Lifting Line Theory

14.1.2.1 Flow Model

Explain, in simple terms, how an inviscid flow model, Prandtl Lifting Line theory, can account for a drag in the flow past a finite wing. Use arguments such as Joukowski Lift theorem with a sketch, or energy equation.

Give the expressions for the lift coefficient CL and induced drag coefficient CDi of an arbitrary wing, in terms of the Fourier coefficients A1, A2,…, An. The area of reference, Aref = S is the wing area.

14.1.2.2 Non Singular Lift Distribution

To lower the risks of tip vortex encounter in airport traffic associated with intense vorticity( dy D = ±сю), the new generation of airplanes is required to have weak tip vortices. This is achieved by enforcing a Г distribution such that dT (±|) = 0.

Подпись: Г [y (t)] Подпись: = 2Ub (A1 sin t + A3 sin 3t) y (t) = ~b cos t Подпись: 0 < t < n

For a given CL = 0, a distribution that satisfies this condition and has a low induced drag is a combination of modes 1 and 3:

Find the relation between A1 and A3 that satisfies dL (-1) = 0 (by symmetry, the other condition will be satisfied).

Make a graph of the corresponding distribution Г(y) or T(t).

14.1.2.3 Drag Penalty

For a given CL (at take-off), find the penalty in induced drag for a given aspect ratio AR compared to the elliptic loading. Work the answer formally (do not plug in numbers).

Find the efficiency factor e — (1 + J)-1.

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