= Wake-plane dependence of a downwash measurement
<Wake-plane dependence of a downwash measurement> means that a measured downward <velocity> depends on where the observation plane lies relative to the lifting wing and its evolving wake. The induced <velocity> entering a wing-section <force> calculation is not generally equal to a downstream probe reading. Classical <lifting-line theory> explicitly adjusts local incidence using the induced flow at the lifting line, under assumptions such as <steady flow>, <incompressible flow> and a nearly planar high-aspect-ratio wing.<Prandtl's Lifting Line Introduction (16.100) by MIT OpenCourseWare>[printed pp. 1–3, assumptions, induced angle and circulation equation]{ref}
An original straight-filament model isolates one reason for the difference. Consider a vortex filament of <circulation> magnitude $\Gamma$ starting at streamwise coordinate zero and extending downstream forever. At a point with streamwise coordinate $x$ and perpendicular distance $r>0$, its induced azimuthal <speed> is
$$
v(x,r)=\frac{\Gamma}{4\pi}\int_0^\infty\frac{r\,ds}{[(s-x)^2+r^2]^{3/2}}
=\frac{\Gamma}{4\pi r}\left(1+\frac{x}{\sqrt{x^2+r^2}}\right).
$$
The integral follows by differentiating $(s-x)/[r\sqrt{(s-x)^2+r^2}]$. At $x=0$, only one half of the corresponding infinite straight filament contributes; far downstream, the local filament extends far in both axial directions relative to $r$.
For two straight trailing filaments at transverse positions $y=\pm a$ with opposite <circulation> signs chosen to induce downward flow between them, their midpoint contributions add:
$$
\boxed{w(x)=\frac{\Gamma}{2\pi a}\left(1+\frac{x}{\sqrt{x^2+a^2}}\right),\quad
w(0)=\frac{\Gamma}{2\pi a},\quad w(+\infty)=\frac{\Gamma}{\pi a}.}
$$
With stipulated $\Gamma=20\,\mathrm{m^2/s}$ and $a=5\,\mathrm m$, this trailing-filament contribution is $0.6366\,\mathrm{m/s}$ at $x=0$, $1.0868\,\mathrm{m/s}$ at $x=5\,\mathrm m$, and approaches $1.2732\,\mathrm{m/s}$. This is an idealized geometry calculation, not a measured wake or a universal factor-of-two correction for every wing.
The model omits finite vortex cores, the bound-vortex contribution away from the lifting line, the complete distributed vortex sheet, wake deformation and <wake contraction in a downwash survey>. Each can change the spatial transfer. In particular, a wing-plane value and a far-wake value must be connected using the same load and wake model before using one to infer the other.
In the small-angle lifting-line model, the wing-plane induced-drag calculation weights downwash by the <circulation> or local <lift>. Its source derivation integrates the local induced contribution along the wing.<Force Calculations for Lifting Line (16.100) by MIT OpenCourseWare>[printed pp. 1–2, local lift, induced angle and induced-drag integration]{ref} Substituting an arbitrary downstream <speed> into that expression can misestimate <drag> even if the <speed> measurement itself is accurate.
A proposed survey measures <velocity> components at several downstream stations with the same probe calibration and freestream reference, covering both vortex cores and the intervening region. Record the wing loading, probe orientation and cross-plane coordinates, and track how vortex separation and <circulation> estimates change. A centerline reading alone does not recover the whole momentum field. Matching an axial trend tests the proposed transfer model only over the surveyed region; it does not establish that a rolled-up real wake remains a pair of straight mathematical filaments.
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