On Global Linear Instability Mechanisms of Flow Around Airfoils at Low Reynolds Number and High Angle of Attack

Abstract

The existence of modal and non-modal linear three-dimensional global instabilities of spanwise-homogeneous low-Reynolds number laminar incompressible flow around two-dimensional airfoils is documented. Work has commenced to examine the relative significance of such instabilities as a function of airfoil thickness and camber at conditions close to stall: the symmetric NACA0009 and NACA0015 airfoils as well as the cambered NACA4415 are used for these analyses at angles of attack 15 ≤ AoA ≤ 20. At all conditions examined, the combination of the Re and AoA parameters is chosen such that steady two-dimensional flow over the airfoil is obtained. Four independent codes, one based on finite-volume, one on finite-element and two on spectral-element spatial discretization have been used to cross-validate the base flow and instability analysis results presented, all delivering consistent pre- dictions. In line with previous findings on the NACA0015 airfoil, two classes of three-dimensional stationary and traveling global eigenmodes are recovered. Contrary to what was previously reported, the traveling wake-mode instability dominates that pertain- ing to the stationary three-dimensional flow eigenmode. Furthermore, neither traveling nor stationary unstable three-dimensional modes have been found in the NACA0015 airfoil, prior to the two-dimensional wake mode undergoing a Hopf bifurcation. Conse- quently, amplified three-dimensional structures akin to the stall cells reported in earlier work have not been observed in the present analyses. Finally, relatively strong transient energy growth is shown here for the first time for flows around airfoils, as known from earlier analyses of flow around bluff bodies. In the context of non-modal instability too, strongest energy amplification is found to be associated with the two-dimensional flow and converts, via the Orr mechanism, optimal initial conditions to wake-mode perturbations.