Turbulence Chapter04
Transcript of Turbulence Chapter04
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
TurbulenceLecture 4
Reynolds Averaging
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
Reminder: DNS is not practical
"Turbulence in a box"
Re: 40,000Grid: 40963 (70 billion cells)
Computer: Earth SimulatorT. Ishihara, T. Gotoh, Y. Kaneda, Study of High-Reynoldds Number Isotropic Turbulence by Direct Numerical Simulation, Annual Reviewof Fluid Mechanics 2009, 41:165-180.
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
•
Do we really need all the details of a DNS? – Time resolved flow field in 3D?
– The entire temporal evolution at every point?
☝ Are time dependent quantities meaningful for other realisations?
– All probability density functions?
– All moments?
– Any other statistical quantity?
• So, what are we really interested in?
– Mean values (first moments)
– Perhaps fluctuation levels (second moments, stresses)
– Perhaps probablity density functions (for combustion)
A less costly approach than DNS is needed!
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
•
We are mainly interested in the mean values.• We need equations for mean quantities!
• Reynolds decomposition can provide the mean quantities.
• It is now sufficient to substitute ui and p in the equations to get the
"Reynolds averaged Navier Stokes equations"
u u u ui i i i
= + ! ! =, 0
p p p p= + ! ! =, 0
Reynolds Decomposition
!"
!
!"
! t
u
x
i
i
+ = 0
! "
" !
"
"
"
"
"
" µ
"
"
"
"
"
" # !
u
t u
u
x
p
x x
u
x
u
x
u
x g i j
i
j i j
i
j
j
i
k
k
ij i+ = $ + + $%
&' (
)*+
,--
.
/00
+2
3
✍
u02
i ≥ 0
p02
i ≥ 0
u0
iu0
j ???
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
Derivation of Reynolds-Averaged Navier Stokes Eq.
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
•
The Conservation equation for mass (constant viscosity and density)
(3.3.1a)
• The Conservation equation for momentum (constant viscosity and
density)
(3.3.2a)
0=
i
i
x
u
!
!
! "
" !
"
"
"
"
"
" µ
"
"
"
"
"
" # !
u
t u
u
x
p
x x
u
x
u
x
u
x g i j
i
j i j
i
j
j
i
k
k
ij i+ = $ + + $%
&'
(
)*
+--
.00
+
2
3
0
i
i j
j
j j
i
i j
i j
i g x x
u
x x
u
x
p
x
uu
t
u !
" "
"
" "
" µ
"
"
"
" !
"
" ! +
##
$
%
&&
'
(++)=+
22 0
i
j j
i
i j
i
j
i g x x
u
x
p
x
uu
t
u++!=+
" "
" #
"
"
$ "
"
"
" 21
Averaged Balance Equations Ia
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
Averaged Balance Equations I
The Conservation equation for mass (instantaneous)
(3.3.1)
The Conservation equation for momentum (instantaneous)
(3.3.2)
!"
!
!"
! t
u
x
i
i
+ = 0
! "
" !
"
"
"
"
"
" µ
"
"
"
"
"
" # !
u
t u
u
x
p
x x
u
x
u
x
u
x g i j
i
j i j
i
j
j
i
k
k
ij i+ = $ + + $%
&'
(
)*
+
,--
.
/00
+
2
3
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
Applying Reynolds decomposition to the conservation equation formass and momentum, and considering ! and µ to be constant yields:
(3.3.5)
with
(3.3.6)
! = µ / " ( )
!
!
!
!
u
x
u
x
i
i
i
i
+
"= 0
( )!
!
!
!
!
!
u
t
u
t xu u u u u u u ui i
j
i j i j i j i j+
"+ + " + " + " " =
! ! "
+ +
"#
$%
&
'( +
1 1 2 2
)
*
* )
*
* +
*
* *
*
* *
p
x
p
x
u
x x
u
x x g i i
i
j j
i
j j
i
Averaged Balance Equations III
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
For constant density, time averaging yields:
After the averaging process, the unknown
Reynolds Stress Tensor arises in the conservation equation
for momentum.
!
Closure problem of turbulence for the Reynolds-Averaged-
Navier Stokes equations.
!
!
u
x
i
i
= 0
!
!
!
! "
!
!
!
! #
!
! !
u
t
u u
x
p
x
u u
x
u
x x g i
i j
j i
i j
j
i
j j
i+ = $ $
% %+ +
1 2
! !u ui j
Reynolds Averaged Navier Stokes Equations
ρu0iu0
j
i =
m2
s21
m3kg = m
2
s21
m3Ns2
m =
N m2
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
Interpretation of Reynolds Stresses
•
Similar to viscous stresses
• Example in a shear layer :
!
!
!
! "
!
!
!
! #
!
! !
u
t
u u
x
p
x
u u
x
u
x x g i
i j
j i
i j
j
i
j j
i+ = $ $
% %+ +
1 2
∂ u1
∂ x2> 0
"21,v
"
21,t
x1
x2
"21,v
viscous stress turbulent stress
u1
u1
u01
u2
u0
2u
0
1
u1
u1
u0
1u2
u0
2u
0
1
molecule (discontinuous) fluid particle (continuous)
n e t f l u
x o f h o r i -
z o n t a l m o m e n t u m
"21,t
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
The result
The Reynolds Averaged equations for Reynolds averaged velocities
are identical to
the Navier Stokes equations for the instantaneous velocities
with the exception of the Reynolds Stresses.
! Similar CFD codes can be used
!
Turbulent flows behave „similar“ to laminar flows
! A turbulent flow might be thought of as „non-Newtonian“
!
!
u
x
i
i
= 0
!
!
!
! "
!
!
!
! #
!
! !
u
t
u u
x
p
x
u u
x
u
x x g i
i j
j i
i j
j
i
j j
i+ = $ $
% %+ +
1 2
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
The sign of Reynolds stresses (example for cosines)
The sign of a Reynolds stress depends on whether u‘ and v‘ arepositively correlated (+), negatively corelated (-), or un-correlated (0).
u0v0
u0 = 0
v0 = 0
u0
, v0
t
v0
u0
u0
v0
u0 v
0
v0
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8/16/2019 Turbulence Chapter04
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Prof. Dr.-Ing. Andreas Kempf, Chair of Fluid Dynamics
Original course material by J. Janicka, TU-Darmstadt
Problem
How to determine turbulent stresses?• Derive transport equations: Second moment closure
• Exploit similarity to viscous stresses
– Eddy viscosity / Bousinesq approach, with turbulent viscosity
(most common turbulence models)
– How to determine turbulent viscosity?