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Nudging-based data assimilation of the turbulent flow around a square cylinder

Published online by Cambridge University Press:  03 March 2022

M. Zauner ,
V. Mons Open the ORCID record for V. Mons [Opens in a new window] ,
O. Marquet Open the ORCID record for O. Marquet [Opens in a new window]  and
B. Leclaire
M. Zauner
Affiliation:
DAAA, ONERA, Université Paris Saclay, F-92190 Meudon, France
V. Mons*
Affiliation:
DAAA, ONERA, Université Paris Saclay, F-92190 Meudon, France
O. Marquet
Affiliation:
DAAA, ONERA, Université Paris Saclay, F-92190 Meudon, France
B. Leclaire
Affiliation:
DAAA, ONERA, Université Paris Saclay, F-92190 Meudon, France
*
†Email address for correspondence: vincent.mons@onera.fr
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Abstract

We investigate the estimation of the turbulent flow around a canonical square cylinder at $Re=22\,000$ based on temporally resolved but spatially sparse velocity data and solving the unsteady Reynolds-averaged Navier–Stokes (URANS) equations. Flow reconstruction from sparse data is achieved through the application of a nudging data assimilation technique. It involves the introduction of a feedback control term in the momentum equations which allows us to drive URANS predictions towards reference data, which are here extracted from a direct numerical simulation. Such a data assimilation approach induces negligible supplementary computational cost compared with that of a standard URANS simulation. The influence of the spatial resolution of the reference data on the reconstruction performances is systematically investigated. Using a spacing of the order of one cylinder length between data points, we already observe synchronisation of the low-frequency vortex shedding between the full reference flow and the one that is estimated by URANS. The present data assimilation procedure allows us to compensate for deficiencies in standard URANS calculations and leads to a significant decrease in temporal and spectral errors as computed by spectral proper orthogonal decomposition. Furthermore, high accuracy in terms of mean-flow prediction by URANS is achieved. When considering spacings between measurements that are of the order of the wavelength of the Kelvin–Helmholtz vortices, such phenomena in the shear layers at the top and bottom of the cylinder are correctly estimated, while they are not self-sustained in standard URANS. The influence of the structure of the feedback control term in the data assimilation procedure is also investigated.

JFM classification

Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulence simulation Wakes/Jets: Vortex streets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , A38
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Azouani, A., Olson, E. & Titi, E.S. 2014 Continuous data assimilation using general interpolant observables. J. Nonlinear Sci. 24, 277–304.CrossRefGoogle Scholar
Azouani, A. & Titi, E.S. 2014 Feedback control of nonlinear dissipative systems by finite determining parameters – a reaction-diffusion paradigm. Evol. Equ. Control Theory 3 (4), 579–594.CrossRefGoogle Scholar
Brooks, A.N. & Hughes, T.J.R. 1982 Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Meth. Appl. Mech. Engng 32, 199–259.CrossRefGoogle Scholar
Buzzicotti, M. & Di Leoni, P.C. 2020 Synchronizing subgrid scale models of turbulence to data. Phys. Fluids 32, 125116.CrossRefGoogle Scholar
Chandramouli, P., Mémin, E. & Heitz, D. 2020 4D large scale variational data assimilation of a turbulent flow with a dynamics error model. J. Comput. Phys. 412, 109446.CrossRefGoogle Scholar
Chaouat, B. 2017 The state of the art of hybrid RANS/LES modeling for the simulation of turbulent flows. Flow Turbul. Combust. 99, 279–327.CrossRefGoogle ScholarPubMed
Chaouat, B. & Schiestel, R. 2005 A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Phys. Fluids 17, 065106.CrossRefGoogle Scholar
Colburn, C.H., Cessna, J.B. & Bewley, T.R. 2011 State estimation in wall-bounded flow systems. Part 3. The ensemble Kalman filter. J. Fluid Mech. 682, 289–303.CrossRefGoogle Scholar
Da Silva, A.F.C. & Colonius, T. 2020 Flow state estimation in the presence of discretization errors. J. Fluid Mech. 890, A10.CrossRefGoogle Scholar
Dandois, J., Mary, I. & Brion, V. 2018 Large-eddy simulation of laminar transonic buffet. J. Fluid Mech. 850, 156–178.CrossRefGoogle Scholar
Di Leoni, P.C., Mazzino, A. & Biferale, L. 2018 Inferring flow parameters and turbulent configuration with physics-informed data assimilation and spectral nudging. Phys. Rev. Fluids 3, 104604.CrossRefGoogle Scholar
Di Leoni, P.C., Mazzino, A. & Biferale, L. 2020 Synchronization to big data: nudging the Navier–Stokes equations for data assimilation of turbulent flows. Phys. Rev. X 10 (1), 011023.Google Scholar
Evensen, G. 2009 Data Assimilation: The Ensemble Kalman Filter, 2nd edn. Springer.CrossRefGoogle Scholar
Franceschini, L., Sipp, D. & Marquet, O. 2020 Mean-flow data assimilation based on minimal correction of turbulence models: application to turbulent high-Reynolds number backward-facing step. Phys. Rev. Fluids 5, 094603.CrossRefGoogle Scholar
Girimaji, S.S. 2006 Partially-averaged Navier–Stokes model for turbulence: a Reynolds-averaged Navier–Stokes to direct numerical simulation bridging method. Trans. ASME J. Appl. Mech. 73, 413–421.CrossRefGoogle Scholar
Hayase, T. 2015 a A review of measurement-integrated simulation of complex real flows. J. Flow Control Measure. Visual. 03 (02), 51–66.CrossRefGoogle Scholar
Hayase, T. 2015 b Numerical simulation of real-world flows. Fluid Dyn. Res. 47, 051201.CrossRefGoogle Scholar
Hayase, T. & Hayashi, S. 1997 State estimator of flow as an integrated computational method with the feedback of online experimental measurement. Trans. ASME J. Fluids Engng 119 (4), 814–822.CrossRefGoogle Scholar
Hecht, F. 2012 New development in FreeFem$++$. J. Numer. Maths 20, 251–265.Google Scholar
Hoke, J.E. & Anthes, R.A. 1976 The initialization of numerical models by a dynamic-initialization technique. Mon. Weath. Rev. 104, 1551–1556.2.0.CO;2>CrossRefGoogle Scholar
Iaccarino, G., Ooi, A., Durbin, P.A. & Behnia, M. 2003 Reynolds averaged simulation of unsteady separated flow. Intl J. Heat Fluid Flow 24, 147–156.CrossRefGoogle Scholar
Imagawa, K. & Hayase, T. 2010 Numerical experiment of measurement-integrated simulation to reproduce turbulent flows with feedback loop to dynamically compensate the solution using real flow information. Comput. Fluids 39 (9), 1439–1450.CrossRefGoogle Scholar
Lakshmivarahan, S. & Lewis, J.M. 2013 Nudging methods: a critical overview. In Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (II) (ed. S.K. Park & L. Xu), vol. II, pp. 27–57. Springer.CrossRefGoogle Scholar
Le Dimet, F.-X. & Talagrand, O. 1986 Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus 38A, 97–110.CrossRefGoogle Scholar
Lewis, F.L., Xie, L. & Popa, D. 2008 Optimal and Robust Estimation: With an Introduction to Stochastic Control Theory, 2nd edn. CRC.Google Scholar
Li, Y., Zhang, J., Dong, G. & Abdullah, N.S. 2020 Small-scale reconstruction in three-dimensional Kolmogorov flows using four-dimensional variational data assimilation. J. Fluid Mech. 885, A9.CrossRefGoogle Scholar
Lyn, D.A., Einav, S., Rodi, W. & Park, J.-H. 1995 A laser-Doppler velocimetry study of ensemble-averaged characteristics of the turbulent near wake of a square cylinder. J. Fluid Mech. 304, 285–319.CrossRefGoogle Scholar
Mary, I. & Sagaut, P. 2002 Large eddy simulation of flow around an airfoil near stall. AIAA J. 40, 1139–1145.CrossRefGoogle Scholar
Menter, F.R. & Egorov, Y. 2010 The scale-adaptive simulation method for unsteady turbulent flow predictions. Part 1: theory and model description. Flow Turbul. Combust. 85, 113–138.CrossRefGoogle Scholar
Menter, F.R., Garbaruk, A., Smirnov, P., Cokljat, D. & Mathey, F. 2010 Scale-adaptive simulation with artificial forcing. In Progress in Hybrid RANS-LES Modelling (ed. S.-H. Peng, P. Doerffer & W. Haase), pp. 235–246. Springer.CrossRefGoogle Scholar
Moldodovan, G., Lehnasch, G., Cordier, L. & Meldi, M. 2021 A multigrid/ensemble Kalman filter strategy for assimilation of unsteady flows. J. Comput. Phys. 443, 110481.CrossRefGoogle Scholar
Mons, V., Chassaing, J.-C., Gomez, T. & Sagaut, P. 2016 Reconstruction of unsteady viscous flows using data assimilation schemes. J. Comput. Phys. 316, 255–280.CrossRefGoogle Scholar
Nakao, M., Kawashima, K. & Kagawa, T. 2009 Application of MI simulation using a turbulent model for unsteady orifice flow. Trans. ASME J. Fluids Engng 131 (11), 1114011–1114016.CrossRefGoogle Scholar
Neeteson, N.J. & Rival, D.E. 2020 State observer-based data assimilation: a PID control-inspired observer in the pressure equation. Meas. Sci. Technol. 31, 014003.CrossRefGoogle Scholar
Nisugi, K., Hayase, T. & Shirai, A. 2004 Fundamental study of hybrid wind tunnel integrating numerical simulation and experiment in analysis of flow field. JSME Intl J. 47 (3), 593–604.CrossRefGoogle Scholar
Olshanskii, M., Lube, G., Heister, T. & Löwe, J. 2009 Grad–div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations. Comput. Meth. Appl. Mech. Engng 198, 3975–3988.CrossRefGoogle Scholar
Palkin, E., Mullyadzhanov, R., Hadziabdić, M. & Hanjalić, K. 2016 Scrutinizing URANS in shedding flows: the case of cylinder in cross-flow in the subcritical regime. Flow Turbul. Combust. 97, 1017–1046.CrossRefGoogle Scholar
Raffel, M., Willert, C.E., Scarano, F., Kähler, C., Wereley, S.T. & Kompenhans, J. 2018 Particle Image Velocimetry: A Practical Guide. Springer.CrossRefGoogle Scholar
Sagaut, P., Deck, S. & Terracol, M. 2013 Multiscale and Multiresolution Approaches in Turbulence. Imperial College Press.CrossRefGoogle Scholar
Saredi, E., Ramesh, N.T., Sciacchitano, A. & Scarano, F. 2021 State observer data assimilation for RANS with time-averaged 3D-PIV data. Comput. Fluids 218, 104827.CrossRefGoogle Scholar
Spalart, P.R. & Allmaras, S.R. 1994 A one-equation turbulence model for aerodynamic flows. Rech. Aerosp. 1, 5–21.Google Scholar
Suzuki, T. 2012 Reduced-order Kalman-filtered hybrid simulation combining particle tracking velocimetry and direct numerical simulation. J. Fluid Mech. 709, 249–288.CrossRefGoogle Scholar
Suzuki, T. & Hasegawa, Y. 2017 Estimation of turbulent channel flow at $Re_{\tau }=100$ based on the wall measurement using a simple sequential approach. J. Fluid Mech. 830, 760–796.CrossRefGoogle Scholar
Suzuki, T., Ji, H. & Yamamoto, F. 2009 Unsteady PTV velocity field past an airfoil solved with DNS: part 1. Algorithm of hybrid simulation and hybrid velocity field at $Re \simeq 10^{3}$. Exp. Fluids 47, 957–976.CrossRefGoogle Scholar
Towne, A., Schmidt, O.T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821–867.CrossRefGoogle Scholar
Trias, F.X., Gorobets, A. & Oliva, A. 2015 Turbulent flow around a square cylinder at Reynolds number 22 000: a DNS study. Comput. Fluids 123, 87–98.CrossRefGoogle Scholar
Wang, M. & Zaki, T.A. 2021 State estimation in turbulent channel flow from limited observations. J. Fluid Mech. 917, A9.CrossRefGoogle Scholar
Yamagata, T., Hayase, T. & Higuchi, H. 2008 Effect of feedback data rate in PIV measurement-integrated simulation. J. Fluid Sci. Technol. 3 (4), 477–487.CrossRefGoogle Scholar