 Surface pressure integration for airfoil lift

Surface pressure integration for airfoil lift reconstructs the pressure contribution to lift by summing the local surface pressure loads with their directions. With $\mathbf{n}$ pointing from a two-dimensional body into the fluid, the force per unit span from its closed contour $\Gamma$ isThe uniform reference pressure cancels on the closed contour. This is a vector line integral, not the sum of positive pressure magnitudes.^[ref]

For upper and lower surfaces that are graphs over $0\le x\le c$, their pressure contribution perpendicular to the chord isVertical end closures have no chord-normal pressure contribution. The pressure force parallel to the chord, $A_p^{\prime }$, still matters when resolving flow-relative lift. With flow direction $(\cos \alpha ,\sin \alpha )$ in aft/chord-normal axes, $L_p^{\prime }=N_p^{\prime }\cos \alpha -A_p^{\prime }\sin \alpha$. Equating $N_p^{\prime }$ with lift needs a suitable angle of attack approximation. Pressure integration also omits the viscous stress contribution.

Finite pressure tappings require numerical integration. A composite trapezoidal rule integrates the linearly interpolated samples exactly; it need not integrate the actual unsampled distribution exactly. For an original example, let $f(\xi )=C_{p,l}-C_{p,u}=0.8$ and place taps at $\xi =0,0.2,\dots ,1$. A second invented profile agrees at every tap but has an additional $1.2{\sin }^2[\pi (\xi -0.4)/0.2]$ on $0.4\le \xi \le 0.6$ and zero addition elsewhere. Both reconstructions give $0.8$, while the second true integral is $0.8+1.2(0.2/2)=0.92$.

At $q_{\infty }=500\mathrm{Pa}$ and $c=0.20\mathrm{m}$, the two stipulated normal pressure forces are $80$ and $92N/m$, although the six samples are identical. This quadrature ambiguity of pressure-tap data is an instance of how finite samples do not certify quadrature accuracy. Additional geometric or regularity information is needed to bound the missed contribution.

A motivating experimental comparison uses a force balance independently of the pressure reconstruction. Robert M. Pinkerton's 54-orifice distribution measurements provide a historical example of resolving the shape of a pressure field.^[ref] A modern proposed test repeats both readings at controlled Reynolds number and angle of attack, checks tap coordinates and pressure-channel response, and refines sampling near sharp features. Agreement in a single integrated load does not prove agreement in the entire pressure distribution or pitching moment.