Geometrical effects on the airfoil flow separation and transition

We present results from direct numerical simulations (DNS) of incompressible flow over two airfoils, NACA-4412 and NACA-0012-64, to investigate the effects of the airfoil geometry on the flow separation and transition patterns at Re=104 and 10 degrees incidence. The two chosen airfoils are geometrically similar except for maximum camber (respectively 4%C and 0 with C the chord length), which results in a larger projection area with respect to the incoming flow for the NACA-4412 airfoil, and a larger leeward surface curvature at the leading edge for the NACA-0012-64 airfoil. The governing equations are discretized using an energy conservative fourth-order spatial discretization scheme. An assessment on the two-point correlation indicates that a spanwise domain size of 0.8C is sufficiently large for the present simulations. We discuss flow separation at the airfoil leading edge, transition of the separated shear layer to three-dimensional flow and subsequently to turbulence. Numerical results reveal a stronger adverse pressure gradient field in the leading edge region of the NACA-0012-64 airfoil due to the rapidly varying surface curvature. As a result, the flow experiences detachment at x/C=0.08, and the separated shear layer transition via Kelvin-Helmholtz mechanism occurs at x/C=0.29 with fully developed turbulent flow around x/C=0.80. These flow development phases are delayed to occur at much downstream positions, respectively, observed around x/C=0.25, 0.71 and 1.15 for the NACA-4412 airfoil. The turbulent intensity, measured by the turbulent fluctuations and turbulent Reynolds stresses, are much larger for NACA-0012-64 from the transition onset until the airfoil trailing edge, while turbulence develops significantly downstream of the trailing edge for the NACA-4412 airfoil. For both airfoils, our DNS results indicate that the mean Reynolds stress u′u′‾/U02 reaches its maximum value at a distance from the surface approximately equal to the displacement thickness, consistent with the experimental observations (Boutilier & Yarusevych, Phys. Fluids, 2012). A quantitative eigen-system analysis on the instantaneous velocity field shows that although the flow over an airfoil is intrinsically anisotropic, the alignments between the vorticity vector and the eigenvectors of Sij and SikSkj+ΩikΩkj are quite similar to those of the homogeneous isotropic turbulent flows due to the formation of vortex tubes.