Why has the bell distribution not become widely used? Is the bell distribution only ideal in inviscid flow scenarios or is there another reason for the lack of use?
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1Are you talking about Figure 2 on page 4 of this NASA document? https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20160003578.pdf – CrossRoads Apr 21 '19 at 15:05
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Yes and also shown in Figure 1 (a -1933) in the same publication. – user38891 Apr 21 '19 at 15:09
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Something like it is widely used, only not by this name.
Although less elegant and general than Prandtl's paper, the same solution was found 17 years later by R.T. Jones when he optimized the ratio of lift to mass of wings in NACA TN 2249. The figure below is from the concluding page of this report.
However, induced drag is not all. By increasing span, wing surface area is also increased which will increase viscous drag. Therefore, the optimum really is a combination of both; having the least amount of wetted surface for the most net lift (total lift minus wing weight). This is somewhere between an elliptical and a triangular load distribution where size also matters. Small aircraft are closer to the elliptical distribution whereas large aircraft tend more to the triangular one. Negative lift at the tips only happens with extreme span to elliptical span ratios (note the curve for S/S$_\text{e}$ = 1.30). Negative lift as in the bell-shaped lift distribution, however, is a prerequisite for the Horten type of flying wings because it allows to create proverse yaw. If you have a multi-engined aircraft and need to size the vertical surface for the engine-out case, covering adverse yaw will be trivial and there is no need for a true bell-shaped lift distribution.
Peter Kämpf
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@ROTOR Better think before you type. Look at it at different angle of attacks. Especially, study the Horten bell distribution. Who is wrong now? – Peter Kämpf May 02 '20 at 05:38
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you are wrong.when flying in design- range,wingtips allways produce small lift..listen at 20:58-22:00 https://www.youtube.com/watch?v=RoT2upDbdUg – ROTOR May 02 '20 at 08:58
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@PeterKämpf, the CFD studies by J Murillo, and previous paper by Al Bowers seems to suggest the lift force reduces to zero at the tips over a wide range of angles of attack (granted the CFD study is interested in stall analysis so shows detail from AoA of 0 degrees to around 20), Figure 20 in the flight validation paper by Bowers et al shows negative loading only at an elevator deflection of -4 (it's a flying wing so those are elevons). I can send links to these papers if that is helpful? What am I missing? – Alastair Wyllie Jan 17 '24 at 12:37
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@AlastairWyllie The lift force will only be zero over a wide angle of attacks if either you measure directly at the tip or if chord is zero. Near the tip the angle of attack will of course change the lift; less than in the center, but still markedly. Or all aerodynamics so far has been wrong. If you want the lowest bending moment, you need to design for the max alpha case. At lower AoA lift will go down over the whole span, and if it was near zero at max alpha, it will become negative at practical AoAs. – Peter Kämpf Jan 17 '24 at 12:51
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@PeterKämpf, that makes sense, that explains fig. 20 in this paper: https://ntrs.nasa.gov/citations/20210014683 this shows elevator deflections at a cl of 1.16 using vortex lattice. given that this paper: https://ntrs.nasa.gov/citations/20180006832 predicts stalling at around cl max near 1, that makes sense I guess. On page 7 of the first paper I think that suggests the design Cls for the Prandtl-D1&2 are about 0.6. Thank you! – Alastair Wyllie Jan 17 '24 at 17:58
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@AlastairWyllie Thank you for the links! Prandtl himself in his paper used the findings to explain the much higher taper ratio of contemporary designs (which do not fit well with an elliptical circulation). This makes a sentence like the creation of aircraft using the Prandtl bell spanload will require considerable new techniques rather odd and sensational. This devalues the whole paper - after all, what it tells us has been common knowledge for almost a century by now. – Peter Kämpf Jan 17 '24 at 20:14
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i think the short answer to your first question is that there aren't very many aircraft without vertical surface. Only a few body-style such as flying wing come to mind, i have a friend who designed one recently.
For your second question no unless you're approaching the speed of sound, "inviscid flow" simplifies equations by omitting complexity with negligible effect where the flow of fluids with low values of viscosity can be assumed. i would say in practice you would have to describe a very specific scope within which your function would be considered ideal.
Jason K.
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