---
title: "Why Two Vortex Rings Leapfrog Interactive Lab | Sandbox Physics"
description: "How does the trailing ring slip through the one ahead? How does the rear ring shrink and pass through the front, and do the three-dimensional shape and numerical histories explain the same exchange?"
canonical_url: "https://sandboxphysics.com/labs/vortex-rings/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/vortex-rings/"
---

Fluid dynamics · three-dimensional filament studio

# Vortex Rings

Two already-formed vortex rings shrink, expand, and exchange axial order in their mutually induced velocity field. Orbit the numerical filaments and passive tracers, then connect the motion to position and radius histories, invariant drift, and an independent single-ring speed baseline.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Inviscid regularized filaments: every segment contributes through the Biot–Savart kernel, with fourth-order Runge–Kutta stepping. The fixed core scale is numerical regularization, not solved smoke density or viscous spreading. The model pauses at core contact or insufficient resolution; it does not simulate nozzle formation, viscosity, turbulence, or reconnection.

Physics tutorial

## Why vortex rings leapfrog

**Background** A smoke ring contains a closed vortex tube. This experiment begins with formed rings, not nozzle roll-up. Each centerline is a closed three-dimensional polygon; every segment induces velocity at all other nodes and passive tracers.

**Why it matters** Two co-directed rings are neither rigid objects nor independent circular tracks. The rear ring can shrink and accelerate while the front expands and slows, then exchange roles. The three-dimensional view, position plot, and radius plot must explain the same exchange.

Start with the essentials

- **Focus question**: When the rear ring passes through the front ring, do their cores cut each other, or do their radii and speeds exchange?
- **One-sentence intuition**: Leapfrogging exchanges axial order, not filament topology. Mutual induction changes each ring’s radius. Closed filaments are never artificially reconnected in this model.

### Core mathematical model

#### Regularized Biot–Savart velocity

$$
\mathbf u(\mathbf x)=\sum_j\frac{\Gamma_j}{4\pi}\oint_{C_j}\frac{d\mathbf X\times(\mathbf x-\mathbf X)}{\left(|\mathbf x-\mathbf X|^2+a^2\right)^{3/2}}
$$

Interpolate nodes with periodic cubic segments, integrate the Rosenhead–Moore kernel with three-point Gauss quadrature per segment, then move nodes with fourth-order Runge–Kutta. The fixed scale $a$ is regularization, not a resolved viscous core radius.

#### Independent smooth-circle speed baseline

$$
U_{\rm ref}=\frac{\Gamma}{4\pi}\int_0^{2\pi}\frac{R^2(1-\cos\theta)}{\left[2R^2(1-\cos\theta)+a^2\right]^{3/2}}\,d\theta
$$

Integrate independently over a smooth circle, rather than the solver’s polygon. Refinement should bring single-ring speed toward this same-kernel result. This tests discretization consistency, not experimental fluid accuracy.

#### Closed-filament impulse

$$
\frac{\mathbf I}{\rho}=\frac12\sum_j\Gamma_j\oint_{C_j}\mathbf X\times d\mathbf X
$$

Report the change in the impulse vector normalized by the sum of the initial individual-ring impulse scales. Opposing rings may have zero total impulse, so dividing by initial total impulse would be invalid.

#### Same-kernel energy integral

$$
\frac{H}{\rho}=\frac{1}{8\pi}\sum_{i,j}\Gamma_i\Gamma_j\oint\!\oint\frac{d\mathbf X_i\cdot d\mathbf X_j}{\sqrt{|\mathbf X_i-\mathbf X_j|^2+a^2}}
$$

Monitor energy with three-point Gauss double quadrature on segments. Fixed node count, polygon geometry, and time stepping all introduce error; drift during strong deformation is not physical viscous dissipation.

### Common difficulties

#### A core tube is not smoke density

**Typical misconception** Brighter or thicker tubes indicate numerically resolved denser smoke.

**Better mental model** Tube thickness marks regularization; colors identify rings. Passive particles follow the computed velocity field without feeding back on it or resolving compressible smoke density. Hiding particles does not change filament dynamics.

#### Opposing rings do not imply a reconnection simulation

**Typical misconception** These rings must eventually merge, annihilate, or reconnect into smaller rings here.

**Better mental model** As cores approach, the thin-filament description becomes unreliable. The experiment pauses explicitly. Reconnection needs an additional viscous or quantum-fluid model, not an invented visual transition.

### Run the experiment

1. 01

   #### Match three readings of one exchange

   Choose Leapfrogging and play or advance. Ring A is teal; ring B is amber.

   **What to observe:** When axial curves cross, the order counter increases and the smaller ring passes through the larger ring. Disable center-following to observe overall translation.

2. 02

   #### Audit segments, not smooth-looking graphics

   Choose Single ring, compare coarse and fine segment counts, and reveal the numerical nodes.

   **What to observe:** The gap between polygon speed and smooth-circle speed decreases. Reducing core scale without refining can increase error instead.

3. 03

   #### Find where the model should stop explaining

   Compare Opposing rings with Wavy ring while keeping other parameters fixed.

   **What to observe:** Opposing rings approach and expand; a wavy seed changes local curvature. Core contact or unresolved bending pauses the model with a reason, without invented reconnection.

Physical conclusion

The visual exchange is the entry point. The deeper test is whether one velocity field drives the shape, histories explain the motion, and benchmarks plus drift reveal which parts deserve trust.

Conceptual inspiration: [FluidMotion](http://www.bugman123.com/FluidMotion/index.html). Original simulations and graphics; the model limits are described above.

Fields & matter · Starter path

## From a field to an electron microscope

Read a field, steer a charge, give an electron a wavelength, then use it to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)