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Dynamics of Inviscid Fluids 77

5.4 Example
In the sequel we will illustrate a particular transformation (mapping),
; ; ; a
namely the Joukovski transformation (see section 2.5.3), z = 24+ —.
By this transformation the complex potential of a uniform flow becomes

f(z) =U (2 + a; ) i.e., the potential for a uniform flow past a circular

cylinder of radius a, U being the magnitude of the velocity at far field.
This transformation, z= 2’ + oe where b? < a”, allows the conformal
transformation of a circle of radius @ centered at P(z'p, yp) from the
second quadrant Oz'y’ onto a so-called Joukovski airfoil (profile) in the
Oxy plane.

Let us now consider a uniform flow of velocity U in the positive Ox
direction past the above Joukovski airfoil. In particular, its sharp trailing
edge at x = 2b, is the image of the point Q at z’ = b where Oz’ is crossed
by the above circle.
The magnitude V of the velocity in the Oxy plane is related to the
magnitude V’ of the velocity in the Oz’y' plane by the relation

ef
dz!

df
3
dz dz
dz!

1.e.,
y!
V= aT: (2.8)
1 (8)”

We remark that if the velocity V’ # 0 at Q where 2’ = b, then the
velocity V at the sharp trailing edge z = 2b becomes infinite, which
is a contradiction with the Joukovski—Kutta condition. Thus, we must

impose that the point Q on the circle be a stagnation point; this goal
may be reached if we create a clockwise circulation I’ on the circle,
and this circulation is then conserved by the conformal mapping. The
magnitude of this circulation is [T = 4xaU sin@ = 4ay,U and the flow
past the circle is then constructed by adding to the uniform stream a
doublet and a point vortex, so that we get the complex potential of the

resultant flow
2 fot
pauls —zp4 — - + i2yplog (# =e).

zp a
Here the constant term —i2y loga has been added but the values of the

stream function WV on the circle do not change after this superposition.