Joukowski Transformations and Aerofoils |
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One of the ways of finding the flow patterns, velocities and pressures about streamlined shapes moving through an inviscid fluid is to apply a conformal mapping to the potential flow solution for a circular cylinder. The cylinder can be mapped to a variety of shapes and by knowing the derivative of the transformation, the velocities in the mapped flow field can be found as a function of the known velocities around the cylinder. |
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A simple mapping which produces a family of elliptical shapes and streamlined aerofoils is the Joukowski mapping. The 2-D cylinder (z1 flow field) is mapped to a streamlined shape (z2 flow field) using the mapping,
The mapping is done in complex arithmetic with z1 and z2 representing the complete coordinate space of each flow field,
The transformation constant 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 will produce a very thin streamlined shape. Values of k greater than the radius of the cylinder produce mappings that are NOT conformal and hence do not represent valid flows. |
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By adjusting the centre of the cylinder relative to the origin of flow field z1 the mapped object can be made streamlined and curved.
As the far-field is undisturbed by the mapping, the stream velocity,
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thus producing a cambered Joukowski aerofoil section. |
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The velocities in flow field z2 can be determined by the derivative of the transformation function (dz2/dz1).
where |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 flow field z2. |
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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.
For streamline shapes with sharp trailing edges, such as Joukowski aerofoil sections, circulation must be added to the flow to obtain the correct lifting solution. The value of circulation applied to the cylinder in flow field 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. This 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.).
Having obtained the correct lifting flow pattern, the lift will be a function of the amount of circulation applied.
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, will be the same for both flow fields.
