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Spectral proper orthogonal decomposition of harmonically forced turbulent flows

Published online by Cambridge University Press:  29 April 2024

Liam Heidt Open the ORCID record for Liam Heidt [Opens in a new window]  and
Tim Colonius Open the ORCID record for Tim Colonius [Opens in a new window]
Liam Heidt*
Affiliation:
Graduate Aerospace Laboratories of the California Institute of Technology, California Institute of Technology, Pasadena, CA 91101, USA
Tim Colonius
Affiliation:
Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91101, USA
*
†Email address for correspondence: lheidt@caltech.edu
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Abstract

Many turbulent flows exhibit time-periodic statistics. These include turbomachinery flows, flows with external harmonic forcing and the wakes of bluff bodies. Many existing techniques for identifying turbulent coherent structures, however, assume the statistics are statistically stationary. In this paper, we leverage cyclostationary analysis, an extension of the statistically stationary framework to processes with periodically varying statistics, to generalize the spectral proper orthogonal decomposition (SPOD) to the cyclostationary case. The resulting properties of the cyclostationary SPOD (CS-SPOD for short) are explored, a theoretical connection between CS-SPOD and the harmonic resolvent analysis is provided, simplifications for the low and high forcing frequency limits are discussed, and an efficient algorithm to compute CS-SPOD with SPOD-like cost is presented. We illustrate the utility of CS-SPOD using two example problems: a modified complex linearized Ginzburg–Landau model and a high-Reynolds-number turbulent jet.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulent Flows

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JFM Papers
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Journal of Fluid Mechanics , Volume 985 , 25 April 2024 , A42
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© The Author(s), 2024. Published by Cambridge University Press

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References

Amaral, F.R., Cavalieri, A.V.G., Martini, E., Jordan, P. & Towne, A. 2021 Resolvent-based estimation of turbulent channel flow using wall measurements. J. Fluid Mech. 927, A17.CrossRefGoogle Scholar
Antoni, J. 2007 Cyclic spectral analysis in practice. Mech. Syst. Signal Process. 21 (2), 597–630.CrossRefGoogle Scholar
Antoni, J. 2009 Cyclostationarity by examples. Mech. Syst. Signal Process. 23 (4), 987–1036.CrossRefGoogle Scholar
Antoni, J., Bonnardot, F., Raad, A. & El Badaoui, M. 2004 Cyclostationary modelling of rotating machine vibration signals. Mech. Syst. Signal Process. 18 (6), 1285–1314.CrossRefGoogle Scholar
Arbabi, H. & Mezić, I. 2017 Study of dynamics in post-transient flows using Koopman mode decomposition. Phys. Rev. Fluids 2 (12), 124402.CrossRefGoogle Scholar
Aubry, N. 1991 On the hidden beauty of the proper orthogonal decomposition. Theor. Comput. Fluid Dyn. 2 (5–6), 339–352.CrossRefGoogle Scholar
Aubry, N., Holmes, P., Lumley, J.L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.CrossRefGoogle Scholar
Bagheri, S., Henningson, D.S., Hoepffner, 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
Bendat, J.S. & Piersol, A.G. 2011 Random Data: Analysis and Measurement Procedures. Wiley.Google Scholar
Boyles, R. & Gardner, W. 1983 Cycloergodic properties of discrete-parameter nonstationary stochastic processes. IEEE Trans. Inf. theory 29 (1), 105–114.CrossRefGoogle Scholar
Braun, S. 1975 The extraction of periodic waveforms by time domain averaging. Acta Acust. United Acust. 32 (2), 69–77.Google Scholar
Brereton, G.J. & Kodal, A. 1992 A frequency-domain filtering technique for triple decomposition of unsteady turbulent flow. J. Fluids Engng 114 (1), 45–51.CrossRefGoogle Scholar
Brès, G.A., Ham, F.E., Nichols, J.W. & Lele, S.K. 2017 Unstructured large-eddy simulations of supersonic jets. AIAA J. 55 (4), 1164–1184.CrossRefGoogle Scholar
Brès, G.A., Jordan, P., Jaunet, V., Le Rallic, M., Cavalieri, A.V.G., Towne, A., Lele, S.K., Colonius, T. & Schmidt, O.T. 2018 Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets. J. Fluid Mech. 851, 83–124.CrossRefGoogle Scholar
Brown, W.A. III 1987 On the Theory of Cyclostationary Signals. University of California.Google Scholar
Chen, K.K. & Rowley, C.W. 2011 $H_2$ optimal actuator and sensor placement in the linearised complex Ginzburg–Landau system. J. Fluid Mech. 681, 241–260.CrossRefGoogle Scholar
Chomaz, J.M., Huerre, P. & Redekopp, L.G. 1988 Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett. 60 (1), 25.CrossRefGoogle ScholarPubMed
Chu, B.-T. 1965 On the energy transfer to small disturbances in fluid flow (part I). Acta Mech. 1 (3), 215–234.CrossRefGoogle 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
Cossu, C. & Chomaz, J.M. 1997 Global measures of local convective instabilities. Phys. Rev. Lett. 78 (23), 4387.CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large–scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.CrossRefGoogle Scholar
Crow, S.C. & Champagne, F.H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48 (3), 547–591.CrossRefGoogle Scholar
Dormand, J.R. & Prince, P.J. 1980 A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6 (1), 19–26.CrossRefGoogle Scholar
Farrell, B.F. & Ioannou, P.J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A: Fluid Dyn. 5 (11), 2600–2609.CrossRefGoogle Scholar
Farrell, B.F. & Ioannou, P.J. 2001 Accurate low-dimensional approximation of the linear dynamics of fluid flow. J. Atmos. Sci. 58 (18), 2771–2789.2.0.CO;2>CrossRefGoogle Scholar
Flandrin, P. 1998 Time-Frequency/Time-Scale Analysis. Academic.Google Scholar
Franceschini, L., Sipp, D., Marquet, O., Moulin, J. & Dandois, J. 2022 Identification and reconstruction of high-frequency fluctuations evolving on a low-frequency periodic limit cycle: application to turbulent cylinder flow. J. Fluid Mech. 942, A28.CrossRefGoogle Scholar
Gardner, W. 1986 a Measurement of spectral correlation. IEEE Trans. Acoust. 34 (5), 1111–1123.CrossRefGoogle Scholar
Gardner, W.A. 1986 b Introduction to Random Processes with Applications to Signals and Systems, p. 447. MacMillan.Google Scholar
Gardner, W.A. 1986 c The spectral correlation theory of cyclostationary time-series. Signal Process. 11 (1), 13–36.CrossRefGoogle Scholar
Gardner, W.A. 1972 Representation and estimation of cyclostationary processes. Tech. Rep. AFOSR-TR-72-2378. Massachusetts University Amherst Engineering Research Institute.Google Scholar
Gardner, W.A. 1994 An introduction to cyclostationary signals. In Cyclostationarity in Communications and Signal Processing, pp. 1–90. IEEE.Google Scholar
Gardner, W.A. 2018 Statistically inferred time warping: extending the cyclostationarity paradigm from regular to irregular statistical cyclicity in scientific data. EURASIP J. Adv. Signal Process. 2018 (1), 1–25.Google Scholar
Gardner, W.A., Napolitano, A. & Paura, L. 2006 Cyclostationarity: half A century of research. Signal Process. 86 (4), 639–697.CrossRefGoogle Scholar
Gardner, W.A. & Robinson, E.A. 1989 Statistical Spectral Analysis—A Nonprobabilistic Theory. Prentice Hall.CrossRefGoogle Scholar
Gladyshev, E.G. 1963 Periodically and almost-periodically correlated random processes with a continuous time parameter. Theory Probab. Appl. 8 (2), 173–177.CrossRefGoogle Scholar
Glezer, A., Kadioglu, Z. & Pearlstein, A.J. 1989 Development of an extended proper orthogonal decomposition and its application to a time periodically forced plane mixing layer. Phys. Fluids A: Fluid Dyn. 1 (8), 1363–1373.CrossRefGoogle Scholar
Gudzenko, L.I. 1959 On periodic nonstationary processes. Radio Engng Electron. Phys. (USSR) 4 (6), 220–224.Google Scholar
Heidt, L., Colonius, T., Nekkanti, A., Schmdit, O., Maia, I. & Jordan, P. 2021 Analysis of forced subsonic jets using spectral proper orthogonal decomposition and resolvent analysis. In AIAA Aviation 2021 Forum, p. 2108.Google Scholar
Hunt, R.E. & Crighton, D.G. 1991 Instability of flows in spatially developing media. Proc. R. Soc. Lond. A 435 (1893), 109–128.Google Scholar
Hurd, H.L. 1969 An investigation of periodically correlated stochastic processes. PhD thesis. Duke University, Durham, North Carolina, USA.Google Scholar
Hussain, A.K.M.F. & Reynolds, W.C. 1970 The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41 (2), 241–258.CrossRefGoogle Scholar
Hussain, A.K.M.F. & Reynolds, W.C. 1972 The mechanics of an organized wave in turbulent shear flow. Part 2. Experimental results. J. Fluid Mech. 54 (2), 241–261.CrossRefGoogle Scholar
Jenkins, G.M. 1968 Spectral Analysis and its Applications. Holden-Day.Google Scholar
Jeun, J., Nichols, J.W. & Jovanović, M.R. 2016 Input-output analysis of high-speed axisymmetric isothermal jet noise. Phys. Fluids 28 (4), 047101.CrossRefGoogle Scholar
Jovanovic, M. & Bamieh, B. 2001 Modeling flow statistics using the linearized Navier–Stokes equations. In Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No. 01CH37228), vol. 5, pp. 4944–4949. IEEE.Google Scholar
Kim, K.-Y. & North, G.R. 1997 EOFs of harmonizable cyclostationary processes. J. Atmos. Sci. 54 (19), 2416–2427.2.0.CO;2>CrossRefGoogle Scholar
Kim, K.-Y., North, G.R. & Huang, J. 1996 EOFs of one-dimensional cyclostationary time series: computations, examples, and stochastic modeling. J. Atmos. Sci. 53 (7), 1007–1017.2.0.CO;2>CrossRefGoogle Scholar
Lebedev, V.L. 1959 On random processes having nonstationarity of periodic character. Nauchn. Dokl. Vysshch. Shchk. Ser. Radiotekh. Elektron. 2, 32–34.Google Scholar
Lumley, J.L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Propagation, pp. 166–178.Google Scholar
Lumley, J.L. 1970 Stochastic tools in turbulence. J. Fluid Mech. 67, 413–415.CrossRefGoogle Scholar
Martin, W. 1982 Time-frequency analysis of random signals. In ICASSP’82. IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 7, pp. 1325–1328. IEEE.Google Scholar
Martin, W. & Flandrin, P. 1985 Wigner-Ville spectral analysis of nonstationary processes. IEEE Trans. Acoust. 33 (6), 1461–1470.CrossRefGoogle Scholar
Martinsson, P.-G. & Tropp, J.A. 2020 Randomized numerical linear algebra: foundations and algorithms. Acta Numer. 29, 403–572.CrossRefGoogle Scholar
McKeon, B.J. & Sharma, A.S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336382.CrossRefGoogle Scholar
Meliga, P., Pujals, G. & Serre, E. 2012 Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability. Phys. Fluids 24 (6), 061701.CrossRefGoogle Scholar
Mezić, I. 2013 Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378.CrossRefGoogle Scholar
Moarref, R., Sharma, A.S., Tropp, J.A. & McKeon, B.J. 2013 Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275–316.CrossRefGoogle Scholar
Morra, P., Semeraro, O., Henningson, D.S. & Cossu, C. 2019 On the relevance of Reynolds stresses in resolvent analyses of turbulent wall-bounded flows. J. Fluid Mech. 867, 969–984.CrossRefGoogle Scholar
Napolitano, A. 2019 Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations. Academic.Google Scholar
Oberleithner, K., Paschereit, C.O. & Wygnanski, I. 2014 On the impact of swirl on the growth of coherent structures. J. Fluid Mech. 741, 156–199.CrossRefGoogle Scholar
Padovan, A., Otto, S.E. & Rowley, C.W. 2020 Analysis of amplification mechanisms and cross-frequency interactions in nonlinear flows via the harmonic resolvent. J. Fluid Mech. 900, A14.CrossRefGoogle Scholar
Padovan, A. & Rowley, C.W. 2022 Analysis of the dynamics of subharmonic flow structures via the harmonic resolvent: application to vortex pairing in an axisymmetric jet. Phys. Rev. Fluids 7 (7), 073903.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
Pickering, E., Rigas, G., Schmidt, O.T., Sipp, D. & Colonius, T. 2021 a Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets. J. Fluid Mech. 917, A29.CrossRefGoogle Scholar
Pickering, E., Towne, A., Jordan, P. & Colonius, T. 2021 b Resolvent-based modeling of turbulent jet noise. J. Acoust. Soc. Am. 150 (4), 2124–2433.CrossRefGoogle ScholarPubMed
Randall, R.B., Antoni, J. & Chobsaard, S. 2001 The relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other cyclostationary machine signals. Mech. Syst. Signal Process. 15 (5), 945–962.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. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
Schmid, P.J., Li, L., Juniper, M.P. & Pust, O. 2011 Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn. 25 (1), 249–259.CrossRefGoogle Scholar
Schmidt, O.T. 2022 Spectral proper orthogonal decomposition using multitaper estimates. Theor. Comput. Fluid Dyn. 36 (5), 741–754.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., Rigas, G., Colonius, T. & Brès, G.A. 2018 Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953–982.CrossRefGoogle Scholar
Shampine, L.F. & Reichelt, M.W. 1997 The MATLAB ODE suite. SIAM J. Sci. Comput. 18 (1), 1–22.CrossRefGoogle Scholar
Sharma, A.S. & McKeon, B.J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.CrossRefGoogle Scholar
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 030801.CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Maths 45 (3), 561–571.CrossRefGoogle Scholar
Sirovich, L. 1989 Chaotic dynamics of coherent structures. Physica D 37 (1–3), 126–145.CrossRefGoogle Scholar
Sonnenberger, R., Graichen, K. & Erk, P. 2000 Fourier averaging: a phase-averaging method for periodic flow. Exp. Fluids 28 (3), 217–224.CrossRefGoogle Scholar
Towne, A., Bres, G.A. & Lele, S.K. 2017 A statistical jet-noise model based on the resolvent framework. In 23rd AIAA/CEAS Aeroacoustics Conference, p. 3706.Google Scholar
Towne, A., Lozano-Durán, A. & Yang, X. 2020 Resolvent-based estimation of space–time flow statistics. J. Fluid Mech. 883, A17.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
Welch, P. 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