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

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 metho...

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Vydané v:	Journal of fluid mechanics Ročník 759; číslo november; s. 404 - 431
Hlavní autori:	Foures, Dimitry P. G., Dovetta, Nicolas, Sipp, Denis, Schmid, Peter J.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cambridge, UK Cambridge University Press 25.11.2014 Elsevier
Predmet:	Algorithms Computational fluid dynamics Computational methods in fluid dynamics Computer simulation Cylinders Data Direct numerical simulation Exact sciences and technology Filtration Flow measurement Fluid dynamics Fluid flow Fluid mechanics Fluids Frameworks Fundamental areas of phenomenology (including applications) Identification methods Lagrange multiplier Mathematical models Mechanics Methods Navier-Stokes equations NMR Nuclear magnetic resonance Physics Problems Reconstruction Reynolds averaged Navier-Stokes method Reynolds number Reynolds stresses Stokes law (fluid mechanics) Two dimensional flow United Kingdom > UK France variational methods general fluid mechanics turbulence modelling Lagrange multiplier Algorithms Computational fluid dynamics Variational methods Digital simulation Modelling System identification Data assimilation Navier-Stokes equations
ISSN:	0022-1120, 0377-0257, 1469-7645
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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.
AbstractList	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 [formula omitted: see PDF] . 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. 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. 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 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.
Author	Dovetta, Nicolas Sipp, Denis Foures, Dimitry P. G. Schmid, Peter J.
Author_xml	– sequence: 1 givenname: Dimitry P. G. surname: Foures fullname: Foures, Dimitry P. G. organization: DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, UK – sequence: 2 givenname: Nicolas surname: Dovetta fullname: Dovetta, Nicolas organization: LadHyX, Ecole Polytechnique, 91128 Palaiseau, France – sequence: 3 givenname: Denis surname: Sipp fullname: Sipp, Denis email: denis.sipp@onera.fr organization: ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France – sequence: 4 givenname: Peter J. surname: Schmid fullname: Schmid, Peter J. organization: Department of Mathematics, Imperial College London, London SW7 2AZ, UK
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Keywords	variational methods general fluid mechanics turbulence modelling Lagrange multiplier Algorithms Computational fluid dynamics Variational methods Digital simulation Modelling System identification Data assimilation Navier-Stokes equations
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Snippet	We present a data-assimilation technique based on a variational formulation and a Lagrange multipliers approach to enforce the Navier–Stokes equations. A... We present a data-assimilation technique based on a variational formulation and a Lagrange multipliers approach to enforce the Navier-Stokes equations. A...
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SubjectTerms	Algorithms Computational fluid dynamics Computational methods in fluid dynamics Computer simulation Cylinders Data Direct numerical simulation Exact sciences and technology Filtration Flow measurement Fluid dynamics Fluid flow Fluid mechanics Fluids Frameworks Fundamental areas of phenomenology (including applications) Identification methods Lagrange multiplier Mathematical models Mechanics Methods Navier-Stokes equations NMR Nuclear magnetic resonance Physics Problems Reconstruction Reynolds averaged Navier-Stokes method Reynolds number Reynolds stresses Stokes law (fluid mechanics) Two dimensional flow
Title	A data-assimilation method for Reynolds-averaged Navier–Stokes-driven mean flow reconstruction
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