Finite element computation of unsteady viscous transonic flows past stationary airfoils

Abstract

Results are presented for computations of unsteady viscous transonic flows past a stationary NACA0012 airfoil at various angles of attack. The Reynolds number, based on the chord-length of the airfoil, is 10,000 and the Mach number is 0.85. Stabilized finite-element formulations are employed to solve the compressible Navier-Stokes equations. The equation systems, resulting from the discretization, are solved iteratively by using the preconditioned GMRES technique. Time integration of the governing equations is carried out for large values of the non-dimensional time to understand the unsteady dynamics and long-term behavior of the flows. The results show interesting flow patterns and a complex interaction between the boundary/shear layers, shock/expansion waves and the lateral boundaries of the computational domain. For transonic flow past an airfoil at various angles of attack in a narrow channel/wind-tunnel one can observe solutions that are qualitatively different from each other. At low angles of attack an unsteady wake is observed. At moderate angles of attack the interaction between the shock system and the lateral walls becomes significant and the temporal activity in the wake decreases and eventually disappears. At high angles of attack a reflection shock is formed. Hysteresis is observed at an angle of attack 8^∘. For the flow in a domain with the lateral boundaries located far away, the unsteadiness in the flow increases with an increase in the angle of attack. Computations for a Mach 2, Re 106 flow past an airfoil at 10^∘ angle of attack compare well with numerical and experimental results from other researchers