Joukowski's airfoils, introduction to conformal mapping

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).

•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.

Two transformations examples w=z2

→ u+iv = (x+iy)2 = x2 - y2 + 2ixy

→so, u= x2 - y2 & v= 2xy

Case 1/a

In w-plane, let u=a.

Then x2 - y2 = 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.

Case 2/a In z-plane let x=c x2 - y2 = u xy=v/2 → y2 = x2 - u →y=v/2c • Eliminating y from both these equations, we have

v2=4c2(c2-u), which is a parabola in w-plane. • Similarly by keeping y=d, in z-plane. We get v2=4d2(c2+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.

Example 2 w=1/z → z=1/w →x+iy = 1/(u+iv) = {(u-iv)/(u2+v2)} • Comparing both sides x= u/(u2+v2) & y=-v/(u2+v2) Now let us see how this transformation works for a

circle. • The most general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0 Substituting x and y from above We get → c(u2 +v2) + 2gu – 2fv + 1=0 in w-plane.

Case 1 c = 0 c(u2 +v2) + 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(u2 +v2) + 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.

Aerodynamic in air foil

• Now we will use a conformal mapping technique to study flow of fluid around a airfoil.

• 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

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.

• 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.

•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.

• 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.

• 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.

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