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8. Turbulence
Summer Term 2011
Werner Aeschbach-Hertig
Institut für Umweltphysik
Lecture Program of MKEP4
Part 1: Introduction and Fundamentals (4 sessions)
1.
2.
3.
4.
Introduction to Environmental Physics and the Earth System
Global energy balance and structure of the atmosphere
Stratification and convection in air and water
Transport
p p
processes
Part 2: Geophysical Fluid Dynamics (7 sessions)
5.
6.
7.
8.
9.
10.
11.
Introduction to Geophysical Fluid Dynamics
Navier-Stokes equation and geostrophic approximation
Geostrophic Flow and Vorticity
Turbulence
Turbulent transport and flow near boundaries
Global circulation of the atmosphere
Global circulation of the ocean
Part 3: Other Compartments and Fields (4 sessions)
12.
13.
14.
15.
Gas and heat transfer between air and water
Freshwater systems
Soil and Groundwater
The cryosphere
2
1
Contents of Today's Lecture
Turbulence
• The phenomenon of turbulence
• Reynolds number
• Reynolds decomposition
• Turbulent kinetic energy and the turbulence spectrum
• Kolmogorov theory
• Autocorrelation
• Taylor's theorem and turbulent diffusion
Literature on Turbulence:
1) Roedel, W., 2000. Physik unserer Umwelt, Die Atmosphäre. Springer Verlag,
Heidelberg, 3rd Edition. (IUP 1780). (Kap. 6)
2) Pedlosky, J., 1987. Geophysical Fluid Dynamics. Springer Verlag, Heidelberg, 2nd
Edition. (IUP 720)
3
Turbulence: Eddies in the Gulf Stream
4
2
Laminar and turbulent flow
Both types of flow
are solutions of a
deterministic
differential
equation
(Navier-Stokes)
5
Unstable Solutions of Differential Equations
Stable Solutions:
Unstable Solutions:
Similar initial conditions
lead to similar solutions
Infinitesimal differences in the initial
conditions lead to very different solutions
"Butterfly effect", chaos
Roedel, W., 2000. Physik unserer Umwelt. Die Atmosphäre. Springer
6
3
Deterministic Chaos - Edward Lorenz
Lorenz-attractor in 3D phase space
Edward Lorenz
1917 - 2008
7
Turbulence: An Unsolved Problem of Physics
A full description of turbulent flow maybe the last unsolved problem in
classical physics. Famous physicist are reported to have been doubtful,
whether it can be solved:
Heisenberg was asked what he would like to know from God, given the
opportunity. His reply was:
"When I meet God, I am going to ask him two questions: Why relativity?
And why turbulence? I really believe he will have an answer for the first."
A similar saying is attributed to Horace Lamb (author of a famous
t tb k in
textbook
i h
hydrodynamics):
d d
i )
"When I die and go to heaven there are two matters on which I hope for
enlightenment. One is quantum electrodynamics, and the other is the
turbulent motion of fluids. And about the former I am rather optimistic."
8
4
Criterion for Turbulence: Reynolds Number
Reynolds number: Ratio of turbulence producing non-linear
term to turbulence destroying friction term
  
v    v U2 L UL

Re 

 
v
U L2

Re > Rec ≈ several 1000: Turbulent flow
Atmosphere, ocean, lakes are usually turbulent:
air
Osborne Reynolds
1842-1912
Critical length scale:
  Re c
Lc 
v
viscosity 
10-5
velocity U
10 m/s
m2/s
length scale L 1000 m
water
10-6
m2/s
0.1 m/s
1000 m
Re
109
108
Lc
10-3 m
10-2 m
Laminar flow only on scales < cm/mm
9
Turbulence and Reynolds Number
Experiment to determine Rec
Flow pattern for various Re
From Stewart, 2003
10
5
Examples for Turbulent Flow
Top: A mixing layer at high Reynolds number. The upper stream is moving at 100 m/s and the lower at 38 m/s, both from
left to right in the image. Two observations: (1) the transition to turbulence is evident on the left side of the image, where
the initially smooth roller structures suddenly develop small-scale detail, and (2) the large-scale organization of the flow is
evident as the flow moves downstream, even though the flow has a lot of small-scale activity. Bottom: The same flow
arrangement as above, but at twice the Reynolds number. There appears to be more small-scale activity, but the large-scale
organization is not greatly affected.
(Images from van Dyke, An Album of Fluid Motion.)
11
Velocities in a Turbulent Flow
y

v
velocity of mean flow
r
x
Eulerian measurement
of transverse velocity
vy
T 2r
4r

 T
2 v
v
t
x
v
12
6
Velocity Fluctuations due to Eddies
Deviation of velocity from mean in a wind channel
big eddy
intermed. eddy
small eddy
from:
Frisch 1995,
Turbulence
13
Turbulent Fluctuations
Laminar flow (stationary):
At each point the velocity of flow
is constant in time.
Turbulent flow (stationary):
Statistical variations of the flow
velocity due to eddies.
Description by fluctuations in flow velocity, i.e. deviations from the average
velocity:
v '(t)
v(t)
Typical wind speed fluctuations (10 m height, Lake Ontario)
Time
14
7
Reynolds Decomposition
Method to analyse turbulence and to parameterise the nonlinear terms in the Navier-Stokes equation
Basic idea: Separate velocity components into a mean
value and (statistical) fluctuations around this mean



v(t)  v(t)  v '(t)
with
T

1 
v(t)   v(t)dt
T0

v : Mean velocity

v ' : Turbulent velocity fluctuations,
fluctuations "turbulent
turbulent velocities"
velocities

From the definition follows: v '  0

Description of turbulence by statistical properties of v '(t)
In the following, v denotes one component of the velocity
15
Frequency Spectrum of Turbulence
Eddies and hence velocity fluctuations exist on various time
and length scales.
Separation of contributions of different frequency  or wave
number k (mathematical: Spectral analysis
analysis, Fourier analysis)

1
2
,k 
T
L
T, L: Period and size of eddies
Frequency spectrum of velocity fluctuations: Fourier transform
F  

(t)  e
 v '(t)
2  i  t
dt

Inverse transform:

v '(t) 
 F  e
2  i  t
d

16
8
Turbulent Kinetic Energy (TKE)
From the separation of the flow field in a mean flow and
statistical fluctuations follows:
Kinetic energy density of the mean flow:
[J/m3]
Emean
1
kin
 v 2
V
2
turb
Ekin
1
  v 2
Kinetic energy density of the turbulent flow:
V
2
[J/m3]
TKE 
Def.: Turbulent kinetic energy (TKE):
[J/kg = m2/s2]
1 2
v
2
TKE = half of the variance of the velocity fluctuations
Dimension: L2/T2
1 T
2
v '(t) 
T 0
v '(t)  v '(t)dt
(Note: Frequently factor ½ is omitted)
17
Power Spectrum of Turbulence
Average turbulent energy density:


dE
1 dE
d  v ' 2  f( )d with f( ) 
d
E d
0
0
E  v '2  
f()d quantifies the fraction of the total energy density
present in the frequency interval [, +d]:
dE  v ' 2 f( )d
One can show ( Roedel) that the power spectrum is
connected
t d tto the
th spectrum
t
off the
th velocity
l it fluctuations
fl t ti
via:
i
f  
F
2
T v'²
and
dE F   

d
T
2
T is the averaging time.
18
9
Turbulent Energy Cascade (Richardson)
Turbulent energy
generation at
large scales
Transport of
turbulent energy
down the eddy
scales
Dissipation of
turbulent energy
at small scales
due to viscosity
Lewis Fry Richardson
(1881 - 1953)
Frisch 1995, Turbulence, Cambridge Univ. Press
Richardson 1922:
Big whirls have little whirls that feed on their velocity,
and little whirls have lesser whirls and so on to viscosity.
19
Kolmogorov’s Theory of Turbulence
Kolmogorov's postulates (1941):
1) For very high Reynolds number, the small scale
turbulent motions are statistically isotropic. The
anisotropy from the geometry of the system at large
scales ((LS) is lost in Richardson's energy
gy cascade.
2) The statistics of the small scales has a universal
character, determined only by the viscosity () and the
rate of energy dissipation (). By dimensional analysis,
 and  uniquely define a dissipation length scale LK.
3) In the "inertial range", i.e. the intermediate range of
scales LS >> LI >> LK between the system scale and
the dissipative
p
scale,, kinetic energy
gy is essentially
y not
dissipated but merely transferred to smaller scales.
Thus viscosity does not play a role and the statistics in
the inertial range is uniquely determined by the length
scale (LI) and the rate of energy dissipation ().
Andrey Kolmogorov
(1903 – 1987)
Kolmogorov A.N. (1941), The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers".
Proc. USSR Academy of Sciences 30: 299–303. (Russian), translated into English by Kolmogorov, The local structure of
turbulence in incompressible viscous fluid for very large Reynolds numbers, Proc Roy. Soc. London, Series A: Mathematical
and Physical Sciences 434 (1980), 9–13.
20
10
Kolmogorov Scales of Turbulence
Turbulence at high Re and small scales is determined by
1) Kinematic viscosity  of the fluid
2) The rate of energy dissipation  in the turbulent flow
Idea: Derive by dimensional analysis the universal, smallest scales
off turbulent
t b l t motion
ti from
f
 and
d .
Note: Dimension of  follows from
Dimensional analysis:
  
2
L
T
 
2
L
T3
TKE 
 
LK   
  
3
length scale LK:

tK   

time scale tK:
1 2
v
2

d  TKE 
dt
1
1
4
 L2 3  L2 1 L6 T 3
4
    3   3  2  L4 
T L
 T   T 

1
2
 L2

 T
1
2
L 
L T
  3    2  T2 
T L

T 
1
2
2
3
21
Kolmogorov Scales: Example
Kinematic viscosity  = 1.510-5
The rate of energy dissipation  = 1 W/kg (= m2/s3)
1
4

 1.5  10 5
 3 
LK     
1
   


1
3
4
  2.5  10 4 m  0.25 mm


1
1
5
   2  1.5  10  2
3
tK     
  4  10 s

1
  

1


v K  LK tK     4  1.5  10 5  1
1
4
 0.06
m
s
22
11
Kolmogorov's Energy Spectrum
In the inertial subrange, the energy density spectrum should depend
only on the length scale (LI) and the rate of energy dissipation ().
The wavenumber k is related to the length scale by k = 2/LI.
Dimensional analysis yields the shape of the power spectrum E(k):
 
L2
T3
k  
E  k   c  a  k b
It follows:
1
L
E k  

d  TKE 
dk
L3 L2a 1

T 2 T 3a Lb

L3
E  k   T 2
2
5
a  ,b  
3
3
E  k   c   2 3  k 5 3
23
Turbulence Spectrum
E(k)
Energy
input
Inertial
subrange
Dissipation
24
12
Autocorrelation of v'(t)
v  t 
v  x 
v  t 
v  t  1 
v  t  2 
t, x
Sh t times
Short
ti
:
v  t   v  t   
Long times :
v  t  has no similarity with
v  t   
25
Autocorrelation Functions
Lagrange's Autocorrelation Function:
Observer moves with fluid parcel and determines velocity fluctuations. The
autocorrelation is determined by spatially averaging over many parcels:
R L ( ) 
v '(t)  v '(t  )
v ' 2 (t)
Euler's Autocorrelation Function:
Velocity fluctuations are measured at a fixed location. The autocorrelation
is determined by temporally averaging over many observations:
R E ( ) 
v '(t)  v '(t  )
v ' 2 (t)
Euler's Autocorrelation Function can be directly determined from
(temporally highly resolved) measurements of the velocity.
26
13
Eulerian Autocorrelation
Examples for the Eulerian
autocorrelation function
(atmosphere, neutral or
weakly labile conditions)
u = 5m/s
Averaging time: 1 hour
x = direction of flow
y = horizontal direction
perpendicular to flow
z = vertical
Roedel, W., 2000. Physik unserer Umwelt. Die Atmosphäre. Springer
27
Autocorrelation and Power Spectrum
Autocorrelation and turbulent energy density are closely linked.
It can be shown ( Roedel) that Euler's autocorrelation function is the
fourier transform of the power spectrum:
RE    

 f()e
2 i
d

Thus the relative spectral energy density can be determined from
the Eulerian autocorrelation (i.e., from highly resolved time series
of velocity measurements):

f( ) 
 R   e
E
2 i
d

28
14
The Theorem of Taylor
How far do fluid parcels move due to turbulence? Over which distance
does turbulence mix constituents (e.g., dissolved conc.) of the fluid?
1-D: We use a (moving) coordinate system where v x  v x  0
i.e. dx/dt = v = vv',, and x(t
x(t=0)
0) = 0
Since fluctuations v' are statistical, the mean displacement is 0: x  t   0
But not the variance:  2x  x 2  t   0
Turbulence produces displacement  x  x 2 analogous to diffusion!
The theorem of Taylor connects the variance of the displacement of a
fluid parcel with the Lagrangian Autocorrelation Function:
We have:
d 2x
d 2
d 2
d

x (t) 
x (t)  2x(t) x(t)  2 x(t)  v ' x (t)
dt
dt
dt
dt
with
x  t    v ' x  t '  dt '
t
0
29
The Theorem of Taylor
Thus we have:
d 2x
2
dt
t

v ' x (t)
t
v ' x (t ') dt '  2 v ' 2x (t)
0
0
 2 v ' 2x (t)  

0
v ' x (t) v ' x (t  )
 t

v ' x (t) v ' x (t  )
 0
v ' 2x (t)
v ' x (t)  v ' x (t ')
v ' 2x (t)
dt '
t'  t  
d
v ' 2x (t)
 t
 2 v ' 2x (t) 

dt '  d
t
d  2 v ' 2x (t)
R
L,x ( )
d
0
Integration yields the Theorem of Taylor:
t t'
 2x (t)  2 v ' 2x (t)
 R
L,x ( ) d dt '
0 0
30
15
The Theorem of Taylor - Discussion
Taylor theorem:
t t'
 2x ((t))  2  v ' 2x    R L,x
L x (  ) d  dt '
0 0
Since RL → 0 for large ,
we have for large t':

t'
0
R L,x     d   L  const.
const
L: Lagrangian time scale
t
t   L R L  1   1 ddt '   t ' dt ' 
0
1 2
t   x (t)  v ' 2  t
2
t
t   L R L  0   R L,x  ddt '    L dt '   L  t   x (t)  2  v ' 2   L  t
0
31
Turbulent „Diffusion“
Molecular diffusion (see lecture 4):
 2 (t)  2Dt    t  
D
2Dt,
1
1
v G  v 2
3
3
mixing length
l' = v' λL.
Turbulent diffusion:
 t   L
1 d2
 
K 
2 dt
 t   L
Analogy:
K x ( t )  v ' 2x  t
K x ( t )  v ' 2x  
molecular

v m2 o lec.

Molecular velocity
Collision interval


Mean free path
 G  v

Molecular diffusivity
D
D from variance
1
v G 
3
2

D
2t
L
 v 'x  l'x
turbulent
v '2
Turbulent velocity
L
l  v    L
Lagrangian time scale
K  v ' l '
Turbulent diffusivity
K 
1 d 2
2 dt
Mixing length
K from variance
32
16
Molecular versus Turbulent Diffusion
Molecular Diffusion
Turbulent Diffusion
t = x/<vx>
Molecular diffusion:
σ2 = 2/3 v  t with v: thermal velocity;  : mean free path
Turbulent diffusion:
2 2
for t   L
v ' t
2  
2
2 v '  L t for t   L
In analogy to molecular diffusion, define mixing length l' = v' λL. For large
t, this yields: 2
  2 v ' 2 L t  2 v ' 2  L t  2 v ' v '  L t  2 v 'l' t
Typical values for Lagrangian scale time in case of exhaust from a chimney:
λL ≈ 15-30 min.
33
Summary
• Turbulence is a result of the instability of fluid flow brought about by the
non-linearity of the Navier-Stokes equation
• Turbulence leads to chaotic, unpredictable behaviour (determin. chaos)
• Turbulence occurs for large Reynolds numbers
• Turbulent energy is passed from larger to smaller eddies, until molecular
friction dominates and energy is dissipated (turbulent energy cascade)
• Reynolds decomposition: Mean and fluctuations of velocity
• Important concepts for a statistical description of turbulent fluctuations:
– Power spectrum (energy density of turbulence)
– Autocorrelation of velocity fluctuations
• Turbulence produces statistical displacements and hence transport:
turbulent diffusion
• For sufficiently large diffusion times, turbulent diffusion is analogous to
molecular diffusion (but with different diffusion coefficients)
34
17