Correlated position errors of pressure taps

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 common-origin pressure-tap position error shifts several inferred tap positions together relative to a fixed physical reference. It is a shared contribution to correlated position errors of pressure taps, not an independent placement error for every hole. Whether it affects a pressure-derived load depends on which references are held fixed.

For dimensionless chord coordinate $\xi$, suppose the uncertain displacement is the same scalar $e$ for every tap. A first-order Taylor expansion of the pressure coefficient gives $\delta C_{p,i}\simeq C_p^{\prime }({\xi }_i)e$. For a reconstruction $Q={\sum }_i{w}_i{C}_{p,i}$ with fixed quadrature weights,The position covariance matrix is $u_e^{2}11^T$, a rank-one matrix. Squaring each tap's contribution before adding would discard the shared covariances. The covariance terms belong in the general propagation of uncertainty.^[ref]

An original locally linear example takes eight equal weights $w_i=1/8$, equal local slopes $C_p^{\prime }=20$ and $u_e=0.002$. The reconstructed coefficient has standard uncertainty $u_Q=0.040$. Incorrect independence would give $0.040/\sqrt{8}\approx 0.0141$. If a two-dimensional load per span is $L^{\prime }=qcQ$ with stipulated dynamic pressure $q=500$ Pa and airfoil chord $c=0.20$ m, these become $4.0$ and $1.41$ N/m. The chosen slopes describe a sensitivity model, not measured airfoil data.

Opposite-signed sensitivities can cancel the shared contribution. Conversely, repeating coordinate readings without independently checking the misplaced datum cannot average away its fixed error. A proposed test measures an independent reference mark and deliberately translates the assumed origin in the reconstruction, comparing the observed load sensitivity with the derivative above. If moving coordinates also changes the integration weights, those additional derivatives must be included.

Pressure-tap datum cancellation is a different situation. When the tap, leading edge and trailing edge are all expressed through the same affine map $u=a+bx$, with $b>0$,The common origin and scale cancel exactly. Merely translating a shared coordinate system causes no physical tap misregistration. Affine pressure-tap coordinate calibration must therefore state whether the datum is shared or independently fixed before assigning a common-origin uncertainty.