Spatial resolution of surface-pressure measurements

Spatial resolution of surface-pressure measurements describes which spatial variations of surface pressure can be distinguished by the measurement arrangement. Tap spacing, opening size and the reconstruction model matter separately from uncertainty in each measured pressure. Repeatedly measuring the same taps more precisely does not supply the missing values between them.

A primary wind tunnel experiment by Robert M. Pinkerton sampled a NACA4412 airfoil with 54 pressure orifices and compared measured distributions with theoretical calculations.^[ref] The observations constrain pressures at the instrumented sites. Interpolation and surface pressure integration for airfoil lift require assumptions about the field between those sites.

An original sampling counterexample makes the limitation exact. On a chordwise coordinate $\xi =x/c\in [0,1]$, place five taps at ${\xi }_j=j/4$. Compare two candidate pressure differences,Both give zero at every tap, butAt $A=400\mathrm{Pa}$ and chord $c=0.20\mathrm{m}$, the integrals differ by $40N/m$ in a simplified force-per-span calculation on a flat strip. The geometry and candidate fields are constructed examples, not solutions for a measured airfoil. They prove that these five readings alone do not determine an arbitrary continuous field or its integral.

This is spatial aliasing: distinct fields produce the same samples. Additional smoothness or spatial-bandwidth assumptions can constrain reconstruction, but must be stated and tested. The pressure-tap spatial footprint adds another consideration: a finite opening can respond differently from an ideal point measurement. Increasing sampling density, characterizing the probe and checking an independent integrated force address different parts of the inference.

A pressure-tap spatial footprint describes the finite region of a surface pressure field that contributes to a tap's response, rather than treating the reading as an exact point value. A pressure-footprint response kernel can represent spatial averaging in a simplified linear model. A real opening can also disturb the flow, so averaging is not a complete physical correction model.

For an ideal one-dimensional strip of width $w$, suppose the reading at $x_0$ isFor a smooth field, a Taylor expansion gives $p_m=p(x_0)+p^{\prime \prime }(x_0)w^2/24+O(w^4)$. Symmetry cancels the first-order gradient term, but not curvature. For an exactly quadratic field this expression through $w^2$ is exact. It shows why a footprint cannot in general be summarized by a displaced point location.

If $p(x)=A\sin (kx)$, direct integration instead givesThe multiplier is a sinc function. For wavelength $\lambda =2\pi /k$, a half-wavelength strip retains $2/\pi \approx 0.637$ of the amplitude, while a full-wavelength strip gives zero. These results concern an ideal strip spatial averaging filter, not an asserted weighting law for a circular pressure hole.

An actual wind tunnel experiment by E. B. Plentovich and Blair B. Gloss used interchangeable orifices in a flat plate. Increasing opening size increased the observed pressure error; a flush porous-metal plug reduced it under the tested conditions.^[ref] This is direct evidence that opening geometry affects the measurement, including through local flow disturbance.

Footprint characterization and tap spacing address separate limitations. Closely spaced readings from broad footprints may still suppress short spatial variations, while narrow footprints sampled too sparsely can suffer spatial aliasing. Neither problem is removed merely by improving the precision of the pressure transducer.