= Surface pressure
{title2=$p_s$}
= Fluid surface pressure
{synonym}
\b[Surface pressure], in fluid-mechanical loading, is the local <pressure> evaluated at a boundary such as an <airfoil> surface. It is a scalar field $p_s(\mathbf x,t)$ over position and time, with units of pascals. The field can vary strongly along the same body even when the surrounding free-stream conditions are fixed.
For a small surface patch with outward unit <normal vector> $\mathbf n$ pointing from the solid into the fluid, the pressure contribution to the force is
$$
d\mathbf F_p=-p_s\mathbf n\,dA.
$$
It acts normally to the surface. Tangential <viscous stress> is a separate contribution. Experiments commonly compare the boundary value with a reference pressure $p_\infty$ using the <pressure coefficient>
$$
C_p=\frac{p_s-p_\infty}{q_\infty},\qquad
q_\infty=\frac12\rho U_\infty^2,
$$
where $q_\infty$ is <dynamic pressure>. A coefficient is dimensionless; reconstructing the pressure difference requires the reference conditions.
For an original patch example, let $p_s=99.8\,\mathrm{kPa}$, $p_\infty=100.0\,\mathrm{kPa}$ and $q_\infty=500\,\mathrm{Pa}$. Then $C_p=-0.4$. On a $0.010\,\mathrm{m^2}$ patch, absolute pressure gives an inward force of $998\,\mathrm N$; relative to the reference pressure, the pressure-force contribution differs by $2\,\mathrm N$ outward. A negative coefficient therefore describes a reference-relative deficit, not a negative absolute pressure.
A primary <wind tunnel> experiment by <Robert M. Pinkerton> used 54 pressure orifices in a NACA4412 model and compared measured distributions at different <angles of attack> with <potential flow> predictions. Such <pressure tapping> observes local boundary pressures through small openings; the distribution is reconstructed from measurements at selected positions.<Calculated and Measured Pressure Distributions over the Midspan Section of the NACA 4412 Airfoil by Robert M. Pinkerton>[NACA-TR-563, NTRS abstract: 54 orifices, tested incidence range and comparison with calculations]{ref}
<Spatial resolution of surface-pressure measurements> limits what that reconstruction can establish between taps. Reference pressure, probe response and flow unsteadiness also affect the result. <Surface pressure integration for airfoil lift> uses the reconstructed field and surface directions to obtain an overall force; one pressure reading alone does not determine <lift>.
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