### JOUKOWSKI AEROFOILS AND FLOW MAPPING

A simple mapping which produces a family of elliptical shapes and streamlined aerofoils s the Joukowski mapping. The 2-D cylinder (**z1** flow field) is mapped to a streamlined shape (**z2** flow field) using the mapping.

**z1 = x1 + i.y1**) and (

**z2 = x2 + i.y2**), and are mapped by,

**k**) is used to control the stretching of the flow field. A small (

**k**) value will produce a near cylindrical shape with large thickness to chord ratio. A large (

**k**) value approaching the radius of the cylinder (

**a**) will produce a very thin streamlined shape. A value of

**k=a**will produce a flat plate. Values of (

**k**) greater than the radius of the cylinder produce mappings that are NOT conformal and hence do not represent valid flows.

By adjusting the centre of the cylinder relative to the origin of flow field (**z1**) the mapped object can be made streamlined and curved, thus producing a cambered Joukowski aerofoil section. To guarantee a valid aerofoil shape the transformation constant must be adjusted to match the circle flow geometry.

**(V**) can be determined by the derivative of the transformation function , (

_{∞}**dz2/dz1**) such that,

**|V1|**is the magnitude of the velocity at a point in flow field (

**z1**) and

**|V2|**(or

**V**) is the magnitude of the velocity at the mapped point in the aerofoil flow field (

**z2**).

Pressure coefficients on the surface of the streamlined shape in flow field (**z2**) can then be found by applying Bernoulli's equation for inviscid incompressible flow.

**z1**) should be specified so that a stagnation point is produced at the point of intersection of the rear of the cylinder and the x-axis.

**z1**) point maps to the trailing edge of the aerofoil and when the correct amount of circulation is applied, the Kutta condition will be satisfied at the trailing edge of the aerofoil in flow field (

**z2**), (ie. vorticity = 0 at trailing edge.).

This means the required amount of circulation is

**a**) is the radius of the original circle and (α) is the stream angle of attack.

Having obtained the correct flow pattern, the lift can be calculated as a function of the amount of circulation applied.

### The flat plate aerofoil.

If the transformation constant is set to be equal to the radius of the circle ( **k = a** ) and no center shift is used the circle maps to a flat plate aerofoil. By applying the velocity mapping and Bernoulli relationships, the pressure field on the plate can be predicted and hence the lift, drag and moment can be calculated.

### General Joukowski Aerofoil solutions.

While Joukowski aerofoils are relatively simple to create and analyse, they are relatively crude in terms of performance. The geometric properties of this family can be described by the following approximations.

Maximum thickness ,

and maximum camber height,

The location of maximum thickness is always at the 30% chord location and the location of the maximum camber point is always 50% chord. This arrangement promotes early boundary layer transition and hence moderate drag. The cusped trailing edge is extremely thin and impractical for real construction purposes.

The performance due to camber is modified such that,

### Software :

The following software application is available to construct and display flow patterns, pressures and list coordinate data for these transformed aerofoil sections.