hypothesis• Differential Reynolds stress models
• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach
• Summary
Folie 7Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
~~~
Folie 8Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
Overview: Differential Reynolds stress models
• Repetition: Boussinesq hypothesis• Differential Reynolds stress models
• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach
• Summary
Folie 9Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
Folie 10Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
Differential Reynolds stress models: Redistribution
Folie 11Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
Differential Reynolds stress models: Redistribution
Folie 12Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
Overview: Differential Reynolds stress models
• Repetition: Boussinesq hypothesis• Differential Reynolds stress models
• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach
• Summary
Folie 13Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
Folie 14Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
Overview: Differential Reynolds stress models
• Repetition: Boussinesq hypothesis• Differential Reynolds stress models
• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach
• Summary
Folie 15Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
Folie 16Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
Overview: Differential Reynolds stress models
• Repetition: Boussinesq hypothesis• Differential Reynolds stress models
• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach
• Summary
Folie 17Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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)
Folie 18Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
Overview: Differential Reynolds stress models
• Repetition: Boussinesq hypothesis• Differential Reynolds stress models
• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach
• Summary
Folie 19Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
Folie 20Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
Overview: Differential Reynolds stress models
• Repetition: Boussinesq hypothesis• Differential Reynolds stress models
• Re-distribution• Diffusion• Dissipation• Length scale• Engineering approach
• Summary
Folie 21Introduction to statistical turbulence modelling. Part IV, RWTH Aachen, 09.03.2010
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
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