Applications of conformal mappings to the fluid flow

Abstract

The complexity of airfoil shapes has brought di culties in modeling the uid ow around an airplane. This di culty is based on uid interactions around aircraft and generally on the aspect that one wants a solution of the practical part of aerodynamics. To model this, a lot of simpli cations must be done. Conformal mapping technique is widely used to transform complicated

uid ow problems into simpler uid ow problems. In our thesis, we reviewed and implemented the model used to model the ideal uid around a circular cylinder. This model was constructed by a superposition of elementary potential uid ows including uniform

ow, doublet ow and vortex ow. The Conformal mapping technique known as Joukowsky transformation was used to map the uid ow around the circular cylinder into the uid ow around the airfoil. This transformation was implemented in MATLAB to be able to visualize the streamlines around the circular cylinder and the corresponding airfoil. The lift force was calculated versus the angle of attack using the Kutta-Joukowsky formula. It was found that there is a strong linear dependence between the lift force and the angle of attack. Again, the lift coe cient as a function of the angle of attack of the Joukowsky airfoil was compared to the lift coe cient for the NACA 0012 airfoil and these almost match better as the maximum absolute error in the lift coe cient for the two airfoils is in the range of 0 to 12 degrees was found to be 2:8%.