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

Abstract

Linear instability mechanisms of steady and time-periodic spanwise homogeneous separated flows over an airfoil profile, placed at large angles of attack against the oncoming flow, have been investigated using linear global theory. Large-scale steady and unsteady separation occurs at the conditions investigated, such that Tollmien-Schlichting (TS) and crossflow mechanisms were not considered in the present analysis. Three-dimensional global modal and non-modal analyses of steady laminar two-dimensional flow, as well as temporal Floquet analysis of the time-periodic flow set up as a consequence of the first two-dimensional bifurcation, have been performed.Non-modal analysis of the steady laminar massively separated flow over the airfoil reveals the potential of the flow to sustain transient growth which becomes stronger with increasing angle of attack and Reynolds number. Optimal initial conditions are computed and are shown to evolve into two-and three-dimensional Kelvin-Helmholtz (KH) modes as the time-horizon is increased. Primary modal analysis of the same steady flow reveals that the leading modal instability is that associated with the KH mechanism, taking the form of the eigenmodes known from analysis of generic bluff bodies. The three-dimensional stationary eigenmode of the two-dimensional separation bubble, associated in previous analyses with the formation of stall-cell patterns on the airfoil surface, is also recovered here, but is shown to be stronger damped than the KH mode. Secondary three-dimensional Floquet instability analysis of the periodic wake ensuing linear amplification of the two-dimensional KH mode delivers long-and short-wavelength modes amplifying in the opposite sense as those of the cylinder. The three-dimensional wall-streamline pattern corresponding to the strongest amplified secondary Floquet eigenmode is akin to the periodic separated flow cellular structures on airfoils close to stall. Unlike the analogous pattern associated with the stationary three-dimensional mode, discussed in earlier work, the mechanism revealed in this work can persist at Reynolds numbers approaching those of gliders in flight.

Transonic buffet instability: From two-dimensional airfoils to three-dimensional swept wings

“…This shape is close to the shape of the physical phenomenon of stall cells which is better known at low speed, for instance, from the work of Rodriguez and Theofilis [19], who showed for incompressible flows that these cells originate from a 3D global unstable mode. The explication of the stall cells phenomenon by an unstable global mode was then questioned by a work of the same authors [20], where the mode appears to move towards unstable values but remains stable.…”

Spatio-temporal flow structures in the three-dimensional wake of a circular cylinder

“…As seen, the maximum value for the Floquet multipliers in modes A and B are: μ crit =1.755 for β crit =1.792 in mode A and μ crit =7.570 for β crit =1.464 in mode B. The base flow computations and Floquet stability analysis followed in this work are described more in detail, in the case of numerical analysis around airfoils, in He et al (2017), Gioria et al (2016).…”