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Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows

Published online by Cambridge University Press:  04 February 2015

Bartosz Protas*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada
Bernd R. Noack
Affiliation:
Institut PPRIME, CNRS - Université de Poitiers - ENSMA, UPR 3346, Départment Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
Jan Östh
Affiliation:
Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
*
Email address for correspondence: bprotas@mcmaster.ca

Abstract

We propose a variational approach to the identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for proper orthogonal decomposition (POD) models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at $\mathit{Re}=500$ (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at $\mathit{Re}=300\,000$ (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark proposed recently by Östh et al. (J. Fluid Mech., vol. 747, 2014, pp. 518–544). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.

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Copyright
© 2015 Cambridge University Press 

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References

Adams, R. A. & Fournier, J. F. 2005 Sobolev Spaces. Elsevier.Google Scholar
Artana, G., Cammilleri, A., Carlier, J. & Mémin, E. 2012 Strong and weak constraint variational assimilations for reduced-order fluid flow modeling. J. Comput. Phys. 231, 32643288.CrossRefGoogle Scholar
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115173.CrossRefGoogle Scholar
Balajewicz, M., Dowell, E. H. & Noack, B. R. 2013 Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation. J. Fluid Mech. 729, 285308.CrossRefGoogle Scholar
Berger, M. S. 1977 Nonlinearity and Functional Analysis. Academic.Google Scholar
Bukshtynov, V. & Protas, B. 2013 Optimal reconstruction of material properties in complex multiphysics phenomena. J. Comput. Phys. 242, 889914.CrossRefGoogle Scholar
Bukshtynov, V., Volkov, O. & Protas, B. 2011 On optimal reconstruction of constitutive relations. Physica D 240, 12281244.CrossRefGoogle Scholar
Cacuci, D. G., Navon, I. M. & Ionescu-Bujor, M. 2013 Computational Methods for Data Evaluation and Assimilation. Chapman & Hall.Google Scholar
Cordier, L., Noack, B. R., Daviller, G., Delvile, J., Lehnasch, G., Tissot, G., Balajewicz, M. & Niven, R. K. 2013 Control-oriented model identification strategy. Exp. Fluids 54, 1580.CrossRefGoogle Scholar
D’Adamo, J., Papadakis, N., Mémin, E. & Artana, G. 2007 Variational assimilation of POD low-order dynamical systems. J. Turbul. 9, 122.Google Scholar
Deane, A. E., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids A 3, 23372354.CrossRefGoogle Scholar
Fletcher, C. A. J. 1984 Computational Galerkin Methods, 1st edn. Springer.CrossRefGoogle Scholar
Galletti, G., Bruneau, C. H., Zannetti, L. & Iollo, A. 2004 Low-order modelling of laminar flow regimes past a confined square cylinder. J. Fluid Mech. 503, 161170.CrossRefGoogle Scholar
Gunzburger, M. D. 2003 Perspectives in Flow Control and Optimization. SIAM.Google Scholar
Hemati, M. S., Eldredge, J. D. & Speyer, J. L. 2014 Improving vortex models via optimal control theory. J. Fluids Struct. 49, 91111.CrossRefGoogle Scholar
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Kaiser, E., Noack, B. R., Cordier, L., Spohn, A., Segond, M., Abel, M. W., Daviller, G., Östh, J., Krajnović, S. & Niven, R. K. 2014 Cluster-based reduced-order modelling of a mixing layer. J. Fluid Mech. 754, 365414.CrossRefGoogle Scholar
Kalnay, E. 2003 Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press.Google Scholar
Kasten, J., Reininghaus, J., Hotz, I., Hege, H.-C., Noack, B. R., Daviller, G., Comte, P. & Morzyński, M.2014 Acceleration feature points of unsteady shear flows. Preprint arXiv:1401.2462 [physics.fui-dyn].Google Scholar
Kutz, J. N. 2013 Data-Driven Modeling and Scientific Computation: Methods for Complex Systems and Big Data. Oxford University Press.Google Scholar
Lamb, S. H. 1945 Hydrodynamics, 6th edn. Dover.Google Scholar
Langford, J. A. & Moser, R. D. 1999 Optimal LES formulations for isotropic turbulence. J. Fluid Mech. 398, 321346.CrossRefGoogle Scholar
Liberzon, A., Lüthi, B., Guala, M., Kinzelbach, W. & Tsinober, A. 2007 On anisotropy of turbulent flows in regions of ‘negative eddy viscosity’. In Progress in Turbulence II, Proceedings of the iTi Conference in Turbulence 2005 (ed. Oberlack, M., Khujadze, G., Günther, S., Weller, T., Frewer, M., Peinke, J. & Barth, S.), Springer Proceedings in Physics, vol. 109. Springer.Google Scholar
Luenberger, D. 1969 Optimization by Vector Space Methods. John Wiley and Sons.Google Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic.Google Scholar
Ma, X. & Karniadakis, G. E. 2002 A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458, 181190.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335363.CrossRefGoogle Scholar
Noack, B. R., Morzyński, M. & Tadmor, G. 2011 Reduced-Order Modelling for Flow Control. CISM Courses and Lectures, vol. 528. Springer.CrossRefGoogle Scholar
Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339365.CrossRefGoogle Scholar
Nocedal, J. & Wright, S. 2002 Numerical Optimization. Springer.Google Scholar
Orfanidis, S. J. 1996 Optimum Signal Processing. An Introduction, 2nd edn. Prentice-Hall.Google Scholar
Östh, J., Noack, B. R., Krajnović, S., Barros, D. & Borée, J. 2014 On the need for a nonlinear subscale turbulence term in pod models as exemplified for a high Reynolds number flow over an ahmed body. J. Fluid Mech. 747, 518544.CrossRefGoogle Scholar
Podvin, B. 2009 A proper-orthogonal-decomposition based model for the wall layer of a turbulent channel flow. Phys. Fluids 21, 015111.CrossRefGoogle Scholar
Press, W. H., Flanner, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes: the Art of Scientific Computations. Cambridge University Press.Google Scholar
Protas, B., Bewley, T. & Hagen, G. 2004 A comprehensive framework for the regularization of adjoint analysis in multiscale PDE systems. J. Comput. Phys. 195, 4989.CrossRefGoogle Scholar
Protas, B., Noack, B. R. & Morzynski, M. 2014 An optimal model identification for oscillatory dynamics with a stable limit cycle. J. Nonlinear Sci. 24, 245275.CrossRefGoogle Scholar
Rempfer, D. & Fasel, F. H. 1994a Evolution of three-dimensional coherent structures in a flat-plate boundary-layer. J. Fluid Mech. 260, 351375.CrossRefGoogle Scholar
Rempfer, D. & Fasel, F. H. 1994b Dynamics of three-dimensional coherent structures in a flat-plate boundary-layer. J. Fluid Mech. 275, 257283.CrossRefGoogle Scholar
Sapsis, T. P. & Majda, A. J. 2013 Statistically accurate low-order models for uncertainty quantification in turbulent dynamical systems. Proc. Natl Acad. Sci. USA 110, 1370513710.CrossRefGoogle ScholarPubMed
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures, Part I: coherent structures. Q. Appl. Maths 45, 561571.CrossRefGoogle Scholar
Van Driest, E. R. 1956 On turbulent flow near a wall. J. Aero. Sci. 23, 1007.CrossRefGoogle Scholar
Wang, Z., Akhtar, I., Borggaard, J. & Iliescu, T. 2011 Two-level discretizations of nonlinear closure models for proper orthogonal decomposition. J. Comput. Phys. 230, 126146.CrossRefGoogle Scholar
Wang, Z., Akhtar, I., Borggaard, J. & Iliescu, T. 2012 Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison. Comput. Meth. Appl. Mech. Engng 237–240, 1026.CrossRefGoogle Scholar
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