Two scale modeling of superfluid turbulence Tomasz Lipniacki

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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.

• Solutions of single vortex motion in LIA
• Fenomenological models of anisotropic
• turbulence

vortex line core diameter 1Å
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Localized Induction Approximation
c - curvature - torsion
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Ideal vortex
Totally integrable, equivalent to non-linear
Schrödinger equation. Countable (infinite) family
of invariants
Quasi static solutions
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Hasimoto soliton on ideal vortex (1972)

F
i
g
u
r
e
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2
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Kida 1981
General solutions of in terms of elliptic
integrals including shapes such as circular
vortex ring helicoidal and toroidal
filament plane sinusoidal filament Eulers
elastica Hasimoto soliton
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Quasi static solutions for quantum vortices ?
Quantum vortex shrinks
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Quasi-static solutions Lipniacki JFM2003
Frenet Seret equations
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In the case of pure translation we get analytic
solution
For
where
Solution has no self-crossings
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Quasi-static solution pure rotation
Vortex asymptotically wraps over cones with
opening angle
for in cylindrical coordinates
where
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Quasi-static solution general case
For vortex wraps over paraboloids
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Results in general case
I. 4-parametric class of solutions, determined
by initial condition
II. Each solution corresponds to a specific
isometric transformation G(t) related
analytically to initial condition. III.
The asymptotic for is related analytically via
transformation G(t) to initial condition.
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Shape-preserving solutions Lipniacki, Phys.
Fluids. 2003
Equation
is invariant under the transformation
If the initial condition is scale-free, than the
solution is self-similar
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

• solutions with
• decreasing scale

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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
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Shape preserving solution general case
Logarithmic spirals on cones
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Results in general case
I. 4-parametric class of solutions, determined
by initial condition
II. Each solution corresponds to a specific
similarity transformation III. The
asymptotics for is given the initial condition
of the original problem (with time).
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Self-similar solution for vortex in Navier-Stokes?
N-S is invariant to the same transformation
The friction coefficient corresponds to
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Wing tip vortices
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Macroscopic description
-- Euler equation for superfluid component --
Navier-Stokes equation for normal component
Coupled by mutual friction force
where
and L is superfluid vortex line density.
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Microscopic description of tangle superfluid
vortices
Aim
Assuming that quantum tangle is close to
statistical equilibrium derive equations for
quantum line length density L and anisotropy
parameters of the tangle.
Statistical equilibrium?
Two cases I rapidly changing counterflow, but
uniform in space II slowly changing
counterflow, not uniform in space
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Model of quantum tangle
Particles segments of vortex line of length
equal to characteristic radius 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.
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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
is
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Evolution of line length density
where
average curvature
average curvature squared
Average binormal vector (normalized)
I
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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 ring of 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.
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Degradation term relaxation due to reconnections
Reconnections produce curvature but not net
binormal
Total curvature of the tangle
Reconnection frequency
Relaxation time
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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
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Quantum turbulence with high net macroscopic
vorticity
Vinen-Schwarz equation
We assume now
Let
denote anisotropy of the tangent
Vinen type equation for turbulence with net
macroscopic vorticity
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Drift velocity
where
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Mutual friction force
drag force
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 velocity U-superfluid velocity
q- anisotropy l-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
  analyze
• dynamic of both components.

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Non-linear Schrodinger equation (Gross,
Pitaevskii)
For
Madelung Transformation
superfluid velocity superfluid density
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Modified Euler equation
With pressure
Quantum stress
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Mixed turbulence Kolgomorov spectrum for each
fluid
- 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