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Is there an obvious, ideal lift distribution for human-powered aircraft?
Elliptic or Bell (unloaded tips) distribution?
For this who will tell the Bell distribution, I have a counter-question ready, why don't put wires to wingtips to reduce bending moment at root and use elliptic distribution?
So what is the purpose of any Bell distribution, if the wing is designed correctly?
The essential feature of a human-powered aircraft is to maximise aerodynamic efficiency, which is to say to minimise drag for a given amount of lift. This is what the bell curve achieves. The elliptical distribution minimises drag for a given span, which says nothing about the lift available. The bell curve also gives you a lighter structure for a given amount of lift. Yes you can brace an elliptical wing for reduced weight, but you can brace a bell wing for an even greater reduction.
Also, the ideal distribution will depend on things like how tightly you want to turn. The inner wing will tend to stall, the outer wing to drag backwards (adverse yaw). Jonathan Bowers at NASA has shown that achieving the bell distribution through significant washout can create negative tip drag and proverse yaw. UK pioneer JW Dunne published as much in 1913. Washout also greatly reduces the problems of tip stalling and Dunne's biplanes proved wholly unstallable.
No doubt computer simulations could help refine the exact optimisation of washout, airfoil variation and span loading for the flight regime envisaged, as they did for Bowers' PRANDTL-D research drones, but it is not going to be far from the classic bell distribution.
All modern HPAs (Human Powered Aircraft) use a bracing wire, so the theory which yields the bell distribution as the optimum cannot be applied directly since it applies to cantilever wings. The best you can do is to use a semi-bell for the part of the wing outboard of the bracing wire attachment point and use an almost constant lift load over span inboard of this point (still with its maximum at the wing root, though).
If you care about ground handling, roll speed and the maximum bank angle in low level flight, you might want to reduce span a bit in order to keep all three up. The last meter of wingspan in case of the ideal bell distribution doesn't save much drag anymore but leaving it out will make hangaring and maneuvering easier.
Note that you can save more induced drag by picking a lower height over ground than by optimizing wing shape. In the end, you need to consider what the HPA is meant to achieve.
what is main reason why this is very hard to achieve
Human powered flight generally? The most basic reason would be insufficient power-to-weight ratio.
In order to hold an airplaine aloft, you have to accelerate air downwards. For any given total weight of flying vehicle, the product of added vertical speed and volume of air per second which is given this acceleration kick is something you can not change. Only choice is either accelerating small amount of air to high speed or big amount to lower speed.
But energy/power you need to put into accelerating this airmass increases with square of added velocity (while provided lift depends on velocity only linearly). So it is always much more efficient to be pushing down big volume of air by only little (compare to induced drag of airfoil).
Theoretically, this energy necessary for staying aloft could be arbitrary low, jast take a sufficiently big amount of air and push it downwards only a little.
Well, not exactly. Important question is where are you getting the air to accelerate from. And here starts all practical problems.
You can either increase wingspan, so aircraft ploughs through more air at same airspeed, or you can increase airspeed. First comes with penalty in mass (so you need even more lift and many related issues), second results in increase in "common" parasitic drag -- power necessary increases with third power of speed -- so there is relatively hard celing (you can not expect being able to pedal big airplane much faster than your common bike regardless of any lift being generated or not, can you?).
