2 D Wind Tunnel — molecular flow over a flat plate

How the numbers are produced

Every particle has mass m = 1. A collision with the (infinitely thin, immovable) plate is specular: v' = v − 2(v·n)n. Conservation of momentum then gives the impulse delivered to the plate as J = m(v − v') = 2(v·n)n. These are counted per iteration: at the end of every timestep the simulation reports the momentum that actually changed hands during that step, in momentum units, not divided by time.

• Δp vertical (lift) = ΣJ_y over the iteration, positive towards the top of the screen.
• Δp horizontal (drag) = ΣJ_x over the iteration, positive towards the right.
• Twist (pitching moment about the plate's mid-point) = Σ(r×J)_z over the iteration, with r = t d, t the hit position measured along the chord from the centre. Because a smooth plate can only take a normal load this collapses to Σ t J·n. Positive twist is nose up: it tends to increase α, i.e. it is destabilising.
• One iteration is 1/90 s and typically contains a handful of collisions, so the instantaneous readings are pure shot noise — they are the raw event stream, not a converged measurement. Divide by the timestep for a force, or read the averaged block below them. Ratios (the coefficients, L/D, centre of pressure) are built from the running sums, since they are undefined on a single step.
• Centre of pressure = the load-weighted mean hit position, Σt Jn / ΣJn, reported as a percentage of the chord from the leading (upstream) edge. 50 % means the load is symmetric and the twist vanishes.
• Coefficients use CL,D = F / (½ρU²c) and CM = M / (½ρU²c²), with ρ = N m / (W H) and c the chord.
• Smoke is a passive tracer, exactly as in a real tunnel. Nothing is created at the rake — a particle is simply marked orange the moment it crosses the rake's plane moving downstream inside the slot, so the streakline shows where that gas goes without adding or removing any momentum. Widening the opening makes the line denser but blunter.
• Particle–particle collisions are always on, because a real gas always collides. They conserve total momentum, so they never contribute to the plate force directly — they only change how the flow arrives. Their effect is visible as the gap between the measured curve and the dashed collisionless reference: at small α they raise lift (the flow is turned over a region rather than molecule by molecule), and at large α they lower it, because the compressed layer on the windward face and the wake behind partly shield the plate. That is what moves the peak of the lift curve down from 55° to about 43° — the beginning of real stall, which a collisionless gas cannot show at all.
• There are no walls. All four sides are open: a particle that crosses any boundary is removed. To hold the ambient density fixed, each one that leaves is replaced by a particle entering from the surrounding reservoir, on a side chosen in proportion to that side's influx, with the inward normal speed drawn from the flux-weighted distribution p(v) ∝ v exp(−(v−m)²/2σ²) — fast molecules cross a line more often than slow ones. In equilibrium influx equals outflux, so this is an exact open boundary rather than a recycling trick, and the plate now sits in free stream with no wall interference.
• Coarse-graining (cell size a). At a = 0 you see individual molecules. Move the slider and the domain is tiled with a × a squares, every particle in a square is replaced by one arrow, and the particles themselves are hidden. The arrow is the mean velocity u = (1/N) Σv over the cell — which is the first velocity moment of the distribution, the quantity written u in the Navier–Stokes equations, computed here by brute force instead of by integration. A checkbox switches it to the raw vector sum Σp, the total momentum in the cell.
• Nothing is averaged over time — every arrow is one instant. So the slider is really a control on how many molecules each arrow speaks for, and the panel reports it. At a = 5 there are 720 000 cells and 0.4 particles in each, so almost every arrow is a single molecule's velocity: a hash of thermal motion with no flow visible in it at all. At a = 25 it is 9 particles per cell, ~33 % noise, and a direction starts to emerge. At a = 50 it is 37 per cell, ~16 % noise, and you get a clean vector field — uniform stream upstream, flow parting around the plate, a slowed wake behind it. Raise the particle count and the same picture arrives at a smaller a.
• Field time-averaging. Thermal noise in a cell falls as 1/√N_cell, and averaging the field over time is a way of buying more samples without making the cells coarser. The control is an exponential time constant; the sums and the counts are averaged separately and divided afterwards, so a cell that is momentarily empty contributes zero to both rather than an undefined velocity. Averaging over 1.5 s is about 135 iterations and divides the noise by roughly 12 — and since vortex shedding at these settings has a period of order 1700 iterations, that is under a tenth of a cycle, so unsteady structure survives while the hash disappears. It is disabled automatically when the net is fine enough that sweeping every cell would cost more than the simulation, and where an average would be meaningless anyway.
• Nothing about the molecules changes between those pictures; only how many of them you average. That is what "continuum" means, and it is why the continuum description is a statement about scale rather than about substance.
• Why there are no eddies behind the plate. Vortex shedding needs low Mach and Reynolds above roughly 47, and in a particle gas those two cannot be chosen independently. Kinematic viscosity is ν ≈ 0.28 σ/(ndd) and the sound speed is √2 σ, so σ cancels and Re = 2.51 M (c/λ): Reynolds number is set by Mach and Knudsen alone, and the temperature slider cannot move it. The only lever is c/λ — a longer chord, or more particles to shorten the mean free path. At the maximum here (24 000 particles, chord 320, λ = 5.4 u) that gives Re ≈ 0.66 M c: about 42 at Mach 0.2 and 63 at Mach 0.3. So the shedding threshold is barely reachable at best, and the settings that look energetic — Mach 5, Re ≈ 660 — are hypersonic, where the wake is shock-dominated and sheds nothing at all. The dark region behind the plate is a ballistic shadow, not a recirculation.
• Even at the favourable end, thermal noise finishes the job. An eddy carries an ordered speed of order 0.2 U while each coarse-graining cell has velocity noise σ/√Ncell. At Mach 0.3 that needs about a thousand particles per cell for a signal-to-noise of 3 — and a thousand particles per cell out of 24 000 leaves you 24 cells in the whole window, far too coarse to resolve an eddy. This is the standard difficulty of particle methods at low speed, and it is why production DSMC codes use billions of particles or ensemble-average many runs.
• Freeze (the button on the picture, or the space bar) halts the simulation and holds the frame. The gas and trace layers are accumulated rather than redrawn each frame, so freezing stops touching them entirely and the picture is preserved exactly; the plate and force vectors stay live, so you can still drag the plate around a frozen flow field.
• Traces fade the particle layer instead of clearing it, so each particle drags a pathline behind it. The smoke and the gas have separate trace-length controls and separate layers, quoted as the time a mark takes to decay. Nothing is computed from a trace — any structure you see is the particle dynamics alone. Push the particle count up to shorten the mean free path (more collisions, lower viscosity) if you are hunting for organised wake motion.
• Why lift is not negligible just off zero incidence. The intuition that a nearly edge-on plate is barely touched is what happens in a cold gas, where lift goes as sin²α. In a gas at finite temperature the plate is hammered from both sides by thermal motion even at α = 0 — the two faces simply balance. Tilting it unbalances them, and that imbalance is first order in α. Writing β = U sinα/σ, the exact free-molecular normal force per unit chord is 2ρσ²[(β²+1) erf(β/√2) + 2β φ(β)], which becomes 4√(2/π) ρσU sinα for β ≪ 1 — linear in α, with the slope set by the thermal speed. The sin²α behaviour only takes over above the crossover at α ≈ arcsin(σ/U). Lower the ambient temperature and the small-angle lift collapses towards the quadratic curve. The panel shows this theory live next to the measurement.
• Ambient speeds follow a distribution. Each velocity component is drawn independently from a Gaussian — vx ~ N(U, σ) and vy ~ N(0, σ) — which makes the speed |v| follow the 2 D Maxwell–Boltzmann law. With no wind that is a Rayleigh distribution peaking at σ; with a drift it becomes a Rice distribution peaking near √(U²+σ²). The "speed distribution" chart plots the live histogram of all N particles against that formula evaluated directly — nothing is fitted, so any disagreement would be a real one. Collisions redistribute energy between particles but preserve the equilibrium, which is why the shape holds up with particle–particle collisions on.
• Particle colour is speed — one species, one law, just coloured by how fast each particle happens to be moving at that instant. Blue particles are slow and white ones are the fast tail of exactly the distribution above; there is no other difference between them, and the toggle in the panel paints them all one colour if you want to check. The scale is: deep blue is at rest, mid blue and cyan are around the thermal speed, and pale yellow to white is the fastest tail. The scale tops out at 2σ + 1.15 U, so it rescales when you change the temperature or the wind. With the wind at zero you see a blue field speckled with the fast tail of the Maxwell distribution; wind it up and the whole field warms as the drift is added to every particle.
• Lift, drag and twist are reported as running averages since the last "Zero averages"; any change of plate position, angle or chord restarts that average, because the geometry it refers to has changed. With particle–particle collisions on, give the flow a few seconds to re-establish after a change, then press "Zero averages" and let it integrate for ten seconds or so — these are statistical quantities and short samples are noisy.