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Looking at lift on a spinning cylinder, the upwash and downwash are identical, so surely there cannot be any net downwash?
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If the cylinder is spinning in still air, there is no lift. Lift is only developed if there is relative airflow over the cylinder - and in which case there will be downwash at the "trailing edge" of the cylinder - check your Navier-Stokes notes...
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Also, one can turn the trailing edge of a wing up and still produce lift, albeit not very efficiently, but without downwash.
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Yes, but only in the two dimensional cross section of the wing that you are examining. There will also be an axial flow towards the wingtip (in the case of a non-infinite aspect ratio!) and at the wing-tip there will be vortices with very large downwash.
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So have I missed something? Is lift = downwash = integrated pressure around the whole body?
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Yes, this is always the case (ignoring the vertical component of engine thrust of course - which is far from negligible especially during take-off).
I'm aware that there are some books around, like Stick and Rudder, that say lift is entirely down to downwash and nothing to do with pressure distribution. I reckon this is wrong - but have I missed something?
The only thing that you are missing is the vertical component of engine thrust, but otherwise you are correct in straight and level flight, that the aerodynamic component of lift is equal to aerodynamic downwash (conservation of momentum) and that the weight of the aircraft is equal to the net vertical force created by the pressure distribution around the entire airframe