Aerodynamic interference

Aerodynamic interference occurs when nearby bodies alter one another's fluid flow, so their combined aerodynamic loads differ from the sum measured separately under a specified comparison. Pressure fields, boundary layers and fluid wakes can all participate. An aircraft assembly therefore needs an operating-condition definition as well as a list of parts.

A factorial contrast for aerodynamic interference makes that definition measurable. At the same dynamic pressure, model orientation and mounting convention, record a balance reading $D_{ij}$ with component A absent/present according to $i$ and component B absent/present according to $j$. The departure from additivity isThe empty-configuration reading $D_{00}$ matters: it prevents a common support background from being subtracted twice. For original illustrative readings $(D_{00},D_{10},D_{01},D_{11})=(2,22,7,32)\mathrm{N}$, the interaction is $5\mathrm{N}$. This aircraft component drag interaction is a property of the stipulated four configurations and conditions.

Kenneth E. Ward's 1931 experiment supplied an early concrete comparison. A Göttingen 387 wing was tested with struts in different upper- and lower-surface positions in a variable-density wind tunnel; separate strut measurements helped distinguish direct strut drag from interaction. Interference could be as large as the struts' own drag.^[ref]

The contrast does not identify the responsible flow mechanism by itself. Surface pressure measurements and observations of flow separation can test where the changes occur. Comparing at fixed angle of attack also differs from comparing at fixed lift: adding a component can change lift, requiring a new angle or an explicitly documented interpolation for the latter comparison. A negative contrast is possible; interaction can reduce the chosen load as well as increase it.

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.

An aircraft component drag interaction is the part of a combined drag measurement that is not accounted for by an additive model of the specified separate components. It is one measurable consequence of aerodynamic interference; its sign and magnitude depend on operating conditions and on what counts as the common baseline.

Use the factorial contrast for aerodynamic interference, $\Delta =D_{11}-D_{10}-D_{01}+D_{00}$. With readings ordered as $\mathbf{d}=(D_{00},D_{10},D_{01},D_{11}{)}^T$, put $\mathbf{a}=(1,-1,-1,1{)}^T$. If $\Sigma$ is their covariance matrix, thenThe second expression propagates the standard uncertainty of the linear contrast; it requires the actual covariance model, not an assumption that every reading is independent.

For a stipulated instrument model, let $\mathbf{d}=\mathbf{\mu }+Z1+\mathbf{ϵ}$, with a shared zero error $Z$ independent of four independent residual errors of standard deviation $0.5\mathrm{N}$. Because ${\mathbf{a}}^{T}1=0$, the shared zero cancels exactly and $u_{\Delta }=1.0\mathrm{N}$, regardless of its variance. This common-zero cancellation in a drag interaction contrast fails for a changing zero or for a shared gain error. At the illustrative contrast $5\mathrm{N}$, a common relative gain error $g$ leaves the factor $(1+g)5\mathrm{N}$.

Kenneth E. Ward distinguished attached-component measurements from isolated-strut measurements in his wind tunnel study, rather than attributing the entire assembly drag increase to the struts alone.^[ref] In a modern repetition, interleave baseline and assembly runs, retain all raw balance channels, and compare after consistent measurement calibration. Repeatability alone does not bound a systematic support interaction shared by every run.

Common-zero cancellation in a drag interaction contrast follows from the coefficients summing to zero. Order the readings as $\mathbf{d}=(D_{00},D_{10},D_{01},D_{11}{)}^T$ and put $\mathbf{a}=(1,-1,-1,1{)}^T$. If
then
because ${\mathbf{a}}^{T}1=0$. The cancellation is pointwise and does not require $Z$ to be independent of the residual errors. It requires the same additive offset to enter all four readings on the same scale.

For finite covariance, the residual contribution to the contrast's variance is ${\mathbf{a}}^T{\Sigma }_{ϵ}\mathbf{a}$. Four independent residual errors with standard deviation $0.5\mathrm{N}$ give contrast standard deviation $1.0\mathrm{N}$, however large the exactly common zero error is. A shared gain error instead multiplies a nonzero true contrast and generally survives.

Run order can also reject some changing zeros. With equally spaced run times $\mathbf{t}=(0,1,2,3{)}^T$ in the displayed configuration order, ${\mathbf{a}}^T\mathbf{t}=0$. An additive linear drift $r\mathbf{t}$ therefore cancels. A quadratic drift $c{\mathbf{t}}^{\circ 2}$ does not: ${\mathbf{a}}^T{\mathbf{t}}^{\circ 2}=4$, leaving $4c$. This drift rejection by contrast orthogonality concerns the actual time assignments; a different order can change it.

Normalization can defeat the common-scale assumption. If every dimensional reading has offset $Z$ but coefficients divide the last reading by twice the dynamic pressure used for the others, the scaled offset no longer has the form $Z1$. Keeping raw loads and reference readings allows the factorial contrast for aerodynamic interference and its measurement uncertainty to be reconstructed consistently.