flow reversal in a simple dynamical model of turbulence
DESCRIPTION
Flow Reversal in a Simple Dynamical Model of Turbulence. Kees Kuijpers, 0632305 Mariska van Rijsbergen, 0636290. Content. Introduction Shell models Pitchfork bifurcation GOY model Conclusions and further research. Introduction. Focus of the article: Rayleigh- Benard convection - PowerPoint PPT PresentationTRANSCRIPT
Flow Reversal in a Simple Dynamical Model of TurbulenceKees Kuijpers, 0632305Mariska van Rijsbergen, 0636290
Content
• Introduction• Shell models• Pitchfork bifurcation• GOY model• Conclusions and further research
04/22/2023 2Flow Reversal in a Simple Dynamical Model of Turbulence
Introduction
• Focus of the article: Rayleigh-Benard convection
• Research also useful for the magnetic polarity of the Earth
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Shell models
• Simple model of turbulence• Find a simplified version of the Navier-Stokes
equations• Shell models
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From Navier-Stokes to the GOY-model
The Navier-Stokes equation:
The Fourier transformed Navier-Stokes equation is
The GOY model is given by
2 pt
u u u u f
* * * * * * 21 2 1 1 1 2 1 2
nn n n n n n n n n n n n n n n
du i a k u u b k u u c k u u k u fdt
2 , , ,n m ij j jk u t ik P u t u t dt
p q k
k k p p p
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Conservative quantities (1)
• Conservation of these quantities require f=0 and ν=0.
• The GOY model reduces to:
• Theorem of Noether: For each symmetry there is a corresponding conservation law
* * * * * *1 2 1 1 1 2 1 2
nn n n n n n n n n n n n
du i a k u u b k u u c k u udt
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Conservative quantities (2)
• Conservation of energy
• Conservation of helicity
• Conservation of enstrophy
22
0
1 12 2
NFourier
nn
E v dx E u
22 2 2
0
NFourier
n nn
d x k u
2
0
' ' 1N
nn n
n
H d H k u
v ω x
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Conservation of energy
• The energy is given by:
• Conservation of energy requires:
• Taking a=1, and b=-eps then c=-(1+eps)
*2 * *
0 0 0
1 1 1 02 2 2
N N Nn n
n n n n nn n n
du dud dE u u u u udt dt dt dt
* * *1 2 1 2 1 2
0
02
N
n n n n n n n n n nn
iE k a b c u u u u u u
1 2 0n n na b c
2
0
12
N
nn
E u
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Conservation of enstrophy
• The enstrophy is given by:
• Conservation of enstrophy requires:
• Take bn=-eps=-5/4, and kn=2n k0 then the enstrophy is conserved
2 2 21 1 2 2 0n n n n n na k b k c k
*22 2 * 2 * 2
0 0 0
0N N N
n nn n n n n n n n n
n n n
du dud dk u k u u k u k udt dt dt dt
22
0
N
n nn
k u
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Conservation of helicity
• The helicity is given by:
• Conservation of helicity requires:
• Take bn=-eps=-1/2, and kn=2n k0 then the helicity is conserved
2
0
1N
nn n
n
H k u
1 1 2 2 0n n n n n na k b k c k
*
2 * *
0 0 0
1 1 1 1 0N N N
n n n nn nn n n n n n n n n
n n n
du dud dH k u k u u k u k udt dt dt dt
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GOY shell model vs. Sabra shell model
• GOY shell model:
• a=1 and c=-(1+b)
• Sabra shell model:
• a=1 and c=(1+b)
* * * * * * 21 2 1 1 1 2 1 2
nn n n n n n n n n n n n n n n
du i a k u u b k u u c k u u k u fdt
* * 21 1 2 1 1 1 1 2
nn n n n n n n n n n n n n n n
dui a k u u b k u u c k u u k u f
dt
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Flow reversal in the model
• GOY shell model:
• Instead of a force, we use a different approach
* * * * * * 21 2 1 1 1 2 1 2
nn n n n n n n n n n n n n n n
du i a k u u b k u u c k u u k u fdt
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Pitchfork bifurcation
Two possible standard solutions:• Subcritical bifurcation:
• Supercritical bifurcation
3xrxx
3xrxx
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Our problem: Goy-model
12
13
120
11 ukuu
uu Imperfect parameter
In our case we have a special form of the supercritical pitchfork bifurcation:
Imperfect supercritical pitchfork bifurcation3xrxhx
14Flow Reversal in a Simple Dynamical Model of Turbulence
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Our problem: Goy-model
Introduce 120
*1 u
uu
Rewriting 12
13
120
11 ukuu
uu
We get the standard form with
3xrxhx 2
1
32120
20
*1
kr
uuaikuu
h
ux
15Flow Reversal in a Simple Dynamical Model of Turbulence
Solutions imperfect supercritical bifurcation• For h=0 we get the same solution as in the normal
supercritical bifurcation
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Solutions imperfect supercritical bifurcation• For h≠0 we get solutions depending on h:
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Solutions imperfect supercritical bifurcation, with r constant
21r k
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Calculation GOY model with bifurcation
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Calculation GOY model with bifurcation
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Calculation GOY model with bifurcation
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Calculation GOY model with bifurcation
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Two state model
• In order to get a two state model, we introduced a pitchfork bifurcation:
• The quantity real(Φu1*) is the amount of energy
transferred by mode u1 to smaller scales (u2, u3)
221 1
1 1 120
1du uu k udt u
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Find the average values for the upper and lower state (1)
• Assume:
where
• Now, we can compute β:
• Our bifurcation formula:
• Combination of the last two equations:
1 'u *1' 0 and 0u
221 1
1 1 120
1du uu k udt u
2 31 2
0
0R Rk U Uu
2*1 1real u u
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Find the average values for the upper and lower state (2)
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Conclusion
• We can work with a simple model for the fluid motion
• We can simulate:• A system with two stable solutions• The reversal of the flow
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Further research
• The influence of k0 needs to be investigated• The influence and physical meaning of b (-eps)• The reliability of the model• Is the energy really conserved?• Why do our solutions sometimes explode?
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