How Aeroplanes Fly: Deriving Equation For Lift Force

[billy92]

I am currently looking into how aeroplanes fly for a report which i am writing.

I understand the theory behind the lift force produced by different pressures above and below the wing. However, i need to use equations which i can show how to derive in the report.

I have found the following equation $\frac{1}{2}ρv^{2}AC_{L}$ but i am not sure how to derive this equation.

Thanks

 

[DeadPhoenix]

If you understand the mechanism by which the aeroplanes fly, understanding this equation is not a very difficult task. First of all, you must realize that this equation only describes the proportional relationship between different factors affecting the lift.

As you pointed out, Lift is produced due to pressure difference on the upper and lower surface of the wing.
Lift (Force) = Aera X Pressure.
The term 1/2 ρ v^2 is the dynamic pressure, which can be thought of as the pressure (rather fall in pressure) due to movement of air over the wing (remember bernoolli's equation: P + 1/2 ρ v^2 = constant). Then you have the area (S) of the wing.

The last term, Cl is basically the proportionality constant which depends only on the shape and the cross-section (airfoil) of the wing. It basically depends on the way the wing redirects the flow to produce lift. So you will have the same Cl for a particular aircraft flying at different speeds or even if the aircraft is scaled down.

 

[billy92]

Thanks for the reply. I am fine with most of the theory behind it its just the derivations and formulas that I am a little stuck on.

Im still not entirely sure how to derive the equation i stated. Also is it this equaion which best describes lift as I've found others which also explain lift in slightly different ways.

 

[rcgldr]

I have found the following equation $\frac{1}{2}ρv^{2}AC_{L}$ but i am not sure how to derive this equation.

It's not derived, it's a rearrangement of the equation used to define C_L

$C_L = \displaystyle \frac {L} {\frac{1}{2}ρv^{2}A}$

where L is lift force.

In the real world C_L is a complex function of angle of attack, velocity, air parameters and wing parameters. Normally velocity, air parameters and the chord length of a wing are combined to create a term called Reynolds number:

http://en.wikipedia.org/wiki/Reynolds_number

2d charts called "polars" for airfoils will show one or more curves for C_L versus angle of attack, C_d (coefficient of drag) versus angle of attack, and/or C_L versus C_d, with separate curves based on Reynolds Number, like RE = 10^5, RE = 10^6, ...

http://www.davincitechnologies.com/AOTour5.htm

Programs used to generate these polar charts use some simplified version of Navier Stokes equations. XFOIL is a freeware version of this type of program:

http://web.mit.edu/drela/Public/web/xfoil

 

[billy92]

Thanks for the reply. Makes a bit more sense. Is the equation for the coefficient of lift derived from anything?

Also is there any other equations for lift which i should be looking in to?