Finite element computation of unsteady viscous compressible flows

In this article we present our results for computation of unsteady viscous transonic flows past cylinders and airfoils. Stabilized finite element methods are employed to solve the compressible Navier-Stokes equations in their conservation law form. The nonlinear equations resulting from the finite element discretizations are solved using GMRES technique. To test the accuracy of the formulation, Mach 2 flow past a circular cylinder is computed with a mesh that is fine enough to resolve the shock structure for Reynolds number 50. The computational results agree quite well with the analytical and theoretical results. The methodology is applied to compute unsteady transonic flows past cylinders and airfoils. Several cases involving flows past a NACA0012 airfoil are computed and compared with numerical results from other researchers. Interesting flow patterns are observed for all the cases. It is observed that flow past an airfoil in a channel, of width 8.5 chord-lengths, at Mach 0.85 and Re 10 000 is unsteady when it is placed at zero incidence to the flow, while it reaches a steady-state when the angle of attack of the airfoil is 10 °