Weakly nonlinear behavior of transonic buffet on airfoils

The occurrence of large scale buffeting flow can result in a reduced operating envelope for an aircraft. In practice, the buffet-onset boundary is defined in terms of finite-amplitude lift fluctuations exceeding a threshold value. For transonic flow conditions, the buffeting flow is associated with oscillations in the shock position that result in large-amplitude lift fluctuations. For both airfoils and wings, these oscillations have been linked to global flow instabilities that arise from a Hopf bifurcation. We employ a combination of numerical simulations and global stability analysis to investigate the near-critical behavior of the oscillatory buffet-onset instability on airfoils. The flow is governed by the unsteady RANS equations, with a basic state provided by a steady RANS solution. In the weakly nonlinear formulation, the disturbance amplitude is described by the Landau equation. The linear growth rate can be determined from either the simulations or the stability analysis, and the Landau constant is derived from simulations resulting in finite-amplitude equilibrium states. The results show that the Landau constant is nearly independent of Mach number and angle of attack for a given airfoil. Using the Landau constant derived from a small number of simulations, the stability analysis can be employed to efficiently capture the essential finite-amplitude behavior needed to estimate the buffet-onset boundary.