# Why is the polar diagram of an airplane depended on weight?

Can anybody answer this question? Although aviation-related, it is in fact a fluid mechanics problem:

https://aviation.stackexchange.com/questions/54192/why-is-polar-curve-of-a-glider-dependent-on-flight-load

So far I got only a lot of unsatisfying answers. Maybe here are experts on fluid mechanics.

For my understanding the polar-curve is a property of geometry only. It provides a $c_{lift}$ and a $c_{descend}$. Given those coefficients, the speed of a glider can be calculated by simple math, since force in direction and perpendicular to air flow is calculated by $F_i=c_i \cdot \rho \cdot v²$.

Even if that is not true because of non-linearities of whatever reason, I always imagine the aircraft to be tested in a wind channel of given air-speed with a given angle of attack: in that case (I assume) the flow of air and associated forces cannot be different from the real case. But then weight cannot be relevant. Of course different weights give rise to different velocity, but then this can be explained fully based on the polar diagram.

So why is the polar diagram depended on weight?

## 1 Answer

I sense a few misconceptions from your question:

1. The polar curve does change with wing loading because a higher wing loading shifts all polar points to a higher speed. This brings with it a higher Reynolds number, so the viscous forces become smaller relative to the inertial forces. In simple words: The aircraft experiences relatively less friction drag when flying with a higher wing loading. For modern gliders the difference between a light wing loading (say, 35 kg/m²) and a heavy wing loading (around 50 kg/m²) is typically as high as an increase in best L/D of 3 to 4%.
2. Windtunnel data always needs to be interpreted. Every tunnel is calibrated to arrive at tunnel factors which must be applied to test data in order to get useable results. The differences come from several effects: Smaller Reynolds and Mach numbers, vicinity of the tunnel walls affecting the vortex structure around the model, the inevitable structure needed for mounting the model inside the tunnel and differences in flow from the model blocking part of the tunnel cross section are the most important ones. Also the elastic deformation between model and original is different if the model's elasticity has not been precisely scaled, and this can only be done for one polar point.
• There was a very big mistake on my side: Lilienthal-polar (C values for given angle of attack) are constant, and from that I inferred wrongly, that speed-polare is also constant - this is of course wrong. The curves for different wing-loads (weights) scale with weight-ratio and are subject to a simple curve transformation. So there is no magic behind, the whole question was kind of "silly". – MichaelW Aug 13 '18 at 5:23
• But now you mention, that Reynolds number is changing, because of speed change. However, this must be a second order effect, because applying the simple scaling transformation to the speed-polar-curve for a change in weight ends up in the right curve - without considering Reynolds number. It seems that this doesn't need to be considered practically. Or did I misunderstand your explanation regarding (1)? – MichaelW Aug 13 '18 at 5:25
• @michael: Yes, it is a second order effect, but flying with or without water ballast does make a noticeable difference. It is not only the L/D which improves, but low speed handling overall. Of course, with water the circles for thermalling become wider, so this cannot be done in any weather. – Peter Kämpf Aug 13 '18 at 6:03