Transcript What is turbulence

‫به نام خدا‬
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Turbulence
is
beautiful
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Turbulence?
Every day phenomena: nature --- administration
Leonardo da Vinci : 1452 –1519
van Gogh
Turbolenza
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Generation of the random motions from
a smooth flow:
•The dynamic systems approach )chaos)
•The development of singular solutions of the ideal
fuid equations
•The excitation of instabilities and their effects
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Two body system (earth and sun)
Play 1
Sun an attractor of the Earth
•s
•E
Three body system as a nonlinear interaction.
1)The trajectory is chaotic
2)Sensitive Dependence on
Initial Conditions.
3) the trajectory never repeats.
4) All chaotic systems are nonlinear.
classical chaotic systems
deterministic.
non-deterministic.
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Quantum chaotic systems
What is turbulence ?
Whorls and eddies of all sizes
Chaotic and unpredictable
Unsteady and irregular
Diffusivity, rapid mixing
The motion is nonlinear
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HD and MHD equations
Viscose-stress tensor
Navier Stokes equation: NL
Magnetohydrodynamics:
NL
Vis
Thermal pressure
Speed of light
Using Faraday’s law we have
Viscosity
Which
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NL
Res
For large Re
non-linear regime
For small Re
dissipative regime
When non-linear terms dominate, energy is
transfer from large to small scales along a
turbulence cascade.
We observe Universality only at high Re.
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Turbulence in a magnetized plasma
Good confinement
Magnetized plasma is
usually unstable and turbulent
ITER
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High magnetic Reynolds number ~ 105
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Conservation equations:
1. The momentum-conservation law
2. The energy-conservation law
3. The cross-helicity-conservation law
4. The magnetic-helicity-conservation law
Where A is the vector potential and
Natural phenomenon is dissipative and
conservation equations are broken
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We introduce two dimensionless variables
which were first introduced by Elsasser, now:
Which
is total pressure
In the presence of a background magnetic field,
and neglecting the dissipation terms the
fields describe
Alfven waves propagating in the two opposite direction of
The interesting property of the Elsasser field is that there is
no self-coupling in the nonlinear term but only crosscoupling
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Turbulence?
A.N. Kolmogorov
1903 - 1987
Energy cascade, input at large scales
– dissipation at small scales
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What is different between the laminar flow and turbulence?
Re
Osborne Reynolds showed that as
increases ,
less than a threshold value for example in Reynolds ‘s
experimental it was 2300, the flow is laminar.
the fluid velocity dose not change with
time and all of streamline are parallel with
the one of the axial system
And in high Re number turbulence state is obtained.
streamlines become non-distinction
because of mixing with the surrounding
flow
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Play 3
High Reynolds numbers in MHD (Re , Rm) lead to
global scale of the order of the system size
down to very small scales, where
dissipation occurs.
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Spectral properties of MHD equations
One of the practical way to study MHD turbulence is direct
numerical simulation (DNS)
The nonlinear terms in MHD equations become
numerically expensive convolution sums. Based on
Pseudospectral method the nonlinear terms are compute
by applying Fast Fourier transforms (FFTs) and shuttle
between real-and Fourier-space.
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velocity and magnetic fields or Elsasser fields, can be written
in Fourier representation
Where
It is evident that the dissipative terms becomes very simple in
k space
while the nonlinear terms obviously are more complicated
•In the high Reynolds number, the nonlinear term be dominant. Since
turbulence dynamics contain many transfer process of certain
quantities, such as, energy or helicity, between different scale, the
Fourier representation which directly yields the corresponding
spectral densities, is very convenient.
•The transfer process of a conserved quantity is called a
cascade.
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Phenomenological models in MHD turbulence
According to the kolmogorov’s theory three range of
lengths
can be identifed
Injection range
inertial range where energy is transferred towards smaller and smaller
lengths, without any production of energy or dissipation
  l  L
Dissipation range
l 
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Viscous dissipation
Energy production
Transfer of energy to
successively smaller scale
the spectrum exhibits a
power law behavior
Dissipation range
Inertial range
Production range
injection
energy
Flux of energy
Dissipation of
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energy
Energy
Large scale
Small scale
structure
structure
Derived
range
Dissipation range
Inertial range
K
k
spectral slope is “UNIVERSAL”
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spectral-transfer process is local, i.e., the mode interactions
are dominated by wavenumbers of the same order of
magnitude.
Divide the inertial range into a discrete number of scales.
A typical turbulence eddy of scale can be represented by
the average difference in velocity
between two points a
distance apart or by the Fourier component
The time taken for transfer of energy between two neighboring
scales
and
or turnover, time of the eddy
is given
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The energy flux is constant across the inertial range
En
n
v
~ n
v n  E n  
2
k n 1
kn
3
ln
vn ~  l n
~
1/ 3
1/ 3
E k dk  E kn k n
Then energy spectrum is obtained
Ek  C K 
2/3
k
5 / 3
The well-known Kolmogorov spectrum
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Iroshnikov-Kraichnan (IK) spectrum:
The process is based on the interaction of eddies of size
with the magnetic field
Alfven waves
For two colliding Alfven waves of extent
the interaction
time to exchange energy nonlinearly is given by
which is much shorter than the nonmagnetic eddy-distortion
time
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inertial range scaling of the Elsasser fields
And energy spectrum
E k  C IK (v A )
1/ 2
k
3 / 2
Iroshnikov-Kraichnan (IK) spectrum
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Magnetic energy
Universality
Total energy
Kinetic energy
Compensated
Energies
with
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Turbulence Structures :
Self-similarity which was proposed by Kolmogorov related to
the lack of any characteristic length in the inertial range
???
Intermittent character
monofractality or multifractality behavior
I will discuss in the next seminar……
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Applications:
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DNA sequences (C.K Peng,PRE,1994- Z. Chen,PRE 2002)
heart rate dynamics (A. Bunde,PRL,2000-Y. Ashkenazy
,PRL,2001)
Neuron spiking(S. Bahar,Europhys. Lett. 2001)
human gait (Hausdorff J.M ,J.Appl.Physiology 1997)
long-time weather records(E.Bounde,PRL,1998)
cloud structure( K. Ivanova,Europhys. Lett.2000)
geology )B.D Malamud,J. Stat. Plan. Infer,1999)
ethnology (C.L. Alados, Ethnology,2000)
economical time series (R.N. Mantegna, Nature 1998,
PRE,2004) )Stock Markets , M.R. Rahimitabar PRE 2007, oil
price, M.Momeni,PRE, Submitted)
solid state physics , Rough Surface of graphene
)morphology(
Geophysics(Earth’s liquid core, Seismic (
Space plasma ( MHD, M.Momeni,PRE 2008, sunspot time
series and the solar wind(
Discharge Current
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The end
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