Computation of compressible flows using finite elements and optimization techniques

A variational formulation for compressible subcritical flows, shocked transonic flows and shockless transonic flows is developed using a special class of fictitious gas to regularize the local supersonic flow of the flow field. A technique to enforce the Kutta-Joukowski condition at the trailing edge of a lifting airfoil is introduced and the variational formulation is extended to treat lifting problems. The existence and uniqueness of the solution for the subcritical problem or the regularized transonic problem follow from an application of standard results from convex analysis. The finite element method is used to approximate the problem and optimization algorithms are introduced to compute the flow field. Numerical examples are computed and compared with either wind tunnel data or those obtained by other numerical methods. The method of characteristics is used in the supersonic pocket as part of the analysis to compute new shockless airfoils for shock-free airfoil designs or shock waves in shocked flow studies.