Initially, a serious work on development of airfoils is done from late 1800's. People at that time know that the flat plates would produce lift when some angle of incidence is given. Later this work is improved by inducing some shapes with curvatures as an experiment for obtaining more lifts. JOUKOWSKI TRANSFORMATION explained the procedure and the working methods for construction of Airfoils. Various airfoils with different shapes are produced based on this transformations. Based on the equation of flows about a circular cylinder's with circulation, solutions which produces a lift force are shown. A circle which is in a quadrant is considered and based on the centre of circle and its location in the quadrant, shape of the airfoils is developed by doing a huge mathematical calculations.
The above figure shows us the Joukowski Transformation in which a airfoil is constructed from a circle by applying various mathematical operations. The location of the centre of the circle is a prime factor in obtaining the airfoil. Many airfoil shapes are based on this. If the centre of the circle is located at the origin, then a straight-line or flat plate airfoil is obtained. Then, this experiments are re-conducted for studying the results for various angles of attack for assumed circulation. Based on these experimental results, various parameters like Lift and Moment coefficients are found. These parameters are now applied to a number of Joukowshi airfoils which are obtained for various centre of the circle positions so that the theoretical characteristics of the particular Airfoils are obtained.
When the centre of the cirlce is located on the y-axis, a thin circular-arc airfoil is obtained from Joukowski transformation. In the same way, when the centre of the circle is located to the right of and above the origin, then a Cambered airfoil is obtained. These Joukowski transformations are simple and direct ways for finding flate-plate and circular-arc airfoils. But these transformations are found to be complicated in the case of cambered airfoils. So to make the construction of cambered airfoils easier, a researcher called TREFFTZ developed a method called Trefftz Graphical Construction. This method is the extension of Joukowski transformation. In this method, each point on the circle is transformed twice and both the transformations yields circles. Now again many mathematical operations are done on the trefftz method for obtaining the cambered airfoils.
The above figure shows the construction of Joukowski airfoil with 4-ft chord, 4 per cent camber, 10 percent thick by the Trefftz method. The same methods are used for Construction of Thin-Airfoils and Thick Airfoils.
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