Joukowski's airfoils, introduction to conformal mapping

What is conformal mapping?

Aerodynamic in air foil


Joukowski’s transformation

2. What is conformal mapping?
•A conformal map is the transformation of a complex
valued function from one coordinate system to
another.
•This is accomplished by means of a transformation
function that is applied to the original complex
function.
For example, consider a
complex plane z shown.
Coordinates in this plane
are defined with the
complex function z=x + iy.
This is mapped to
w=f(z)=u(x,y)+iv(x,y).

3. •A conformal mapping can be used to transform this complex plane z into a new
complex plane given by w = f(z). This figure shows is the example if w = √z.
The variables x and y in the z plane have been transformed to the new variables u
and v.
•Note that while this transformation has changed the relative shape of the
streamlines and equi-potential curves, the set of curves remain perpendicular.
This angle preserving feature is the essential component of conformal mapping.

4. Two transformations examples
w=z
2
→ u+iv = (x+iy)
2
= x
2
- y
2
+ 2ixy
→so, u= x
2
- y
2
& v= 2xy
Case 1/a
In w-plane, let u=a.
Then x
2
- y
2
= a (This is a rectangular hyperbola.)
Case 1/b
In w-plane, let v=b.
b=2xy → xy=b/2 (This is a rectangular hyperbola.)
Both are rectangular hyperbola…and they are orthogonal.
So lines u=a & v=b(parallel to the axis) in w-plane is mapped
to orthogonal hyperbolas in z-plane.

5. Case 2/a
In z-plane let x=c
x
2
- y
2
= u xy=v/2
→ y
2
= x
2
- u →y=v/2c
•Eliminating y from both these equations, we have
v
2
=4c
2
(c
2
-u), which is a parabola in w-plane.
•Similarly by keeping y=d, in z-plane.
We get v
2
=4d
2
(c
2
+u), which is also a parabola.
•Both these parabolas are again orthogonal.
So the straight line parallel to the axis in z-plane is
mapped to orthogonal parabolas in w-plane.

6. Example 2
w=1/z
→ z=1/w
→x+iy = 1/(u+iv) = {(u-iv)/(u
2
+v
2
)}
•Comparing both sides
x= u/(u
2
+v
2
) & y=-v/(u
2
+v
2
)
Now let us see how this transformation works for a
circle.
•The most general equation of a circle is
x
2
+ y
2
+ 2gx + 2fy + c = 0
Substituting x and y from above
We get
→ c(u
2
+v
2
) + 2gu – 2fv + 1=0 in w-plane.

7. Case 1
c = 0
c(u
2
+v
2
) + 2gu – 2fv + 1=0 is a equation of a circle in w-
plane.
•So the circle in z-plane is mapped to another circle in
w-plane.
Case 2
c=0 (i.e. the circle is passing through the origin with
center (-g,-f) in z-plane )
c(u
2
+v
2
) + 2gu – 2fv + 1=0
→ 2gu – 2fv + 1=0 which is a straight line in w-plane

So the function w=1/z maps a circle in z-plane onto a
circle in w-plane provided that the circle in z-plane
should not pass through origin.

8. Aerodynamic in air foil
•Now we will use a conformal mapping
technique to study flow of fluid around a
airfoil.

9. •Using this technique, the fluid flow around the
geometry of an airfoil can be analyzed as the flow
around a cylinder whose symmetry simplifies the
needed computations. The name of the
transformation is Joukowski’s transformation

10. Joukowski’s transformation
•The joukowski's transformation is used because it has the
property of transforming circles in the z plane into
shapes that resemble airfoils in the w plane.
•The function in z-plane is a circle given by


Where b is the radius of the circle and ranges from 0 to 2∏.
• The joukowski's transformation is given by the function


Where w is the function in the transformed w-plane, and λ
is the transformation parameter that determines the
resulting shape of the transformed function.

11. •For λ = b, the circle is mapped into a at plate
going from -2b to 2b.
•Setting the transformation parameter larger
than b causes the circle to be mapped into an
ellipse.

12. •The airfoil shape is realized by creating a circle in the z plane with a
centre that is offset from the origin, If the circle in the z plane is
offset slightly, the desired transformation parameter
is given by
Where s is the coordinates of the centre of the circle.
•The transformation in the w plane resembles the shape of an
airfoil symmetric about the x axis. The x coordinate of the circle
origin therefore determines the thickness distribution of
the transformed airfoil.

13. •If the centre of the circle in the z plane is also
offset on the y axis, the joukowski's
transformation yields an unsymmetrical airfoil.
This shows that the y coordinate of the circle
centre determines the curvature of the
transformed airfoil.

14. •In addition to the circle in the z plane being
transformed to air foils in w-plane, the flow
around the circle can also be transformed
because of the previously mentioned angle
preserving feature of conformal mapping
functions.
•This requires that the velocity potential and
stream function should be expressed as a
complex function. This is accomplished by
expressing the velocity potential and stream
function in a complex potential, given by


Where ɸ is velocity potential function and Ψ is
streamline function.