MAE 360. Aerodynamics I.
Homework Assignment 2. Due September 21.
1. (Similar to problems 5—7 in Chapter 2.) Consider the case of a cylinder of radius awith circulation Γin a
uniform flow. The potential of the resulting flow field is given in polar coordinates by
ϕ=V∞rcos θ+Bcos θ
r−
Γ
2πθ.
Find:
(a) The dimensions of Band Γ.
(b) An expression for the cylinder radius, a,intermsofBand V∞.
(c) The stagnation points.
(d) The pressure coefficient as a function of θ.PlotCpvs. θfor 0 ≤θ≤2π. Use Bernoulli’s equation to
find pin terms of the (known) velocities.
(e) The lift on the cylinder. Find this by integrating over the surface, and show that =ρV∞Γ.
(f) The drag on the cylinder. Find this by integrating over the surface.
2. Rankine Oval.Asource,representedby
ϕ=Λ
2πln r,ψ=Λ
2πθ,
where Λisthesourcestrength,satisfies the condition of irrotationality. Define the dimensionless values,
Λ=Λ/V∞band y=y/b where bis a reference length. Consider a source-sink combination in a uniform
flow. (A sink is simply the negative of a source.) Place a source of strength Λ=πat y=−1and a sink
of strength Λ=πat y=1. Thus, the dimensionless stream function can be written as
ψ=y+1
2·tan−1y
x+1 −tan−1y
x−1¸.
Use MATLAB to do the following:
(a) Find and plot the stagnation points in this flow.
(b) Find and plot the stagnation streamline for this body.
(c) Plot several streamlines for this flow.
Turn in your MATLAB code, your plot and a complete explanation of your work. You may wish to utilize
the function fzero to complete this problem, but you will probably have to find a way to force fzero to
find the correct root for -1 ≤¯x≤1. Feel free to work with other members of the class on this assignment,
but please acknowledge the help of your teammates.
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