Session MN: Quantized Vortices and Vortex Dynamics

Quantum length scale to distinguish between Kolmogorov and Vinen forms of quantum turbulence

Quantum turbulence (QT) is the motion of a tangle of thin vortex filaments spontaneously generated in superfluids. Once created, the self-interaction of vortex lines in the tangle will itself create conditions for the decay of its energy. Two robust asymptotic decay laws appear in past experiments, with the vortex line density decaying either as the -1 or the -3/2 power of time. We define a new quantum length-scale which demarcates these two types of decaying QT, denoted as the Vinen (or ultraquantum) QT and the Kolmogorov (or quasi-classical) QT, respectively. We discuss spectral characteristics of energy for the two decay laws and relate them to the decay exponents. While the Vinen QT has no counterpart in classical turbulence, the Kolmogorov QT does, and we discuss this similarity.   

Particles for flow visualization and velocimetry in liquid nitrogen

Liquid nitrogen may be used to generate, in a facility of a given size, Reynolds numbers which are substantially larger than that in water because its kinematic viscosity is one fifth that of water at 25$^{\circ}$ C. We present a simple technique, previously used in liquid helium [1,2], to create solid tracers for visualization and velocimetry in turbulent liquid nitrogen. These tracers are created by injecting a gaseous mixture of room-temperature nitrogen and an additional gas (element or compound) into the flow. The latter is selected such that, when cooled below 77 K (nitrogen boiling point), it freezes into solid particles with the highest mismatch in the index of refraction and the lowest mismatch in density compared to the surrounding liquid nitrogen. We discuss the formation process of the particles, and characterize the effects of the dilution ratio and gas selection on their size, brightness and fidelity. Possibilities of using this technique for fluid dynamics experiments that require visualization of high Reynolds number flows are reviewed. [1] G. P. Bewley, D. P. Lathrop, and K. R. Sreenivasan, Nature 441, 588 (2006). [2] M. S. Paoletti, R. B. Fiorito, K. R. Sreenivasan, and D. P. Lathrop, J. Phys. Soc. Jpn. 77, 111007 (2008).   

Reconnection of quantized vortices and quantum turbulence

Turbulence in superfluid 4-Helium is dominated by reconnection and ring collapse. We utilize micron and nano-scale ice particles to visualize the dynamics of quantized vortices and the normal component. After briefly reviewing our observations of these phenomena, I will discuss reconnection dynamics at large and small scales. Those dynamics can be understood using scaling solutions and some ideas from dynamical systems. There is one underlying question we work to address: is there a single universal reconnection dynamics, do we need to consider a one or two parameter family of reconnection events?   

Visualization of the Quantized Vortex Lattice Dynamics in $^{4}$He

We study the lattice structure and dynamics of quantized vortices in superfluid helium using a new rotating experiment. This setup includes control of the entire apparatus from the rotating frame as well as implementation of a novel isolation cell, which permits investigation into new phenomena such as differential rotation in helium-II. Our documentation of the vortex lattice dynamics in the (r, $\varphi )$ plane (i.e. longitudinal to the vortices) includes real-time visualization of Tkachenko waves as well as evidence of differential rotation with distinct Stewartson layer boundaries. We also present possible Kelvin-Helmholtz instabilities and the formation and propagation of superfluid vortex bundles. We show that the angular velocity is a function of radius and may be driven by the geometry of the isolation cell.   

Initial conditions for reconnection calculations of quantized vortices

Vortex reconnection occurs when two vortices intersect and then rejoin with exchanged tails. This process occurs in both classical fluids and superfluids, in superconductors, and in magnetized plasmas. In helium II this phenomenon has been investigated numerically via the Gross-Pitaevskii equation. Although a simplified model, the Gross-Pitaevskii equation is interesting since it naturally embodies vortex reconnection, as first numerically shown by Koplik {\&} Levine [Phys. Rev. Lett. 71, 1375 (1993)]. A crucial issue in such computations is the selection of initial conditions. The question we address is how initial wave functions may effect the outcome of reconnection. Traditionally the initial conditions have been generated by multiplying approximate wave functions for a single vortex and imposing, to some degree, periodic boundary conditions. An alternative approach will be presented. It consists in selecting an initial configuration that minimizes the total energy. The differences between the results obtained with this approach and previous ones will be discussed especially with respect to the dispersion of energy seen in quantum turbulence.   

Motion of a Vortex Filament in the Local Induction Approximation: Reformulation of the Da Rios-Betchov Equations in the Extrinsic Filament Coordinate Space

In recognition of the highly non-trivial task of computation of the inverse Hasimoto transformation mapping the intrinsic geometric parameter space onto the extrinsic vortex filament coordinate space a reformulation of the Da Rios-Betchov equations in the latter space is given (Shivamoggi and van Heijst [1]). The nonlinear localized vortex filament structure solution given by the present formulation is in detailed agreement with the Betchov-Hasimoto solution in the small- amplitude limit and is also in qualitative agreement with laboratory experiment observations of helical-twist solitary waves propagating on concentrated vortices in rotating fluids. The present formulation also provides for a discernible effect of the slipping motion of a vortex filament on the vortex evolution via an amplitude change in the vortex kink.\\[4pt] [1] B. K. Shivamoggi and G. J. F. van Heijst: {\it Physics Letters A}, (2010).   

On Lagrangian and vortex-surface fields in Taylor-Green and Kida-Pelz flows

A methodology is developed for constructing smooth scalar fields $\phi$ for Taylor-Green and Kida-Pelz velocity fields that, at $t=0$, satisfy $\omega\cdot\nabla\phi=0$. We refer to such fields as vortex-surface fields. Iso-surfaces of $\phi$ then define vortex surfaces. Given the vorticity, our definition of a vortex-surface field is shown to admit nonuniqueness, and this is resolved numerically using an optimization approach. Equations describing the evolution of vortex-surface fields are obtained for both inviscid and viscous incompressible flows. For the former, the Helmholtz vorticity theorem shows that Lagrangian material surfaces which are vortex (or vorticity) surfaces at the initial time remain so for later times. By tracking $\phi$ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes, with subsequent rolling up of structures near the interface. The non- local geometry in the evolution is quantified by differential geometry properties.   

Quasi-steady linked vortices with chaotic streamlines

We study the dynamics of two or more toroidal filamentary vortices ---i.e. thin tubular vortices coiled on an immaterial torus--- in an otherwise quiescent, ideal fluid. Assuming that the vortices are identical and equally spaced on a meridional section of the torus, the flow evolution depends on the torus aspect ratio ($R$), the number of vortices ($N$), and the vortex topology ($V_{p,q}$, where $p$ and $q$ are coprime integers such that the $V_{p,q}$ vortex winds $p$ times round the torus symmetry axis and $q$ times round the torus centerline). The evolution of sets of $V_{1,1}$ and $V_{1,2}$ vortices was computed using the Rosenhead--Moore approximation to evaluate the velocity field and a fourth-order Runge-Kutta scheme to advance in time. It was found that vortex sets with $N<6$ and $R<0.15$ progressed along and rotated around the torus symmetry axis in an almost steady manner while each vortex in the set approximately preserved its shape. The velocity field, observed in the comoving frame, has two stagnation points. The stream tube starting at the forward stagnation point and the stream tube ending at the backward stagnation point transversely intersect along a finite number of streamlines. The three-dimensional chaotic tangle that arises has a geometry which depends primarily on the number of vortices $N$.   

Dynamics of Oscillatory Vortex Multipoles Generated by Electromagnetic Forcing

Vortices formed by the concurrent effect on a localized magnetic field distribution and two alternate electric currents perpendicular to each other in a shallow (4mm) layer of an electrolyte are analyzed. Alternate currents with frequencies and amplitude in the range of 1-500 mHz and 80 mA, respectively, are explored. For a single dipolar magnetic field and a single electric current, the dominant structure of the flow is a pair of alternating lobes located co-linear with the generated Lorenz force. The flow presents a resonant behavior when the forcing frequency is around 10 mHz. When multipoles are used to generate the magnetic field, more complicated lobe distributions are obtained. The flow patterns were successfully described using a quasi-two-dimensional numerical model. A tridimensional numerical models corroborates the theoretical results. Flow visualization and numerical Lagrangian particle tracking indicate that multipolar flows present symmetries according to the magnetic field distributions. Although in some regions the flow patterns efficiently mix the fluid, the mixing is inhomogeneous due to symmetry conditions of the flows. Mixing is enhanced when symmetries are destroyed by the use of a random array of magnets or by injecting two electric currents.   

Vortical structures in the shallow flow past a magnetic obstacle in an electrolytic layer

It is known that the interaction of electric currents (induced or injected) and a localized magnetic field produces a Lorentz force that inhibits the motion of the fluid and acts as an obstacle for the flow (a magnetifc obstacle). In this work, the flow in a shallow electrolytic layer produced by a uniform injected current and a localized non-uniform magnetic field is simulated numerically using quasi-twodimensional and three-dimensional models, with a parallelized version of the numerical code. Different vortex patterns that have been observed experimentally in the wake of the magnetic obstacle are obtained, including steady vortex dipoles and vortex shedding flow. The three dimensional structure of the flow is explored and, particularly, the velocity profiles in the layer depth showing the appearance of inflection points that determine the stability properties of the flow, are analyzed.