I wish the tone in here didn't have to get nasty, though I understand (and experience myself) the frustration when the other side doesn't see things one's own way. Hopefully it can return to something more civil, where we can all simply address the other side's arguments.
I will reiterate the following: There is no preferred frame of reference, and as long as you do all the math right, you should be able to calculate any physics in any frame. It's just that lots of variables drop out and the calculations get a whole lot easier in some frames, namely the airmass frame when calculating things regarding aircraft. The danger comes when you mix frames within the same calculation without doing the necessary transformations.
And
Brercrow, I'm afraid that's just what you've done. In the Downwind Turn page Figure 1b, you resolve a component Ft of force Fc (itself the horizontal component of lift, perpendicular to the fuselage) parallel to the ground track. You are correct that in the ground frame, that component exists and accelerates the ground velocity. However, my first hint of your trouble is that you don't likewise complete the resolution and take a component of Fc perpendicular to the ground track.
You have: Ft = Fc sin d (this is the force that accelerates the ground velocity forward when turning downwind)
You're missing: Fm (m stands for missing) = Fc cos d. (This is the weakening of the centripetal force, which accounts for the widening radius when turning downwind)
That would be the component of the horizontal component of lift that is centripetal to the curve, in the ground frame;
not Fc. I'm not sure what the blue triangle is supposed to represent, but I am sure that it inappropriately mixes frames, as it connects Fc (which exists only in the airmass frame) and Ft (which exists only in the ground frame).
You are right Fc gives the curvature of the air path and Fc * cos d gives the curvature of the ground track. But the ground track is a throw-away item because we are not interested in the ground track. What we are discussing is the effect on airspeed.
Forces exist in all frames of reference and for an aircraft the effect of tangential and centripetal forces are different in each frame. The forces described are acting on the aircraft not on the ground. Hence a component Ft = Fc .sin d causes a tangential acceleration of the ground -velocity. That then causes a rate of change of component K which is almost exactly balanced by the opposite rate of change of component H. The difference between the two gives the slight acceleration of airspeed.
Didn't you read the whole thing? I suspect not!
I've actually been revisiting your site for a few weeks, and this is the only hole I can poke in your math so far. The rest of the things you go on to do with your Ft seem to be valid. So why haven't I changed my mind? Maybe a small part of being an obstinate ass given to the natural human inclination to hang on to an opinion, but I would guess a larger part of lending more credibility to there being an undiscovered hole in your math, than that Newtonian relativity is wrong.
My work complies with Newtonian physics but obviously there are several levels of complexity in this problem.
I do have a main suspicion, which is that instead of an analytical solution you have a numerical one based on an Excel spreadsheet, and that your resulting change in airspeed is an artifact of accumulations of rounding errors in the increments. I would please like to ask you to do 2 things:
1. I assume the resolution of your spreadsheet is 1 degree, therefore 360 increments. Can you rerun it with more increments, say half a degree and a quarter of a degree? If the airspeed change trends to zero with an increasing resolution, there's our problem.
I have tried different resolutions. The end result is the same but the curves are smoother with fine resolution.
2. With apologies to the huge number of counterexamples thrown at you in the thread, I'd like you to reread my post #133?
https://www.pprune.org/tech-log/607454-windward-turn-theory.html#post10204374
and plug the constants of the second example into your spreadsheet, and see what answer it gives for an airspeed change; and decide if you've ever felt such a tendency while turning around in an airliner.