A contravariant velocity based implicit multilevel method for simulating the whole process of incompressible flow transition arround Joukowsky airfoils

A contravanant velocity based incompressible Navier-Stokes governing system is derived to generate an implicit multi-level method so that we can simulate the receptivity and flow transition in cases with complex geometnes. The two- and three- dimensional development of leading-edge receptivity and flow transition in a 2-D Joukowsky airfoil boundary layer are investigated by direct numencal simulation (DNS) using this system. The numencal investigation is based on the so-called spatial approach. The numencal results agree very well with linear stability theory (LST) and the experimental results for the flat plate case. They also agree well with the result obtained by other researchers for the 2-D elliptic leadingedge receptivity case. Some new phenomena for the transition around Joukowsky airfoils are abserved. The details of this approach are described.