Releasing a Smoke Streak Made of Numbers

I spent this afternoon trying to photograph a filament of smoke peeling off a wing at the stall angle, and mostly I produced grey mush under flat light. So tonight I did the thing I always do when the physical version defeats me: I built the digital twin and let it work.

A smoke streak is deceptively simple to describe. You release tracer at a fixed point, and the moving air carries each puff downstream. Where the smoke is at any instant is just the answer to a running sum: take the velocity at your current spot, multiply by a small slice of time, move, repeat. That’s forward-Euler integration, the oldest trick in numerical analysis and the workhorse of every desktop flow simulation that appeared once 1980s micros got fast enough to loop a few thousand times without you making tea.

Let the tunnel be a gentle vortex — velocity always perpendicular to the radius — and drop one particle at the edge:

x, y = 1.0, 0.0
dt = 0.1
for _ in range(6):
    vx, vy = -y, x          # vortex: velocity runs perpendicular to the radius
    x += vx * dt            # one forward-Euler step downstream
    y += vy * dt
    print(f"{x:6.3f} {y:6.3f}")

const std = @import("std");
pub fn main() !void {
    const out = std.io.getStdOut().writer();
    var x: f64 = 1.0;
    var y: f64 = 0.0;
    const dt: f64 = 0.1;
    var i: usize = 0;
    while (i < 6) : (i += 1) {
        const vx = -y; // vortex field
        const vy = x;
        x += vx * dt; // forward-Euler step
        y += vy * dt;
        try out.print("{d:6.3} {d:6.3}\n", .{ x, y });
    }
}

Both print the same trail:

 1.000  0.100
 0.990  0.200
 0.970  0.299
 0.940  0.396
 0.901  0.490
 0.851  0.580

The particle should ride a perfect circle. Instead the radius creeps past 1.0 — by the sixth step it’s out to 1.03. Euler always steps along the tangent, so on anything curved it flings the point slightly outward every time. My smoke drifts wide, and no fan or dye is to blame; the arithmetic itself leaks.

Which is oddly reassuring. In the tunnel, real smoke smears from turbulent diffusion I can’t fully control. In the code, it smears from truncation error I can measure exactly — halve dt and the drift roughly halves with it. One of these two failures I can fix from the keyboard, and tonight that’s the one I’ll take.