Topological Structure of Knotted Vortex Lines in Liquid Crystals

In this paper, a novel decomposition expression for the U（1） gauge field in liquid crystals （LCs） is derived. Using this decomposition expression and the b-mapping topological current theory, we investigate the topological structure of the vortex lines in LCs in detail. A topological invariant, i.e., the Chern-Simons （CS） action for the knotted vortex lines is presented, and the CS action is shown to be the total sum of all the self-linking and linking numbers of the knot family. Moreover, it is pointed out that the CS action is preserved in the branch processes of the knotted vortex lines.

Vortex lines in a ferromagnetic spin-triplet superconductor

Based on Duan＇s topological current theory,we show that in a ferromagnetic spin-triplet superconductor there is a topological defect of string structures which can be interpreted as vortex lines.Such defects are different from the Abrikosov vortices in one-component condensate systems.We investigate the inner topological structure of the vortex lines.The topological charge density,velocity,and topological current of the vortex lines can all be expressed in terms of 未 function,which indicates that the vortices can only arise from the zero points of an order parameter field.The topological charges of vortex lines are quantized in terms of the Hopf indices and Brouwer degrees of-mapping.The divergence of the self-induced magnetic field can be rigorously determined by the corresponding order parameter fields and its expression also takes the form of a 未-like function.Finally,based on the implicit function theorem and the Taylor expansion,we conduct detailed studies on the bifurcation of vortex topological current and find different directions of the bifurcation.

Vortex lines in a ferromagnetic spin-triplet superconductor

Based on Duan’s topological current theory,we show that in a ferromagnetic spin-triplet superconductor there is a topological defect of string structures which can be interpreted as vortex lines.Such defects are different from the Abrikosov vortices in one-component condensate systems.We investigate the inner topological structure of the vortex lines.The topological charge density,velocity,and topological current of the vortex lines can all be expressed in terms of δ function,which indicates that the vortices can only arise from the zero points of an order parameter field.The topological charges of vortex lines are quantized in terms of the Hopf indices and Brouwer degrees of-mapping.The divergence of the self-induced magnetic field can be rigorously determined by the corresponding order parameter fields and its expression also takes the form of a δ-like function.Finally,based on the implicit function theorem and the Taylor expansion,we conduct detailed studies on the bifurcation of vortex topological current and find different directions of the bifurcation.

Topological aspect of vortex lines in two-dimensional Gross-Pitaevskii theory

Using the C-mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross Pitaevskii theory in （3＋l）-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the function indicates that the vortices can only be generated from the zero points of Ф and are quantized in terms of the Hopf indices and Brouwer degrees. The C-mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.

Vortex Lines and Monopoles in Electrically Conducting Plasmas

Based on the φ-mapping topological current theory and the decomposition of gauge potential theory, the vortex lines and the monopoles in electrically conducting plasmas are studied. It is pointed out that these two topological structures respectively inhere in two-dimensional and three-dimensional topological currents, which can be derived from the same topological term n^→·（Эin^→×Эjn^→）, and both these topological structures axe characterized by the φ-mapping topological numbers-Hopf indices and Brouwer degrees. Furthermore, the spatial bifurcation of vortex lines and the generation and annihilation of monopoles are also discussed. At last, we point out that the Hopf invaxiant is a proper topological invaxiant to describe the knotted solitons.

Exploration of the mechanism of cavitation vortex rope and vortex development in the draft tube of tubular turbine units

Draft tube vortex rope is considered a special cavitation flow phenomenon in tubular turbine units.Cavitation vortex rope is one of the most detrimental factors affecting the safety of hydraulic turbines.In this study,ANSYS CFX software was utilized to numerically simulate the internal cavitation flow of a hydraulic turbine draft tube.The evolution of the cavitation vortex core was characterized by vortex line distribution and vorticity transport equation.The shape and number of blades influenced the revolving direction and distribution characteristics of the vortex close to the runner cone,which formed a counterclockwise-clockwise-counterclockwise distribution pattern.Simultaneously,there were many secondary flows in the draft tube.Mutual cancellation and dissipation between the flows was one of the reasons for reduction in vorticity.When the cross-sectional shape of the draft tube was changed,the vorticity was distributed from the center of the vortex rope to all parts of the cross-sectional draft tube,with extreme values at the center and at the walls.The vortex stretching and dilatation terms played a major role in the change in vorticity,with the baroclinic torque having an effect at the center of the vortex rope,this study is helpful to understand the flow of water in the draft tube and guide the design and optimization of the draft tube in engineering application.

Prediction of the precessing vortex core in the Francis-99 draft tube under off-design conditions by using Liutex/Rortex method

The turbulent flow in the draft tube of a Francis turbine is very complicated while working under off-design conditions. Although the off-design conditions were widely studied, the vortex core line in the draft tube of a Francis turbine with splitter blades is not well understood, especially the vortex rope property. This letter presents a prediction of the behavior of the vortex rope in the draft tube of the Francis-99 turbine obtained by the computational fluid dynamics (CFD), where the Liutex/Rortex method, as the most recent vortex definition, is applied to analyze the periodical precession of the vortex rope in the draft tube cone. The advantage of this Liutex/Rortex method is shown by its enhanced ability to represent the vortex rope structurewith the vortex-core lines. Furthermore, since it seems to be very hard to define a sharp boundary surface for the whole vortex structure, it is advantageousfocusing only on the vortex core line,by which different vortex structures can be clearly differentiated. The evolution of the vortex core and the process of the vortex breakdown in the draft tube are revealed, which might help to comprehend the development of the turbulent flow in the draft tube.

A modified-Liutex-based vortex-core-line extractor and its application

Vortices,which appear as swirling behaviour of a flow field,are an important phenomenon in fluid dynamics,and the extraction of vortex cores is necessary for the research of vortex evolution in many areas of fluid mechanics.The Liutex method is a milestone in vortex identification and provides a reasonable mathematical definition for a vortex core.Based on this definition,a novel integration-based method is presented,which can reduce the numerical error in the integration process through location optimization.Two typical test cases,the wake vortices of an A320 in the near-ground stage and a helicopter rotor,are examined to show that the proposed method can extract continuous vortex core lines with accuracy and efficiency for vortex parameter study.

POD analysis on vortical structures in MVG wake by Liutex core line identification

The Liutex core line method, first combined with the snapshot proper orthogonal decomposition (POD), is utilized in a supersonic micro-vortex generator (MVG) wake flow at Ma = 2.5 and Reθ = 5 760 to reveal the physical significance of each POD mode of the flow field. Compared with other scalar-based vortex identification methods, the Liutex core line identification is verified to be the most appropriate approach that is threshold-free and provides full information of a fluid rotation motion. Meanwhile, the Liutex integration is employed to quantitatively track the evolution of the vortices in MVG wake and is applied to the determination of the effective control section of the MVG wake for the optimization study of MVG design. The physical mechanism of each POD mode for multi-scale and multi-frequency vortical structures is investigated by using Liutex core line identification to give some revelations. For the mean mode (mode 0) indicating the time-averaged velocity flowfield of the MVG wake flow, a pair of primary counter-rotating streamwise vortices and another pair of secondary vortices is uniquely identified by two pairs of Liutex core lines with Liutex magnitude. In contrast, mode 1 is featured by a fluctuated roll-up motion of streamwise vortex, and the streamwise component of the MVG wake is demonstrated to be dominant in terms of the total kinetic energy contribution. Meanwhile, a dominant shedding frequency of St = 0.072 is detected from the temporal behavior of mode 2, which has the organized arc-shaped vortex structures shedding from MVG induced by the K-H instability. Additionally, mode 4 subjects to low-frequency oscillations of the wall vortices and thus takes a relatively lower frequency of St = 0.044.