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Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. (English) Zbl 1404.76145
Summary: We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space-time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of J. L. Lumley [Stochastic tools in turbulence. Mineola, NY: Dover Publications, Inc. (1970)] but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg-Landau equation and a turbulent jet.

MSC:
76F65 Direct numerical and large eddy simulation of turbulence
76-04 Software, source code, etc. for problems pertaining to fluid mechanics
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[1] Abreu, L. I., Cavalieri, A. V. G. & Wolf, W. R.2017 Coherent hydrodynamic waves and trailing-edge noise. AIAA Paper 2017-3173.
[2] 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) · Zbl 1383.76299
[3] Arbabi, H.; Mezić, I., Study of dynamics in unsteady flows using Koopman mode decomposition, Phys. Rev. Fluids, 2, (2017)
[4] 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)
[5] Aubry, N., On the hidden beauty of the proper orthogonal decomposition, Theoret. Comput. Fluid Dyn., 2, 5, 339-352, (1991) · Zbl 0732.76044
[6] 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) · Zbl 0643.76066
[7] Bagheri, S.; Henningson, D. S.; Hoepffner, J.; Schmid, P. J., Input – output analysis and control design applied to a linear model of spatially developing flows, Appl. Mech. Rev., 62, 2, (2009)
[8] Bendat, J. S.; Piersol, A. G., Random Data: Analysis and Measurement Procedures, (2000), John Wiley & Sons · Zbl 0953.62128
[9] Beneddine, S.; Sipp, D.; Arnault, A.; Dandois, J.; Lesshafft, L., Conditions for validity of mean flow stability analysis, J. Fluid Mech., 798, 485-504, (2016) · Zbl 1422.76070
[10] Berkooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, 1, 539-575, (1993)
[11] Braud, C.; Heitz, D.; Arroyo, G.; Perret, L.; Delville, J.; B., J.-P., Low-dimensional analysis, using POD, for two mixing layer – wake interactions, Intl J. Heat Fluid Flow, 25, 3, 351-363, (2004)
[12] Brès, G. A.; Ham, F. E.; Nichols, J. W.; Lele, S. K., Unstructured large-eddy simulations of supersonic jets, AIAA J., 55, 4, 1164-1184, (2017)
[13] Brès, G. A., Jaunet, J., Le Rallic, M., Jordan, P., Colonius, T. & Lele, S. K.2015 Large eddy simulation for jet noise: the importance of getting the boundary layer right. AIAA Paper 2015-2535.
[14] Brès, G. A., Jaunet, V., Le Rallic, M., Jordan, P., Towne, A., Schmidt, O., Colonius, T., Cavalieri, A. V. G. & Lele, S. K.2016 Large eddy simulation for jet noise: azimuthal decomposition and intermittency of the radiated sound. AIAA Paper 2016-3050.
[15] Brès, G. A., Jordan, P., Jaunet, V., Le Rallic, M., Cavalieri, A. V. G., Towne, A., Lele, S. K., Colonius, T. & Schmidt, O. T.2017b Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets. J. Fluid Mech. (in press). · Zbl 1415.76555
[16] Cammilleri, A.; Guéniat, F.; Carlier, J.; Pastur, L.; Mémin, E.; Lusseyran, F.; Artana, G., POD-spectral decomposition for fluid flow analysis and model reduction, Theoret. Comput. Fluid Dyn., 27, 6, 787-815, (2013)
[17] Cavalieri, A. V. G.; Agarwal, A., Coherence decay and its impact on sound radiation by wavepackets, J. Fluid Mech., 748, 399-415, (2014) · Zbl 1416.76272
[18] Cavalieri, A. V. G.; Jordan, P.; Agarwal, A.; Gervais, Y., Jittering wave-packet models for subsonic jet noise, J. Sound Vib., 330, 18, 4474-4492, (2011)
[19] Chatterjee, A., An introduction to the proper orthogonal decomposition, Curr. Sci., 78, 7, 808-817, (2000)
[20] Chen, K. K.; Rowley, C. W., H_2 optimal actuator and sensor placement in the linearised complex Ginzburg-Landau system, J. Fluid Mech., 681, 241-260, (2011) · Zbl 1241.76164
[21] Chen, K. K.; Tu, J. H.; Rowley, C. W., Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses, J. Nonlinear Sci., 22, 6, 887-915, (2012) · Zbl 1259.35009
[22] Chen, X.; Kareem, A., Proper orthogonal decomposition-based modeling, analysis, and simulation of dynamic wind load effects on structures, J. Engng Mech., 131, 4, 325-339, (2005)
[23] Chu, B.-T., On the energy transfer to small disturbances in fluid flow (Part I), Acta Mech., 1, 3, 215-234, (1965)
[24] 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) · Zbl 1103.76333
[25] Cordier, L. & Bergmann, M.2008Proper orthogonal decomposition: an overview. In Lecture Series 2002-04, 2003-03 and 2008-01 on Post-Processing of Experimental and Numerical Data. Von Karman Institute for Fluid Dynamics.
[26] Delville, J.; Ukeiley, L.; Cordier, L.; Bonnet, J. P.; Glauser, M., Examination of large-scale structures in a turbulent plane mixing layer. Part 1. Proper orthogonal decomposition, J. Fluid Mech., 391, 91-122, (1999) · Zbl 0995.76030
[27] Dergham, G.; Sipp, D.; Robinet, J.-Ch., Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow, J. Fluid Mech., 719, 406-430, (2013) · Zbl 1284.76317
[28] Farrell, B. F.; Ioannou, P. J., Stochastic forcing of the linearized Navier-Stokes equations, Phys. Fluids, 5, 11, 2600-2609, (1993) · Zbl 0809.76078
[29] Farrell, B. F.; Ioannou, P. J., Generalized stability theory. Part I: autonomous operators, J. Atmos. Sci., 53, 14, 2025-2040, (1996)
[30] Farrell, B. F.; Ioannou, P. J., Accurate low-dimensional approximation of the linear dynamics of fluid flow, J. Atmos. Sci., 58, 18, 2771-2789, (2001)
[31] Garnaud, X.; Lesshafft, L.; Schmid, P. J.; Huerre, P., The preferred mode of incompressible jets: linear frequency response analysis, J. Fluid Mech., 716, 189-202, (2013) · Zbl 1284.76149
[32] George, W. K.1988Insight into the dynamics of coherent structures from a proper orthogonal decomposition dy. In International Seminar on Wall Turbulence.
[33] George, W. K.2017A 50-year retrospective and the future. In Whither Turbulence and Big Data in the 21st Century? pp. 13-43. Springer. doi:10.1007/978-3-319-41217-7_2
[34] George, W. K., Beuther, P. D. & Lumley, J. L.1978Processing of random signals. In Proceedings of the Dynamic Flow Conference 1978 on Dynamic Measurements in Unsteady Flows, pp. 757-800. Springer. doi:10.1007/978-94-009-9565-9_43
[35] Glauser, M. N., Leib, S. J. & George, W. K.1987Coherent Structures in the Axisymmetric Turbulent Jet Mixing Layer, pp. 134-145. Springer.
[36] Gómez, F., Blackburn, H. M., Rudman, M., Sharma, A. S. & Mckeon, B. J.2016aOn the coupling of direct numerical simulation and resolvent analysis. In Progress in Turbulence VI, pp. 87-91. Springer. doi:10.1007/978-3-319-29130-7_16
[37] Gómez, F.; Blackburn, H. M.; Rudman, M.; Sharma, A. S.; Mckeon, B. J., A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator, J. Fluid Mech., 798, (2016)
[38] 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) · Zbl 0949.76511
[39] Gudmundsson, K.; Colonius, T., Instability wave models for the near-field fluctuations of turbulent jets, J. Fluid Mech., 689, 97-128, (2011) · Zbl 1241.76203
[40] Heinzel, G., Rüdiger, A. & Schilling, R.2002 Spectrum and spectral density estimation by the discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flat-top windows. https://holometer.fnal.gov/GH_FFT.pdf (unpublished).
[41] Hellström, L. H. O.; Smits, A. J., The energetic motions in turbulent pipe flow, Phys. Fluids, 26, 12, (2014)
[42] Hilberg, D.; Lazik, W.; Fiedler, H. E., The application of classical POD and snapshot POD in a turbulent shear layer with periodic structures, Appl. Sci. Res., 53, 3, 283-290, (1994)
[43] Holmes, P.; Lumley, J. L.; Berkooz, G.; Rowley, C. W., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, (2012), Cambridge University Press · Zbl 1251.76001
[44] Holmes, P. J.; Lumley, J. L.; Berkooz, G.; Mattingly, J. C.; Wittenberg, R. W., Low-dimensional models of coherent structures in turbulence, Phys. Rep., 287, 4, 337-384, (1997)
[45] Hunt, R. E.; Crighton, D. G., Instability of flows in spatially developing media, Proc. R. Soc. Lond. A, 435, 109-128, (1991) · Zbl 0731.76032
[46] Jeun, J.; Nichols, J. W.; Jovanović, M. R., Input – output analysis of high-speed axisymmetric isothermal jet noise, Phys. Fluids, 28, 4, (2016)
[47] Jordan, P., Zhang, M., Lehnasch, G. & Cavalieri, A. V. G.2017 Modal and non-modal linear wavepacket dynamics in turbulent jets. AIAA Paper 2017-3379.
[48] Jovanović, M. & Bamieh, B.2001Modeling flow statistics using the linearized Navier-Stokes equations. In Decision and Control, 2001. Proceedings of the 40th IEEE Conference, vol. 5, pp. 4944-4949. IEEE.
[49] Jovanović, M. R.; Bamieh, B., Componentwise energy amplification in channel flows, J. Fluid Mech., 534, 145-183, (2005) · Zbl 1074.76016
[50] Landahl, M. T.; Mollo Christensen, E., Turbulence and Random Processes in Fluid Mechanics, (1992), Cambridge University Press · Zbl 0772.76004
[51] Liang, Y. C.; Lee, H. P.; Lim, S. P.; Lin, W. Z.; Lee, K. H.; Wu, C. G., Proper orthogonal decomposition and its applications. Part I: theory, J. Sound Vib., 252, 3, 527-544, (2002) · Zbl 1237.65040
[52] Lumley, J. L.1967The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Propagation (ed. Yaglom, A. M. & Tatarski, V. I.), pp. 166-178. Nauka.
[53] Lumley, J. L., Stochastic Tools in Turbulence, (1970), Academic Press · Zbl 0273.76035
[54] Mckeon, B. J.; Sharma, A. S., A critical-layer framework for turbulent pipe flow, J. Fluid Mech., 658, 336-382, (2010) · Zbl 1205.76138
[55] Mezić, I., Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dyn., 41, 1, 309-325, (2005) · Zbl 1098.37023
[56] Moarref, R.; Jovanović, M. R., Model-based design of transverse wall oscillations for turbulent drag reduction, J. Fluid Mech., 707, 205-240, (2012) · Zbl 1275.76152
[57] Moarref, R.; Jovanović, M. R.; Tropp, J. A.; Sharma, A. S.; Mckeon, B. J., A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization, Phys. Fluids, 26, 5, (2014)
[58] Moin, P.; Moser, R. D., Characteristic-eddy decomposition of turbulence in a channel, J. Fluid Mech., 200, 471-509, (1989) · Zbl 0659.76062
[59] Mula, S. M. & Tinney, C. E.2014 Classical and snapshot forms of the POD technique applied to a helical vortex filament. AIAA Paper 2015-3257.
[60] 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) · Zbl 1067.76033
[61] Noack, B. R.; Stankiewicz, W.; Morzyński, M.; Schmid, P. J., Recursive dynamic mode decomposition of transient and post-transient wake flows, J. Fluid Mech., 809, 843-872, (2016) · Zbl 1383.76122
[62] Picard, C.; Delville, J., Pressure velocity coupling in a subsonic round jet, Intl J. Heat Fluid Flow, 21, 3, 359-364, (2000)
[63] Pinier, J. T.; Ausseur, J. M.; Glauser, M. N.; Higuchi, H., Proportional closed-loop feedback control of flow separation, AIAA J., 45, 1, 181-190, (2007)
[64] Pope, S. B., Turbulent Flows, (2000), Cambridge University Press · Zbl 0966.76002
[65] Rowley, C. W., Model reduction for fluids, using balanced proper orthogonal decomposition, Intl J. Bifurcation Chaos, 15, 3, 997-1013, (2005) · Zbl 1140.76443
[66] Rowley, C. W.; Colonius, T.; Murray, R. M., Model reduction for compressible flows using POD and Galerkin projection, Physica D, 189, 1, 115-129, (2004) · Zbl 1098.76602
[67] Rowley, C. W.; Dawson, S. T. M., Model reduction for flow analysis and control, Annu. Rev. Fluid Mech., 49, 387-417, (2017) · Zbl 1359.76111
[68] Rowley, C. W.; Mezić, I.; Bagheri, S.; Schlatter, P.; Henningson, D. S., Spectral analysis of nonlinear flows, J. Fluid Mech., 641, 115-127, (2009) · Zbl 1183.76833
[69] Schmid, P. J., Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656, 5-28, (2010) · Zbl 1197.76091
[70] Schmid, P. J.; Henningson, D. S., Stability and Transition in Shear Flows, vol. 142, (2001), Springer Science & Business Media · Zbl 0966.76003
[71] Schmid, P. J. & Sesterhenn, J.2008Dynamic mode decomposition of numerical and experimental data. In Bull. Am. Phys. Soc., 61st APS meeting, p. 208. San Antonio.
[72] Schmid, P. J.; Violato, D.; Scarano, F., Decomposition of time-resolved tomographic PIV, Exp. Fluids, 52, 6, 1567-1579, (2012)
[73] Schmidt, O. T.2017 An efficient streaming algorithm for spectral proper orthogonal decomposition. arXiv:1711.04199.
[74] Schmidt, O. T.; Towne, A.; Colonius, T.; Cavalieri, A. V. G.; Jordan, P.; Brès, G. A., Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability, J. Fluid Mech., 825, 1153-1181, (2017) · Zbl 1374.76074
[75] Schmidt, O. T., Towne, A., Rigas, G., Colonius, T. & Brès, G. A.2017b Spectral analysis for jet turbulence. arXiv:1711.06296.
[76] Semeraro, O.; Bellani, G.; Lundell, F., Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes, Exp. Fluids, 53, 5, 1203-1220, (2012)
[77] Semeraro, O., Jaunet, V., Jordan, P., Cavalieri, A V. G. & Lesshafft, L.2016a Stochastic and harmonic optimal forcing in subsonic jets. AIAA Paper 2016-2935.
[78] Semeraro, O.; Lesshafft, L.; Jaunet, V.; Jordan, P., Modeling of coherent structures in a turbulent jet as global linear instability wavepackets: theory and experiment, Intl J. Heat Fluid Flow, 62, 24-32, (2016)
[79] Shampine, L. F.; Reichelt, M. W., The Matlab ODE suite, SIAM J. Sci. Comput., 18, 1, 1-22, (1997) · Zbl 0868.65040
[80] Sharma, A. S.; Mckeon, B. J., On coherent structure in wall turbulence, J. Fluid Mech., 728, 196-238, (2013) · Zbl 1291.76173
[81] Sharma, A. S.; Mezić, I.; Mckeon, B. J., Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations, Phys. Rev. Fluids, 1, 3, (2016)
[82] Sieber, M.; Paschereit, C. O.; Oberleithner, K., Spectral proper orthogonal decomposition, J. Fluid Mech., 792, 798-828, (2016) · Zbl 1381.76133
[83] Sinha, A.; Rodriguez, D.; Bres, G. A.; Colonius, T., Wavepacket models for supersonic jet noise, J. Fluid Mech., 742, 71-95, (2014)
[84] Sipp, D.; Marquet, O.; Meliga, P.; Barbagallo, A., Dynamics and control of global instabilities in open-flows: a linearized approach, Appl. Mech. Rev., 63, 3, (2010)
[85] Sirovich, L., Turbulence and the dynamics of coherent structures. I. Coherent structures, Q. Appl. Maths, 45, 3, 561-571, (1987) · Zbl 0676.76047
[86] Sirovich, L., Chaotic dynamics of coherent structures, Physica D, 37, 1, 126-145, (1989)
[87] Suzuki, T.; Colonius, T., Instability waves in a subsonic round jet detected using a near-field phased microphone array, J. Fluid Mech., 565, 197-226, (2006) · Zbl 1104.76023
[88] Taira, K.; Brunton, S. L.; Dawson, S.; Rowley, C. W.; Colonius, T.; Mckeon, B. J.; Schmidt, O. T.; Gordeyev, S.; Theofilis, V.; Ukeiley, L. S., Modal analysis of fluid flows: An overview, AIAA J., 55, 12, 4013-4041, (2017)
[89] Tinney, C. E.; Jordan, P., The near pressure field of co-axial subsonic jets, J. Fluid Mech., 611, 175-204, (2008) · Zbl 1151.76356
[90] Towne, A., Brès, G. A. & Lele, S. K.2016Toward a resolvent-based statisitical jet-noise model. In Annual Research Briefs, Center for Turbulence Research, Stanford University.
[91] Towne, A., Brès, G. A. & Lele, S. K.2017 A statistical jet-noise model based on the resolvent framework. AIAA Paper 2017-3406.
[92] Towne, A., Colonius, T., Jordan, P., Cavalieri, A. V. G. & Brès, G. A.2015 Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet. AIAA Paper 2015-2217.
[93] Trefethen, L.; Trefethen, A.; Reddy, S.; Driscoll, T., Hydrodynamic stability without eigenvalues, Science, 261, 5121, 578-584, (1993) · Zbl 1226.76013
[94] Tu, J. H.; Rowley, C. W.; Luchtenburg, D. M.; Brunton, S. L.; Kutz, J. N., On dynamic mode decomposition: theory and applications, J. Comput. Dyn., 1, 2, 391-421, (2014) · Zbl 1346.37064
[95] Tutkun, M.; George, W. K., Lumley decomposition of turbulent boundary layer at high Reynolds numbers, Phys. Fluids, 29, 2, (2017)
[96] Tutkun, M.; Johansson, P. B. V.; George, W. K., Three-component vectorial proper orthogonal decomposition of axisymmetric wake behind a disk, AIAA J., 46, 5, 1118, (2008)
[97] 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)
[98] Zare, A.; Jovanović, M. R.; Georgiou, T. T., Colour of turbulence, J. Fluid Mech., 812, 636-680, (2017) · Zbl 1383.76303
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