A Study on Topological Vortex Ring Interactions Based on...

A Study on Topological Vortex Ring Interactions Based on Möbius Loop and Hopf Link Concepts (2)

Статьи пользователей Literature

Сообщение 2025-10-29 01:21:19

977

2. Theoretical Foundation: Topological Invariants

2.1 Linking Number (Lk) For two simple, unknotted but mutually entwined closed curves (like an ideal Hopf link), the linking number Lk is a topological invariant representing the total number of times (with sign) one loop winds around the other. For two vortex rings, Lk quantitatively describes the topological strength of their mutual threading.

2.2 Writhe and Twist Number (Wr & Tw) According to the Călugăreanu–White formula, for a twisted ring, its total linking number (if linked with another ring) or self-linking number can be decomposed as [1,8]:

a. Writhe (Wr): Describes the degree of distortion and coiling of the curve's global geometric configuration, related to the crossing points in its projection onto a plane. For two vortex rings, their mutual spatial configuration contributes to the writhe.

b. Twist (Tw): Describes the degree to which the curve rotates around its own centerline. This is precisely where the concept of the "Möbius loop" comes into play. A vortex ring with high Tw has its vorticity field direction rotating along the circumference, analogous to the abstract concept of the direction flipping after traversing a Möbius strip once. Tw is an intrinsic property of the vortex core.

3. Superposition and Interaction of Co-rotating Vortex Rings

When two vortex rings with the same circulation direction (co-rotating) collide head-on or move side-by-side [6]:

(1) Topological Description: Their interaction can be seen as the superposition of two co-oriented Hopf links. The total topological charge (total circulation) of the system increases.

(2) Dynamical Process: Due to their mutually reinforcing induced velocity fields, the two rings tend to repel each other or orbit a common axis. Under specific conditions (e.g., matched size, speed), they can:

a. Stable Linking: Form a stable, higher-order linked state (linking number Lk > 1), analogous to the "Solomon's link" in topology.

b. Merging: Through viscous dissipation or vortex reconnection, the two rings merge into a single, larger vortex ring potentially carrying higher Tw (twist). This process may reduce the overall topological complexity (the link disappears) but increase the intrinsic twist.

(3) Energy and Topology: Co-rotating superposition generally leads to a decrease in the system's kinetic energy (more stable), and the topological structure (although possibly more complex in linking) remains overall ordered and stable [7].

Войдите, чтобы отмечать, делиться и комментировать!

秋天要来了，洗车时候遇到了美丽的晚霞

入秋可以开始玩摩托了

От 2025-08-25 09:37:36 0 2Кб

CELINE 皮夾選購指南：從短夾到長夾的經典款式解析

提到法國精品品牌，celine 以極簡優雅與高級質感聞名。從經典的凱旋門包到日常必備的皮件配飾，CELINE 一直深受時尚女性喜愛。除了包款外，celine 皮夾...

От 2025-09-19 02:40:25 0 1Кб

山系機能革命！Nike ACG巔峰解密

當玉山雲海漫過稜線，Nike ACG Mountain Fly 2 米黑橙色的鋸齒大底正撕裂碎石坡！這雙被山友封為「移動安全艙」的巔峰戰靴，以軍規級nike acg gore...

От 2025-06-17 01:12:34 0 2Кб

The Physical Connection Between Topological Vortex Theory (TVT) and Point Defects (2)

4. Dynamics and Thermodynamic Evolution of Point Defects TVT links the vortex rotation rate to...

От 2025-10-17 05:43:54 0 986

A complete analysis of overseas direct investment filing (ODI)

Driven by the wave of economic globalization, more and more Chinese companies are turning their...

От 2025-06-19 01:53:02 0 3Кб