Finite Element Solutions for Turbulent Flow over the NACA 0012 Airfoil

A high-order Petrov–Galerkin finite element scheme is used to compute turbulent flow over a NACA 0012 airfoil at a freestream Mach number of 0.15, an angle of attack of 10 deg, and a Reynolds number based on the airfoil chord of 6 million. Results are obtained on a series of grids available on the NASA Turbulence Modeling Resource Web site and are compared with reference solutions that have been obtained using the FUN3D and CFL3D finite volume solvers on meshes with as many as 14.6 million degrees of freedom. Forces, moments, pressure distributions, skin friction, and profiles of velocity and turbulence working variable for the Spalart–Allmaras turbulence model are compared between the finite element and the finite volume solutions. It is demonstrated that the finite element scheme shows similar results as the finite volume schemes for most of the comparisons, but demonstrates significantly less dissipation of the wake profiles downstream of the airfoil. It is shown that, when the same number of degrees of freedom is used in the simulations, solutions obtained with quadratic elements are more accurate than those obtained using linear elements with only minor increases in computational cost. Finally, initial results with an adjoint-based hp-adaptive methodology are obtained that further demonstrate that the high-order finite element framework can efficiently yield accurate results.