Floquet stability analysis in the wake of a NACA0015 airfoil at post-stall angles of attack

Abstract

We perform a Floquet stability analysis in the wake of a NACA0015 airfoil, with four angles of attack, α=20°, 17.5°, 15°, and 12.5°, considered. The central aim is to predict the secondary instabilities at the fixed moderate angles of attack, which are sufficiently large for massive separation from the airfoil, while at the same time allowing the formation and travelling of surface vortex on the airfoil, differentiating from the cases with very large angles of attack. We divide the angles of attack into two groups. We report that mode C is the first unstable mode for both groups, and which is also the only unstable mode for the small angles of attack, i.e., α=15° and α=12.5°, while for the high angles of attack, i.e., α=20° and α=17.5°, four unstable modes are observed. They are mode A, mode quasi-periodic, mode SL, and mode SS. The modes SL and SS are both subharmonic but with different wave numbers. We conjecture that these two subharmonic modes are resulted from the splitting of mode C. A comparison in the base flow topologies between the two groups shows that the different instability behaviors are probably due to their different flow patterns of the base flows. Three-dimensional direct numerical simulations (DNSs) have also been employed to study the physical realizability of the dominant unstable modes. A good consistency between the Floquet analysis and the 3D DNS results is achieved, indicating that the dominance of the linear instability is responsible for such a three-dimensional flow as the Reynolds number is not far from the critical value. Moreover, we find that the critical Reynolds numbers for the onset of three-dimensional instability fall into the range of 159.7–234.2 by defining this new Reynolds number according to the width of the flow wake. These values are very close to that of a bluff body. Furthermore, we note that the corresponding Strouhal numbers are around 0.17 for all the angles of attack, implying the relevance of three-dimensional instabilities to the wake dynamics or more specifically the fluctuation in the wake.