How can coefficient of lift on the ground with flaps in the takeoff position be calculated using airfoil data?

Question

In essence I'm working on a design project and I'm specifically looking at takeoff analysis at the moment. I already have access to great textbook resources, but I'm struggling to determine the coefficient of lift $(C_{L_G})$ on the ground with flaps in take-off position for the following equation taken from Appendix K-4 of Synthesis of Subsonic Airplane Design by Torenbeek:

$$\frac{a}{g} = \frac{T}{W} - \mu - \left(C_{D_G} - \mu C_{L_G}\right) \frac{\frac{1}{2}\rho V^2 S}{W}$$

Similar equations can be found in other textbooks, but I haven't found any reference to how $C_{L_G}$ can be determined from airfoil data. Currently I'm considering just using the zero geometric angle (ie compensating for the AoA the wing is mounted at) coefficient of lift for the airfoil, but how would I then account for the position of the flaps?

Answer 1

There is quite a lot written on the topic, especially as part of "ground effect" vehicles research. For example, Cui and Zhang cite two different types of lift coefficient. The first one (simpler and given here) is based on flat plate airfoils and is a modification to the well-known $\frac{dC_l}{d\alpha}=2\pi$:

$$C_L=2\pi\alpha(1+\delta^2)(1-2\zeta)$$

with

$$\zeta=\frac{\sin\alpha}{4(h/c)}$$

and

$$\delta=\frac{\cos\alpha}{4(h/c)}$$

In the above expressions, $h$ is the height of the wing above the surface (ground, water, etc.), $c$ is the chord and $\alpha$ is the angle of attack.

I strongly recommend you check out "ground effect"/"Ekranoplan" literature - should be useful.

Answer 2

The question is asking a principal analysis of aircraft aerodynamics. The aircraft is a 3-dimensional object, and 2D airfoil calculations or data ($C_l$) can be used to approximately define the 3D coefficients ($C_L$).

The main methodology is the wing-strip approach. This is simple to comprehend and simple to put into calculations (excel, code, etc).

Basically the wing is divided span wise into several sections. Each section is assumed to have a constant chord, so the 2d data is quite useful to define the force at this section. After combining all forces, one can conclude the total force and then conclude the calculations by non-dimensionalizing the total force.

In helicopter rotors the same methodology is named as blade element model.

There's also literature on approximating the flap effects using empirical formulas, if you do not wish to analyze different foil geometries using CFD. Refer to ESDU, DATCOM, or other references, or Hoerner's "lift" book, for these empirical approaches.