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)

Blogs Literature

Posté 2025-10-29 01:21:19

2KB

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].

Connectez-vous pour aimer, partager et commenter!

Michael Kors 台灣門市人氣托特包全攻略｜經典老花色與實用尺寸一次看懂

作為美國知名時尚品牌，**Michael Kors（MK）**一直以優雅大方的設計與高實用性聞名。在台灣，透過 michael kors...

Par 2025-08-15 03:18:26 0 2KB

Rising Opportunities and Digital Transformation in the Corporate Lending Market

The Corporate Lending Market Share is expanding rapidly as global businesses seek flexible...

Par 2025-10-31 06:22:04 0 1KB

電子菸煙彈選購攻略：如何為你的主機挑選最佳伴侶

在電子菸的世界裡，煙彈的選擇直接影響使用體驗。許多新手常會困惑於空倉和預注式煙彈的差別，其實這就像咖啡機的膠囊與現磨咖啡的選擇，各有其獨特優勢。今天我們就來深入探討如何為您的Relx...

Par 2025-07-09 02:33:57 1 2KB

拋棄式電子菸與 ILIA、SHAXIAO 熱門機型全指南

隨著電子菸市場的多樣化，消費者在選購時更傾向於便利、口感多變且設計時尚的產品。其中拋棄式電子菸因為「開盒即用、用完即丟」的特性，在台灣年輕族群間備受青睞。而在眾多品牌中，ILIA...

Par 2025-08-15 03:22:46 0 2KB

test

商品

Par 2025-12-15 11:21:08 0 1KB