Skip to main content
SpringerLink
Log in

Spectral proper orthogonal decomposition using multitaper estimates

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

The use of multitaper estimates for spectral proper orthogonal decomposition (SPOD) is explored. Multitaper and multitaper-Welch estimators that use discrete prolate spheroidal sequences (DPSS) as orthogonal data windows are compared to the standard SPOD algorithm that exclusively relies on weighted overlapped segment averaging, or Welch’s method, to estimate the cross-spectral density matrix. Two sets of turbulent flow data, one experimental and the other numerical, are used to discuss the choice of resolution bandwidth and the bias-variance tradeoff. Multitaper-Welch estimators that combine both approaches by applying orthogonal tapers to overlapping segments allow for flexible control of resolution, variance, and bias. At additional computational cost but for the same data, multitaper-Welch estimators provide lower variance estimates at fixed frequency resolution or higher frequency resolution at similar variance compared to the standard algorithm.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Lumley, J.L.: Stochastic tools in turbulence. Academic Press, New York (1970)

    MATH  Google Scholar 

  2. Glauser, M.N., Leib, S.J., George, W.K.: Coherent structures in the axisymmetric turbulent jet mixing layer. Turbul. Shear Flows 5, 134–145 (1987)

    Article  Google Scholar 

  3. Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010). https://doi.org/10.1017/S0022112010001217

    Article  MathSciNet  MATH  Google Scholar 

  4. McKeon, B.J., Sharma, A.S.: A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382 (2010). https://doi.org/10.1017/S002211201000176X

    Article  MathSciNet  MATH  Google Scholar 

  5. Towne, A., Schmidt, O.T., Colonius, T.: Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821–867 (2018). https://doi.org/10.1017/jfm.2018.283

    Article  MathSciNet  MATH  Google Scholar 

  6. Schmidt, O.T., Towne, A., Rigas, G., Colonius, T., Brès, G.A.: Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953–982 (2018). https://doi.org/10.1017/jfm.2018.675

    Article  MathSciNet  MATH  Google Scholar 

  7. Arndt, R.E.A., Long, D.F., Glauser, M.N.: The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet. J. Fluid Mech. 340, 1–33 (1997)

    Article  Google Scholar 

  8. Citriniti, J.H., George, W.K.: Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J. Fluid Mech. 418, 137–166 (2000)

    Article  Google Scholar 

  9. Nidhan, S., Chongsiripinyo, K., Schmidt, O.T., Sarkar, S.: Spectral proper orthogonal decomposition analysis of the turbulent wake of a disk at Re=50000. Phys. Rev. Fluids 5(12), 124606 (2020). https://doi.org/10.1103/PhysRevFluids.5.124606

    Article  Google Scholar 

  10. Gordeyev, S.V., Thomas, F.O.: Coherent structure in the turbulent planar jet. Part 1. extraction of proper orthogonal decomposition eigenmodes and their self-similarity. J. Fluid Mech. 414, 145–194 (2000)

    Article  Google Scholar 

  11. Gudmundsson, K., Colonius, T.: Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97–128 (2011)

    Article  Google Scholar 

  12. Hellström, L.H.O., Smits, A.J.: The energetic motions in turbulent pipe flow. Phys. Fluids 26(12), 125102 (2014)

    Article  Google Scholar 

  13. Tutkun, M., George, W.K.: Lumley decomposition of turbulent boundary layer at high reynolds numbers. Phys. Fluids 29(2), 020707 (2017)

    Article  Google Scholar 

  14. Araya, D.B., Colonius, T., Dabiri, J.O.: Transition to bluff-body dynamics in the wake of vertical-axis wind turbines. J. Fluid Mech. 813, 346–381 (2017)

    Article  MathSciNet  Google Scholar 

  15. Abreu L.I., Cavalieri A.V.G., Wolf W.: Coherent hydrodynamic waves and trailing-edge noise. In: 23rd AIAA/CEAS aeroacoustics conference. vol. AIAA 2017-3173. (2017). https://doi.org/10.2514/6.2017-3173

  16. He, X., Fang, Z., Rigas, G., Vahdati, M.: Spectral proper orthogonal decomposition of compressor tip leakage flow. Phys. Fluids 33(10), 105105 (2021)

    Article  Google Scholar 

  17. Li, X.B., Chen, G., Liang, X.F., Liu, D.R., Xiong, X.H.: Research on spectral estimation parameters for application of spectral proper orthogonal decomposition in train wake flows. Phys. Fluids 33(12), 125103 (2021)

    Article  Google Scholar 

  18. Schmidt O.T., Mengaldo G., Balsamo G., Wedi N.P.: Spectral empirical orthogonal function analysis of weather and climate data. Monthly Weather Review (2019) 147(8), 2979–2995. https://doi.org/10.1175/MWR-D-18-0337.1. arXiv:https://doi.org/10.1175/MWR-D-18-0337.1

  19. Sanjose, M., Towne, A., Jaiswal, P., Moreau, S., Lele, S., Mann, A.: Modal analysis of the laminar boundary layer instability and tonal noise of an airfoil at reynolds number 150,000. Int. J. Aero. 18(2–3), 317–350 (2019)

    Article  Google Scholar 

  20. Nekkanti A., Schmidt O.T.: Modal analysis of acoustic directivity in turbulent jets. AIAA Journal, 0(0), 1–12 (2020). https://doi.org/10.2514/1.J059425. arXiv:https://doi.org/10.2514/1.J059425

  21. Baars, W.J., Tinney, C.E.: Proper orthogonal decomposition-based spectral higher-order stochastic estimation. Phys. Fluids 26(5), 055112 (2014)

    Article  Google Scholar 

  22. Nekkanti, A., Schmidt, O.T.: Frequency-time analysis, low-rank reconstruction and denoising of turbulent flows using spod. J. Fluid Mech. 926, A26 (2021). https://doi.org/10.1017/jfm.2021.681

    Article  MATH  Google Scholar 

  23. Ghate, A.S., Towne, A., Lele, S.K.: Broadband reconstruction of inhomogeneous turbulence using spectral proper orthogonal decomposition and gabor modes. J. Fluid Mech. 888, R1 (2020). https://doi.org/10.1017/jfm.2020.78

    Article  MathSciNet  MATH  Google Scholar 

  24. Chu, T., Schmidt, O.T.: A stochastic spod-galerkin model for broadband turbulent flows. Theoretical and Computational Fluid Dynamics (2021). https://doi.org/10.1007/s00162-021-00588-6

  25. Towne A.: Space-time galerkin projection via spectral proper orthogonal decomposition and resolvent modes. In: AIAA Scitech 2021. vol. AIAA 2021-1676. pp 1676. (2021)

  26. Pain, R., Weiss, P.E., Deck, S., Robinet, J.C.: Large scale dynamics of a high reynolds number axisymmetric separating/reattaching flow. Phys. Fluids 31(12), 125119 (2019)

    Article  Google Scholar 

  27. Welch, P.: 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 (1967)

    Article  Google Scholar 

  28. Schmidt, O.T., Colonius, T.: Guide to spectral proper orthogonal decomposition. AIAA J. 58(3), 1023–1033 (2020). https://doi.org/10.2514/1.J058809

    Article  Google Scholar 

  29. Thomson, D.J.: Spectrum estimation and harmonic analysis. Proc. IEEE 70(9), 1055–1096 (1982)

    Article  Google Scholar 

  30. Bronez, T.P.: On the performance advantage of multitaper spectral analysis. IEEE Trans. Signal Proc. 40(12), 2941–2946 (1992)

    Article  Google Scholar 

  31. Geoga, C.J., Haley, C.L., Siegel, A.R., Anitescu, M.: Frequency-wavenumber spectral analysis of spatio-temporal flows. J. Fluid Mech. 848, 545–559 (2018). https://doi.org/10.1017/jfm.2018.366

    Article  MathSciNet  MATH  Google Scholar 

  32. Slepian, D., Pollak, H.O.: Prolate spheroidal wave functions, fourier analysis and uncertainty-I. Bell Syst. Tech. J. 40(1), 43–63 (1961)

    Article  MathSciNet  Google Scholar 

  33. Slepian, D.: Prolate spheroidal wave functions, fourier analysis, and uncertainty-v: The discrete case. Bell Syst. Techn. J. 57(5), 1371–1430 (1978)

    Article  Google Scholar 

  34. Heinzel G., Rüdiger A., Schilling R.: Spectrum and spectral density estimation by the discrete fourier transform (dft), including a comprehensive list of window functions and some new at-top windows (2002)

  35. Brès, G., Jordan, P., Le Rallic, M., Jaunet, V., Cavalieri, A.V.G., Towne, A., Lele, S., Colonius, T., Schmidt, O.T.: Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets. J. Fluid Mech. 851, 83–124 (2018). https://doi.org/10.1017/jfm.2018.476

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang Y., Cattafesta L., Ukeiley L.: A spectral analysis modal method applied to cavity flow oscillations. In: TSFP11, Southampton, UK. (2019)

  37. Zhang Y., Cattafesta L., Ukeiley L.: Identification of coherent structures in cavity flows using stochastic estimation and dynamic mode decomposition. In: TSFP10, Chicago, USA. pp 3. (2017)

  38. Rossiter J.E.: Wind tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. RAE Technical Report No 64037 (1964)

  39. Lii, K.S., Rosenblatt, M.: Prolate spheroidal spectral estimates. Stat. Probab. Lett. 78(11), 1339–1348 (2008)

    Article  MathSciNet  Google Scholar 

  40. Schmidt O.T., Towne A.: An efficient streaming algorithm for spectral proper orthogonal decomposition. Computer Physics Communications (2018) http://www.sciencedirect.com/science/article/pii/S0010465518304016. https://doi.org/10.1016/j.cpc.2018.11.009

  41. Citriniti, J.H., George, W.K.: Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J. Fluid Mech. 418, 137–166 (2000)

    Article  Google Scholar 

  42. Schmidt O.T.: Bispectral mode decomposition of nonlinear flows. Nonlinear Dynamics (102(4)), 2479–2501. (2020) https://doi.org/10.1007/s11071-020-06037-z

Download references

Acknowledgements

OTS gratefully acknowledges support from Office of Naval Research grant N00014-20-1-2311 and NSF grant CBET 2046311. I would like to thank Lou Cattafesta and Yang Zhang for providing the TR-PIV data, Gregg Abate for connecting me to Lou, Guillaume Brès for generating the turbulent jet database, and Eduardo Martini and Akhil Nekkanti for their constructive feedback. The TR-PIV data was created with support from AFOSR Award Number FA9550-17-1-0380. Creation of the LES data was supported by NAVAIR SBIR under the supervision of J. T. Spyropoulos, with computational resources provided by DoD HPCMP at the ERDC DSRC supercomputer facility.

Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, 92093, USA

    Oliver T. Schmidt

Authors
  1. Oliver T. Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oliver T. Schmidt.

Ethics declarations

Data and code availability

The datasets analyzed in this study are currently not publicly available as they were generated by other researchers (see Sect. 2.4, Table 1) as part of DoD funded independent studies (see Acknowledgments), but are available from the corresponding author on reasonable request. A reduced version of the turbulent jet data set is publicly available as part of the open-source MATLAB code SPOD that now supports multitaper-Welch estimates for SPOD.

Additional information

Communicated by Denis Sipp.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schmidt, O.T. Spectral proper orthogonal decomposition using multitaper estimates. Theor. Comput. Fluid Dyn. 36, 741–754 (2022). https://doi.org/10.1007/s00162-022-00626-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-022-00626-x

Keywords

Search

Navigation