Example: Prandtl Lifting Line calculation

A rectangular wing has an aspect ratio of 10 (AR = 10), and does not have any geometric or aerodynamic twist (α, αL=0 are constant on the span). The airfoil profile used on the wing is symmetric, αL=0 = 0. The wing has a constant angle of attack along its length of 10o. Use a Prandtl lifting line analysis with 5 stations to find: CL, CDi, αi at mid span.

The stations are selected as to be evenly spaced along the wing. The axis along the span of the wing is chose to be the y axis. The total span of the wing is b. Stations are located as fractions of b as indicated in the figure above. Since the Prandtl Lifting Line approach uses a Fourier transform with a transformation function,

2 cos

where the value of θo can range from 0 to π, the location of each station must be written in the same form. As an example, the location of station 2 at y = -b/6 in terms of θo can be found using,

6 2 cos

giving

1.23

Values of θo for the other stations are found in the same manner. The Fourier lift distribution function used in a general wing analysis is,

Γ 2 sin

The Prandtl Lifting Line (PLL) equation is,

Y=‐b/2          y=‐b/3        y=‐b/6           y = 0          y=b/6       y=b/3         y=b/2 

     θ=0         θ=0.841       θ=1.23      θ=1.57        θ=1.91      θ=2.30       θ=3.14 

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2sin

sinsin

At station 1, the PLL equation is,

2sin sin 2 sin 5

2sin 2sin 3

sin 3sin 5

sin 5sin

0.17452

10 0.745 0.994 0.579 0.2205 0.8742 1.33 3 0.781 4 0.2975 5 1.173

Similar equations can be created at the other stations. This will lead to a system of 5 linear equations in 5 unknowns (An). The system can be written as,

5.745 9 6.02 2.6 11.47 5.35 4.97 10.4 1.55

7.36 0.0133 9.36 0.033 11.367 5.32 5.01 10.36 1.457

5.75 9 6.01 2.61 11.44

0.17450.17450.17450.17450.1745

Solved, the solution is, A1 = 0.0275, A2 = 0.0, A3 = 0.003508, A4 = 0.0, A5 = 0.000412.

The Lift coefficient of the wing is then, CL = A1 π AR = 0.0275(π)(10) = 0.8643

For an infinite wing without induced effects at the same angle of attack, CL = 2πα = 1.096. The lift coefficient of the finite wing should always be less than or equal to the CL of an infinite wing. The two values will become equal as the aspect ratio of the finite wing goes to infinite.

Induced drag coefficient,

1

10 0.0275 1 30.003508

0.0275 50.000412

0.0275 0.0249

Induced angle of attack at mid span (θo = 1.57),

1.57sinsin

sin 1.57sin 1.57 2

sin 2 1.57sin 1.57 0.01905 1.09