# Burgers Vortex

Flow · (ẋ, ẏ, ż) = f(x, y, z, t)

`(− α · x/2 − κ · z · (1 − exp(−(x² + z²) · α /(4 · ν)))/(x² + z²),  α · y,  − α · z/2 + κ · x · (1 − exp(−(x² + z²) · α /(4 · ν)))/(x² + z²))`

[Open in the app](https://www.wavelace.com/app#p=161) · [This page](https://www.wavelace.com/presets/burgers-vortex)

### The flow

This is a real solution of the Navier–Stokes equations, the equations of a viscous fluid, and each seed is a speck carried by it. Wavelace draws `y` upward, so the vortex stands on the `y` axis. Its distance from the axis is `r = √(x² + z²)`.

The flow has three parts. It draws fluid inward at `−αr/2` and throws it out along the axis at `αy`, upward above the middle plane and downward below it. Around the axis it swirls at `κ·(1 − e^(−r²/δ²))/r`, with the core radius `δ` given by `δ² = 4ν/α`.

The first two parts balance exactly: whatever comes in along the plane leaves along the axis, so no fluid piles up anywhere. Here `α` is the stretching, `ν` the viscosity and `κ` the circulation divided by 2π.

### Why the core holds

The inflow drags the swirl toward the axis and would squeeze it into a line. Viscosity smears it outward. The two balance at the radius `δ`, which is why this vortex is steady and why its width is set by `ν`.

At the opening values `α = ν = 0.1` the core has radius `δ = 2`. Inside it the fluid turns like a solid disc at `κ/δ² = 1.5` radians a second, one turn every 4.2 seconds at `Speed` 1. Far outside it the swirl falls off as `κ/r`, like a whirlpool.

Along the axis each speck's height doubles every `ln 2/α ≈ 6.9` seconds while its distance from the axis shrinks, keeping `r²·y` fixed. That is the funnel the trails draw.

### History

Johannes Burgers published it in 1948 as a model of the thin, intense vortices inside turbulence. It remains one of the few exact solutions of the full equations that show vortex stretching, the mechanism by which turbulence feeds small whirls.

### Try

- Raise `ν` to 0.4. The core doubles to `δ = 4` and spins four times slower: the fluid is thicker.
- Raise `α` to 0.3. The core narrows to `δ ≈ 1.15`, spins three times faster, and specks leave the axis in a third of the time.
- Set `κ` to 0. The swirl vanishes and what is left is the stagnation flow alone, specks sliding in and out along hyperbolas.
- Press `Top` in the deck to look down the axis: tight spirals in the core, wide loose ones outside it.

### Read more

- [Burgers vortex](https://en.wikipedia.org/wiki/Burgers_vortex)
- [Navier–Stokes equations](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations)
- [Vortex stretching](https://en.wikipedia.org/wiki/Vortex_stretching)
- [Rankine vortex](https://en.wikipedia.org/wiki/Rankine_vortex)