What is the most simple plane lift equation that gives realistic solutions?

Question

I need the most simple lift equation that once solved with Mathcad gives a realistic vertical speed of a plane and implicitly the altitude h(t).

Also the drag eq. (1) leads to a good solution, a Vh(t) that increases and finally reaches a limit (the maximum horizontal speed) and stays there during the duration of the flight, the lift eq. (2) stabilizes at a Lift(t) - m * g = ct. > 0 and in consequence Vh(t) keeps growing indefinitely because an eq. of the type m * dVv(t)/dt = ct. leads to a solution Vv(t) that rises linearly with time.

Question: It is quite clear that the vertical speed of a plane, Vh(t), can not grow indefinitely. How can I stabilize it to o constant value. What do I have to add in the lift eq.?

Drag eq.: m * dVh(t)/dt = T - Drag(t) (1),

Drag(t) = 0.5 * Cd * r * S * ( Vh(t) + Vw(t) )^2,

Lift eq.: m * dVv(t)/dt = Lift(t) - m * g (2),

Lift(t) = 0.5 * Cl * r * S * ( Vh(t) + Vw(t) )^2,

where:

• Vh(t) = horizontal speed, Vv(t) = vertical speed, both of them have to be determined being unknown functions.
• known parameters: m = the mass of the plane, r = air density, S = the wing surface, T = thrust = ct., Cd, Cl are the drag and lift coefficients, g = 9.81 m/s^2, Vw(t) = the wind speed, that is usually a known constant but can have other forms given as functions of time, t.

Answer 1

First, you need to add the effect of density $\rho$: Air density changes with altitude, and this affects both lift and drag (at least if you model airbreathing engines).

For lift $L$, use $$L = c_L\cdot\rho\cdot\frac{v_h^2+v_v^2}{2}\cdot S$$

For drag $D$, you should only simplify the equation as much as possible. Your equation is simpler still and doesn't model the increase in drag with more lift. The simplest practical equation will look like this: $$D = \left(c_{D0} + \frac{c_L^2}{\pi\cdot AR\cdot\epsilon}\right)\cdot\rho\cdot\frac{v_h^2+v_v^2}{2}\cdot S$$

More lift will need more thrust and will limit the upward acceleration. Climbing higher will reduce thrust in proportion with density and limit the possible speed range.

Nomenclature:
$c_L \:\:\:$ lift coefficient (normally between 0 and 1.5)
$\pi \:\:\:\:\:$ 3.14159$\dots$
$AR \:\:$ aspect ratio of the wing (ratio of span to mean chord)
$\epsilon \:\:\:\:\:$ the wing's Oswald factor (use 0.8 when in doubt)
$c_{D0} \:$ zero-lift drag coefficient (use 0.02 when in doubt)