Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T07:28:23.315Z Has data issue: false hasContentIssue false
  • English
  • Français

Improved convergence of the spectral proper orthogonal decomposition through time shifting

Published online by Cambridge University Press:  17 October 2022

Diego C.P. Blanco Open the ORCID record for Diego C.P. Blanco [Opens in a new window] ,
Eduardo Martini Open the ORCID record for Eduardo Martini [Opens in a new window] ,
Kenzo Sasaki Open the ORCID record for Kenzo Sasaki [Opens in a new window]  and
André V.G. Cavalieri Open the ORCID record for André V.G. Cavalieri [Opens in a new window]
Diego C.P. Blanco*
Affiliation:
Divisão de Engenharia Aeroespacial, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos/SP, Brazil
Eduardo Martini
Affiliation:
Département Fluides, Thermique et Combustion, Institut Pprime, CNRS, Université de Poitiers, ENSMA, 86000 Poitiers, France
Kenzo Sasaki
Affiliation:
Divisão de Engenharia Aeroespacial, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos/SP, Brazil
André V.G. Cavalieri
Affiliation:
Divisão de Engenharia Aeroespacial, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos/SP, Brazil
*
†Email address for correspondence: diegodcpb@ita.br
Get access Rights & Permissions [Opens in a new window]

Abstract

Spectral proper orthogonal decomposition (SPOD) is an increasingly popular modal analysis method in the field of fluid dynamics due to its specific properties: a linear system forced with white noise should have SPOD modes identical to response modes from resolvent analysis. The SPOD, coupled with the Welch method for spectral estimation, may require long time-resolved datasets. In this work, a linearised Ginzburg–Landau model is considered in order to study the method's convergence. Spectral proper orthogonal decomposition modes of the white-noise forced equation are computed and compared with corresponding response resolvent modes. The quantified error is shown to be related to the time length of Welch blocks (spectral window size) normalised by a convective time. Subsequently, an algorithm based on a temporal data shift is devised to further improve SPOD convergence and is applied to the Ginzburg–Landau system. Next, its efficacy is demonstrated in a numerical database of a boundary layer subject to bypass transition. The proposed approach achieves substantial improvement in mode convergence with smaller spectral window sizes with respect to the standard method. Furthermore, SPOD modes display growing wall-normal and spanwise velocity components along the streamwise direction, a feature which had not yet been observed and is also predicted by a global resolvent calculation. The shifting algorithm for the SPOD opens the possibility for using the method on datasets with time series of moderate duration, often produced by large simulations.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 950 , 10 November 2022 , A9
Check for updates
Copyright
© The Author(s), 2022. Published by 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

Abreu, L.I., Cavalieri, A.V.G., Schlatter, P., Vinuesa, R. & Henningson, D.S. 2020 a Resolvent modelling of near-wall coherent structures in turbulent channel flow. Intl J. Heat Fluid Flow 85, 108662.CrossRefGoogle Scholar
Abreu, L.I., Cavalieri, A.V.G., Schlatter, P., Vinuesa, R. & Henningson, D.S. 2020 b Spectral proper orthogonal decomposition and resolvent analysis of near-wall coherent structures in turbulent pipe flows. J. Fluid Mech. 900, A11.CrossRefGoogle Scholar
Abreu, L.I., Tanarro, A., Cavalieri, A.V.G., Schlatter, P., Vinuesa, R., Hanifi, A. & Henningson, D.S. 2021 Spanwise-coherent hydrodynamic waves around flat plates and airfoils. J. Fluid Mech. 927, A1.CrossRefGoogle Scholar
Andersson, P., Berggren, M. & Henningson, D.S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 134–150.CrossRefGoogle Scholar
Aranson, I.S. & Kramer, L. 2002 The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143.CrossRefGoogle Scholar
Araya, D.B., Colonius, T. & Dabiri, J.O. 2017 Transition to bluff-body dynamics in the wake of vertical-axis wind turbines. J. Fluid Mech. 813, 346–381.CrossRefGoogle Scholar
Arndt, R.E.A., Long, D.F. & Glauser, M.N. 1997 The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet. J. Fluid Mech. 340, 1–33.CrossRefGoogle Scholar
Aubry, N. 1991 On the hidden beauty of the proper orthogonal decomposition. Theor. Comput. Fluid Dyn. 2 (5), 339–352.CrossRefGoogle Scholar
Bagheri, S., Henningson, D.S., Hœpffner, J. & Schmid, P.J. 2009 Input–output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62 (2), 020803.CrossRefGoogle Scholar
Beneddine, S., Sipp, D., Arnault, A., Dandois, J. & Lesshafft, L. 2016 Conditions for validity of mean flow stability analysis. J. Fluid Mech. 798, 485–504.CrossRefGoogle 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.CrossRefGoogle Scholar
Bonnet, J.P., et al. 1998 Collaborative testing of eddy structure identification methods in free turbulent shear flows. Exp. Fluids 25 (3), 197–225.CrossRefGoogle Scholar
Brandt, L. 2014 The lift-up effect: the linear mechanism behind transition and turbulence in shear flows. Eur. J. Mech. B/Fluids 47, 80–96.CrossRefGoogle Scholar
Brandt, L., Schlatter, P. & Henningson, D.S. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167–198.CrossRefGoogle Scholar
Cavalieri, A.V.G., Jordan, P. & Lesshafft, L. 2019 Wave-packet models for jet dynamics and sound radiation. Appl. Mech. Rev. 71 (2), 020802.CrossRefGoogle Scholar
Chevalier, M., Lundbladh, A. & Henningson, D.S. 2007 Simson – a pseudo-spectral solver for incompressible boundary layer flow. Tech. Rep. TRITA-MEK.Google Scholar
Citriniti, J.H. & George, W.K. 2000 Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J. Fluid Mech. 418, 137–166.CrossRefGoogle Scholar
Delville, J., Ukeiley, L., Cordier, L., Bonnet, J.P. & Glauser, M. 1999 Examination of large-scale structures in a turbulent plane mixing layer. Part 1. Proper orthogonal decomposition. J. Fluid Mech. 391, 91–122.CrossRefGoogle Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18 (4), 487–488.CrossRefGoogle Scholar
Glauser, M.N. & George, W. 1987 Orthogonal decomposition of the axisymmetric jet mixing layer including azimuthal dependence. In Advances in Turbulence (ed. F. Durst et al.), pp. 357–366. Springer.CrossRefGoogle Scholar
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97–128.CrossRefGoogle Scholar
He, X., Fang, Z., Rigas, G. & Vahdati, M. 2021 Spectral proper orthogonal decomposition of compressor tip leakage flow. Phys. Fluids 33 (10), 105105.CrossRefGoogle Scholar
He, G., Jin, G. & Yang, Y. 2017 Space–time correlations and dynamic coupling in turbulent flows. Annu. Rev. Fluid Mech. 49 (1), 51–70.CrossRefGoogle Scholar
Hellström, L.H.O. & Smits, A.J. 2014 The energetic motions in turbulent pipe flow. Phys. Fluids 26 (12), 125102.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22 (1), 473–537.CrossRefGoogle Scholar
Iqbal, M.O. & Thomas, F.O. 2007 Coherent structure in a turbulent jet via a vector implementation of the proper orthogonal decomposition. J. Fluid Mech. 571, 281–326.CrossRefGoogle Scholar
Jaunet, V., Jordan, P. & Cavalieri, A.V.G. 2017 Two-point coherence of wave packets in turbulent jets. Phys. Rev. Fluids 2, 024604.CrossRefGoogle Scholar
Jovanović, M.R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Jung, D., Gamard, S. & George, W.K. 2004 Downstream evolution of the most energetic modes in a turbulent axisymmetric jet at high Reynolds number. Part 1. The near-field region. J. Fluid Mech. 514, 173–204.CrossRefGoogle Scholar
Kaplan, O., Jordan, P., Cavalieri, A.V.G. & Brès, G.A. 2021 Nozzle dynamics and wavepackets in turbulent jets. J. Fluid Mech. 923, A22.CrossRefGoogle Scholar
Lesshaft, L., Semeraro, O., Jaunet, V., Cavalieri, A.V.G. & Jordan, P. 2019 Resolvent-based modeling of coherent wave packets in a turbulent jet. Phys. Rev. Fluids 4, 063901.CrossRefGoogle Scholar
Li, X.-B. , Chen, G., Liang, X.-F., Liu, D.-R. & Xiong, X.-H. 2021 Research on spectral estimation parameters for application of spectral proper orthogonal decomposition in train wake flows. Phys. Fluids 33 (12), 125103.CrossRefGoogle Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.CrossRefGoogle Scholar
Lugrin, M., Beneddine, S., Leclercq, C., Garnier, E. & Bur, R. 2021 Transition scenario in hypersonic axisymmetrical compression ramp flow. J. Fluid Mech. 907, A6.CrossRefGoogle Scholar
Lumley, J.L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. A.M. Yaglom & V.I. Tatarski). Nauka.Google Scholar
Lumley, J.L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
Martini, E., Cavalieri, A.V.G., Jordan, P. & Lesshafft, L. 2020 Accurate frequency domain identification of ODEs with arbitrary signals. arXiv:1907.04787.Google Scholar
Martini, E., Rodríguez, D., Towne, A. & Cavalieri, A.V.G. 2021 Efficient computation of global resolvent modes. J. Fluid Mech. 919, A3.CrossRefGoogle Scholar
Matsubara, M. & Alfredsson, P.H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.CrossRefGoogle Scholar
McKeon, B.J. & Sharma, A.S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Monokrousos, A., Åkervik, E., Brandt, L. & Henningson, D. 2010 Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181–214.CrossRefGoogle Scholar
Nidhan, S., Chongsiripinyo, K., Schmidt, O.T. & Sarkar, S. 2020 Spectral proper orthogonal decomposition analysis of the turbulent wake of a disk at $Re = 50\,000$. Phys. Rev. Fluids 5, 124606.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, 335–363.CrossRefGoogle Scholar
Nogueira, P.A.S., Cavalieri, A.V.G., Jordan, P. & Jaunet, V. 2019 Large-scale streaky structures in turbulent jets. J. Fluid Mech. 873, 211–237.CrossRefGoogle Scholar
Picard, C. & Delville, J. 2000 Pressure velocity coupling in a subsonic round jet. Intl J. Heat Fluid Flow 21 (3), 359–364.CrossRefGoogle Scholar
Pickering, E., Rigas, G., Nogueira, P.A.S., Cavalieri, A.V.G., Schmidt, O.T. & Colonius, T. 2020 Lift-up, Kelvin–Helmholtz and Orr mechanisms in turbulent jets. J. Fluid Mech. 896, A2.CrossRefGoogle Scholar
Ravindran, S.S. 2000 A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Intl J. Numer. Meth. Fluids 34 (5), 425–448.3.0.CO;2-W>CrossRefGoogle Scholar
Reiss, J., Schulze, P., Sesterhenn, J. & Mehrmann, V. 2018 The shifted proper orthogonal decomposition: a mode decomposition for multiple transport phenomena. SIAM J. Sci. Comput. 40 (3), A1322–A1344.CrossRefGoogle Scholar
Rowley, C.W. & Dawnson, S.T. 2017 Model reduction for flow analysis and control. Annu. Rev. Fluid Mech. 49 (1), 387–417.CrossRefGoogle Scholar
Sano, A., Abreu, L.I., Cavalieri, A.V.G. & Wolf, W.R. 2019 Trailing-edge noise from the scattering of spanwise-coherent structures. Phys. Rev. Fluids 4, 094602.CrossRefGoogle Scholar
Sasaki, K., Morra, P., Cavalieri, A.V.G., Hanifi, A. & Henningson, D.S. 2020 On the role of actuation for the control of streaky structures in boundary layers. J. Fluid Mech. 883, A34.CrossRefGoogle Scholar
Schmidt, O.T. & Colonius, T. 2020 Guide to spectral proper orthogonal decomposition. AIAA J. 58 (3), 1023–1033.CrossRefGoogle Scholar
Schmidt, O.T. & Towne, A. 2019 An efficient streaming algorithm for spectral proper orthogonal decomposition. Comput. Phys. Commun. 237, 98–109.CrossRefGoogle Scholar
Schmidt, O.T., Towne, A., Colonius, T., Cavalieri, A.V.G., Jordan, P. & Brès, G.A. 2017 Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability. J. Fluid Mech. 825, 1153–1181.CrossRefGoogle Scholar
Schmidt, O.T., Towne, A., Rigas, G., Colonius, T. & Bres, G.A. 2018 Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953–982.CrossRefGoogle Scholar
Sieber, M., Paschereit, C.O. & Oberleithner, K. 2016 Spectral proper orthogonal decomposition. J. Fluid Mech. 792, 798–828.CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Part I: coherent structures. Q. Appl. Maths 45 (3), 561–571.CrossRefGoogle Scholar
Taira, K., Hemati, M.S., Brunton, S.L., Sun, Y., Duraisamy, K., Bagheri, S., Dawson, S.T.M. & Yeh, C.-A. 2020 Modal analysis of fluid flows: applications and outlook. AIAA J. 58 (3), 998–1022.CrossRefGoogle Scholar
Tinney, C.E. & Jordan, P. 2008 The near pressure field of co-axial subsonic jets. J. Fluid Mech. 611, 175–204.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
Tutkun, M. & George, W.K. 2017 Lumley decomposition of turbulent boundary layer at high Reynolds numbers. Phys. Fluids 29 (2), 020707.CrossRefGoogle Scholar
Welch, P.D. 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.CrossRefGoogle Scholar
Zhang, B., Ooka, R. & Kikumoto, H. 2021 Identification of three-dimensional flow features around a square-section building model via spectral proper orthogonal decomposition. Phys. Fluids 33 (3), 035151.CrossRefGoogle Scholar