= Pressure-tap position error
{title2=$\delta x$}
A \b[pressure-tap position error] assigns a measured <surface pressure> to the wrong point on an <airfoil>. This differs from a <pressure transducer>'s pressure-reading error: an accurate instrument can faithfully measure the pressure at a misplaced or incorrectly labeled hole. A steep spatial <pressure> gradient makes a small location error important.
Let $\xi=x/c$ be the nominal chord coordinate and let the actual <pressure tapping> lie at $\xi+\varepsilon$. For a differentiable <pressure coefficient> profile,
$$
\delta C_p=C_p(\xi+\varepsilon)-C_p(\xi)\simeq C_p'(\xi)\varepsilon.
$$
If the <derivative> is bounded by $K$ throughout the interval between the two positions, the <mean value theorem> gives the rigorous bound $|\delta C_p|\le K|\varepsilon|$. Knowing only an endpoint slope does not establish that bound across an unresolved peak.
In an original local example, take $c=0.20\,\mathrm m$, a physical position error $\delta x=0.40\,\mathrm{mm}$, and a profile locally linear with $C_p'=20$. Then $\varepsilon=0.002$ and \b[$\delta C_p=0.040$]. At <dynamic pressure> $q=500\,\mathrm{Pa}$, the corresponding pressure difference is $q\delta C_p=20\,\mathrm{Pa}$. These are stipulated values illustrating sensitivity, not measurements of a particular <airfoil>.
For fixed <quadrature weights> $\mathbf w$ and small errors in several tap positions, put $J=\operatorname{diag}(C_p'(\xi_i))$. If $\Sigma_\xi$ is their position-error <covariance matrix>, the first-order variance contribution to a reconstructed coefficient is
$$
u_{\rm position}^2=\mathbf w^TJ\Sigma_\xi J^T\mathbf w.
$$
This formula retains <correlated position errors of pressure taps>, including a shared coordinate-origin error. It holds the reconstruction weights and surface geometry fixed; uncertainty in those also needs accounting when geometry is inferred from the same coordinates.
<Pressure-tap position calibration> measures the actual hole coordinates and their reference origin, alongside pressure-channel calibration. <Robert M. Pinkerton>'s distributed-pressure experiment used 54 orifices on an <airfoil> model to compare a measured distribution with <potential flow> predictions.<Calculated and Measured Pressure Distributions over the Midspan Section of the NACA 4412 Airfoil by Robert M. Pinkerton>[Abstract, pressure-orifice distribution and theoretical comparison]{ref} Such a comparison tests both where a feature occurs and how large it is; pressure-only calibration cannot fix a displaced coordinate.
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