Virtual Wind Tunnel

The body is a closed, counter-clockwise contour of $N$ straight panels. The airfoil uses the NACA four-digit thickness/camber construction with camber position $p=0.4$ and a closed trailing-edge coefficient $-0.1036$. The cylinder has unit normalized diameter. Geometry coordinates are normalized by chord or diameter.

For a target in panel-local coordinates $(a,b)$ and panel length $\ell$, the integrated unit-source velocity is

Rotating these velocities and projecting onto outward normals and tangents gives the dense matrices $A$ and $A_t$. Exterior self-influences are $A_{ii}=1/2$ and $(A_t)_{ii}=0$. A positive, counter-clockwise vortex sheet has normal influence $\mathbf g_n=-A_t\mathbf1$ and tangential influence $\mathbf g_t=A\mathbf1$.

The no-penetration system and trailing-edge Kutta condition are

where $\mathbf k$ is the sum of the first and last rows of $A_t$ and $h=g_{t,1}+g_{t,N}$. With LU solves of $A$,

For the cylinder, circulation is prescribed as zero. Surface velocities are $V_t=V_{\infty,t}+A_t\boldsymbol\sigma+\gamma\mathbf g_t$. The residual is $\max|A\boldsymbol\sigma+\gamma\mathbf g_n-\mathbf b|$.

The lift polar uses superposition: the sources for any angle are $\cos\alpha\,\boldsymbol\sigma_x+\sin\alpha\,\boldsymbol\sigma_y$, where $\boldsymbol\sigma_x$ and $\boldsymbol\sigma_y$ solve $A$ for unit winds along $x$ and $y$; the Kutta condition then gives the circulation at each of 49 angles from $-12^\circ$ to $12^\circ$ exactly, without solving again.

Bernoulli gives $C_p=1-(V_t/U_\infty)^2$. Pressure is integrated panel by panel to give $C_L$. Independently, circulation gives $C_{L,\Gamma}=-2\Gamma/(U_\infty c)$ with counter-clockwise circulation positive. Their finite-panel discrepancy is a discretization effect; it need not vanish to machine precision. Dimensional lift per span is $L'=\tfrac12\rho U_\infty^2cC_L$.

The picture is in the tunnel frame: the wind blows along $x$ and the body is turned by $-\alpha$. Velocities anywhere follow from the solved source and vortex strengths. The colours (a scale diverging from near black at $C_p=0$: blue suction, orange overpressure) are the pressure coefficient $C_p=1-(V/U_\infty)^2$ on a $52\times28$ grid, blended between its points; points inside the body (winding number 1) take the value of the nearest panel, so the map meets the surface. The same scale colours the surface. Streamlines start at the inlet at equal spacing, so equal flow passes between neighbours and their crowding shows the speed; they are traced with steps along the flow (second-order Adams–Bashforth on the direction, shorter near the body). Particles move along them at the local speed (by the time of flight), so they spread out where the air is fast and bunch up where it slows. The picture shows the steady flow; it is not an unsteady simulation.

For readability, the pressure chart shows separate upper/lower PCHIP curves over $0.02\le x/c\le0.98$. Cards and force integrals use the original panel values, including the leading-edge extrema.

Steady, two-dimensional, incompressible, inviscid, irrotational exterior flow, with circulation selected by the Kutta condition for an airfoil. The cylinder is noncirculating. No boundary layer, turbulence, wake shedding, stall, viscous drag or finite-span correction is modeled. Speed of sound is fixed at 343 m/s, approximately air at 20 °C. The equations are derived here from the cited source/vortex panel formulation. Reference checks are assistant-added limiting-case checks, not an external certification.

Use 32–512 panels, thickness/chord 0.06–0.24, camber/chord 0–0.06 and angles within ±12°. These are exploration limits, not a claim that real flow stays attached. A peak surface Mach warning flags a breakdown of the incompressible approximation. 128 panels resolve the lift coefficient to about 0.5 %; 512 take several seconds on a phone. Streamlines very close to the surface are approximate where the steps are long compared with the panels. Do not interpret the small computed pressure-drag residual as a prediction of real aerodynamic drag. Odd panel counts slightly unbalance airfoil sampling.