= Surface pressure integration for airfoil lift
\b[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 $\Gamma$ is
$$
\mathbf F'_p=-\oint_\Gamma (p-p_\infty)\mathbf n\,ds.
$$
The uniform reference pressure cancels on the closed contour. This is a vector <line integral>, not the sum of positive pressure magnitudes.<Aerodynamic Forces by NASA Glenn Research Center>[Variation in Pressure and Definitions of Lift and Drag]{ref}
For upper and lower surfaces that are graphs over $0\le x\le c$, their pressure contribution perpendicular to the chord is
$$
N'_p=\int_0^c[p_l(x)-p_u(x)]\,dx=q_\infty c\int_0^1[C_{p,l}(\xi)-C_{p,u}(\xi)]\,d\xi.
$$
Vertical end closures have no chord-normal pressure contribution. The pressure force parallel to the chord, $A'_p$, still matters when resolving flow-relative <lift>. With flow direction $(\cos\alpha,\sin\alpha)$ in aft/chord-normal axes, \b[$L'_p=N'_p\cos\alpha-A'_p\sin\alpha$]. Equating $N'_p$ 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 $\xi=0,0.2,\ldots,1$. A second invented profile agrees at every tap but has an additional $1.2\sin^2[\pi(\xi-0.4)/0.2]$ on $0.4\le\xi\le0.6$ and zero addition elsewhere. Both reconstructions give $0.8$, while the second true integral is $0.8+1.2(0.2/2)=0.92$.
\Image[/pressure-tap-aliasing.png]
{title=Two invented pressure-difference profiles agree at every tap but have different chord integrals. Equal readings alone do not bound an unsampled peak.}
At $q_\infty=500\,\mathrm{Pa}$ and $c=0.20\,\mathrm m$, the two stipulated normal pressure forces are \b[$80$ and $92\,\mathrm{N/m}$], 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.<Calculated and Measured Pressure Distributions over the Midspan Section of the NACA 4412 Airfoil by Robert M. Pinkerton>[Abstract, apparatus and measured/theoretical distributions]{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>.
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