An implicit finite element method implementing recent advances in algorithm development for the solution of compressible viscous flow is presented. The numerical algorithm formulated in the present paper is based on the general family of implicit Taylor-Galerkin methods which make use of a combination of second-order Tayloer series expansion in time with a Galerkin approximation in space. The present scheme is applied to h-adaptive mesh refinement on the unstructured quadrilateral elements.
Numerical applications include several examples of two-dimensional inviscid and viscous transonic and supersonic flows. For validation of the present scheme, computational results of the viscous flows are compared with other numerical solutions. Inviscid test problems include supersonic and transonic internal flows over arc bump in a channel. The accuracy of results on unstructured grids turned out as good as that of the initially structured grids. For viscous flow calculation, supersonic laminar flow over a flat plate and transonic laminar flow around a NACA0012 airfoil are presented. The airfoil flow is calculated for the Reynolds numbers 500, 2000, and 10000 at zero angle of attack, and for Reynolds number 500 at 10 degrees angle of attack. At Reynolds number 10000 and zero angle of attack which is known as a critical case, unsteady motion of the wake occured and vortex shedding was accompanied by oscillation of shock wave in the near wake of the airfoil.
