Skip to main content
SpringerLink
Log in

Bispectral mode decomposition of nonlinear flows

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Triadic interactions are the fundamental mechanism of energy transfer in fluid flows. This work introduces bispectral mode decomposition as a direct means of educing flow structures that are associated with triadic interactions from experimental or numerical data. Triadic interactions are characterized by quadratic phase coupling which can be detected by the bispectrum. The proposed method maximizes an integral measure of this third-order statistic to compute modes associated with frequency triads, as well as a mode bispectrum that identifies resonant three-wave interactions. Unlike the classical bispectrum, the decomposition establishes a causal relationship between the three frequency components of a triad. This permits the distinction of sum- and difference-interactions, and the computation of interaction maps that indicate regions of nonlinear coupling. Three examples highlight different aspects of the method. Cascading triads and their regions of interaction are educed from direct numerical simulation data of laminar cylinder flow. It is further demonstrated that linear instability mechanisms that attain an appreciable amplitude are revealed indirectly by their difference-self-interactions. Applicability to turbulent flows and noise-rejection is demonstrated on particle image velocimetry data of a massively separated wake. The generation of sub- and ultra-harmonics in large eddy simulation data of a transitional jet is explained by extending the method to cross-bispectral information.

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
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Aubry, N., Holmes, P., Lumley, J.L., Stone, E.: The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173 (1988)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  3. Brès, G.A., Lele, S.K.: Modelling of jet noise: a perspective from large-eddy simulations. Philos. Trans. R. Soc. A 377(2159), 20190081 (2019)

    MathSciNet  Google Scholar 

  4. Brillinger, D.R.: An introduction to polyspectra. Ann. Math. Stat. 36, 1351–1374 (1965)

    MathSciNet  MATH  Google Scholar 

  5. Cheung, L.C., Zaki, T.A.: An exact representation of the nonlinear triad interaction terms in spectral space. J. Fluid Mech. 748, 175–188 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Collis, W., White, P., Hammond, J.: Higher-order spectra: the bispectrum and trispectrum. Mech. Syst. Signal Process. 12(3), 375–394 (1998)

    Google Scholar 

  7. Corke, T., Shakib, F., Nagib, H.: Mode selection and resonant phase locking in unstable axisymmetric jets. J. Fluid Mech. 223, 253–311 (1991)

    Google Scholar 

  8. Corke, T.C., Mangano, R.A.: Resonant growth of three-dimensional modes in trnsitioning Blasius boundary layers. J. Fluid Mech. 209, 93–150 (1989)

    Google Scholar 

  9. Craig, S.A., Humble, R.A., Hofferth, J.W., Saric, W.S.: Nonlinear behaviour of the mack mode in a hypersonic boundary layer. J. Fluid Mech. 872, 74–99 (2019)

    MathSciNet  MATH  Google Scholar 

  10. Craik, A.: Non-linear resonant instability in boundary layers. J. Fluid Mech. 50(2), 393–413 (1971)

    MathSciNet  MATH  Google Scholar 

  11. Craik, A.D.D.: Wave Interactions and Fluid Flows. Cambridge University Press, Cambridge (1988)

    MATH  Google Scholar 

  12. Deane, A., Kevrekidis, I., Karniadakis, G.E., Orszag, S.: Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids A 3(10), 2337–2354 (1991)

    MATH  Google Scholar 

  13. Domaradzki, J.A.: Nonlocal triad interactions and the dissipation range of isotropic turbulence. Phys. Fluids A 4(9), 2037–2045 (1992)

    MATH  Google Scholar 

  14. Domaradzki, J.A., Rogallo, R.S.: Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence. Phys. Fluids A 2(3), 413–426 (1990)

    Google Scholar 

  15. Drivas, T.D., Wunsch, S.: Triad resonance between gravity and vorticity waves in vertical shear. Ocean Model. 103, 87–97 (2016)

    Google Scholar 

  16. Farazmand, M., Sapsis, T.P.: A variational approach to probing extreme events in turbulent dynamical systems. Sci. Adv. 3(9), e1701533 (2017)

    Google Scholar 

  17. Farazmand, M., Sapsis, T.P.: Closed-loop adaptive control of extreme events in a turbulent flow. Phys. Rev. E 100(3), 033110 (2019)

    MathSciNet  Google Scholar 

  18. Farrell, B.F., Ioannou, P.J.: Structural stability of turbulent jets. J. Atmos. Sci. 60(17), 2101–2118 (2003)

    Google Scholar 

  19. Farrell, B.F., Ioannou, P.J., Jiménez, J., Constantinou, N.C., Lozano-Durán, A., Nikolaidis, M.A.: A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane poiseuille flow. J. Fluid Mech. 809, 290–315 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Gee, K.L., Atchley, A.A., Falco, L.E., Shepherd, M.R., Ukeiley, L.S., Jansen, B.J., Seiner, J.M.: Bicoherence analysis of model-scale jet noise. J. Acoust. Soc. Am. 128(5), EL211–EL216 (2010)

    Google Scholar 

  21. Giannetti, F., Luchini, P.: Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Goza, A., Colonius, T.: A strongly-coupled immersed-boundary formulation for thin elastic structures. J. Comput. Phys. 336, 401–411 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Hammack, J.L., Henderson, D.M.: Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25(1), 55–97 (1993)

    MathSciNet  Google Scholar 

  24. He, C., Watson, G.A.: An algorithm for computing the numerical radius. IMA J. Numer. Anal. 17(3), 329–342 (1997)

    MathSciNet  MATH  Google Scholar 

  25. Herring, J.R.: Theoretical calculations of turbulent bispectra. J. Fluid Mech. 97(1), 193–204 (1980)

    Google Scholar 

  26. Herring, J.R., Métais, O.: Spectral transfer and bispectra for turbulence with passive scalars. J. Fluid Mech. 235, 103–121 (1992)

    MATH  Google Scholar 

  27. Horn, R.A., Johnson, C.: Topics in matrix analysis. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  28. Janssen, P.A.E.M.: Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33(4), 863–884 (2003)

    MathSciNet  Google Scholar 

  29. Kim, K.I., Powers, E.J.: A digital method of modeling quadratically nonlinear systems with a general random input. IEEE Trans. Acoust. Speech Signal Process. 36(11), 1758–1769 (1988)

    MATH  Google Scholar 

  30. Kim, Y.C., Powers, E.J.: Digital bispectral analysis and its applications to nonlinear wave interactions. IEEE Trans. Plasma Sci. 7(2), 120–131 (1979)

    Google Scholar 

  31. Kim, Y.C., Beall, J.M., Powers, E.J., Miksad, R.W.: Bispectrum and nonlinear wave coupling. Phys. Fluids 23(2), 258–263 (1980)

    Google Scholar 

  32. Kraichnan, R.H.: Inertial ranges in two-dimensional turbulence. Phys. Fluids 10(7), 1417–1423 (1967)

    Google Scholar 

  33. Kraichnan, R.H.: Inertial-range transfer in two-and three-dimensional turbulence. J. Fluid Mech. 47(3), 525–535 (1971)

    MATH  Google Scholar 

  34. Lii, K.S., Rosenblatt, M., Van Atta, C.: Bispectral measurements in turbulence. J. Fluid Mech. 77(1), 45–62 (1976)

    Google Scholar 

  35. Lumley, J.L.: Stochastic Tools in Turbulence. Academic Press, New York (1970)

    MATH  Google Scholar 

  36. Manz, P., Ramisch, M., Stroth, U., Naulin, V., Scott, B.D.: Bispectral experimental estimation of the nonlinear energy transfer in two-dimensional plasma turbulence. Plasma Phys. Control. Fusion 50(3), 035008 (2008)

    Google Scholar 

  37. Marston, J., Chini, G., Tobias, S.: Generalized quasilinear approximation: application to zonal jets. Phys. Rev. Lett. 116(21), 214501 (2016)

    MathSciNet  Google Scholar 

  38. McKeon, B.J.: The engine behind (wall) turbulence: perspectives on scale interactions. J. Fluid Mech. 817, P1 (2017)

  39. Moffatt, H.K.: Note on the triad interactions of homogeneous turbulence. J. Fluid Mech. 741, R3 (2014)

  40. Nikias, C.L., Mendel, J.M.: Signal processing with higher-order spectra. IEEE Signal Process. Mag. 10(3), 10–37 (1993)

    Google Scholar 

  41. Nikias, C.L., Raghuveer, M.R.: Bispectrum estimation: a digital signal processing framework. Proc. IEEE 75(7), 869–891 (1987)

    Google Scholar 

  42. Noack, B.R., Afanasiev, K., Morzyński, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003)

    MathSciNet  MATH  Google Scholar 

  43. Phillips, O.M.: On the dynamics of unsteady gravity waves of finite amplitude part 1. The elementary interactions. J. Fluid Mech. 9(2), 193–217 (1960)

    MathSciNet  MATH  Google Scholar 

  44. Pope, S.B.: Turbulent Flows, 1st edn. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  45. Rigas, G., Oxlade, A., Morgans, A., Morrison, J.: Low-dimensional dynamics of a turbulent axisymmetric wake. J. Fluid Mech. 755 (2014)

  46. Rigas, G., Sipp, D., Colonius, T.: Non-linear input/output analysis: application to boundary layer transition.. arXiv:2001.09440 (2020)

  47. Ritz, C.P., Powers, E.: Estimation of nonlinear transfer functions for fully developed turbulence. Physica D 20(2–3), 320–334 (1986)

    MATH  Google Scholar 

  48. Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)

    MathSciNet  MATH  Google Scholar 

  49. Sapsis, T.P.: Statistics of extreme events in fluid flows and waves. Annu. Rev. Fluid Mech. 53 (2020), 85–111

  50. Schmid, P., Henningson, D.S.: Stability and transition in shear flows. In: Applied Mathematical Sciences vol. 142, 1st edn. Springer, New York (2001)

  51. 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 

  52. Schmidt, O.T., Colonius, T.: Guide to spectral proper orthogonal decomposition. AIAA J. 58(3), 1023–1033 (2020)

    Google Scholar 

  53. Sefusatti, E., Crocce, M., Pueblas, S., Scoccimarro, R.: Cosmology and the bispectrum. Phys. Rev. D 74(2), 023522 (2006)

    Google Scholar 

  54. Sirovich, L.: Turbulence and the dynamics of coherent structures. Q. Appl. Math. 45(3), 561–571 (1987)

    MathSciNet  MATH  Google Scholar 

  55. Soomere, T.: Rogue waves in shallow water. Eur. Phys. J. Spec. Top. 185(1), 81–96 (2010)

    Google Scholar 

  56. Strykowski, P.J., Sreenivasan, K.R.: On the formation and suppression of vortex ‘shedding’at low reynolds numbers. J. Fluid Mech. 218, 71–107 (1990)

    Google Scholar 

  57. Thomas, V.L., Binh, K., Jovanović, M.R., Farrell, B.F., Ioannou, P.J., Gayme, D.F.: Self-sustaining turbulence in a restricted nonlinear model of plane couette flow. Phys. Fluids 26(10), 105112 (2014)

    Google Scholar 

  58. 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 

  59. Waleffe, F.: The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4(2), 350–363 (1992)

    MathSciNet  MATH  Google Scholar 

  60. Watson, G.A.: Computing the numerical radius. Linear Algebra Appl. 234, 163–172 (1996)

    MathSciNet  MATH  Google Scholar 

  61. 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)

    Google Scholar 

  62. Williamson, C.: Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28(1), 477–539 (1996)

    MathSciNet  Google Scholar 

  63. Yamada, T., Itoh, S.I., Inagaki, S., Nagashima, Y., Shinohara, S., Kasuya, N., Terasaka, K., Kamataki, K., Arakawa, H., Yagi, M., et al.: Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma. Phys. Plasmas 17(5), 052313 (2010)

    Google Scholar 

Download references

Acknowledgements

I would like to thank Tim Colonius for pointing me to higher-order spectra, Ethan Pickering for pointing out the need to compute difference-interactions, Aaron Towne, Peter Schmid, Georgios Rigas and Tim Colonius for many helpful discussions, and the two anonymous referees for their very insightful comments. I gratefully acknowledge Karen Mulleners for providing the PIV data, Guillaume Brès for the LES data, and Andres Goza for helping with the DNS.

Author information

Authors and Affiliations

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

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Appendices

Appendix A: Computation of \(\mathbf{a }_1= \text {arg max}_{\Vert \mathbf{a }\Vert =1}|\mathbf{a} ^*\mathbf{B }\mathbf{a }|\)

A slightly modified version of the algorithm for the computation of the numerical radius by He and Watson [24] is used. He and Watson’s algorithm requires two nested iterations. The first, or so-called simple iteration [60], converges to a local solution of Eq. (56). A tolerance of \( tol =10^{-8}\) was found to be a good compromise between accuracy and compute time for both iterations. The number of iterations was limited to \(k=300\).

figure a

Building on the simple iteration to find local solutions, the purpose of the main algorithm is to find the global solution. Double precision arithmetic with machine precision \(\epsilon =2^{-52}\) was used to compute the results in Sect. 4. If the algorithm did not converge within 500 iterations, it was restarted up to five times with a new random initial guess for \(\mathbf{a} _0\). This procedure was necessary to ensure that all results are fully converged.

figure b

Appendix B: Convergence

Figure 16 demonstrates the convergence of all three cases discussed in Sect. 4 in terms of summed mode spectra. The convergence of the results is tested by recomputing the BMD for smaller subsets of the available data. In particular, Fig. 16a shows that the summed mode spectrum obtained for the first 3 and 5 blocks of the cylinder flow DNS data is very similar to the one obtained for all 7 block. Similarly, the full data of the flat plate PIV and the jet LES are compared to the spectra obtained for approximately one-third and two-thirds of the full data in Fig. 16b and c, respectively. It can be seen that the summed mode spectra for the latter cases, too, remain largely unaltered by the data reduction. The summed mode spectrum was chosen as the most compact representation of the results. A comparison of the bispectral modes and mode bispectra (not shown) obtained from the full and reduced data sets confirmed that the results are also well-converged with respect to mode spectra and mode shapes.

Fig. 16
figure 16

Convergence of summed mode spectra for different \(N_{\mathrm {blk}}\): a cylinder DNS; b flat plate PIV; c \(\mathopen {[\![}0,0,0\mathclose {]\!]}\) azimuthal triplet of the transitional jet LES. Solid lines correspond to the full data

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-06037-z

Keywords

Search

Navigation