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 — and the second group is the interesting one. A molecule arriving at height $y$ with vertical velocity $\xi_y$ last touched the airfoil at the retarded time

so it reports the plate’s temperature and motion as they were in the past. Summing over all molecular velocities, 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 at their own retarded times). For the density,

The velocity components and the full stress tensor $P_{xx},P_{yy},P_{zz}$ follow from the same recipe — higher 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 is 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).

You can inspect that mechanism directly. Move a probe point through the gas below: the shaded fan marks the directions of plate-reflected molecules that reach it, and the timeline underneath shows which past states of the plate it is currently sampling — because the delay is $y/\xi_y$, a point higher above the plate hears older news.

The clearest way to see the retarded time is to let the plate do nothing but breathe — no freestream at all, just a 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.

Those stresses hold a subtlety a continuum never shows. In an ordinary gas the pressure is a single number — the normal stresses $P_{xx}$, $P_{yy}$, $P_{zz}$ are equal, so the “pressure ellipse” is a circle. Here the plate re-emits molecules as a directed beam, and near the plate the normal stresses split apart: the pressure acquires a direction. Far away the gas relaxes back to isotropy. Add a steady freestream and watch the stress ellipses stretch near the plate and in the trailing wake, and round out into circles far from it.

Results — lift, drag, and the 45° optimum

Add a steady freestream at speed $U_\infty$ and an angle of attack $\alpha$ (the plate stays on the axis; the oncoming flow tilts, so $U_x = U_\infty\cos\alpha$, $U_y = U_\infty\sin\alpha$). Evaluating the wall stresses in the limit $y\to 0^{\pm}$ gives closed forms for 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, both the lift and the lift-to-drag ratio peak near

far higher than the few-degree optimum of a classical thin airfoil — because here “lift” is produced by asymmetric molecular reflection, not by a bound circulation that stalls. Sweep the angle of attack below and watch the density field reorganize while the inset traces $L$, $D$ and $L/D$.

Results — the far-field density wedge

Looking farther from the plate in steady flow reveals a distinct density wedge: a “V” trailing the airfoil, marking where the wake of reflected molecules concentrates. The paper derives an analytic far-field model for it (via a Taylor expansion of the error functions and an Abramowitz asymptotic estimate of the resulting integrals) that matches the exact quadrature to under 1% error. Its shape is governed by the freestream speed: as $U_\infty$ rises the wedge sharpens and swings toward the axis. Let the speed sweep and watch it form.

Results — far-field waves, and a closed form that nails them

A second far-field result lives directly above the mid-chord. Switch off the flow and let only the plate temperature oscillate, $T_{\mathrm{af}}(t) = 1 + \varepsilon\sin(\omega t)$: the gas overhead answers with density waves that propagate up, decay, and phase-shift with height. The paper evaluates the governing integral with an Abramowitz asymptotic estimate — the complex integrals $J_n(z) = \int_0^\infty s^n e^{-s^2 - z/s}\,ds$ with $z = i\omega y$ — and lands a closed form (Eq. 3.17) whose decay $e^{-\frac{3}{2}\Omega}$ and phase $\frac{3\sqrt3}{2}\Omega$ are set by $\Omega = (\omega y/2)^{2/3}$.

The clearest way to judge it is a space–time diagram: height up, time across, shade the density fluctuation. Wave crests become diagonal bands that fade with height. The left panel evaluates the field integral directly; the right is the closed form — and they are, to the eye, the same picture, which is the whole point.

Every view here evaluates the free-molecular field integrals live — a simplified, real-time cousin of the MATLAB research code — and loops on its own; drag the sliders while they play. Together they trace the paper’s arc: from the retarded-time mechanism that makes an oscillating plate radiate (probe it, watch it breathe), through the directional stresses and the aerodynamic forces they imply, to the far-field signatures — the density wedge and the abeam waves — it leaves in the gas.