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Jul 18, 2019, 03:31 AM
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Braced wings?


Something I've been thinking about is the benefit of BSLD is primarily reduced induced drag. Prandtl's derivation is assuming the constraint is wing bending moment of a cantilevered wing. And you end up with a longer span wing, less drag, and the same root bending moment.

What I'm now wondering is what does the ideal lift distribution look like for non-cantilevered wings? For example a wire braced wing, or something with a strut? Because in this case, the maximum bending moment occurs at a spanwise station where the strut attaches. Seems like boeing has also been thinking about this, as their hybrid aircraft concept has a very interesting strut-braced design.
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Jul 18, 2019, 12:10 PM
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Originally Posted by ornitech
Something I've been thinking about is the benefit of BSLD is primarily reduced induced drag. Prandtl's derivation is assuming the constraint is wing bending moment of a cantilevered wing. And you end up with a longer span wing, less drag, and the same root bending moment.

What I'm now wondering is what does the ideal lift distribution look like for non-cantilevered wings? For example a wire braced wing, or something with a strut? Because in this case, the maximum bending moment occurs at a spanwise station where the strut attaches. Seems like boeing has also been thinking about this, as their hybrid aircraft concept has a very interesting strut-braced design.
Actually the BSLD has increased induced drag compared to an elliptical spanwise distribution of the same span. It is a reduction in this increased induced drag with rolling moment that results in the proverse yaw characteristic that Bowers values so much.

The ideal lift distribution for non-cantilevered wings can be determined by specifying the downwash distribution along the span, in accordance with the constraints on bending moment. Figure 3 in the attached paper answers precisely the question you're asking. Inboard of the struts, there are no bending moment constraints, so the best thing to do is to have a uniform downwash in that region. Outboard of the struts, you want a linearly tapered downwash distribution. Minimum drag for a fixed span will be obtained when the downwash is uniform over the outboard portion, and you're back to the elliptical lift distribution. But if you allow a greater span, or if the structural constraint is less than the bending moment associated with the elliptical distribution, then the tapered downwash distribution is the best approach. The resulting planform shape will have little taper inboard of the struts and be much more tapered outboard of the struts.

This spreadsheet was originally written to design sail rigs, but it can just as easily be used to calculate wing planforms. You can set the gap to zero so the ground plane corresponds to the plane of symmetry of the wing. You can put in an arbitrary downwash distribution and calculate the planform shape that will produce it. Or, you can specify the planform shape and calculate the twist distribution that will give you the desired downwash at that operating condition. There is a separate sheet that you can use to calculate the off-design characteristics of a given planform and twist. Use the Design sheet to see what sort of planform or twist has minimum drag, and use the Analysis sheet to see how close more practical choices (like linear twist or straight taper) come to the minimum drag solution.
Jul 19, 2019, 05:35 AM
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Ah Yes, this is exactly what I was looking for. I have the RT Jones paper, strangely missing the exact diagram I needed! This is basically how I figured the downwash would look for this situation, but was unsure of the inboard portion.

Thanks!
Jul 19, 2019, 12:20 PM
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Originally Posted by ornitech
Ah Yes, this is exactly what I was looking for. I have the RT Jones paper, strangely missing the exact diagram I needed! This is basically how I figured the downwash would look for this situation, but was unsure of the inboard portion.

Thanks!
Yeah, the lesson from Munk and Jones is go with uniform downwash everywhere if you can. If there's some kind of non-aerodynamic constraint, use a linear variation in the downwash.

Where it gets really interesting is when you start to apply the same principles to nonplanar wings. When you have more than one surface, invariably you're looking at trying to design the optimum shape for one (or more) surface in the presence of other surfaces whose design is fixed. I thought the mixed analysis-inverse problem would be difficult, but it turned out to be simple. In both the analysis problem (what will a specified geometry do) and the inverse problem (what should the shape be for specified aerodynamics) you are solving for the spanwise distribution of circulation. Because the unknowns are the same, you can simply mix and match rows from the matrices and right-hand-sides of both problems.

Here's an example where I applied this principle to a nonplanar planform. It is the horizontal stabilizer hydrofoil on the rudder of a 72 ft sailing catamaran. The Design Rule placed a constraint on the span outboard of the rudder, but a larger span was allowed inboard. There was a structural limit to how much moment could be generated at the junction with the rudder shaft. The rudder shaft was a fixed surface that I couldn't change, and even though I was mainly concerned with vertical lift on the stabilizer, the rudder shaft had an influence on the stabilizer due to the asymmetrical lift distribution and planform. I used a linear downwash distribution across the span, with zero wake wash along the vertical winglet. The slope of the wash distribution was tuned to put the center of effort at the distance from the rudder that matched the structural constraint. The maximum downwash was at the outboard end and a winglet there was worth the extra wetted area. I have a version of the spreadsheet in which I've expanded it to handle five surfaces, so I used one surface each for the inboard and outboard stabilizer panels, one for the winglet and one for the rudder. The planform shape would have had a discontinuity at the rudder junction because of the discontinuity in the spanwise lift distribution from rudder interference, but we smoothed through that. It was definitely not your typical elliptical planform!


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