A Newton–Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics

The determination of limit-cycles plays an important role in characterizing complex dynamical systems, such as unsteady fluid flows. In practice, dynamical systems are described by models equations involving parameters which are seldom exactly known, leading to parametric uncertainties. These parameters can be suitably modeled as random variables, so if the system possesses almost surely a stable time periodic solution, limit-cycles become stochastic, too. This paper introduces a novel numerical method for the computation of stable stochastic limit-cycles based on the spectral stochastic finite element method with polynomial chaos (PC) expansions. The method is designed to overcome the limitation of PC expansions related to convergence breakdown for long term integration. First, a stochastic time scaling of the model equations is determined to control the phase-drift of the stochastic trajectories and allowing for accurate low order PC expansions. Second, using the rescaled governing equations, we aim at determining a stochastic initial condition and period such that the stochastic trajectories close after the completion of one cycle. The proposed method is implemented and demonstrated on a complex flow problem, modeled by the incompressible Navier-Stokes equations, consisting in the periodic vortex shedding behind a circular cylinder with stochastic inflow conditions. Numerical results are verified by comparison to deterministic reference simulations and demonstrate high accuracy in capturing the stochastic variability of the limit-cycle with respect to the inflow parameters.