How would I calculate CL and CD for a VTOL aircraft while it is in it's takeoff/landing stage?
The particular aircraft is a fixed wing/quadcopter hybrid so I know that I can treat this as a thin rectangular plate problem. However, because the aircraft takes off vertically, I do not understand what formulas to use to calculate these coefficients.
If I use CL=2$\pi$$\alpha$, where $\alpha$=$\pi$/2, then I end up with CL=9.87, which just doesn't make sense.
At $\alpha$=$\pi$/2 the flat plate is perpendicular to the air stream.
CL is defined as the force perpendicular to airstream, CD as the force in direction of the air stream. When the craft takes off vertically, its CD is high and its CL is zero.
A graph and a most excellent explanation here
"Lift" (vertical takeoff) would be the aircraft weight + drag at the desired rate of ascent. The aircraft will accelerate upwards until thrust (vertically) = weight + drag.
Drag coefficient can be modelled from flat plate or other appropriate flat shape (top view of the aircraft would be best), but keep in mind the ascent rate will be relatively slow and the majority of engine power will be used to overcome weight.
The power required curve, with an adequate safety margin for downdraft, would be at maximum for liftoff and ascent, reducing as aircraft begins to move forward and the wing begins to generate lift. This power/thrust reduction could be from over the weight of the aircraft at lift off to less than 25% of it in level flight.
Interestingly, all aircraft do this transitioning from climb to level flight as they reduce power and "level off", again using the more efficient wing for lift instead of their prop. VTOL aircraft simply do not use their wings at all for lift off.
Power/thrust requirements considered, a STOL may be a happy compromise.
Since you listed it as a quadcopter with a fixed wing, I'm going to assume (at least in the takeoff/landing stage) that the propulsors are perpendicular to your wing. The simple answer is, your total thrust is equal to weight; consider your lift to be close to zero and your lift coefficient is undefined.
Angle of attack only really makes sense when the forward relative airflow (that is, incoming flow more or less parallel to your wing) is greater than the vertical relative airflow. For reference, a one-to-one relative airflow is 45 deg AOA.
Furthermore, the lift slope of $2\pi$ is for an ideal thin airfoil section (2D); for a 3D wing, it will be less due to induced drag. Even for a section, it is only true for small AOAs (certainly less than 15deg). Above that, the lift slope is no longer constant. Close to 90deg AOA, the concept of airfoil no longer applies and you'd be looking at blunt-body aerodynamics.
