Stochastic low-dimensional modelling of a random laminar wake past a circular cylinder

• Published in: Journal of fluid mechanics Vol. 606; pp. 339 - 367
• Main Authors: VENTURI, DANIELE, WAN, XIAOLIANG, KARNIADAKIS, GEORGE EM
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.07.2008
• Subjects: Computational methods in fluid dynamics; Eigenvalues; Exact sciences and technology; Flow system; Fluid dynamics; Fluid mechanics; Fundamental areas of phenomenology (including applications); Laminar flow; Laminar flows; Mathematics; Physics; Stability of laminar flows; Stochastic models; Turbulence; Stochastic model; Modal analysis; Laminar flow; Orthogonal function; Digital simulation; Wakes; Circular cylinder; Eigenfunctions; Modelling; Unsteady flow; Mesh generation
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S0022112008001821

We present a new compact expansion of a random flow field into stochastic spatial modes, hence extending the proper orthogonal decomposition (POD) to noisy (non-coherent) flows. As a prototype problem, we consider unsteady laminar flow past a circular cylinder subject to random inflow characterized as a stationary Gaussian process. We first obtain random snapshots from full stochastic simulations (based on polynomial chaos representations), and subsequently extract a small number of deterministic modes and corresponding stochastic modes by solving a temporal eigenvalue problem. Finally, we determine optimal sets of random projections for the stochastic Navier–Stokes equations, and construct reduced-order stochastic Galerkin models. We show that the number of stochastic modes required in the reconstruction does not directly depend on the dimensionality of the flow system. The framework we propose is general and it may also be useful in analysing turbulent flows, e.g. in quantifying the statistics of energy exchange between coherent modes.