Introduction to Conformal Mapping in Aerodynamics

IntroductionConformal mapping is a method used to extend the application of potential flow theory to practical aerodynamics. Standard potential flow theory begins with an ideal flow to show that lift on a body is proportional to the circulation about a closed path encompassing an object. Potential flows start with flows over cylinders since the mathematics is more tractable. However, to use potential flow theory on usable airfoils one must rely on conformal mapping to show a relation between realistic airfoil shapes and the knowledge gained from flow about cylinders.

Brief review of complex numbers:Conformal mapping relies entirely on complex mathematics. Therefore, a brief review is undertaken at this point.

A complex number z is a sum of a real and imaginary part; z = real + iimaginary

The term i, refers to the complex number

so that; Complex numbers can be presented in a graphical format. If the real portion of a complex number is taken as the abscissa, and the imaginary portion as the ordinate, a two-dimensional plane is formed.

z = real + iimaginary = x + iy

L = V Lift = ?

y, imaginary

x, real

A complex number can be written in polar form using Euler's equation;

z = x + iy = rei = r(cos + isin)

where r2 = x2 + y2

Complex multiplication: z1z2 = (x1+iy1)(x2+iy2) = (x1x2 - y1y2) + i(x1y2 + y1x2)

Complex representation of potential flowsThe basic flows used in potential flow theory such as uniform flow, source, sink, doublet and vortex, can all be represented using complex numbers. For example, if a complex number w with both real and imaginary parts represents a potential flow, then the form of the number is;

w(z) = + i = (velocity potential) + i(stream function)

Here, both velocity potential and stream function are themselves complex numbers. As an example, the uniform flow can be written;

Uniform flow: w(z) = Vz = + i = V(x+iy) = Vx + Vy

as seen previously, = Vx = V rcos = Vy = V rsin

Source flow:

Vortex flow:

Doublet flow:

In complex terms the flow past a cylinder with lift is written:

Velocity Components:

When a potential flow is represented in complex form, the velocity components can be found using one of two methods;

1. Re-write the expression from the complex variable z form into its separate real and complex components. The form of this expression will be w = + i. The individual velocity components are found by completing the appropriate differentiation on or . to obtain u or v. As an example consider the complex form of the source flow;

2. An alternative method would be to differentiate on the complex expression directly and then separate the real and complex portions to obtain the velocity components according to;

Conformal MappingA conformal mapping is performed through the transformation of a complex function from one coordinate system to another. A transformation function is applied to the original function to perform the mapping. For aerodynamics applications the Joukowski transform is the most commonly used function;

Here, b is a constant. Graphically, a conformal mapping will transform a complex plane in z (z = x+iy) into a complex plane in a new variable w (w = +i).

In the diagram a uniform flow in the z plane is transformed into an equivalent form in the w plane using a transform of the form w = f(z). As an example consider a circle drawn in the z plane, z = bei. The Joukowski transform maps the circle into a flat plate,

= C1

= C2

= C3

= K1 = K3

z plane

x

y w plane

= C3

= C2

= K3

A circle of radius b is mapped into a straight line in the w plane entirely on the real axis between -2b and 2b. If a uniform flow had been drawn over the circle, the transform would have mapped that flow into the flow over a flat plate in the w plane. If the circle originally had a radius slightly larger than the transform constant b, z = aei, with a > b, the circle would have formed an ellipse instead of the flat plate.

Which can be written,

If the flow over a cylinder had been transformed it would have created the flow over an ellipse.

x

y

b

z plane w plane

2b-2b

zbzw

2

x

y

2b-2b

Joukowski AirfoilsFrom and aerodynamics point of view, the most interesting application of the Joukowski transform is to an offset circle. If we consider a circle slightly offset from the origin along the negative real axis, one obtains a symmetric Joukowski airfoil.

The equation of the offset circle is z = aei-eb where the constant e is a small number. If the cylinder is displaced slightly along the complex axis as well, one obtains a cambered airfoil shape.

Here, the points A and B are the intercepts of the displaced circle on the real axis and their corresponding points in the transformed plane. The angle is the angle formed by the line joining the point A (or B) and the origin with the real axis. If lifting flow about the original circle had been imposed, the Joukowski transformation would have generated a lifting flow about the Joukowski airfoil;

b

a

eb-2b 2bx

y

z plane w plane

zbzw

2

w plane

x

y

z plane

zbzw

2

b

a

eb -2b 2bx

y

z plane w plane

zbzw

2

A

B BA

Although such a flow is mathematically possible, in reality it may not be realistic. The stagnation points on the cylinder map to stagnation points that are not always realistic. For instance the stagnation point on the top surface of the airfoil cannot exist is steady flight since the velocity would tend to infinity as one moves very close to the trialing edge. The only means of making a realistic flow is to impose the Kutta condition where the stagnation point is forced to exist at the trailing edge thus making the streamlines flow smoothly from this point. This is done by adjusting the value of vorticity strength , such that the stagnation points on the cylinder reside at the cylinder’s intercepts of the real axis. In this case, when the cylinder is transformed, one stagnation point will be forced to the trailing edge.

The lift force generated by the lifting flow over the cylinder is proportional to the circulation about the cylinder imposed by the added vortex flow according to the Kutta-Joukowski relation, L’ = V . The lifting force on the resulting Joukowski airfoil is not clear. To evaluate the lift, the circulation is needed and therefore the velocity field. The velocity fields in each plane can be related to each other through the chain rule of differentiation. If the lifting flow about the cylinder is defined as function Q where Q = Q(z) in the z plane and Q = Q(w) in the w plane, the velocities in each plane are;

By chain rule:

Using the Joukowski transformation;

Clearly, the velocity field very close to the cylinder and its transformed counterpart are dissimilar as one would expect. However, farther away from these objects the velocity fields become identical as the magnitude of z becomes larger than the constant value of b.

w plane

x

y

z plane

zbzw

2

Since the circulation can be calculated about any closed path, including paths very far from the object surface, the circulations must be the same in both planes.

Vortex strengthThe appropriate vortex strength to impose the Kutta condition must be determined. Consider the lifting flow about a cylinder. The velocity in the direction is,

Here, R is the radius of the cylinder surface. This velocity is zero on the surface of the cylinder at the stagnation points. At the these points =-.

If the field is rotated by to simulate an angle of attack,

Since the cord length of the Joukowski airfoil is 4b, the lift coefficient can be written,

Making the assumption that b R,

ExampleA Joukowski airfoil is formed by displacing a circle of radius 1 by x = -0.08 (real axis) and y = 0.05 (imaginary axis). Find,a) Vortex strength if = 0o, and V = 10 m/sb) CL at = 0o and = 10o

x

yz plane

-

x

y

0.05

-0.08

cylinder

stagnationpoint

b

b = 0.9187

a) = 4VRsin(+) = 4(10)(1)sin(2.87) = 6.2831

b) CL = 2sin(2.87) = 0.31415

CL = 2sin(10 + 2.87) = 1.40