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AA200 - Applied AerodynamicsWinter Quarter 2011-12Juan J. Alonso Durand 252 Tel: 3-9954 email: jjalonso@stanford.edu

PROBLEM SET 4 Due Date: February 17th, 2012, 5pm

Problem 1. Supersonic Thin Airfoil Theory. The main objective of this problem is tounderstand the effects of angle of attack, thickness, and camber distributions of a supersonic thinairfoil on its performance (lift, drag, and moment coefficients, as well as the location of the centerof pressure and the aerodynamic center).

Consider a family of supersonic airfoils (for illustration purposes only: you would normallydesign them differently) given by the following camber and thickness distributions. The camberline

is that of the NACA four-digit series of airfoils that you used in PS1, and is given by the followingtwo parabolas:

Y(x) =

xp2

2p xc

for 0 < x

c< p

(cx)(1p)2

1 + x

c 2p

for p < xc

< 1,

where is the maximum camber ratio and p is the chordwise location of the maximum camber.The thickness distribution is parabolic and given by:

T(x) = 4x

c(c x),

where is the thickness-to-chord ratio. As usual, you can construct the upper and lower surfaces ofthe airfoil by simply adding/subtracting one half the thickness to/from the camberline (vertically:no need to add the thickness normal to the camberline):

yu(x) = Y +1

2T(x)

yl(x) = Y 1

2T(x).

Using thin airfoil theory, derive closed-form expressions (using Mathematica or MATLABs Sym-bolic Toolbox is probably a good idea, although it can also be done by hand) for:

The coefficient of lift, Cl, of the airfoil, as a function of , p, , and .

The coefficient of wave drag, Cd

, of the airfoil, as a function of , p, , and .

The coefficient of moment about the leading edge, Cmle, of the airfoil, as a function of , p,, and .

Use these expressions to compute the location of the center of pressure, xcp, for the family of airfoils.Where is the aerodynamic center located? Is it a function of any of the parameters in this problem(, p, , and )?

Finally, for an airfoil of this family with 1% maximum camber located at the quarter chord,and 2% thickness, please plot the variation of Cl, Cd, xcp, and Cmle as a function of angle of attack

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in what you would consider a logical range of . What is this airfoils optimum lift-to-drag ratioand at what and Cl is it achieved?

Problem 2. Optimal Thickness Distribution for a Supersonic Thin Airfoil. Assume that

we have a supersonic airfoil with no camber and at zero angle of attack in a free stream M

= 2.0.The thickness distribution is given by an arbitrary function, T(x). We are interested in thicknessdistributions that minimize the wave drag of the airfoil, while remaining practical. In other words,the airfoil cannot be a flat plate.

In order to properly pose this shape optimization problem assume that the area enclosed by theairfoil A =

c0 T(x)dx must be greater than a specified amount, A0.

What is the thickness distribution, T(x), that minimizes the wave drag of the airfoil whilesatisfying the area constraint?

Hint: you may want to quickly review the theory of calculus of variations or, alternatively,review how to use the concept of Lagrange multipliers for constrained optimization. Pointers willbe given in lecture on Monday.

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