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Koopman-mode decomposition of the cylinder wake

Published online by Cambridge University Press:  11 June 2013

Shervin Bagheri
Shervin Bagheri*
Affiliation:
Linné Flow Centre, Department of Mechanics KTH, SE-10044 Stockholm, Sweden
*
†Email address for correspondence: shervin@mech.kth.se
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Abstract

The Koopman operator provides a powerful way of analysing nonlinear flow dynamics using linear techniques. The operator defines how observables evolve in time along a nonlinear flow trajectory. In this paper, we perform a Koopman analysis of the first Hopf bifurcation of the flow past a circular cylinder. First, we decompose the flow into a sequence of Koopman modes, where each mode evolves in time with one single frequency/growth rate and amplitude/phase, corresponding to the complex eigenvalues and eigenfunctions of the Koopman operator, respectively. The analytical construction of these modes shows how the amplitudes and phases of nonlinear global modes oscillating with the vortex shedding frequency or its harmonics evolve as the flow develops and later sustains self-excited oscillations. Second, we compute the dynamic modes using the dynamic mode decomposition (DMD) algorithm, which fits a linear combination of exponential terms to a sequence of snapshots spaced equally in time. It is shown that under certain conditions the DMD algorithm approximates Koopman modes, and hence provides a viable method to decompose the flow into saturated and transient oscillatory modes. Finally, the relevance of the analysis to frequency selection, global modes and shift modes is discussed.

JFM classification

Instability: Instability Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Vortex Flows: Vortex shedding
Type
Papers
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , pp. 596 - 623
Copyright
©2013 Cambridge University Press 

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References

Artuso, R., Hugh, R. & Cvitanović, P. 2013 ‘Why does it work?’ In Chaos: Classical and Quantum (ed. Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G. & Vattay, G.). Niels Bohr Institute, chaosBook.org/version14.Google Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75, 750–756.CrossRefGoogle Scholar
Chen, K., Tu, J. H. & Rowley, C. W. 2012 Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22 (6), 887–915.CrossRefGoogle Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Cvitanović, P. 2013 ‘Trace formulas’. In Chaos: Classical and Quantum (ed. Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G. & Vattay, G.). Niels Bohr Institute, chaosBook.org/version14.Google Scholar
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G. & Vattay, G. (Eds) 2013 Chaos: Classical and Quantum. Niels Bohr Institute, chaosBook.org.Google Scholar
Cvitanović, P. & Eckhardt, B. 1991 Periodic orbit expansions for classical smooth flows. J. Phys. A: Math. Gen. 24, L237–L241.Google Scholar
Duke, D., Soria, J. & Honnery, D. 2012 An error analysis of the dynamic mode decomposition. Exp. Fluids 52, 529–542.CrossRefGoogle Scholar
Gaspard, P. 1998 Chaos, Scattering, and Statistical Mechanics. Cambridge University Press.CrossRefGoogle Scholar
Gaspard, P., Nicolis, G., Provata, A. & Tasaki, S. 1995 Spectral signature of the pitchfork bifurcation: Liouville equation approach. Phys. Rev. E 51 (1), 74–94.CrossRefGoogle ScholarPubMed
Gaspard, P. & Tasaki, S. 2001 Liouvillian dynamics of the Hopf bifurcation. Phys. Rev. E 64, 056232.CrossRefGoogle ScholarPubMed
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical System and Bifuracations of Vector Fields. Springer.Google Scholar
Huerre, P. & Monkewitz, P. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Koopman, B. 1931 Hamiltonian systems and transformations in Hilbert space. Proc. Natl Acad. Sci. USA 17, 315–318.Google Scholar
Lasota, A. & Mackey, C. M. 1994 Chaos, Fractals and Noise: Stochastic Aspects of Dynamics. Springer.Google Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic.Google Scholar
Marquet, O., Lombardi, M., Chomaz, J. M., Sipp, D. & Jacquin, L. 2009 Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 1–21.Google Scholar
Mezić, I. 2005 Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41 (1), 309–325.Google Scholar
Mezić, I. 2013 Analysis of fluid flows via spectral properties of the koopman operator. Annu. Rev. Fluid Mech. 45 (1), 357–378.CrossRefGoogle Scholar
Muld, T., Efraimsson, G. & Henningson, D. 2012 Mode decomposition on surface-mounted cube. Flow Turbul. Combust. 88, 279–310.CrossRefGoogle Scholar
Noack, B., Afanasiev, K., Morzynski, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.Google Scholar
Perot, J. B. 1993 An analysis of the fractional step method. J. Comput. Phys. 108 (1), 51–58.Google Scholar
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407–417.CrossRefGoogle Scholar
Provansal, M., Mathis, C. & Boyer, L. 1987 Bénard–von Kármán instability: transient and forcing regimes. J. Fluid Mech. 182, 1–22.CrossRefGoogle Scholar
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54 (2), 263–288.CrossRefGoogle Scholar
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.Google Scholar
Schmid, P. 2011 Application of the dynamic mode decomposition to experimental data. Exp. Fluids 50, 1123–1130.Google Scholar
Schmid, P., Li, L., Juniper, M. & Pust, O. 2011 Applications of the dynamic mode decomposition. Theoret. Comput. Fluid Dyn. 25, 249–259.CrossRefGoogle Scholar
Seena, A. & Sung, H. J. 2011 Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations. Intl J. Heat Fluid Flow 32 (6), 1098–1110.Google Scholar
Semeraro, O., Bellani, G. & Lundell, F. 2012 Analysis of time-resolved piv measurements of a confined turbulent jet using pod and Koopman modes. Exp. Fluids, 53 (5), 1203–1220.Google Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.Google Scholar
Tadmor, G., Lehmann, O., Noack, B. & Morzynski, M. 2010 Mean field representation of the natural and actuated cylinder wake flow. Phys. Fluids 22, 034102.Google Scholar
Taira, K. & Colonius, T. 2007 The immersed boundary method: a projection approach. J. Comput. Phys. 225, 2118–2137.Google Scholar
Thiria, B. & Wesfreid, J. E. 2007 Stability properties of forced wakes. J. Fluid Mech. 579, 137–161.CrossRefGoogle Scholar