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Correlated position errors of pressure taps occur when uncertainty in one tap coordinate is statistically related to uncertainty in another. A shared origin, common scale or common image transformation can couple all inferred positions. Treating every tap as independent can either overestimate or underestimate the uncertainty in a reconstructed pressure load, depending on the sensitivities and their signs.

For fixed reconstruction weights $w_i$ and a differentiable pressure coefficient profile, the linearized position contribution isHere ${\epsilon }_i$ is dimensionless coordinate error and ${\Sigma }_{\xi }$ its covariance matrix. The general covariance propagation law is the same one used for other correlated inputs in a measurement model.^[ref]

An original two-tap example takes equal coordinate standard uncertainties $s=0.002$ and correlation coefficient $r=0.9$. With sensitivities $\mathbf{b}=(1,-1)$,Discarding correlation gives $u_Q=\sqrt{2}s\simeq 0.002828$, which is too large here. For $\mathbf{b}=(1,1)$ instead, $u_Q=\sqrt{2s^2(1+r)}\simeq 0.003899$; the same independence assumption is now too small. Positive correlation alone does not determine the direction of the error in the output uncertainty.

A common-origin pressure-tap position error can be written ${\epsilon }_i=\delta +{\eta }_i$, with an independent shared offset of variance $s_0^2$ and independent residual errors of variances $s_i^2$. ThenThe common contribution cancels to first order when the sensitivities sum to zero. Repeatedly reading the same misplaced coordinate origin does not remove its uncertainty. Pressure-tap position calibration must therefore retain shared reference information, not just a list of per-hole scatters. If the coordinates also determine the quadrature weights or surface geometry, their changes belong in the full sensitivity calculation; the fixed-weight formula alone is incomplete.

A 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 $\xi +\epsilon$. For a differentiable pressure coefficient profile,If the derivative is bounded by $K$ throughout the interval between the two positions, the mean value theorem gives the rigorous bound $|\delta C_p|\le K|\epsilon |$. 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 $C_p^{\prime }=20$. Then $\epsilon =0.002$ and $\delta C_p=0.040$. 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 $J=\mathrm{diag}(C_p^{\prime }({\xi }_i))$. If ${\Sigma }_{\xi }$ is their position-error covariance matrix, the first-order variance contribution to a reconstructed coefficient isThis 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.^[ref] Such a comparison tests both where a feature occurs and how large it is; pressure-only calibration cannot fix a displaced coordinate.

Quadrature ambiguity of pressure-tap data means that the same finite set of measured surface pressure values can be consistent with different integrated forces. A chosen quadrature rule fixes an estimate from those readings; it does not, by itself, establish the unsampled distribution. This is an instance of how finite samples do not certify quadrature accuracy.

For an original smooth example, use the normalized airfoil chord coordinate $\xi \in [0,1]$ and let $f=C_{p,l}-C_{p,u}$ be the lower-minus-upper pressure coefficient difference. Put three sample locations at $0$, $1/2$ and $1$, and considerThese polynomials agree at all three taps. Both the composite trapezoidal rule and any other deterministic rule using only these same values produce identical answers for them. Nevertheless,The discrepancy is not caused by a discontinuity or mislocated tap: both profiles are smooth at every sample. At stipulated dynamic pressure $q=400\mathrm{Pa}$ and airfoil chord $c=0.25\mathrm{m}$, the chord-normal pressure loads are $60$ and $70N/m$. Their relation to flow-relative lift still requires the geometric resolution explained in surface pressure integration for airfoil lift.

Additional assumptions can make an error bound possible. If a twice differentiable $f$ has a known bound $|f^{\prime \prime }|\le M$ over the entire chord, the local linear-interpolation remainder gives an integral error at most $M{h}_i^3/12$ on an interval of width $h_i$. Summing givesFor uniform spacing $h$ on this unit interval it becomes $M{h}^2/12$. A value of $M$ inferred only from the same sparse readings is not an independently established bound. Conversely, a justified polynomial model of fixed degree would supply different information and could remove the ambiguity.

A proposed experiment compares the pressure reconstruction with an independently corrected force-balance measurement at matched angle of attack and Reynolds number, then adds taps near suspected sharp features. Pressure-tap position calibration and repeated pressure readings address other error sources. Agreement of two sparse-data rules is a check on their chosen interpolants, not proof that an unresolved contribution is absent.