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

Published online by Cambridge University Press:  07 October 2013

T. P. Sapsis ,
M. P. Ueckermann  and
P. F. J. Lermusiaux
T. P. Sapsis*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA
M. P. Ueckermann
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
P. F. J. Lermusiaux
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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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.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Instability: Nonlinear instability Turbulent Flows: Turbulent transition
Type
Papers
Information
Journal of Fluid Mechanics , Volume 734 , 10 November 2013 , pp. 83 - 113
Copyright
©2013 Cambridge University Press 

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