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A data-assimilation method for Reynolds-averaged Navier–Stokes-driven mean flow reconstruction

Published online by Cambridge University Press:  04 November 2014

Dimitry P. G. Foures ,
Nicolas Dovetta ,
Denis Sipp  and
Peter J. Schmid
Dimitry P. G. Foures
Affiliation:
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, UK
Nicolas Dovetta
Affiliation:
LadHyX, Ecole Polytechnique, 91128 Palaiseau, France
Denis Sipp*
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: denis.sipp@onera.fr
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Abstract

We present a data-assimilation technique based on a variational formulation and a Lagrange multipliers approach to enforce the Navier–Stokes equations. A general operator (referred to as the measure operator) is defined in order to mathematically describe an experimental measure. The presented method is applied to the case of mean flow measurements. Such a flow can be described by the Reynolds-averaged Navier–Stokes (RANS) equations, which can be formulated as the classical Navier–Stokes equations driven by a forcing term involving the Reynolds stresses. The stress term is an unknown of the equations and is thus chosen as the control parameter in our study. The data-assimilation algorithm is derived to minimize the error between a mean flow measurement and the measure performed on a numerical solution of the steady, forced Navier–Stokes equations; the optimal forcing is found when this error is minimal. We demonstrate the developed data-assimilation framework on a test case: the two-dimensional flow around an infinite cylinder at a Reynolds number of $\mathit{Re}=150$. The mean flow is computed by time-averaging instantaneous flow fields from a direct numerical simulation (DNS). We then perform several ‘measures’ on this mean flow and apply the data-assimilation method to reconstruct the full mean flow field. Spatial interpolation, extrapolation, state vector reconstruction and noise filtering are considered independently. The efficacy of the developed identification algorithm is quantified for each of these cases and compared with more traditional methods when possible. We also analyse the identified forcing in terms of unsteadiness characterization, present a way to recover the second-order statistical moments of the fluctuating velocities and finally explore the possibility of pressure reconstruction from velocity measurements.

JFM classification

Mathematical Foundations: General fluid mechanics Turbulent Flows: Turbulence modelling Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 759 , 25 November 2014 , pp. 404 - 431
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Copyright
© 2014 Cambridge University Press 

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