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Two scale modeling of superfluid turbulence Tomasz Lipniacki
He II: -- normal component,-- superfluid component,-- superfluid vortices.
Relative proportion of normal fluid and superfluid as a function of temperature.
vortex line core diameter ~ 1Å
• Solutions of single vortex motion in LIA
• Fenomenological models of anisotropic turbulence
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Localized Induction Approximation
nbssss cct ),(
binormal - ,normal-
,tangent-
, and where
bssns
ts
ssss
cc
dd
dtd
c - curvature- torsion
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sss ),( tIdeal vortex
)()()(),( utttWt ss
Quasi static solutions:
Totally integrable, equivalent to non-linear Schrödinger equation.Countable (infinite) family of invariants:
)(),(),(),( uttutctc
dcccdcdcddL 222
222 '
4,,,,
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Figure 1
:
Hasimoto soliton on ideal vortex (1972)
sss ),( t
20
20
0
000
4 ,2
,2/)(sech
cu
utccc
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Kida 1981
sss ),( tGeneral solutions of in terms of elliptic integrals including shapes such as
circular vortex ringhelicoidal and toroidal filamentplane sinusoidal filamentEuler’s elasticaHasimoto soliton
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Quasi static solutions for quantum vortices ?
Quantum vortex shrinks:
0
2dc
)(),( nbssss ct
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Quasi-static solutionsLipniacki JFM2003
)0,()()(),( sWs ttt
)(),( ),(),( tctc
)0,()(0
2
sωVtnb dcc
0
2222
0
232
222
0
20
dccc
cccccc
dccccccc
nbbtnnt , , ccFrenet Seret equations
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In the case of pure translation we get analytic solution
220
220
0
,
)tanh( ),sech(
cBcA
ABAcc
AAtcdRdRt /))ln(cosh(,)(qsin,)cos(q),( 220
s
where 220q ),sech( AcAR
Vtnb
0
2)( dcc
0
22
22
dccc
cc
2/
For 254.2/ Solution has no self-crossings
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Quasi-static solution: pure rotation
Vortex asymptotically wraps over coneswith opening angle
0
3/13/2
00
3/1 )cot()(,23)(,||)(
KzK
Kr
for in cylindrical coordinates ),,( zr
/)cot(,2
where
00
303 K
)0('),0(),0('),0( 0000 cccc
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Quasi-static solution: general case
For vortex wraps over paraboloids
0
3/203/2
00
3/1 3)(,23)(,||)(
KTzK
Kr
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Results in general case
I. 4-parametric class of solutions, determined by initial condition
)0( ),0( ),0( ),0( cc
II. Each solution corresponds to a specific isometric transformation (t)related analytically to initial condition.
III. The asymptotic for
is related analytically via transformation (t) to initial condition.
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Shape-preserving solutionsLipniacki, Phys. Fluids. 2003
ssss ),( t
, ,2 tt
Equation
is invariant under the transformation
If the initial condition is scale-free, than the solution is self-similar
t
ttt Ss )(),(
In general scale-free curve is a sum of two logarithmic spirals on coaxial cones; there is four parametric class of such curves.
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Shape-preserving solutions
t
ttt Ss )(),(
. ,1 ,1t
lt
Ttt
Kt
c
solutions withdecreasing scale
222
2
22
0
2222
0
232
TlTldKTTKK
TKTKKK
KTK
KlKldKKKKTKTKTK
l
l
ssss ),( t
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Analytic result: shape-preserving solution
In the case when transformation is a pure homothety we get analytic solution in implicit form:
KK
KKpKKKKdKl
K
K
T ,
)()/ln()(
0
2200
22
scale growing 1p 0T scale, decreasing 1 p
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Shape preserving solution: general case
scale growing 1p scale decreasing 1pLogarithmic spirals on cones
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Results in general case
I. 4-parametric class of solutions, determined by initial condition
)0( ),0( ),0(K ),0( TTK
II. Each solution corresponds to a specific similarity transformation
III. The asymptotics for
is given the initial condition of the original problem (with time).
l
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Self-similar solution for vortex in Navier-Stokes?
ωωvω 2)( v
t
N-S is invariant to the same transformation
,/, ,2 rVVrrtt
The friction coefficient corresponds to
b
n
b
n
VVv
VV
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Wing tip vortices
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-- Euler equation for superfluid component
-- Navier-Stokes equation for normal component
Macroscopic description
Coupled by mutual friction force LVF nsns ~
where snns VVV and L is superfluid vortex line density.
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Microscopic description of tangle superfluid vortices
Assuming that quantum tangle is “close” to statistical equilibriumderive equations for quantum line length density L and anisotropy parameters of the tangle.
Aim
Two cases
I – rapidly changing counterflow, but uniform in space
II – slowly changing counterflow, not uniform in space
Statistical equilibrium?
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Model of quantum tangle“Particles” = segments of vortex line of length equal to characteristicradius of curvature. Particles are characterized by their tangent t, normal n, and binormal b vectors. Velocity of each particle is proportional to its binormal (+collective motion).Interactions of particles = reconnections.
STAdtdQVn /
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Reconnections
1. Lines lost their identity: two line segments are replaced by two new line segments
2. Introduce new curvature to the system
3. Remove line-length
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Motion of vortex line in the presence of counterflow
Evolution of its line-length
snns VVV
dl is
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Evolution of line length density
where
I
average curvature
average curvature squared
Average binormal vector (normalized)
dL 1
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Rapidly changing counterflow
Generation term: polarization of a tangle by counterflow
Degradation term: relaxation due to reconnections
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Generation term: polarization of a tangle by counterflow
evolution of vortex ringof radius R and orientation in uniform counterflow
unit binormal
Statistical equilibrium assumption: distribution of f(b) is the most probable distribution giving right I i.e.
I
bIexp)( Cbf
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Degradation term: relaxation due to reconnections
Reconnections produce curvature but not net binormal
Total curvature of the tangle
Reconnection frequency 2/2/51 Lcfr
Relaxation time: LfCT
rr
2
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Finally
Lipniacki PRB 2001
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Model prediction: there exists a critical frequency of counterflow above which tangle may not be sustained
max78.0 L max94.0 L
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Quantum turbulence with high net macroscopic vorticity
nsns I VVI 0We assume now
Let denote anisotropy of the tangent
Vinen type equation for turbulence with net macroscopic vorticity
Vinen-Schwarz equation
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Drift velocity LV
where
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Mutual friction force
drag force
nsF
where
finally
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Helium II dynamical equations
Lipniacki EJM 2006
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Specific cases: stationary rotating turbulence
Pure heat driven turbulence
Pure rotation
„Sum”
Fast rotation Slow rotation
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Plane Couette flow
V- normal velocityU-superfluid velocity
q- anisotropyl-line length density
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“Hypothetical” vortex configuration
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Summary and Conclusions
• Self-similar solutions of quantum vortex motion in LIA
• Dynamic of a quantum tangle in rapidly varying counterflow
• Dynamic of He II in quasi laminar case.
In small scale quantum turbulence, the anisotropy of the tangle in addition to its density L is key to analyzedynamic of both components.
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bEVmt
i *0
22
2
Non-linear Schrodinger equation (Gross, Pitaevskii)
*2
)/(
mmA
mV
s
s
Madelung Transformation
superfluid velocity
superfluid density
)exp( iAFor
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k
jk
jjss
ss xx
PVVt
V
2
20
2mVP s
kj
ssjk xxm
)ln(
4
2
2
2
Modified Euler equation
With
pressure
Quantum stress
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Mixed turbulence
Kolgomorov spectrum for each fluid
3/5
3/2
)(k
CkE
- energy flux per unit mass
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Two scale modeling of superfluid turbulence Tomasz Lipniacki
He II: -- normal component,-- superfluid component,-- superfluid vortices.
Relative proportion of normal fluid and superfluid as a function of temperature.
vortex line