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Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis

Published online by Cambridge University Press:  15 August 2017

Valentin Resseguier Open the ORCID record for Valentin Resseguier [Opens in a new window] ,
Etienne Mémin ,
Dominique Heitz Open the ORCID record for Dominique Heitz [Opens in a new window]  and
Bertrand Chapron
Valentin Resseguier*
Affiliation:
Fluminance, Inria/Irstea/IRMAR, Campus de Beaulieu, 35042 Rennes, France LOPS, Ifremer, Pointe du Diable, 29280 Plouzané, France
Etienne Mémin
Affiliation:
Fluminance, Inria/Irstea/IRMAR, Campus de Beaulieu, 35042 Rennes, France
Dominique Heitz
Affiliation:
Fluminance, Inria/Irstea/IRMAR, Campus de Beaulieu, 35042 Rennes, France Irstea, UR OPAALE, F-35044 Rennes, France
Bertrand Chapron
Affiliation:
LOPS, Ifremer, Pointe du Diable, 29280 Plouzané, France
*
†Email address for correspondence: [email protected]
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Abstract

We present here a new stochastic modelling approach in the constitution of fluid flow reduced-order models. This framework introduces a spatially inhomogeneous random field to represent the unresolved small-scale velocity component. Such a decomposition of the velocity in terms of a smooth large-scale velocity component and a rough, highly oscillating component gives rise, without any supplementary assumption, to a large-scale flow dynamics that includes a modified advection term together with an inhomogeneous diffusion term. Both of those terms, related respectively to turbophoresis and mixing effects, depend on the variance of the unresolved small-scale velocity component. They bring an explicit subgrid term to the reduced system which enables us to take into account the action of the truncated modes. Besides, a decomposition of the variance tensor in terms of diffusion modes provides a meaningful statistical representation of the stationary or non-stationary structuration of the small-scale velocity and of its action on the resolved modes. This supplies a useful tool for turbulent fluid flow data analysis. We apply this methodology to circular cylinder wake flow at Reynolds numbers $Re=100$ and $Re=3900$. The finite-dimensional models of the wake flows reveal the energy and the anisotropy distributions of the small-scale diffusion modes. These distributions identify critical regions where corrective advection effects, as well as structured energy dissipation effects, take place. In providing rigorously derived subgrid terms, the proposed approach yields accurate and robust temporal reconstruction of the low-dimensional models.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulence modelling Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , pp. 888 - 917
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© 2017 Cambridge University Press 

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