LIFT COEFFICIENT

Variation of Lift at least depends on

1.       Free-stream velocity (V∞)

2.       Free stream density(r)

3.       Size of an aerodynamic surface. We will use the wing area S to indicate the size

4.       Angle of attack

5.       The shape of the airfoil

6.       Viscosity coefficient µ_∞(because the aerodynamic forces are generated in part from skin friction distribution)

7.       Compressibility of the airflow, compressibility effects are governed by the value of the free-stream number M_∞=V_∞/a_∞ , hence L, D&M depend on the speed of flow also the speed of sound

....1

 Using dimensional analysis

...2

...3

Consider mass m. The exponent of m on the left side is 1. Thus, the exponents of m on the right must add to 1, hence  

...4

Similarly, for time t

...5

And for length l,

...6

Solving equation 4, 5&6 for a,b and d in terms of e and f

...7

...8

...9

Substituting equation 7 and 9

Rearranging equation 10

We know that

 Equation 11 becomes

We will now define a new quantity called as coefficient c_l as

Dynamic pressure is

..15

Equation number 15 is a very important the result, as it tells us  that lift, is directly proportional to dynamic pressure, hence the square root of velocity, it is also directly proportional to wing area and to the lift coefficient  

The lift coefficient is a function of M_∞, Re as reflected in equation 12

Since, M_∞, Re is dimensionless and we assumed initially Z as dimensionless from equation 12 c_l is dimensionless.