Presentation on theme: "Two scale modeling of superfluid turbulence Tomasz Lipniacki"— Presentation transcript:
1 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 LIAFenomenological models of anisotropicturbulencevortex line core diameter ~ 1Å
2 Localized Induction Approximation c - curvature- torsion
3 Quasi static solutions: Ideal vortexTotally integrable, equivalent to non-linear Schrödinger equation.Countable (infinite) family of invariants:Quasi static solutions:
4 Hasimoto soliton on ideal vortex (1972) :Figure12
5 Kida 1981General solutions ofin terms of elliptic integrals including shapes such ascircular vortex ringhelicoidal and toroidal filamentplane sinusoidal filamentEuler’s elasticaHasimoto soliton
6 Quasi static solutions for quantum vortices ? Quantum vortex shrinks:
8 In the case of pure translation we get analytic solution ForwhereSolution has no self-crossings
9 Quasi-static solution: pure rotation Vortex asymptoticallywraps over coneswith opening anglefor in cylindrical coordinateswhere
10 Quasi-static solution: general case For vortex wraps over paraboloids
11 Results in general case I. 4-parametric class of solutions, determined byinitial conditionII. Each solution corresponds to a specific isometric transformation G(t) related analytically to initial condition.III. The asymptotic foris related analytically via transformation G(t) to initial condition.
12 Shape-preserving solutions Lipniacki, Phys. Fluids. 2003Equationis invariant under the transformationIf the initial condition is scale-free, than the solution is self-similarIn general scale-free curve is a sum of two logarithmic spirals on coaxial cones; there is four parametric class of such curves.
14 Analytic result: shape-preserving solution In the case when transformation is a pure homothety we get analytic solution in implicit form:
15 Shape preserving solution: general case Logarithmic spirals on cones
16 Results in general case I. 4-parametric class of solutions, determined by initial conditionII. Each solution corresponds to a specific similarity transformationIII. The asymptotics foris given the initial condition of the original problem (with time).
17 Self-similar solution for vortex in Navier-Stokes? N-S is invariant to the same transformationThe friction coefficient corresponds to
19 Macroscopic description -- Euler equation for superfluid component-- Navier-Stokes equation for normal componentCoupled by mutual friction forcewhereand L is superfluid vortex line density.
20 Microscopic description of tangle superfluid vortices AimAssuming that quantum tangle is “close” to statistical equilibriumderive equations for quantum line length density L and anisotropyparameters of the tangle.Statistical equilibrium?Two casesI – rapidly changing counterflow, but uniform in spaceII – slowly changing counterflow, not uniform in space
21 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.
22 Reconnections 1. Lines lost their identity: two line segments are replaced by two new line segments2. Introduce new curvature to the system3. Remove line-length
23 Motion of vortex line in the presence of counterflow Evolution of its line-lengthis
24 Evolution of line length density whereaverage curvatureaverage curvature squaredIAverage binormal vector (normalized)
25 Rapidly changing counterflow Generation term: polarization of a tangle by counterflowDegradation term: relaxation due to reconnections
26 Generation term: polarization of a tangle by counterflow evolution of vortex ringof radius R and orientationin uniform counterflowunit binormalStatistical equilibrium assumption: distribution of f(b) is the mostprobable distribution giving right I i.e.
27 Degradation term: relaxation due to reconnections Reconnections produce curvature but not net binormalTotal curvature of the tangleReconnection frequencyRelaxation time:
37 Summary and Conclusions Self-similar solutions of quantum vortex motion in LIADynamic of a quantum tangle in rapidly varying counterflowDynamic of He II in quasi laminar case.In small scale quantum turbulence, the anisotropy ofthe tangle in addition to its density L is key to analyzedynamic of both components.
44 Two scale modeling of superfluid turbulence Tomasz Lipniacki He II:-- normal component,-- superfluid component,-- superfluid vortices.Relative proportion of normal fluid andsuperfluid as a function of temperature.vortex line