Algebraic transitional extensions for the accurate computation of laminar separation bubbles are developed for use with k-ω models. The transitional model, based on integral boundary-layer parameters and transition criteria, introduces streamwise-variable weighting of the production terms in the k-ω equations and delay of the shear-stresslimiter activation. Calibration to fit available DNS data yields satisfactory results for a flat-plate reference case, both for skin-friction and velocity-profiles. It is shown that the model can be adapted to different k-ω variants by appropriate calibration of coefficients. The model is then validated for 2 airfoil test-cases, both for very low and higher Reynolds numbers, a NACA 0012 airfoil at chord-based Reynolds number Re c = 10 5 and angle-of-attack α = 10.55 deg and a S809 airfoil at Re c = 2 × 10 6 and α = 1 deg. For both application cases comparison with available data is satisfactory. Nomenclature c = airfoil chord length (m) C lim = shear-stress limiter delay function Cp = pressure coefficient

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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.