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Turbulent Flows
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Flow visualisation of a turbulent round jet
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Turbulence
Leonardo da Vinci
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Turbulence
Laminär
Turbulent
Osborne Reynolds (1883)
ULReReynolds number:
O. Reynolds (1883)
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Turbulence
• Random
• 3D
• Diffusive
• Dissipative
• Property of the flow
• High Reynolds number
• Continuum
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Turbulence
Pulsed flow
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Impinging Jet
Impingement wall
inlopp
utlopp
Mean Sherwood number Sherwood number fluctuation
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Turbulence
Big whirls have little whirls
Which feed on their velocity
Little whirls have lesser whirls
And so on to viscosity – in the molecular
sense
L F Richardson
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Kolmogorov’s hypotheses
• At sufficiently high Reynolds number, the small scale
turbulent motions are statistically isotropic.
• In every turbulent flow, at sufficiently high Reynolds
number, the statistics of the small scale motions have
a universal form and are uniquely determined by
viscosity () and dissipation rate (e).
• In every turbulent flow, at sufficiently high Reynolds
number, there is a range of scales, much smaller than
the largest scales and much larger than the smallest
scales, where the statistics of the motions have a
universal form and are uniquely determined by e
idependent of .
Isotropic=equal in all directions
Homogeneous=equal at all locations
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Kolmogorov microscales
Length
Time
Velocity
Reynolds number
e : dissipation rate, i.e. the
rate at which turbulent
kinetic energy is dissipated
to heat
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Turbulent kinetic energy
spectrum
(Kolmogorov theory for isotropic & homogenous turbulence)
log(E
(k))
log(k)
Dissipation
subrange
Inertial
subrange
Large
scales
Universal range
-5/3
k: wave number = (2*p)/l l=wave length
production
dissipationtransport
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Lorenz equations
xyzdt
dz
xzyxdt
dy
xydt
dx
10
3
8
Two cases:23 28
Initial values:
1.0)0(
1.0)0(
1.0)0(
z
y
x
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Lorenz equations
Two cases:23 28
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Lorenz equations
231.0)0( x 1000001.0)0( x
Difference
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Lorenz equations
281.0)0( x 1000001.0)0( x
Difference
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Lorenz equations
Observations:
23 28
What can we learn from this exercise regarding flows?
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There are always perturbations
originating from boundary conditions,
initial conditions etc. present in a flow.
Turbulent flows are acutely sensitive to
perturbations
Turbulence is only meaningful to
describe in a statistical sense
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Turbulence modelling
Direct simulation of isotropic turbulence
Domain: Cubic box of size 118L
8.048
211
11
110 pp
k LL
L
Required resolution: 5.1max k
1.25.1
p
xIn physical space
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Turbulence modelling
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Turbulence modelling
Direct simulation of isotropic turbulence
Required number of grid
nodes in each direction: pk
k
k
k L
L
LL
L
L
LN 1111
110
max
0
max 12
43
Re6.16.1 L
LN
2
94
9
06.0Re4.43lRN L In 3D:
e
2
RekLk
L
ll
guR
e
23
kL
Lengthscale of
large eddies:
Turbulence Reynolds number:
Taylor scale Reynolds number:
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Turbulence modelling
Direct simulation of isotropic turbulence
Required temporal resolution
x
tkC
e
kTurbulence time scale:
The Courant number:
Assume that sampling over at least 4 turbulence
time scales is needed, then the number of time
steps is:
23
2.9120
80804
lpe
R
L
x
Lk
x
k
tM
20
1C
Computational work:633 66.0Re160 lRMN L
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Turbulence modelling
Direct simulation of isotropic turbulence
Required time in days at a computing rate of 82 Gflop
Re N N3 M N3M CPU
time
Memory
94 104 1.1E06 1.2E03 1.3E09 14s 18 Mb
375 214 1.0E07 3.3E03 3.2E10 6.6 min 150 Mb
1500 498 1.2E08 9.2E03 1.1E12 3.8 h 2 Gb
6000 1260 2.0E09 2.6E04 5.2E13 7.3 days 30 Gb
24000 3360 3.8E10 7.4E04 2.8E15 1.1 years 565 Gb
96000 9218 7.8E11 2.1E05 1.6E17 61 years 11 Tb
N3= number of grid points
M= number of time steps
N3M= total work required
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Averaging
Time average: duT
tu
Tt
t
1
Ensemble average: tuN
tu
N
n
n
1
)(1
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Averaging
Variance: duuT
u
Tt
t
22 1
N
n
n uuN
u
1
2)(2 1
Standard deviation:rmsuu 2
uu
u
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Averaged equations
Average:
Fluctuation:
Instantaneous: u
u
'u
Reynolds’ decomposition
'uuu
u
'u
Notation:
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Averaged equations
0
y
v
x
u
2
2
2
2
2
2
2
2
1
1
y
v
x
v
y
p
y
vv
x
uv
t
v
y
u
x
u
x
p
y
uv
x
uu
t
u
Properties of the
averaging:
v'u'vuvu
vuvu
x
u
x
u
´(x,t)u
(x,t) u (x,t) u
0
Mass
Momentum
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0
y
v
x
u
First mass conservation:
uuxx
u
Decomposition:
x
u
x
u
x
u
x
u
x
uuu
xx
u
Also note
Averaged equations
Similarly in y-dir:
0
y
v
x
u
In x-dir:
y
v
y
v
0
y
v
x
u
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Momentum equation
t
u
t
u
2
2
2
2
x
u
x
u
x
p
x
p
11
Averaged equations
2
2
2
21
y
u
x
u
x
p
y
uv
x
uu
t
u
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vuy
vuy
vuvuy
vuuvvuvuy
vvuuyy
uv
Convective terms
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Reynolds Averaged Navier-Stokes (RANS)
equations
In 2D:
•3 equations
•3+3 unknowns
Leads to the closure problem.
0
y
v
x
u
y
vv
x
vu
y
v
x
v
y
p
y
vv
x
uu
t
v
y
vu
x
uu
y
u
x
u
x
p
y
uv
x
uu
t
u
2
2
2
2
2
2
2
2
1
1
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Reynolds stress tensor
wwwvwu
wvvvvu
wuvuuu
wux
w
z
u
vux
v
y
u
uux
u
turbxz
viskxz
totxz
turbxy
viskxy
totxy
turbxx
viskxx
totxx
)()()(
)()()(
)()()( 2
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jj
i
ij
jii
xx
u
x
p
x
uu
t
u
21
0
i
i
x
u
0
i
i
x
u
ji
jjj
i
ij
i
j
i uuxxx
u
x
p
x
uu
t
u
21
Reynolds
stress tensor
Averaged equations
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Turbulence Modelling
N-eq. Models (e.g. k-e)- Short computational time
- Simple, robust
- Limited range of validity
Reynolds Stress Models, RSM- More general, still not universal
- More complex, seven PDE:s
- Extensive modeling
- Longer computational time
Reynolds Averaged Navier-Stokes:
”RANS”
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Turbulence model
i
j
j
i
Tijjix
u
x
ukuu
3
2Boussinesq’s
hypothesis
Turbulent kinetic enegryii
uuwvu
k
2
1
2
222
Eddy viscosityMean rate of strain
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Turbulence model
jk
T
jj
jx
k
xP
x
ku
t
k
e
j
T
jj
jxxk
CPk
Cx
ut
e
eeee
eee
2
21
e
2kCT
Turbulent kinetic energy
Dissipation rate
Production Dissipation rate
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Turbulence model
jk
T
jj
jx
k
xP
x
ku
t
k
e
j
T
jj
jxxk
CPk
Cx
ut
e
eeee
eee
2
21
e
2kCT
The whole system of equations0
i
i
x
u
i
j
j
iT
jij
ij
i
x
u
x
u
xx
p
x
uu
t
u
1
ijijT SSP 2
i
j
j
iij
x
u
x
uS
2
1