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

0
1كيلو بايت

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]:

Lk = Wr + Tw

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].

البحث
الأقسام
إقرأ المزيد
Literature
A Funny Official History of Contemporary Physics (Volume Two: Verification)
Time flowed quietly, and several months passed in the blink of an eye. One day, PRL suddenly...
بواسطة Bao-hua ZHANG 2025-10-16 01:34:15 1 1كيلو بايت
Shopping
卖疯了
承接各类割草机刀片供应,接激光打标定制 Undertake the supply of all kinds of lawn mower blades, and connect laser...
بواسطة Jishen Kog 2025-07-10 04:37:16 1 3كيلو بايت
أخرى
国产用于单导联和六导联的心电算法及API服务
随着心电图应用的逐步推广,智能手表、体脂仪、跑步机等智能设备都逐渐加入心电图功能。心电图作为诊断心脏问题的金标准数据,具备很大的健康监测价值。苹果、欧姆龙等厂商都已经加入。但是心电图的分析算法相...
بواسطة 志峰 吴 2025-07-21 12:35:30 0 2كيلو بايت
Shopping
CELINE儀式感日常:束口後背包領銜時尚革命
在極簡奢華的宇宙裡,CELINE 從不追逐喧囂,卻總能掀起無聲革命。當Triomphe凱旋門老花成為新世代的信仰符碼,品牌將經典靈魂注入日常配件,從實用主義的celine...
بواسطة Ahr Alice 2025-06-17 01:22:39 0 3كيلو بايت
Shopping
SAMBA ตัวจริง เสียงจริง ไอคอนสนีกเกอร์ที่ต้องมี
จุดเริ่มต้นของการเลือกซื้ออย่างมีสไตล์ การตามหารองเท้าคู่ใจให้ “ใช่”...
بواسطة Ahr Alice 2026-02-06 01:24:34 0 257