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Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation

Published online by Cambridge University Press:  18 July 2013

Maciej J. Balajewicz ,
Earl H. Dowell  and
Bernd R. Noack
Maciej J. Balajewicz*
Affiliation:
Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA
Earl H. Dowell
Affiliation:
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA
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
*
†Email address for correspondence: maciej.balajewicz@stanford.edu
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Abstract

We generalize the POD-based Galerkin method for post-transient flow data by incorporating Navier–Stokes equation constraints. In this method, the derived Galerkin expansion minimizes the residual like POD, but with the power balance equation for the resolved turbulent kinetic energy as an additional optimization constraint. Thus, the projection of the Navier–Stokes equation on to the expansion modes yields a Galerkin system that respects the power balance on the attractor. The resulting dynamical system requires no stabilizing eddy-viscosity term – contrary to other POD models of high-Reynolds-number flows. The proposed Galerkin method is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer. Generalizations for more Navier–Stokes constraints, e.g. Reynolds equations, can be achieved in straightforward variation of the presented results.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulence modelling
Type
Papers
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , pp. 285 - 308
Copyright
©2013 Cambridge University Press 

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References

Amsallem, D. & Farhat, C. 2011 Stabilization of projection-based reduced-order models. Intl J. Numer. Meth. Engng 91 (4), 358–377.Google 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 (115), 115–173.Google Scholar
Bailon-Cuba, J., Shishkina, O., Wagner, C. & Schumacher, J. 2012 Low-dimensional model of turbulent mixed convection in a complex domain. Phys. Fluids 24, 107101.CrossRefGoogle Scholar
Bogey, C. 2000 Calcul direct du bruit aérodynamique et validation de modèles acoustiques hybrides. PhD thesis, Ecole Centrale Lyon, France.Google Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.Google Scholar
Brent, R. P. 2002 Algorithms for Minimization without Derivatives. Dover.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1991 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2006 Spectral Methods: Fundamentals in Single Domains. Springer.Google Scholar
Cavalieri, A., Daviller, G., Comte, P., Jordan, P., Tadmor, G. & Gervais, Y. 2011 Using large eddy simulation to explore sound-source mechanisms in jets. J. Sound Vib. 330, 4098–4113.Google Scholar
Cazemier, W., Verstappen, R. W. C. P. & Veldman, A. E. P. 1998 Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids 10 (7), 1685–1699.CrossRefGoogle Scholar
Colonius, T., Lele, S. K. & Moin, P. 1993 Direct computation of the sound generated by two-dimensional shear layer. In 15th AIAA Aeroacoustics Conference. Long Beach.Google Scholar
Cordier, L., Noack, B. R., Daviller, G., Delville, J., Lehnasch, G., Tissot, G., Balajewicz, M. & Niven, R. K. 2013 Control-oriented model identification strategy. Exp. Fluids (submitted).Google Scholar
Daviller, W. 2010 Numerical study of temperature effects in single and coaxial jets. PhD thesis, École Nationale Supérieure de Mécanique et d’Aérotechnique (ENSMA), Poitiers, France.Google Scholar
Demmel, J. W. 1997 Applied Numerical Linear Algebra. SIAM.CrossRefGoogle Scholar
Dixon, L. C. W. & Szegö, G. P. 1975 Towards Global Optimisation: Proceedings of a Workshop at the University of Cagliari, Italy, October 1974. North Holland.Google Scholar
Fletcher, C. A. J. 1984 Computational Galerkin Methods. Springer.Google Scholar
Galletti, B., Bruneau, C. H., Zannetti, L. & Iollo, A. 2004 Low-order modelling of laminar flow regimes past a confined square cylinder. J. Fluid Mech. 503, 161–170.Google Scholar
Gottlieb, D. & Turkel, E. 1976 Dissipative two–four method for time dependent problems. Maths Comput. 30 (136), 703–723.Google Scholar
Hayder, M. E. & Turkel, E. 1993 High-order accurate solutions of viscous problem. In AIAA paper 93-3074.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.Google Scholar
Iliescu, T. & Wang, Z. 2012 Variational multiscale proper orthogonal decomposition: Navier–Stokes equations. Preprint arXiv: 1210.7389.Google Scholar
Iollo, A., Lanteri, S. & Désidéri, J.-A. 2000 Stability properties of Pod–Galerkin approximations for the compressible Navier–Stokes equations. Theor. Comput. Fluid Dyn. 13 (6), 377–396.CrossRefGoogle Scholar
Joseph, D. D. 1976 Stability of Fluid Motions, I and II. New York, Springer.Google Scholar
Kraichnan, R. H. & Chen, S. 1989 Is there a statistical mechanics of turbulence? Physica D: Nonlinear Phenomena 37 (1–3), 160–172.Google Scholar
Ladyzhenskaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach.Google Scholar
Lodato, G., Domingo, P. & Vervisch, L. 2008 Three-dimensional boundary conditions for direct and large-eddy simulation of compressible viscous flows. J. Comput. Phys. 227 (10), 5105–5143.Google Scholar
Lumnley, J. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Wave Propagation (ed. Yaglom, A. M. & Tatarski, V. I.), pp. 166–178. Nauka.Google Scholar
Moin, P. & Mahesh, K. 1998 Direct numerical simulation: a tool in turbulence research. Annu. Rev. Fluid Mech. 30, 539–578.CrossRefGoogle Scholar
Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.CrossRefGoogle Scholar
Noack, B. R., Morzynski, M. & Tadmor, G. 2011 Reduced-Order Modelling for Flow Control. Springer.Google 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 (1), 339–365.CrossRefGoogle Scholar
Noack, B. R., Schlegel, M., Ahlborn, B., Mutschke, G., Morzyński, M., Comte, P. & Tadmor, G. 2008 A finite-time thermodynamics of unsteady fluid flows. J. Non-Equilib. Thermodyn. 33 (2), 103–148.Google Scholar
Nocedal, J. & Wright, S. J. 1999 Numerical Optimization. Springer.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows, 6th edn. Cambridge University Press.CrossRefGoogle Scholar
Rempfer, D. & Fasel, H. F. 1994 Dynamics of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 275, 257–284.Google Scholar
Rowley, C. W., Mezic, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127.Google Scholar
Schittkowski, K. 1986 NLPQL: a FORTRAN subroutine solving constrained nonlinear programming problems. Ann. Oper. Res. 5 (1), 485–500.CrossRefGoogle Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.Google Scholar
Shankar, P. N. & Deshpande, M. D. 2000 Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech. 32 (1), 93–136.Google Scholar
Sirisup, S. & Karniadakis, G. E. 2004 A spectral viscosity method for correcting the long-term behavior of POD models. J. Comput. Phys. 194 (1), 92–116.Google Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures, Part I: coherent structures. Q. Appl. Maths 45, 561–571.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. The MIT Press.CrossRefGoogle Scholar
Terragni, F., Valero, E. & Vega, J. M. 2011 Local POD plus Galerkin projection in the unsteady lid-driven cavity problem. SIAM J. Sci. Comput. 33 (6), 3538–3561.Google Scholar
Ukeiley, L., Cordier, L., Manceau, R., Delville, J., Glauser, M. & Bonnet, J. P. 2001 Examination of large-scale structures in a turbulent plane mixing layer. Part 2. Dynamical systems model. J. Fluid Mech. 441 (1), 67–108.Google 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, 126–146.Google 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, 10–26.Google Scholar
Welch, P. 1967 The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15 (2), 70–73.Google Scholar