Turn Rate/Radius Management
This is an academic discussion. I'm not advocating for any specific maneuver in the event of power loss during takeoff phase other than to maintain a safe angle of attack all thee way to the landing site, wherever that may be.
When we look at aircraft turn performance, we are concerned with turn rate (degrees per second) and turn radius (feet). Only two factors effect turn performance: true airspeed and radial G. Maximum instantaneous turn performance occurs at the aerodynamic limit of the airplane and maximum sustained turn rate occurs ONSPEED (a zero Ps condition). In the technical resources in the previous post, Dr. Rogers develops the thesis to support the use of a 45 degree bank at Vs + 5 for optimum turn performance in the conduct of a turn back. Similar performance with a better aerodynamic margin can be achieved at an ONSPEED AOA condition for a very small penalty in turn performance.
One way to picture that is to use a rate/radius diagram which is a "snapshot in time" of aircraft turn performance. Here is sea-level turn performance for my RV-4 equipped with a 160 HP engine:
In a horizontal turn, it is not possible to maintain energy if you exceed maximum sustained turn (the region above the solid red line in the chart), the airplane "bleeds" energy and the harder you pull, the slower you go. An ONSPEED AOA cue allows the pilot to precisely fly the solid red line. At lower speeds, note that the difference between maximum instantaneous turn performance and maximum sustained turn performance is negligible.
The turn back maneuver is performed in an orthogonal plane, i.e., you have vertical turning room available, and with the engine out, your thrust is being generated by gravity, so the chart isn't 100% applicable; but the point is that optimum turn, glide and approach and landing AOA are all coincident with ONSPEED alpha. This provides more aerodynamic margin than slowing to Vs + 5 for the turn for a very small penalty in turn performance. Here are some tests at altitude using an ONSPEED turn and a "buffet turn" (i.e., right at the aerodynamic limit of the airplane):
https://youtu.be/ZQrOskngSlQ
Here are a couple of data plots of an actual low-altitude turn back test to a full-stop landing. The first plot shows pressure altitude and actual angle of attack (measured relative to the fuselage reference line in the RV-4):
The second plot substitutes KIAS for AOA:
The important thing to note is that the AOA is consistent throughout the maneuver. It is necessary to "unload" the airplane when turning (i.e., adjust pitch to maintain AOA). This maneuver is more difficult to accomplish without an accurate AOA/energy cue.
The closer to the end of the runway you are, the more turning you have to do. In Dr. Roger's papers, the airplane is a sufficient distance from the runway to allow a "teardrop" pattern to be flown. If you are closer to the runway, more turn will be required--much closer to 250-260 degrees with an additional 60-70 degrees required to align with the runway. You can see this in the video:
https://youtu.be/jI_o41SkKOw. The wind was 4-5 knots, 080-090 during these tests, so there was no appreciable cross-wind to turn into to reduce turn radius. Turns are made in both directions.
My RV-4 is equipped with a fixed pitch propeller and the test depicted in the charts is performed at IDLE, residual thrust is not measured. A nominal 45-50 bank is used for turn back and re-alignment. Flaps 20 is selected after a 3" delay after power is reduced to idle and before commencing maneuvering. This flap setting improves turn performance and provides more lift than drag; but will reduce straight line gliding distance. An ONSPEED condition provides maximum ENDURANCE glide, flaps up. As flaps are deployed ONSPEED and L/Dmax
airspeeds begin to marry up.