Rarefied Gas over a Flat Airfoil

My second undergraduate research project (Technion, supervised by Prof. Avshalom Manela) works out, in closed form, the flow of a rarefied gas over a thin flat airfoil — the regime of very low pressure or very small scales, relevant to high-altitude flight, spacecraft, and MEMS devices, where the classical continuum equations no longer hold.

The problem

At ordinary densities a gas molecule collides constantly with its neighbours, and the flow behaves like a continuum governed by Navier–Stokes. As the gas is rarefied — thinned out, or confined to a micro-scale — the mean free path $\lambda$ between collisions grows until it is comparable to the airfoil itself. The relevant measure is the Knudsen number

and this study lives in the ballistic limit $Kn \gg 1$, where molecules essentially never collide with one another between visits to the airfoil. The gas is described by the distribution function $f(t,x,y,\boldsymbol\xi)$ — the density of molecules with velocity $\boldsymbol\xi$ near $(x,y)$ — linearized about a Maxwellian $F$ as $f = F\,[1+\phi]$. Dropping the collision integral from the Boltzmann equation leaves a pure advection equation for the perturbation,

A unit-chord plate sits on the $x$-axis from $x=0$ to $x=1$. It is fully diffuse: molecules that strike it are re-emitted with a Maxwellian set by the airfoil’s own temperature $T_{\mathrm{af}}(t)$ and normal velocity $V_{\mathrm{af}}(t)$, both allowed to oscillate in time. An impermeability condition — no net flux through the plate — fixes the density of that re-emitted stream, $\rho_{\mathrm{af}}$.

How it works

Because molecules fly in straight lines between wall visits, the state of the gas at a point is just an accounting of who is passing through. Every molecular trajectory carries information either straight from the far field or from its last bounce off the plate, so every hydrodynamic field splits into a free part (molecules straight from infinity, carrying only the freestream) and a hitted part (molecules reflected off the plate, carrying the plate’s state). Density, the velocity components, and the full stress tensor $P_{xx},P_{yy},P_{zz}$ all come from the same recipe — moments $\int \boldsymbol\xi\,f\,d\boldsymbol\xi$ and $\int c_i c_j\,f\,d\boldsymbol\xi$ of the distribution — and the static pressure is $P = \tfrac{2}{3}(P_{xx}+P_{yy}+P_{zz})$. Each field reduces to a one-dimensional integral over $\xi_y$, which is all the simulations below actually do: they evaluate these quadratures on a grid, in your browser, and render the result as a monochrome heatmap (compression dark, rarefaction light).

The steady state

Start with nothing changing in time: a steady freestream at speed $U_\infty$ and angle of attack $\alpha$ (the plate stays on the axis; the oncoming flow tilts, so $U_x = U_\infty\cos\alpha$, $U_y = U_\infty\sin\alpha$), with the plate at rest at the ambient temperature. Every reflected molecule then reports the same plate state, and the accounting closes in closed form. Evaluating the wall stresses in the limit $y\to 0^{\pm}$ gives the normal and tangential forces, and hence the lift and drag. The lift works out to

and maximizing over incidence gives the striking result that, in this free-molecular regime, the lift peaks near

because here “lift” is produced by asymmetric molecular reflection, not by a bound circulation that stalls. The efficiency peaks earlier: the drag keeps climbing past 45° while the lift has already turned over, so the lift-to-drag ratio is best a good deal sooner, around 25° — still far higher than the few-degree optimum of a classical thin airfoil. Sweep the angle of attack below and watch the density field reorganize while the inset traces $L$, $D$ and $L/D$.

The retarded time

Now let the plate itself change — heat it, cool it, shake it — and a new ingredient appears. A molecule arriving at height $y$ with vertical velocity $\xi_y$ spent $y/\xi_y$ in flight since its last bounce, so it last touched the airfoil at the retarded time

and it reports the plate’s temperature and motion as they were in the past. Slow molecules carry old news, fast ones carry fresh news, and a single point in the gas hears all of them at once.

The picture below shows only this timing — no flow field yet. The plate’s temperature oscillates, and its shade follows it (hot dark, cool light). Drag the probe point $Q$ through the gas: the shaded fan marks the directions of plate-reflected molecules that can reach it, and the numbered rays follow three of them, leaving different stations on the plate at the thermal speed. The timeline underneath marks the past moment each one left the plate: for a molecule at the thermal speed the delay $y/\xi_y$ is simply its flight distance, so the station farthest from $Q$ delivers the oldest news, and dragging $Q$ away from the plate pushes all three markers deeper into the past.

The unsteady state

To get from this bookkeeping to hydrodynamics, sum the delayed messages over all molecular velocities: the hitted part of every field becomes an integral over the plate’s past. For the density,

Let the plate do nothing but breathe — no freestream at all, just the temperature oscillating as $T_{\mathrm{af}}(t) = 1 + \varepsilon\sin(\omega t)$. Because each reflected molecule carries the plate temperature from a different moment in its past, the density disturbance propagates outward as waves in a gas that never collides with itself — an acoustic-like response with no acoustics. Below is the same probe again, now placed inside that wave field (compression dark, rarefaction light): the timing diagram is unchanged, and you can watch what the delayed messages add up to around $Q$.