MAE3241: Aerodynamics and Flight MechanicsName: ______

April 9, 2010 Exam #2

Short problems 1-3 are worth 5 points each.

Question 1 (5 points):

Anderson clearly articulates that without friction there could be no lift on airfoils and wings in ‘real life’, and he goes on to say that, ‘Nature enforces the Kutta condition by means of friction’. In our development of thin airfoil theory we mathematically enforced a Kutta condition at the trailing edge of the airfoil, (TE)=0. Was a Kutta condition also enforced for the potential flow model of a rotating cylinder generating lift (why or why not)?

Question 2 (5 points):

Show below is a picture of a thick airfoil being modeled by a suitable distribution of vorticies located along the top and bottom surfaces.

In Chapter 3 the Kutta-Joukowski Lift Theorem (KJLT) was derived exactly for the case of the lifting cylinder. We also stated (without much proof) that the KJLT also applies in general to a two-dimensional body of arbitrary shape. Although this result can be proven mathematically, how would you argue that the KJLT is completely true for the airfoil model shown above?

Question 3 (5 points):

In thin airfoil theory applied for cambered airfoils, a Fourier cosine series expansion is used to represent the slope of the mean camber line. In general, The Fourier cosine series representation of a function f() over an interval 0 ≤  ≤  is given by:

Why are all coefficients greater than n ≥ 3 = 0?

Question 4 (10 points):

The lift coefficient and moment coefficient of an airfoil at the quarter cord point at -6° angle of attack are -0.39 and -0.045, respectively. At 4° angle of attack, these coefficients are 0.65 and -0.037, respectively. Calculate the location of the aerodynamic center for this airfoil. How does it compare with thin airfoil theory?

Question 5 (20 points):

A symmetric airfoil with a cord of 1 m is flying at 100 m/s at an angle of attack of 5 degrees.

What is the value of the total circulation, , around this airfoil?

Now consider a cambered airfoil at the same conditions. The slope of the cambered airfoil is given by:

/ Valid between 0 and c/2
/ Valid between c/2 and c

What is the total circulation, , around this airfoil?

How much lift per unit span does this airfoil produce.

Question 6 (30 Points):

Thus far, all of the vortex examples that we have examined involved stationary vortices. A more useful representation, such as that used to model moving blades in an aircraft engine, employ potential flow models of vortices with motion.

Suppose a vortex is in steady motion (non-accelerating) with a velocity V in the vertical direction as shown in the sketch below:

  1. Write an expression for the stream function for this uniformly translating vortex with respect to the fixed point A. Note that the perpendicular (horizontal) distance between the moving vortex and point A is a constant and may be denoted as a.
  2. Write down the Cartesian velocity components u and v.
  3. If the vortex strength, =2 m2/s, the perpendicular distance a=1 m, and the vortex translational velocity, V=1 m/s, sketch the magnitude of the velocity at point A induced by the vortex. Another way of saying this would be, sketch the velocity than an observer at point A would see with time. Assume that at time t=0 the vortex and the observer at point A are located on the same horizontal datum.

Question7 (30 points):

A thin airfoil has a mean camber line described by:

Part 1: Find the circulation distribution along the chord, () for 0 ≤  ≤ 

Part 2: Find the pressure difference between the upper and lower surfaces, CP()

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