Stochastic stability and dynamics of a two-dimensional structurally nonlinear airfoil in turbulent flow

The present study considers the application of stochastic dimensional reduction (low-dimensional approximation of stochastic dynamical systems) to a 11-dimensional nonlinear aeroelastic problem exhibiting a Hopf bifurcation, with one critical mode and several stable modes. The analysis is performed close to the critical value of the bifurcation parameter (the freestream airspeed) that induces flutter in a 2-D airfoil. The system is excited by multiplicative and additive real noise processes whose power spectral densities are given by the Dryden wind turbulence model. The homogenization procedure yields a two dimensional Markov process characterized by a generator. Further simplification yields a one dimensional stochastic differential equation that characterizes the amplitude of the critical mode of the original system. This simplified low-dimensional coarse-grained model, which captures the essential stochastic dynamics close to flutter instability, is used to efficiently simulate the long-term statistics of the slow variables. The explicit forms of the homogenized drift and diffusion coefficients of the reduced stochastic differential equation are determined. The explicit formulas contain both the stochastic perturbations in the unstable and stable modes as well as the action of the nonlinear terms. The reduced order (coarse-grained) model is verified by comparison of distribution functions, obtained computationally, with the original system. Additionally, the top Lyapunov exponent found analytically compares well with the exponent obtained by numerical experiments using the original system. This analysis provides a transparent medium for applying the homogenization procedure and may be of interest to aircraft designers.