3D Flow

Painting the Wind

The built-in rubber duck behind a leaning fence, the tracers coloured by gustiness: the "Floating Ducky" preset.

Air flows through a wind tunnel, drawn as tens of thousands of glowing bits. Each bit follows the air round a body you choose. It can be a cylinder, an oval, a wing, a sphere, or a model of your own. In open air, the flow round the cylinder, the sphere and the oval is exact for frictionless air. The wing, a model and most tunnel walls are solved with many small panels. Every part carries an honesty label. The wake behind the body is drawn. Its rhythm follows measured shedding rates. The scene can react to the sound of your music. This is art drawn from real physics, not scientific wind modelling.

Air and water are both fluids. At wind tunnel speeds air hardly squeezes, so the same flow rules serve for both. One old rule pretends the fluid has no friction and does not spin. Then the flow comes from one simple equation, and its answers add up. A wind plus a doublet makes a cylinder. A wind, a source and a sink make an oval. A sphere works the same way in three dimensions. The rules with friction came later and are far harder to solve. Other shapes are covered with many small panels, each with a little source or swirl. The computer sets their strengths so the air goes round the surface, not through it. This scene works from those rules, not from a fluid simulation.

The shortcut has a famous flaw. It says the air closes up neatly behind the body. Then the body would feel no drag. That is called d'Alembert's paradox. Real air peels off the back and rolls up into swirls. Behind a cylinder, they come off one side and then the other. That row of swirls is called a vortex street. The swirls come off at about 0.2 times the wind speed, divided by the body's width. In 1954 Anatol Roshko measured that rate closely for cylinders. This scene uses his numbers. A sphere sheds loops instead, at its own measured rate. The wake can be a street of swirls or a smooth, stirred trail. When the scene reacts to music, the sound moves the wind, and the wake follows.

Real wind tunnels show the air with smoke from a rake of small tubes upstream. Where the lines squeeze together, the air is fast. Where they spread apart, it is slow. In the wake the air keeps changing, so a smoke line there shows the past, not the present flow. This scene can fill the whole tunnel with drawn air, or use a smoke rake. Engineers also map pressure on a body. Pressure follows speed, so fast air has low pressure. That number is called Cp. It is exactly 1 where the air stops on the nose. In the frictionless flow, it drops to minus 3 at the sides of a cylinder. The scene can color the air by speed, Cp, swirl, gustiness, where it came in, or age.

A note on honesty. In open air, the flow round the cylinder, the sphere and the oval is exact for frictionless air. The wing is solved with many small panels. So are most tunnel walls, the baffle and a loaded model. A model gets up to 1,500 panels. The info bar shows a model's panel leak. The wake is drawn, with measured timing for the cylinder and the sphere only. The gusts are smooth noise that never bunches up, like slow real air. Their strength matches what engineers call turbulence intensity. But they are not real turbulence. A spinning body gets the ideal swirl, and a real one carries less. No drag is shown, because this air has none. This is art drawn from real physics, not scientific wind modelling.

The Math

$$\nabla^2\phi = 0, \qquad \mathbf{u} = \nabla\phi$$

Frictionless air that does not spin obeys Laplace's equation. Its answers can be added together. That is why a wind plus a doublet makes a cylinder.

$$u_\theta = -2U\sin\theta, \qquad C_p = 1 - 4\sin^2\theta$$

In frictionless flow, the air at a cylinder's sides moves at twice the wind speed. The pressure coefficient is 1 on the nose and minus 3 at the sides. On a sphere, the air at the sides moves at one and a half times the wind speed.

$$S = \dfrac{f\,D}{U} \approx 0.2$$

This is the Strouhal number, named after Vincenc Strouhal, who studied singing wires in 1878. Multiply the shedding rate by the width, then divide by the wind speed. For a cylinder you get about 0.2, the value Roshko measured. A flat plate across the wind gives about 0.14.

$$U_2 = U_1\,\dfrac{A_1}{A_2}, \qquad U_{\text{gap}} = \dfrac{U}{1 - c\sin\alpha_b}$$

Squeeze air into a smaller area and it speeds up by the same ratio. The funnel uses this. So does the gap above the baffle.

$$\gamma_0 + \gamma_n = 0$$

The wing's panels obey the Kutta condition. The air must leave the sharp back edge smoothly. That one rule sets how much air goes round the wing, and so its lift.

$$L' = \rho\, U\, \Gamma$$

Circulation measures how much the air goes round the body. With it, the air moves faster over one side. The lift per metre is the air's density times the speed times the circulation. This is the only force shown. A spinning cylinder here gets the ideal circulation for its spin. Real air round a spinning cylinder carries less.

$$\mathbf{v} = \nabla\times\boldsymbol{\psi}$$

The gusts are the curl of a noise field. The curl of a smooth field has no sources and no sinks. So the gusts do not pile up anywhere.

How Spiralyst Lab draws it

Spiralyst Lab does not run a fluid simulation here. Each bit of air asks for the air speed where it is. Then it takes a small step. Near the body it takes several smaller steps. The tunnel walls, the wing, the baffle and a model are solved once, when you change them. The answer is stored on a grid. A big solve is cut into small pieces. The app works on one piece each frame. The old scene keeps flowing until the new one is ready. The wake is a short list of swirls that the wind carries away. The gusts are noise stored on a small grid that drifts with the wind. Changing the wind speed, the spin or the gusts rebuilds nothing. So music can drive them for free. The scene follows the bands of the sound, like the bass and the hi-hats. The info bar always shows the wind speed and a reminder that this is art. When there is room, it also shows the Reynolds number, the shedding rate and the blockage.

Requires a Spiralyst Lab license.

Did you know?

The back of a cylinder is where the textbook flow goes badly wrong. The theory says the pressure there climbs back to the nose value. Roshko measured between about minus 0.7 and minus 0.95 instead. It stayed nearly flat over the back. That low pressure at the back gives a real cylinder most of its drag.

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