= Hot-wire spatial averaging
<Hot-wire spatial averaging> is the loss or alteration of small-scale flow information because a heated sensing wire has finite spatial extent. Even a perfectly fast electronic readout observes a spatially distributed heat-transfer response. The manufacturer's practical guide relates spatial resolution to wire dimensions, flow and instrument <frequency> response.<How to measure turbulence with hot-wire anemometers - a practical guide by Finn E. Jørgensen>[§15.3, printed p. 49]{ref}
An original small-perturbation model treats the inferred <velocity> fluctuation as a uniform line average over wire length $\ell$. With $s$ measuring distance along the wire,
$$
\overline u_\ell(s_0,t)=\frac1\ell\int_{-\ell/2}^{\ell/2}u(s_0+s,t)\,ds.
$$
For a spatial sinusoid $u(s,t)=\widehat u\cos(ks-\omega t+\phi)$, direct integration gives
$$
\boxed{\overline u_\ell(s_0,t)=H(k)\widehat u\cos(ks_0-\omega t+\phi),\qquad
H(k)=\frac{\sin(k\ell/2)}{k\ell/2},\quad H(0)=1.}
$$
This is a <sinc function> transfer factor. It follows from the stipulated uniform weighting and local linearity; a real wire's nonuniform <temperature> and nonlinear <heat transfer> require a more detailed spatial kernel.
For spatial <wavelength> $\lambda_s=2\pi/|k|$, a fluctuation with $\lambda_s=4\ell$ has amplitude factor about $0.9003$ and <variance> or <power> factor $|H|^2\approx0.8106$. At $\lambda_s=2\ell$, the factors are $2/\pi\approx0.6366$ and $4/\pi^2\approx0.4053$. At $\lambda_s=\ell$, one complete sinusoidal cycle lies along the wire and its line average is exactly zero. Raising the sample rate cannot restore a component that this spatial average has canceled.
\Image[/hot-wire-spatial-transfer.png]
{title=Small spatial scales attenuated by an ideal finite-length hot wire}
{description=Derived <power> transfer of a uniform line-average model versus wire length divided by spatial <wavelength>. Marked points show wavelengths four, two and one times the wire length; this is a mathematical <sensor> model, not measured manufacturer performance.}
For a $1\,\mathrm{mm}$ wire, these illustrative wavelengths are $4$, $2$ and $1\,\mathrm{mm}$. The result depends on the component of the spatial wavevector along the wire, rather than only on a temporal <frequency>. <Orientation-dependent spatial filtering by a hot wire> matters because a moving three-dimensional structure can have different projections along the wire and the mean flow. Under a frozen-advection model, temporal <frequency> obeys $\omega=\mathbf k\cdot\mathbf U$; substituting $k=\omega/U$ into the wire-length factor requires an additional alignment assumption.
Spatial and temporal filters can both reduce measured fluctuation <variance>. In this single-sinusoid model, the measured <variance> is $|H|^2$ times the true <variance> even with noiseless electronics. A correction obtained by dividing by $H$ becomes unstable near its zeros, amplifying measurement noise without recovering independently lost information.
A proposed test compares probes of different active lengths in the same statistically repeatable flow, after separate steady calibrations and matched electronic filtering. Record wire orientation, support geometry, ambient <temperature> and traverse location; compare fluctuation spectra as well as mean <velocity>. A difference that tracks wire length supports a spatial-filtering interpretation, but changed thermal response or probe disturbance must also be checked. The observation establishes the filtered response of the actual apparatus over its characterized range, not an unqualified point measurement at every turbulent scale.
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