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As we know, Oswald Efficiency ($\epsilon$) is caused by change in downwash over the span of the wing, thus causing a change in effective angle of attack over the wing and therefore induced drag variation across the wing.

Formula: $C_{di}=\frac{C_l^2}{\pi \epsilon R}$

For this reason, old planes and some hobby planes use elliptical wings (best seen by the Spitfire), so that smaller chord wingtips generate the same downwash as high-chords mid-wing -- keeping constant downwash and effective angle of attack throughout the plane. Spitfire Elliptic Wing

I'm curious about how varying different factors has an effect on Oswald Efficiency, for example for an RC-Airplane, lets say roughly 1.5m span, 25m/s max velocity.

My intuition:

On one hand, having such little space between the wing tips and fuselage means less distance for vortices formed by downwash at mid-wing to form, which should allow downwash to be somewhat even. Further, the air at the wing tip would also "drag along" air closer in the center, kind of balancing out any differences in downwash and increasing Oswald Efficiency.

On the other hand, the pressure gradient above/below the wing grows by velocity squared, and would be much lower for a 25 m/s plane. I'm not sure the force/pressure difference would be strong enough to induce strong vortices mid wing. Thus, there would only be small vortices at the wingtips, resulting in a very large downwash gradient, thus lower Oswald Efficiency.

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Yes and no. All the physics still hold. The absolute distance from wingtip to fuselage doesn't matter -- flow structures scale down to match the aircraft.

However, most R/C aircraft are not flown like full scale aircraft. That is -- most RC aircraft are not flown in trimmed equilibrium flight for long straight segments.

Instead, most RC aircraft are flown at a high throttle setting and the aircraft frequently are turning as we generally enjoy poking holes in the sky.

Maximizing fuel economy, range (or even endurance) is usually not the point of an RC aircraft, so achieving maximum flight efficiency is usually not a problem.

In addition, most RC aircraft are substantially overpowered as compared to their full-scale counterparts. So places where low induced drag might be noticed are generally over-run with sheer might.

Where I have most noticed high induced drag (from Oswald type effects, but also low-span (or low aspect ratio)) -- is with an aircraft that barely has enough power that also has to make relatively sharp turns. Such an under-powered aircraft will lose substantial altitude in a turn (when induced drag is increased). If the aircraft is flying a racetrack pattern, it will lose altitude in the turn, struggle to climb down the straight, only to lose the altitude again in the turn.

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  • $\begingroup$ Very good answer! $\endgroup$ Jul 5, 2023 at 16:24
  • $\begingroup$ Thanks for you answer. You mention that most RC aircraft are not flown like full scale aircraft, and while I agree with you, there are plenty of applications for RC aircraft such as delivery drones as well as research/competitions where maximizing range/endurance does matter, so you're not really answering the question... Further, on RC planes, typically induced drag is a far bigger factor in max speed than pressure drag, and actually is asymptotic because pitching the plane angles your thrust, thus it's not something that can be "over-run with sheer might" as you say. $\endgroup$
    – Ankit
    Jul 5, 2023 at 20:06
  • $\begingroup$ @Ankit, I stand by my statements. I don't consider an aircraft (drone) that is controlled by an autopilot to be an R/C aircraft anymore. The important difference is that an autopilot may actually attempt to throttle back and trim the aircraft to fly in equilibrium flight. A true visually tracked and thumb-stick controlled radio controlled aircraft, the pilot has no airspeed, altimeter, or rate of climb instrumentation. $\endgroup$ Jul 6, 2023 at 5:49
  • $\begingroup$ The pilot can not trim the aircraft to a target airspeed (best endurance, best range, etc) because they don't have the information. Also, when flying within visual range as R/C aircraft do, you never fly in a straight line before needing to turn. I have advised many student groups who have built RC aircraft for competition, I know how those usually work out. Induced drag is never a big factor in max speed -- by definition. Calculate the thrust-to-weight or power-to-weight of most R/C aircraft and compare them to their general aviation equivalents, you will find RC aircraft are overpowered. $\endgroup$ Jul 6, 2023 at 5:52
  • $\begingroup$ I understand that you might not have felt that I answered your question. Your intuition for what happens to aerodynamics when an aircraft is scaled down is wrong. One of the fundamental ideas of aerodynamics is that you can scale a body up or down and the flow structures remain essentially changed. This is why we can test large aircraft with smaller models in wind tunnels. You need to match Reynolds number (Mach number is low enough here to ignore). So any difference you see between model and full scale should be a viscous drag / boundary layer thickness / Reynolds number related. $\endgroup$ Jul 6, 2023 at 5:56

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