Global analysis of Navier–Stokes and Boussinesq stochastic flows using dynamical orthogonality

Abstract We provide a new framework for the study of fluid flows presenting complex uncertain behaviour. Our approach is based on the stochastic reduction and analysis of the governing equations using the dynamically orthogonal field equations. By numerically solving these equations, we evolve in a fully coupled way the mean flow and the statistical and spatial characteristics of the stochastic fluctuations. This set of equations is formulated for the general case of stochastic boundary conditions and allows for the application of projection methods that considerably reduce the computational cost. We analyse the transformation of energy from stochastic modes to mean dynamics, and vice versa, by deriving exact expressions that quantify the interaction among different components of the flow. The developed framework is illustrated through specific flows in unstable regimes. In particular, we consider the flow behind a disk and the Rayleigh–Bénard convection, for which we construct bifurcation diagrams that describe the variation of the response as well as the energy transfers for different parameters associated with the considered flows. We reveal the low dimensionality of the underlying stochastic attractor.

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