so regardless of the direction of the force on the blade by the wind (which will vary with the number of blades, airfoil, and TSR), the wind turbine is going to exert an axial force on the wind in accordance with its power output. and its going to be somewhat proportional, and rising with the square of the windspeed assuming the percentage of power generated is constant.
really is no way to get around it, the axial thrust on a prop is the same as a wind turbine. just a different direction.
Sorta roughly, yep. (Energy goes with the cube, momentum with the square, forces with the change in momentum, assuming the same efficiency approximates assuming the equivalent change in momentum. Not dead on because of aerodynamic effects that vary differently and these deviations amount to a small, but nontrivial, effect.)
Also regardless of what OTHER forces it applies to the wind. The retarding force (along the axis in a "horizontal" {"into the wind"} axis turbine) is how the turbine extracts momentum, and thus energy, from the wind. Other forces (e.g. sideways) may happen from details of how the particular turbine disposes of the slowed air and creates torque out of this momentum, (incidentally returning more of the energy to the departing air stream than the Betz limit demands).
A propeller-style HAWT deflects the air so it is rejected sideways and off-center (as viewed along the axis). So this generates a torque on the shaft, much like that generated by a lawn sprinkler's off-axis water jets. This also applies a counter-torque to the air, leaving a significant amount of energy (beyond what is required by Betz) in the downstream wind. ("Spinning the wind" as I like to say.)
The blades' deflection of the airstream imparts both a radial-outward component (which spreads out the downstream air, as Betz' analysis shows is required to "make room" for the slower air stream leaving the turbine, as its venturi-principle higher pressure requires in free space) and an around-the-axis circular component (which deposits the counter-torque, along with still more of the energy, in the departing air).
The "torquing" of the departing air represents a substantial amount of loss, much of the shortfall from Betz. So you'd like to keep it low. Shaft horsepower is proportional to RPM x torque, so to lower the torque you raise the RPM. Thus the higher the TSR, the less of your collected energy you throw back into the wind in order to generate shaft torque. (also: Higher RPM operation also allows lower costs and losses in the generator/alternator.)
Now raising TSR also raises your blades' (relative) windspeed, and thus your air friction and similar losses (tip vortices, etc.). You might think this is a balancing act and there'd be an optimum design TSR (for a given airfoil) for a fixed-pitch turbine. And indeed there would be. (You can see this in the graphs of Cp vs. TSR, with Cp rising with TSR but curving down to rise more slowly at higher TSR.) Except that it's high enough that another design factor comes into play first:
With a high enough TSR and a high wind, the airflow near the tip starts to go supersonic. As it approaches that speed (and starts to go beyond it in patches) you get terrible vibration and all sorts of problems - like tearing the mill apart and throwing some of the pieces around at a substantial fraction of sonic speed. So you'd like to keep it below that. And you'd like to keep it below that even when a gust arrives before your furling system can react. (Even if they didn't have this issue, spinning that fast would also require them to be very strong to keep the blades from tearing lose from the hub or breaking off, and flying away radially)
So homemade mills tend to be designed with TSRs as high as build strength and the risk of near-supersonic airflow allow. 6, for instance, still keeps you subsonic if you're freewheeling in a sustained 100 MPH wind and your structure will stand the RPM.
