How Aeroplanes Fly: Deriving Equation For Lift Force

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In summary, the conversation discusses the equation \frac{1}{2}ρv^{2}AC_{L} and its derivation for the report on how aeroplanes fly. The equation is a rearrangement of the equation used to define the coefficient of lift, which is a complex function of various factors such as angle of attack and Reynolds number. Other equations and charts, such as polar charts, can also be used to explain lift.
  • #1
billy92
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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 [itex]\frac{1}{2}ρv^{2}AC_{L}[/itex] but i am not sure how to derive this equation.

Thanks
 
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  • #2
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.
 
  • #3
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.
 
  • #4
billy92 said:
I have found the following equation [itex]\frac{1}{2}ρv^{2}AC_{L}[/itex] but i am not sure how to derive this equation.
It's not derived, it's a rearrangement of the equation used to define CL

[itex]C_L = \displaystyle \frac {L} {\frac{1}{2}ρv^{2}A}[/itex]

where L is lift force.

In the real world CL 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 CL versus angle of attack, Cd (coefficient of drag) versus angle of attack, and/or CL versus Cd, 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
 
  • #5
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?
 

1. How do aeroplanes generate lift?

Aeroplanes generate lift through the Bernoulli's principle, which states that as the speed of a fluid increases, its pressure decreases. This is achieved by the shape of the wing, known as an airfoil, which is designed to create a pressure difference between the upper and lower surfaces of the wing.

2. What is the equation for lift force?

The equation for lift force is L = ½ρv²SCL, where L is the lift force, ρ is the density of the air, v is the velocity of the air, S is the surface area of the wing, and CL is the lift coefficient.

3. How does airspeed affect lift force?

Airspeed is directly related to lift force, as shown in the lift force equation. As airspeed increases, the lift force also increases. This is because a higher airspeed results in a lower pressure on the upper surface of the wing, creating a greater pressure difference and thus a greater lift force.

4. Can you explain the role of the angle of attack in lift force?

The angle of attack is the angle between the wing and the direction of the airflow. As the angle of attack increases, the lift force also increases, up to a certain point. This is because a higher angle of attack results in a greater pressure difference between the upper and lower surfaces of the wing, but if the angle becomes too high, the airflow can become separated from the wing and reduce lift force.

5. How does the shape and size of the wing affect lift force?

The shape and size of the wing play a crucial role in generating lift force. A curved, or cambered, wing shape is most efficient at creating lift, as it creates a larger pressure difference. Additionally, a larger wing surface area allows for a greater amount of air to flow over the wing, resulting in a higher lift force.

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