Measuring Lift Curve Slope on a 1/2-Wing vs Full Wing

[ChrisHarvey]

If I take an infinitely long, thin wing and measure its lift curve slope I should get 2pi.

Now if I take a 1/2-wing and fit 1 end to a plate and test it in a wind tunnel, it should simulate the flow around a full span wing, am I wrong? Therefore measuring lift and working out the CL values using the area of the half wing, air density, and flow speed I should get approximately the same results for CL as for a full wing (same section) twice the area and twice the aspect ratio (with no plates fitted to the ends).

I've done that experiment for a NACA 0015 aerofoil section and my results are confusing me. I understand the theory and the limitations of Prandtl's lifting line theory but this doesn't seem to be helping me here.

I was expecting the theoretical and experimental results to match up fairly well (looking at Prandtl's graphs they are pretty much identical).

The 1/2-wing I used was AR 3. Therefore it simulated the full wing AR 6. If I use AR 3 for the theoretical calculations the graphs match perfectly, but I can't do this, because the 1/2-wing AR 3 was simulating a wing AR 6 with that attachment plate on 1 end. So... experimentally the lift curve slope is 3.5rad-1 and theoretically it's 4.5rad-1.

Can anyone just point me in the direction of the reason for this massive difference? I'm guessing it's something to do with the 1/2-wing with a centre-line plate mount not properly simulating the flow around a full span wing, but this to me would suggest a steeper lift curve slope than theoretically predicted because there's 1 less wing tip vortex to cause more downwash over the rest of the wing.

I hope someone can make sense of that - I'm sure the only reason I can understand what I've written is because I know what I'm trying to say!

 

[ChrisHarvey]

As is usually the case I answer my own question: the main (obvious reason) is that the theory I was using (Prandtl's lifting line theory) assumes invsicid flow and of course there are viscous effects (even if small).

 

[ravenprp]

Your understanding of the concept of lift curve slope is correct. In theory, the lift curve slope for an infinitely long, thin wing should be 2pi. However, when using a 1/2-wing with a plate attached to one end, the lift curve slope may not exactly match the theoretical value due to various factors such as flow separation, wing tip effects, and the presence of the plate.

One possible explanation for the difference in your experimental and theoretical results could be the effect of the plate on the 1/2-wing. The plate may cause flow separation and affect the overall lift and lift curve slope of the wing. Additionally, the plate may also create additional vortices and alter the flow around the wing, which could also affect the lift curve slope.

Another factor to consider is the aspect ratio of the wing. In your experiment, you used a 1/2-wing with an aspect ratio of 3, which theoretically simulates a full wing with an aspect ratio of 6. However, the presence of the plate may alter the aspect ratio and affect the lift curve slope. This could explain why using an aspect ratio of 3 in your theoretical calculations matches your experimental results, while using an aspect ratio of 6 does not.

It is also important to remember that Prandtl's lifting line theory is a simplified model and may not accurately predict the lift curve slope for all wing configurations. Other factors such as wing shape, angle of attack, and Reynolds number may also affect the lift curve slope.

In conclusion, the difference in your experimental and theoretical results could be due to the presence of the plate, altered aspect ratio, and other factors that may not be fully captured by Prandtl's lifting line theory. Further experimentation and analysis may be needed to fully understand the discrepancies in your results.