Analytical Solutions to the Flow Field Induced by Uniformly Moving Multiple Helical Vortex Filaments

This paper presents analytic solutions for the flow field of inviscid fluid induced by uniformly and rigidly moving multiple helical vortex filaments in a cylindrical pipe. The relative coordinate system is set on the moving vortex filaments. The analytical solutions of the flow field are obtained on the assumption that the relative velocity field induced is time-independent and helically symmetrical. If the radius of the cylindrical pipe approaches infinity, these solutions are also available for unbounded space. The results show that both the absolute velocity field and pressure field are periodical in time, and may reduce to time-independent when the helical vortex filaments are immobile or slip along the filaments themselves. Furthermore, the solution of velocity field is reduced to Okulov's formula for the case of a single static vortex filament in a cylindrical pipe. The calculated locations of pressure peak and valley on the pipe wall agree with experimental results.

On the Dynamics of Vortex Filaments and Their Core Structure

We give a brief review of the asymptotic theory of slender vortex filaments with emphases on (i) the choices of scalings and small parameters characterizing the physical problem,(ii) the key steps in the formulation of the theory and (iii) the assumptions and/or restrictions on the theory of Callegari and Ting (1978).We present highlights of an extension of the 1978 asymptotic theory:the analyses for core structures with axial variation.Making use of the physical insights gained from the analyses,we present a new derivation of the evolution equations for the core structure.The new one is simpler and straightforward and shows the physics clearly.

Equilibrium Energy and Entropy of Vortex Filaments in the Context of Tornadogenesis and Tornadic Flows

In this work, we study approximations of supercritical or suction vortices in tornadic flows and their contribution to tornadogenesis and tornado maintenance using self-avoiding walks on a cubic lattice. We extend the previous work on turbulence by A. Chorin and collaborators to approximate the statistical equilibrium quantities of vortex filaments on a cubic lattice when both an energy and a statistical temperature are involved. Our results confirm that supercritical (smooth, “straight”) vortices have the highest average energy and correspond to negative temperatures in this model. The lowest-energy configurations are folded up and “balled up” to a great extent. The results support A. Chorin’s findings that, in the context of supercritical vortices in a tornadic flow, when such high-energy vortices stretch, they need to fold and transfer energy to the surrounding flow, contributing to tornado maintenance or leading to its genesis. The computations are performed using a Markov Chain Monte Carlo approach with a simple sampling algorithm using local transformations that allow the results to be reliable over a wide range of statistical temperatures, unlike the originally used pivot algorithm that only performs well near infinite temperatures. Efficient ways to compute entropy are discussed and show that a system with supercritical vortices will increase entropy by having these vortices fold and transfer their energy to the surrounding flow.

Branch Processes of Vortex Filaments and Hopf Invariant Constraint on Scroll Wave

In this paper, by making use of Duan＇s topological current theory, the evolution of the vortex filaments in excitable media is discussed in detail. The vortex filaments are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of a complex function Z（x, t）. [t is also shown that the Hopf invariant of knotted scroll wave filaments is preserved in the branch processes （splitting, merging, or encountering） during the evolution of these knotted scroll wave filaments. Furthermore, it also revealed that the ＂exclusion principle＂ in some chemical media is just the special case of the Hopf invariant constraint, and during the branch processes the ＂exclusion principle＂ is also protected by topology.

Breather-induced quantised superfluid vortex filaments and their characterisation

We study and characterise the breather-induced quantised superfluid vortex filaments which correspond to the Kuznetsov-Ma breather and super-regular breather excitations developing from localised perturbations.Such vortex filaments,emerging from an otherwise perturbed helical vortex,exhibit intriguing loop structures corresponding to the large amplitude of breathers due to the dual action of bending and twisting of the vortex.The loop induced by the Kuznetsov-Ma breather emerges periodically as time increases,while the loop structure triggered by the superregular breather—the loop pair—exhibits striking symmetry breaking due to the broken reflection symmetry of the group velocities of the super-regular breather.In particular,we identify explicitly the generation conditions of these loop excitations by introducing a physical quantity—the integral of the relative quadratic curvature—which corresponds to the effective energy of breathers.Despite the nature of nonlinearity,it is demonstrated that this physical quantity shows a linear correlation with the loop size.These results will deepen our understanding of breather-induced vortex filaments and be helpful for controllable ring-like excitations on the vortices.

TOPOLOGICAL STRUCTURES OF VORTEX AND HELICITY ANALYSIS

The topological structures of the vortex filaments and vortex tubes with an exact solution of a straight spiral vortex tube are discussed. It is found that there are some confusions about the calculation of the helicity of a knotted vortex filament and some linked vortex filaments by using different methods. How to unify these methods is explained and the right results are given. (Edited author abstract) 5 Refs.

ANALYTICAL SOLUTION FOR THE VELOCITY FIELD INDUCED BY A UNIFORMLY MOVING HELICAL VORTEX FILAMENT IN CYLINDRICAL TUBE

A derivation of an analytical expression for the inviscid velocity fieldinduced by a single right-handed helical vortex filament is presented. The vortex filament movesuniformly and rigidly without change of form in a cylindrical tube, where the vortex filamentrotates around its axis with a constant angular velocity and translates along its axis with aconstant translational velocity. The key to solve the problem is to set up a moving cylindricalcoordinate system fixed on the vortex filament. The result shows that the velocity field is atime-periodic function, and may degenerate into Okulovs' s formula when the helical vortex filamentslips along the filament itself or stays immobile.

Dynamic modeling and analysis of vortex filament motion using a novel curve-fitting method

Applications of a novel curve-fitting technique are presented to efficiently predict the motion of the vortex filament, which is trailed from a rigid body such as wings and rotors. The gov- erning equations of the motion, when a Lagrangian approach with the present curve-fitting method is applied, can be transformed into an easily solvable form of the system of nonlinear ordinary dif- ferential equations. The applicability of Bezier curves, B-spline, and Lagrange interpolating polyno- mials is investigated. Local Lagrange interpolating polynomials with a shift operator are proposed as the best selection for applications, since it provides superior system characteristics with minimum computing time, compared to other methods. In addition, the Gauss quadrature formula with local refinement strategy has been developed for an accurate prediction of the induced velocity computed with the line integration of the Biot-Savart law. Rotary-wing problems including a vortex ring problem are analyzed to show the efficiency, accuracy, and flexibility in the applications of the pro- posed method.