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A Newton-Galerkin method for fluid flow exhibiting uncertain periodic dynamics

Bibliography:

M. Schick, V. Heuveline and O.P. Le Maître, A Newton-Galerkin method for fluid flow exhibiting uncertain periodic dynamics, J. Uncertainty Quantification, SIAM-ASA Journal of Uncertainty Quantification, Vol 2, pp. 153-173, (2014).

Authors:

M. Schick, V. Heuveline and O.P. Le Maître

Keywords:

uncertainty quantification, stochastic limit-cycle, stochastic Navier-Stokes equations, stochastic period, polynomial chaos, long term integration

Year:

2014

Abstract:

The determination of stable limit-cycles plays an important role for quantifying the characteristics of dynamical systems. In practice exact knowledge of model parameters is rarely available leading to parameter uncertainties, which can be modeled as an input of random variables. This has the effect that the limit-cycles become stochastic themselves resulting in almost surely time-periodic solutions with a stochastic period. In this paper we introduce a novel numerical method for the computation of stable stochastic limit-cycles based on the Spectral-Stochastic-Finite-Element-Method using Polynomial Chaos (PC). We are able to overcome the difficulties of Polynomial Chaos regarding its well known convergence breakdown for long term integration. To this end, we introduce a stochastic time scaling which treats the stochastic period as an additional random variable and controls the phase-drift of the stochastic trajectories, keeping the necessary PC order low. Based on the re-scaled governing equations, we aim at determining an initial condition and a period such that the trajectories close after completion of one stochastic cycle. Furthermore, we verify the numerical method by computation of a vortex shedding of a ow around a circular domain with stochastic inflow boundary conditions as a benchmark problem. The results are verifid by comparison to
purely deterministic reference problems and demonstrate high accuracy up to machine precision in
capturing the stochastic variations of the limit-cycle.

ISSN:

DOI. 10.1137/130908919