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 predictions. 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 pertaining 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. Consequently, 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.

Modal and non-modal global instability analyses of low-Re massively separated flow around a NACA 0015 airfoil

“…Motivation thus exists to revisit this problem and quantify the relative importance of the possible linear instability mechanisms as a function of flow parameters. Finally, first transient growth analysis results of separated flow around two-dimensional airfoils have been reported by Gioria et al 11,12 on trailing-edge separation at Re = O(10 2 ) and by Loh et al 13 on leading-edge separation bubbles at Re = O(10 4 ). It is interesting to complete the analyses here firstly by assessing the relative importance of modal and non-modal instability mechanisms for this class of flows.…”

“…Ample evidence exists already that the transient growth scenario is active in both lifting surfaces 3 and bluff-bodies, 6 while first non-modal stability analysis results of flow around airfoils at large angles of attack have also been obtained recently. [11][12][13] The steady base flows around the NACA 0015 airfoil monitored in the present work are analyzed with respect to their capacity to sustain transient energy growth at Reynolds numbers slightly below the primary transition to a periodic wake. The choice of range of spanwise wavenumbers monitored, 0 ≤ β ≤ 2π, was based on results of, 12 and the direct-adjoint iteration procedure 16 is performed for a short-time parameter, τ , values, such that the maximum of the growth function G(τ ) is well defined in all spanwise wavenumbers β results.…”

A robust approach for stability analysis of complex flows using high-order Navier-Stokes solvers

“…The main interest in this flow is to identify the primary linear mechanism for unsteadiness, as there are two competing dynamics: amplification of Kelvin-Helmholtz wake mode, and self-excitation of laminar separation bubble appearing as a stationary mode [40]. Though a detailed investigation into the latter requires a three-dimensional stability study, for the current validation purpose we restrict ourselves to two-dimensions in which the wake mode is found to be independent of spanwise wavenumber parameters [41].…”

“…Results from both matrix-forming as well as time-stepper approaches [42,43,41,44] are available for comparison, albeit in an incompressible framework. While earlier studies [42,43] have used a BiGlobal code with conformally mapped curvilinear co-ordinates to obtain eigenmodes, subsequent studies [41,44] use open-source codes, including Nektar++ and Nek5000 for the time-stepper approach, and FreeFEM++ for matrix-forming. In these studies, the Arnoldi subspace iteration is used for the relevant eigenspectrum computation, with the goal of identifying the smallest modulus modes.…”