Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-25T14:20:47.778Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral proper orthogonal decomposition

Published online by Cambridge University Press:  04 March 2016

Moritz Sieber Open the ORCID record for Moritz Sieber [Opens in a new window] ,
C. Oliver Paschereit  and
Kilian Oberleithner
Moritz Sieber*
Affiliation:
Institut für Strömungsmechanik und Technische Akustik, Hermann-Föttinger-Institut, Technische Universität Berlin, Müller-Breslau Str. 8, D-10623 Berlin, Germany
C. Oliver Paschereit
Affiliation:
Institut für Strömungsmechanik und Technische Akustik, Hermann-Föttinger-Institut, Technische Universität Berlin, Müller-Breslau Str. 8, D-10623 Berlin, Germany
Kilian Oberleithner
Affiliation:
Institut für Strömungsmechanik und Technische Akustik, Hermann-Föttinger-Institut, Technische Universität Berlin, Müller-Breslau Str. 8, D-10623 Berlin, Germany
*
†Email address for correspondence: moritz.sieber@fd.tu-berlin.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The identification of coherent structures from experimental or numerical data is an essential task when conducting research in fluid dynamics. This typically involves the construction of an empirical mode base that appropriately captures the dominant flow structures. The most prominent candidates are the energy-ranked proper orthogonal decomposition (POD) and the frequency-ranked Fourier decomposition and dynamic mode decomposition (DMD). However, these methods are not suitable when the relevant coherent structures occur at low energies or at multiple frequencies, which is often the case. To overcome the deficit of these ‘rigid’ approaches, we propose a new method termed spectral proper orthogonal decomposition (SPOD). It is based on classical POD and it can be applied to spatially and temporally resolved data. The new method involves an additional temporal constraint that enables a clear separation of phenomena that occur at multiple frequencies and energies. SPOD allows for a continuous shifting from the energetically optimal POD to the spectrally pure Fourier decomposition by changing a single parameter. In this article, SPOD is motivated from phenomenological considerations of the POD autocorrelation matrix and justified from dynamical systems theory. The new method is further applied to three sets of PIV measurements of flows from very different engineering problems. We consider the flow of a swirl-stabilized combustor, the wake of an airfoil with a Gurney flap and the flow field of the sweeping jet behind a fluidic oscillator. For these examples, the commonly used methods fail to assign the relevant coherent structures to single modes. The SPOD, however, achieves a proper separation of spatially and temporally coherent structures, which are either hidden in stochastic turbulent fluctuations or spread over a wide frequency range. The SPOD requires only one additional parameter, which can be estimated from the basic time scales of the flow. In spite of all these benefits, the algorithmic complexity and computational cost of the SPOD are only marginally greater than those of the snapshot POD.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 792 , 10 April 2016 , pp. 798 - 828
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adrian, R. J. & Moin, P. 1988 Stochastic estimation of organized turbulent structure: homogeneous shear flow. J. Fluid Mech. 190, 531–559.Google Scholar
Bach, A. B., Berg, R., Pechlivanoglou, G., Nayeri, C. & Paschereit, C. O. 2015a Experimental investigation of the aerodynamic lift response of an active finite gurney flap. In 53rd AIAA Aerospace Sciences Meeting, AIAA SciTech. American Institute of Aeronautics and Astronautics.Google Scholar
Bach, A. B., Lennie, M., Pechlivanoglou, G., Nayeri, C. N. & Paschereit, C. O. 2014 Finite micro-tab system for load control on a wind turbine. J. Phys.: Conf. Ser. 524, 012082.Google Scholar
Bach, A. B., Pechlivanoglou, G., Nayeri, C. & Paschereit, C. O. 2015b Wake vortex field of an airfoil equipped with an active finite Gurney flap. In 53rd AIAA Aerospace Sciences Meeting, AIAA SciTech. American Institute of Aeronautics and Astronautics.Google Scholar
Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575.Google Scholar
Boree, J. 2003 Extended proper orthogonal decomposition: a tool to analyse correlated events in turbulent flows. Exp. Fluids 35 (2), 188–192.Google Scholar
Bourgeois, J. A., Noack, B. R. & Martinuzzi, R. J. 2013 Generalized phase average with applications to sensor-based flow estimation of the wall-mounted square cylinder wake. J. Fluid Mech. 736, 316–350.CrossRefGoogle Scholar
Durgesh, V. & Naughton, J. W. 2010 Multi-time-delay LSE-POD complementary approach applied to unsteady high-Reynolds-number near wake flow. Exp. Fluids 49 (3), 571–583.Google Scholar
Gray, R. M. 2005 Toeplitz and circulant matrices: a review. Foundations Trends Commun. Inform. Theory 2 (3), 155–239.Google 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
Hosseini, Z., Martinuzzi, R. J. & Noack, B. R. 2015 Sensor-based estimation of the velocity in the wake of a low-aspect-ratio pyramid. Exp. Fluids 56 (1), 1–16.Google Scholar
Huang, H. T., Fiedler, H. E. & Wang, J. J. 1993 Limitation and improvement of PIV: part II: particle image distortion, a novel technique. Exp. Fluids 15 (4–5), 263–273.Google Scholar
Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yen, N.-C., Tung, C. C. & Liu, H. H. 1998 The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A 454 (1971), 903–995.Google Scholar
Lasagna, D., Orazi, M. & Iuso, G. 2013 Multi-time delay, multi-point linear stochastic estimation of a cavity shear layer velocity from wall-pressure measurements. Phys. Fluids 25 (1), 017101.Google Scholar
Luchtenburg, D. M., Günther, B., Noack, B. R., King, R. & Tadmor, G. 2009 A generalized mean-field model of the natural and high-frequency actuated flow around a high-lift configuration. J. Fluid Mech. 623, 283–316.Google Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence, Applied Mathematics and Mechanics, vol. 12. Academic Press.Google Scholar
Oberleithner, K., Rukes, L. & Soria, J. 2014 Mean flow stability analysis of oscillating jet experiments. J. Fluid Mech. 757, 1–32.Google Scholar
Oberleithner, K., Sieber, M., Nayeri, C. N., Paschereit, C. O., Petz, C., Hege, H.-C., Noack, B. R. & Wygnanski, I. 2011 Three-dimensional coherent structures in a swirling jet undergoing vortex breakdown: stability analysis and empirical mode construction. J. Fluid Mech. 679, 383–414.Google Scholar
Oberleithner, K., Stöhr, M., Seong, H. I., Arndt, C. M. & Steinberg, A. M. 2015 Formation and flame-induced supression of the precessing vortex core in a swirl combustor: experiments and linear stability analysis. Combust. Flame 162 (8), 3100–3114.CrossRefGoogle Scholar
Ostermann, F., Woszidlo, R., Gaertlein, S., Nayeri, C. & Paschereit, C. O. 2015a Phase-averaging methods for the natural flowfield of a fluidic oscillator. AIAA J. 53 (8), 2359–2368.Google Scholar
Ostermann, F., Woszidlo, R., Nayeri, C. & Paschereit, C. O. 2015b Experimental comparison between the flow field of two common fluidic oscillator designs. In 53rd AIAA Aerospace Sciences Meeting, AIAA SciTech. American Institute of Aeronautics and Astronautics.Google Scholar
Raffel, M., Kompenhans, J., Wereley, S. T. & Willert, C. E. 2007 Particle Image Velocimetry: A Practical Guide, 2nd edn. Springer.Google Scholar
Raiola, M., Discetti, S. & Ianiro, A. 2015 On PIV random error minimization with optimal POD-based low-order reconstruction. Exp. Fluids 56 (4), 1–15.Google Scholar
Reichel, T. G., Terhaar, S. & Paschereit, O. 2015 Increasing flashback resistance in lean premixed swirl-stabilized hydrogen combustion by axial air injection. Trans. ASME J. Engng Gas Turbines Power 137 (7), 071503.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
Schlegel, M., Noack, B. R., Jordan, P., Dillmann, A., Gröschel, E., Schröder, W., Wei, M., Freund, J. B., Lehmann, O. & Tadmor, G. 2012 On least-order flow representations for aerodynamics and aeroacoustics. J. Fluid Mech. 697, 367–398.CrossRefGoogle Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.Google Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures: part I: coherent structures. Q. Appl. Maths 45 (3), 561–571.Google Scholar
Soria, J. 1996 An investigation of the near wake of a circular cylinder using a video-based digital cross-correlation particle image velocimetry technique. Exp. Therm. Fluid Sci. 12 (2), 221–233.Google Scholar
Terhaar, S.2015 Identification and modeling of coherent structures in swirl stabilized combustors at dry and steam diluted conditions. PhD thesis, Technische Universität Berlin.Google Scholar
Terhaar, S., Oberleithner, K. & Paschereit, C. O. 2015 Key parameters governing the precessing vortex core in reacting flows: an experimental and analytical study. Proc. Combust. Inst. 35 (3), 3347–3354.CrossRefGoogle Scholar
Troolin, D. R., Longmire, E. K. & Lai, W. T. 2006 Time resolved PIV analysis of flow over a NACA 0015 airfoil with Gurney flap. Exp. Fluids 41 (2), 241–254.CrossRefGoogle Scholar
Tu, J. H., Rowley, C. W., Luchtenburg, D. M., Brunton, S. L. & Kutz, J. N. 2014 On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1 (2), 391–421.Google Scholar
Willert, C. E. & Gharib, M. 1991 Digital particle image velocimetry. Exp. Fluids 10 (4), 181–193.Google Scholar
Williams, M. O., Kevrekidis, I. G. & Rowley, C. W. 2015 A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25 (6), 1307–1346.Google Scholar
Wise, J. 1955 The autocorrelation function and the spectral density function. Biometrika 42 (1–2), 151–159.Google Scholar
Woszidlo, R., Ostermann, F., Nayeri, C. N. & Paschereit, C. O. 2015 The time-resolved natural flow field of a fluidic oscillator. Exp. Fluids 56 (6), 1–12.CrossRefGoogle Scholar
Woszidlo, R., Stumper, T., Nayeri, C. & Paschereit, C. O. 2014 Experimental study on bluff body drag reduction with fluidic oscillators. In 52rd AIAA Aerospace Sciences Meeting, AIAA SciTech. American Institute of Aeronautics and Astronautics.Google Scholar