 Quadrature ambiguity of pressure-tap data

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.