Hopf-Bifurcation Analysis of Airfoil Flutter at Transonic Speeds

Abstract

Hopf-bifurcation analysis is used to determine flutter onset for a pitch-and-plunge airfoil at transonic Mach number conditions. The pitch-and-plunge model is a coupling of the Euler equations and a twodegree-of-freedom structural model composed of linear and torsional springs. The Euler equations are discretized using the upwind total variation diminishing scheme of Harten and Yee. Equilibrium solutions of the aeroelastic model are computed using Newton's method, and dynamic solutions are explicitly integrated in time with first-order accuracy. The Hopf-bifurcation point, which models the flutter condition, is computed directly using a modified form of the Griewank and Reddien algorithm. A path of Hopf points is computed as a function of Mach number to produce a Mach flutter boundary. The flutter boundary is validated by time integration. Flutter boundaries are also obtained through variation of static pretwist and pitch and plunge damping. The direct, Hopf-point method is found to be precise and efficient for grids typical of inviscid, transonic airfoil calculations.