Seeking Expert Advice: Verifying Bell-Shaped Lift Distribution (BSLD) in OpenVSP

[Shakil Salim]

Hi everyone,

I am currently working on a wing design aimed at achieving a Bell-Shaped Lift Distribution (BSLD) based on Prandtl’s theory, and I’m looking for some guidance on verifying my results.

I’ve run my model through VSPAERO and generated the spanwise lift distribution. I have a few questions regarding how to confirm that I’ve successfully hit the BSLD target:

1. Verification: When I plot $C_l \cdot c/C_{ref}$ in the VSPAERO Results Manager, I notice the shape of the curve changes significantly with the Angle of Attack ($\alpha$). At $\alpha = -5^\circ$, the curve matches the "bell" shape well, but at $\alpha = +5^\circ$, it begins to look more elliptical. Is this transition expected behavior for a wing with fixed geometric washout, or does this indicate that my twist distribution is not optimized for a wider range of flight conditions?

2. Best Practices: What is the standard process or "litmus test" in OpenVSP/VSPAERO to definitively prove that a wing is operating in a BSLD state? Are there specific parameters in the .lod files (like induced drag or local wash) that I should be monitoring to confirm the Prandtl-ideal loading?

I have attached my results showing the lift distribution across various flow conditions. Any feedback on how to interpret these results or how to refine the twist profile in OpenVSP to maintain the BSLD shape across a larger $\alpha$ range would be greatly appreciated!

Thanks in advance for your time and expertise.

[Rob McDonald]

Any wing planform can be twisted to match any desired lift distribution at one operating condition (CL).  However, in most cases, that distribution will change as you change angle of attack.

In potential flow, the lift distribution will obey superposition.  This means that lift distributions are commonly broken into two components.  The basic lift distribution and the additional lift distribution.

The basic lift distribution is the lift distribution when the wing is at zero lift (this will not happen at zero alpha in general).

The additional lift distribution is the change in the lift distribution of a wing due to a one degree change in the angle of attack.  When you spanwise integrate the additional lift distribution, you get the lift curve slope for the wing.

The lift distribution at any angle of attack is simply a linear combination of these.

Digging a little deeper, you will find that the basic lift distribution is determined by the wing planform, twist, and camber.

However, the additional lift distribution is only dependent on the planform -- it is independent of the twist and camber.

So, when you twist a wing to match a target lift distribution, you are changing the basic lift distribution.  You achieve this match at a single CL.

If you want to maintain your target distribution shape at other operating conditions, you also need the additional lift distribution to match your desired shape.  This can only be achieved by changing the planform.

So, to study this, construct a wing with zero twist and zero camber.  Then, perform the analysis at 1deg alpha.  That will show you the additional lift distribution.  Adjust the planform (chord distribution, but also sweep) until you achieve your desired distribution.

If you succeed, then this wing will not need any camber or twist to achieve your target distribution across all angles of attack.

So, if  you use twist and camber to match a lift distribution, you will have a non-zero basic distribution and any additional distribution will cause your distribution to vary with operating condition.

Or, if you use planform to match a basic distribution, you must achieve a zero basic distribution (no camber and no twist) in order to maintain the lift distribution over operating condition.

All of this applies to elliptical distributions (or any other target distribution) and it has particular relevance to airplanes with an elliptical planform.

Rob