(a) although it will vary a bit with Type, I think you will find that 5 deg bank to the operating engine(s) results in some slip to the low side.
Well I did write "say 5 degrees". So if it's 3 degrees, alter my numbers to 18 degrees into the live and 12 degrees into the dead. The point stands.
Back around Vmca you will have run out of rudder ? .. otherwise the Vmca generally could be lower ?
OK, so to find the
absolute Vmca we maintain the no-slip condition with appropriate bank and reduce speed until I run out of rudder. Now I've got a no slip condition with max rudder opposing the asymmetric thrust -- let's call that
no-slip Vmca. But wait -- I can reduce speed still further if I'm prepared to slip towards the live engine. Directional stability will provide yawing moment towards the live engine and help out the rudder, which is maxed out towards the live side.
What eventually stops us from decreasing speed still further? I'm not sure: could be running out of aileron, could be mainplane stall as we demand more and more lift, or it could be a fin stall. But something will. So I find
absolute Vmca.
So far so good: we're still going straight.
(b) re the turns .. it will depend strongly on what the actual slip is doing as to what happens with control (keep in mind that we are concerned here with turns at/near Vmca .. somewhat faster and the problems are far less a concern).
Agreed. My previous comments applied to no-slip Vmca, which was a kind of offset equilibrium condition and I haven't seen anything that leads me to believe that they were incorrect. So let's think about absolute Vmca... (rudder maxed out to live, banked into live engine, slipping into live engine).
Regardless of the origin of the slip, if you start slipping toward the dead engine you will have a turning moment increase toward the dead engine .. and need more rudder to control it .. consider it a bit like crosswind generated turning moments on takeoff or landing ? There will also need to be some degree of rudder delta for the turn in any case.
OK, so let's try to turn towards the dead engine and reduce the bank into the live engine. I won't do anything with the rudder because it was maxed out 10 knots ago. I've still got my live foot flat on the floor.
So we develop a yaw rate into the dead engine. I can't check that yaw with rudder, but I can check it with bank angle. I've moved further from mainplane stall (less bank), further from fin stall (less slip). I'm not certain what would happen to the aileron input -- we've reduced the slip so we're not having to fight the lateral stability, but we are yawing towards the dead so we may need to hold that off a little.
What about a turn toward the live engine? So I bank still further into the live engine. I'm now closer to mainplane stall because there's still more bank. I'm also closer to fin stall as I'm demanding more yaw from the fin. And of course if I was limited by running out of aileron, I
can't bank towards the live engine.
So I still can't see why a turn towards the dead engine is more problematic.
(d) main problem if (real world) "Vmca" runs away is that things can happen fairly quickly and the pilot's options can disappear while he/she watches. The action might be intuitive .. eg apply a roll moment input .. but the consequence of the action might not align well with the pilot's expectation.
Noted, but I'm just trying to work out why the unexpected results are more likely with a turn towards the dead.
(e) yaw and slip are much misunderstood. One way to consider it is that yaw is something which a pilot generally does or creates by control inputs while slip is the aerodynamic airflow consequence of the pilot input ?
I agree. In the context of spin entry, I was think of
yaw as yaw rate (regardless of whether it's pilot induced or not.