Bifurcation scenario for a two-dimensional static airfoil exhibiting trailing edge stall

Abstract

We numerically investigate stalling flow around a static airfoil at high Reynolds numbers using the Reynolds-averaged Navier–Stokes equations (RANS) closed with the Spalart–Allmaras turbulence model. An arclength continuation method allows to identify three branches of steady solutions, which form a characteristic inverted S-shaped curve as the angle of attack is varied. Global stability analyses of these steady solutions reveal the existence of two unstable modes: a low-frequency mode, which is unstable for angles of attack in the stall region, and a high-frequency vortex shedding mode, which is unstable at larger angles of attack. The low-frequency stall mode bifurcates several times along the three steady solutions: there are two Hopf bifurcations, two solutions with a two-fold degenerate eigenvalue and two saddle-node bifurcations. This low-frequency mode induces a cyclic flow separation and reattachment along the airfoil. Unsteady simulations of the RANS equations confirm the existence of large-amplitude low-frequency periodic solutions that oscillate around the three steady solutions in phase space. An analysis of the periodic solutions in the phase space shows that, when decreasing the angle of attack, the low-frequency periodic solution collides with the unstable steady middle-branch solution and thus disappears via a homoclinic bifurcation of periodic orbits. Finally, a one-equation nonlinear stall model is introduced to reveal that the disappearance of the limit cycle, when increasing the angle of attack, is due to a saddle-node bifurcation of periodic orbits.

Global stability, sensitivity and passive control of low-Reynolds-number flows around NACA 4412 swept wings

“…This result was found also by Iorio, Gonzalez & Martínez-Cava (2016) and, recently, by Busquet et al. (2021) via global stability analysis in the unsteady Reynolds-averaged Navier-Stokes framework. Similarly, they revealed the existence of two unstable modes: a low-frequency mode, which is unstable for angles of attack in the stall region, and a high-frequency vortex shedding mode, which is unstable at larger angles of attack.…”

Connecting transonic buffet with incompressible low-frequency oscillations on aerofoils

“…As noted in § 1, the results from previous studies show that both flow phenomena occur as global linear instabilities (Crouch et al 2007;Iorio et al 2016;Busquet et al 2021) and are associated with periodic boundary layer separation and reattachment (i.e. switching between stalled and unstalled states), which further corroborate this conclusion.…”

“…Coherent oscillations that occur at a low frequency have also been observed in incompressible flows on aerofoils -the eponymous LFO (Zaman et al 1989;Bragg, Heinrich & Khodadoust 1993;Sandham 2008;Almutairi, Jones & Sandham 2010;Busquet et al 2021). Interestingly, Zaman et al, who were among the first to experimentally and numerically study this phenomenon, noted that these oscillations are hydrodynamic in nature, distinct from a Kàrmàn vortex street, and involve a quasi-periodic switching between stalled and unstalled conditions.…”

“…Almutairi & AlQadi 2013;Almutairi, ElJack & AlQadi 2017) by performing simulations at low free-stream Mach numbers using the same compressible flow solver as the present study. By performing global linear stability analysis of RANS results, Iorio, González & Martínez-Cava (2016) have shown that LFO arise as unstable modes while Busquet et al (2021) have studied hysteresis features. It is interesting to note that LFO has been observed for a wide range of free-stream Reynolds numbers, Re, based on aerofoil chord.…”

“…It is interesting to note that LFO has been observed for a wide range of free-stream Reynolds numbers, Re, based on aerofoil chord. For example, Zaman et al (1989) examined LFO at Re ∼ O(10 5 ), while Bragg et al (1993), Busquet et al (2021) and Iorio et al (2016) studied LFO at Re ∼ O(10 6 ), with the latter exploring a Reynolds number as high as Re = 6 × 10 6 . Although the characteristics of LFO, (i) low frequency, (ii) occurrence close to stall conditions and a behaviour of periodic switching between stalled and unstalled states and (iii) origins as a global instability, are also characteristic of transonic buffet, the connection between the two phenomena has not been clearly explored.…”