The Figures:
The first Figure, Figure 1, Cl v AOA, is a map of Cl versus AOA with a variation of the flap angle of a 25% chord flap. Each curve is an AOA sweep of Cl for a 5 degree incremental deflection of the flaps. The lowest curve is of an airfoil from 0 to some range near stall. Each dot on the curve is 1 degree change in AOA. Following a singular curve, such as the lowest curve would be representative of a non-flapped airplane. The lift would work as an increase in AOA along that curve. We’ll see that the idea of the 1-cos(x) gain works equally well for this configuration.
I’ve arbitrarily placed two lines on the curve chart to illustrate approximately how a coupled flap airplane would generate the lift as function of the combined effect of flap deflection and pitch angle. You could think of the lines as the ratio of elevator authority to flap pitch moment, kinda, sorta. The reason being is there is some velocity dependency as well. From testing I know that the AOA increases at times to around 15 degrees with flaps deployed.
Each line illustrates how the lift increases with control input the line being where control would be with respect to the lift curve which I’ll use the term Operational line. This is kind of how we control inlet guide vanes on turbine engine compressors so the term fits. The control angle, I make an assumption of 1:1 control to flap, from the origin to the crossing at the highest curve would be 45 degrees. As we trace each of the Op lines every curve crossing is an increase of 5 degrees in control position.
Figure 1, Cl v AOA - Lift
Depending on the velocity and the elevator ratio the AOA would be somewhere between zero and allot wherever trim would be. I labeled one line Op line at constant velocity and the other Op line at decaying velocity. This is for illustration purposes and if there were a gain function based on control position the thrust would be increasing to close the difference. This condition wouldn’t likely be static. Whether the condition is or not is dependent upon the difference between the power available, that which is being commanded, and the power required to fly.
The next Figure and the following are of the Lift verses Drag or Cl v Cd. In the same manor I placed multiple curves on each chart for each of the flap positions and AOA sweeps. The interesting result is the kind of fractal growth appearance of the curves. The usefulness of these curves is that the thrust / power requirement can be derived from them. Both power and thrust are a direct function of the coefficient of drag Cd. On the first chart I have done some calculating of the required lift coefficients and from that we can get an idea of the drag coefficient which would result from a combination of flap position and AOA. Notice that for most of these there are more than one possible solution. I’ve made these same charts for many flap configurations from 15% flap to 40% flap. When I’m heard making some seemly brash statement, it is the result of this kind of analysis.
Figure 2 Cl v Cd - Drag
A first pass walk through the analysis of power required to fly is the following. We begin with the first chart and move along the Op line for constant velocity. As we move upward along that line we cross the flap =0 degree line Cl curve and an AOA and Cl value. That Cl value is then translated to Figure 2. I have plotted an intersection for level flight I could have equally used an observed AOA, which I did, and made a point on the corresponding line. I was going to present that on a separate chart but it is so close to the blue line 1-Cos line on Figure 3 it is just as easy to say that is what it looks like.
The next interesting condition is the level flight at 45 degrees condition. This is an interesting analysis and one that absolutely must be done in order to draw the conclusion I have regarding the need to know and use attitude and G in the thrust control algorithm. Like indicated before, I don’t think so provided not too much energy is lost in the corner and after the derivation here, you’ll see that it is very likely to be able to do exactly that. The limiting condition is how steady do we really want airspeed to remain constant as I think that is entirely achievable. Suffice it to say the G force, lift vector, required to fly tethered at 45 degrees constant speed for a given bank is equal to the G force of the tension in level flight. The weight of the airplane reduces the tension of the line. I’ll derive that later but it’s confirmable in the video and you can see on the captured photo above.
Moving up the scale to 6 G loops we cross the calculated Cl of about 0.98 and can see there is an intersection with the 10 degree flap deflection curve around 4 ish degrees AOA. Looking at the AOA video this seems to be nearly exactly the condition. It repeats again for the 10 G square corners.
Figure 3 Cl v Cd - Drag - Annotated