MAE 104, FALL 2013HOMEWORK 5

Due by 5PM Mon 12-09-2013 by email or in professor’s office.Solutions will be posted online that night.

Guidelines

Please turn in a neat homework that gives all the formulae that you have used as well as details that are requiredfor the grader to understand your solution. Required plots should be generated using computer software such asMatlab or Excel. Remember to specify all the units of your results.

Problem 1A NACA 2412 airfoil has a camber line given by the equations:

yc(x) =

{110

(xc

)− 2

16

(xc

)2, 0 ≤ x

c ≤ 410

190 + 2

45

(xc

)− 2

36

(xc

)2, 4

10 ≤ xc ≤ 1

1. Determine the lift coefficient cl(α) as a function of α.

2. Determine the moment coefficient about the aerodynamic center cmAC.

3. Plot the two coefficients just found and compare them with the experimental results shown in Figure 1.

Figure 1: Aerodynamic coefficients of the NACA2412 airfoil from experimental wind tunnel data. [Summary ofairfoil data Ira H. Abbott, A. E. von Doenhoff].

1

Problem 2Consider a wing with elliptical planform shape and aspect ratio Λ = 10, formed by equal airfoils (∂cl∂α = 2π). Itsnon-dimensional circulation distribution is:

Gb(θ) =Γ

bU∞= − 3ε

πΛsin (θ) − ε

2πΛsin (3θ) ,

where yb = 1

2 cos(θ)

1. Calculate the lift coefficient, cL , of the wing when the unperturbed free-stream is parallel to the zero-liftdirection (αL=0) of the central airfoil.

2. Calculate the induced drag coefficient of the wing, cDi.

3. Calculate the angle that the zero-lift direction of the wing forms with the zero-lift direction of the centralairfoil. Make a sketch indicating clearly their relative position.

4. Knowing that the maximum lift coefficient of each airfoil, for the considered Reynolds number, is cLmax = 1.4,determine the airfoil in which the stall begins and the maximum cL of the wing.

Problem 3 (2009 Final)A wing of span b, chord

cy = c0

√1 −

(2y

b

)2

and angle of attack of zero lift

αL′=0 =

(4y

b

)2

− 1

is flying at angle of attack α and speed U∞ on air of density ρ∞ at rest. Using incompressible lifting line theory,

1. Obtain the coefficients An of the circulation distribution Γ(θ).

2. Determine then the lift distribution per unit span L′(θ).

hint: 4 cos2 θ − 1 = sin 3θ/sin θ.

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