Factorial contrast for aerodynamic interference ($D_{11}-D_{10}-D_{01}+D_{00}$)

A factorial contrast for aerodynamic interference compares four configurations formed by independently including or excluding two components. Write their matched-condition mean drag readings as $D_{ij}$, where $i,j\in \{0,1\}$. The interaction contrast is
It measures nonadditivity between the specified configurations, not a universal attribute of either part alone.

The two-factor representation
reproduces any four means. Substitution gives $\Delta ={\beta }_{AB}$. In the original illustrative dataset $(D_{00},D_{10},D_{01},D_{11})=(2,22,7,32)\mathrm{N}$, the baseline is $2$, the separate increments are $20$ and $5$, and the interaction is $5\mathrm{N}$. The four-parameter representation does not by itself explain the responsible fluid flow.

With repeated measurements, retain the factorial design and run order. A change in dynamic pressure, orientation, mounting system or balance zero can otherwise enter the contrast. Common-zero cancellation in a drag interaction contrast handles a truly common additive offset, while drift rejection by contrast orthogonality determines which time-varying offsets a chosen run order removes.

Kenneth E. Ward's strut experiments motivate the physical comparison: separate strut drag was compared with changes measured when struts were attached to a wing, revealing aerodynamic interference beyond a sum of isolated loads.^[ref] The algebra supplies a controlled comparison; pressure and wake observations are needed to test a mechanism.

Drift rejection by contrast orthogonality removes specified additive time trends from a load comparison by choosing contrast weights orthogonal to those trends. In an aerodynamic interference experiment, it prevents an assumed form of balance-zero drift from being confused with nonadditivity between component configurations.

For observations at times $t_r$, writeThe columns of $B$ specify nuisance trends and $\mathbf{\gamma }$ their unknown amplitudes. The drift contribution vanishes for every amplitude exactly whenFor polynomial instrumental drift through degree $p$, the columns are $1,t,\dots ,t^p$, so the conditions are ${\sum }_r{w}_r{t}_r^j=0$ for $j=0,\dots ,p$. This orthogonality is a statement about actual observation times, not merely the number of repetitions.

An original eight-run construction uses equally spaced times $t=0,1,\dots ,7$ in a fixed time unit and the configuration orderFor the factorial contrast for aerodynamic interference, chooseEach configuration occurs twice and its total weight is the coefficient required for $\Delta =D_{11}-D_{10}-D_{01}+D_{00}$. Direct summation givesThe schedule rejects constant, linear and quadratic additive drift, but a cubic term $c{t}^3$ leaves bias $-24c$. If equal run spacing changes, the moments must be recomputed. If independent residual errors have common variance ${\sigma }^2$, the contrast variance is ${\sigma }^2{\mathbf{w}}^T\mathbf{w}=2{\sigma }^2$; correlated errors instead require the full covariance matrix.

Rejection must preserve the physical signal. If a desired signal column lies in the nuisance column space, every vector rejecting that space also rejects the signal. Orthogonal projection exposes this identifiability limit; collecting more readings in the same confounded arrangement need not fix it. Gain drift, configuration-dependent tare and abrupt offsets are also outside the stated additive polynomial model.

Kenneth E. Ward's original attached- and isolated-strut measurements motivate separating combined load from component drag.^[ref] The eight-run schedule here is a constructed design, not a reconstruction of that historical experiment.