turbulent flows...turbulent flows are acutely sensitive to perturbations turbulence is only...
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Turbulent Flows
Flow visualisation of a turbulent round jet
Turbulence
Leonardo da Vinci
Turbulence
Laminär
Turbulent
Osborne Reynolds (1883)
ULReReynolds number:
O. Reynolds (1883)
Turbulence
• Random
• 3D
• Diffusive
• Dissipative
• Property of the flow
• High Reynolds number
• Continuum
Turbulence
Pulsed flow
Impinging Jet
Impingement wall
inlopp
utlopp
Mean Sherwood number Sherwood number fluctuation
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
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
Kolmogorov microscales
Length
Time
Velocity
Reynolds number
e : dissipation rate, i.e. the
rate at which turbulent
kinetic energy is dissipated
to heat
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
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
Lorenz equations
Two cases:23 28
Lorenz equations
231.0)0( x 1000001.0)0( x
Difference
Lorenz equations
281.0)0( x 1000001.0)0( x
Difference
Lorenz equations
Observations:
23 28
What can we learn from this exercise regarding flows?
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
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
Turbulence modelling
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:
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
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
Averaging
Time average: duT
tu
Tt
t
1
Ensemble average: tuN
tu
N
n
n
1
)(1
Averaging
Variance: duuT
u
Tt
t
22 1
N
n
n uuN
u
1
2)(2 1
Standard deviation:rmsuu 2
uu
u
Averaged equations
Average:
Fluctuation:
Instantaneous: u
u
'u
Reynolds’ decomposition
'uuu
u
'u
Notation:
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
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
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
vuy
vuy
vuvuy
vuuvvuvuy
vvuuyy
uv
Convective terms
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
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
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
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”
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
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
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