Lecture+7

1Introduction to Aerodynamics Lecture # 7: Finite Wings (II)

Pejman Akbari, Ph.D.Spring 2015, Columbia University

2Lecture # 7: Finite Wings (II)

The induced drag coefficient increasesrapidly as CL increases, e.g., when theairplane is flying slowly such as ontakeoff or landing.

For the Elliptical Circulation Distribution


Geometric twist: the wing has the samesection shape from root to tip at everyspanwise station, but the wing is physicallydeformed by twisting the tip relative to theroot . This has the effect of changing thegeometric angle of attack along the span sothat = (y).Aerodynamic twist: the geometric angle ofattack for each wing section is the sameacross the span, but the wing sectionschange. Aerodynamic twist has the effectof making L0 take on a different value ata different spanwise station of interest;that is, L0 = L0 (y)

6Lecture # 7: Finite Wings (II)Induced Drag for an Arbitrary Finite Wing

Read pages 435-437 for detailed derivations.

is called the induced drag factor

7Example 1

Lecture # 7: Finite Wings (II)

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In the previous example to obtain the profile drag coefficient from the airfoildata (cd), we used the section lift coefficient on the abscissa, cl , as the samevalue of the wing lift coefficient, CL. This is a reasonable approximation,especially for a wing with a high span efficiency factor, e , very near unity.

The question is how much this approximation was appropriate? To answer thisquestion we have to compare the lift curves for infinite and finite wings. Butbefore than, lets again review the relation between the geometric, effective, andinduced angles of attack.


We see that the (effective) angle of attack seen by the airfoil section is smallerthan the (geometric) angle of attack of the wing, the difference being theinduced angle of attack. Thus, we say that the flow over a finite wing at anangle of attack is essentially the same as the flow over an infinite wing at anangle of attack eff .

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1. The effect of a finite wing is to reduce thelift curve slope.

2. The angle of attack for zero lift L=0 is thesame for the finite and infinite wings. So forfinite wings, L=0 can be obtained from airfoildate/plots.


At zero lift, there are no inducedeffects; that is, i = CD,i = 0.Thus, when CL = 0, = eff . As aresult, L=0 is the same for thefinite and the infinite wings.

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Lecture # 7: Finite Wings (II)You can show that:

1. For an elliptical finite wing:

2. For a finite wing of general platform:

is called the induced lift factor. Its values typically range between 0.01 and 0.25

Example 2:

Note that for low-AR wings, a substantial difference can exist between a0 and a.However, as AR , a a0 .

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Or

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Lecture # 7: Finite Wings (II)Example 3:

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Study Chapter 5 (pages 435-453) of the text book. Also review the posted two article about finite wings in incompressible

flow regimes.

Then, solve the following 4 problems:

Homework (6), due next week

Problem (1)

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Problem (2)

Problem (3)


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Problem (4)


Lecture+7

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