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Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

Published online by Cambridge University Press:  29 May 2018

Aaron Towne Open the ORCID record for Aaron Towne [Opens in a new window] ,
Oliver T. Schmidt Open the ORCID record for Oliver T. Schmidt [Opens in a new window]  and
Tim Colonius
Aaron Towne*
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
Oliver T. Schmidt
Affiliation:
California Institute of Technology, Pasadena, CA 91125, USA
Tim Colonius
Affiliation:
California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: [email protected]
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Abstract

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 Lumley (Stochastic Tools in Turbulence, Academic Press, 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.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulent Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 847 , 25 July 2018 , pp. 821 - 867
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© 2018 Cambridge University Press 

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References

Abreu, L. I., Cavalieri, A. V. G. & Wolf, W. R.2017 Coherent hydrodynamic waves and trailing-edge noise. AIAA Paper 2017-3173.Google 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
Arbabi, H. & Mezić, I. 2017 Study of dynamics in unsteady flows using Koopman mode decomposition. Phys. Rev. Fluids 2, 124402.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. Theoret. Comput. Fluid Dyn. 2 (5), 339–352.Google 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. 2000 Random Data: Analysis and Measurement Procedures, 3rd edn. John Wiley & Sons.Google 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 (1), 539–575.Google Scholar
Braud, C., Heitz, D., Arroyo, G., Perret, L., Delville, J. & B., J.-P. 2004 Low-dimensional analysis, using POD, for two mixing layer–wake interactions. Intl J. Heat Fluid Flow 25 (3), 351–363.CrossRefGoogle Scholar
Brès, G. A., Ham, F. E., Nichols, J. W. & Lele, S. K. 2017a Unstructured large-eddy simulations of supersonic jets. AIAA J. 55 (4), 1164–1184.CrossRefGoogle Scholar
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.Google Scholar
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.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.2017b Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets. J. Fluid Mech. (in press).CrossRefGoogle Scholar
Cammilleri, A., Guéniat, F., Carlier, J., Pastur, L., Mémin, E., Lusseyran, F. & Artana, G. 2013 POD-spectral decomposition for fluid flow analysis and model reduction. Theoret. Comput. Fluid Dyn. 27 (6), 787–815.CrossRefGoogle Scholar
Cavalieri, A. V. G. & Agarwal, A. 2014 Coherence decay and its impact on sound radiation by wavepackets. J. Fluid Mech. 748, 399–415.CrossRefGoogle Scholar
Cavalieri, A. V. G., Jordan, P., Agarwal, A. & Gervais, Y. 2011 Jittering wave-packet models for subsonic jet noise. J. Sound Vib. 330 (18), 4474–4492.CrossRefGoogle Scholar
Chatterjee, A. 2000 An introduction to the proper orthogonal decomposition. Curr. Sci. 78 (7), 808–817.Google Scholar
Chen, K. K. & Rowley, C. W. 2011 H2 optimal actuator and sensor placement in the linearised complex Ginzburg–Landau system. J. Fluid Mech. 681, 241–260.CrossRefGoogle Scholar
Chen, K. K., Tu, J. H. & Rowley, C. W. 2012 Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22 (6), 887–915.CrossRefGoogle Scholar
Chen, X. & Kareem, A. 2005 Proper orthogonal decomposition-based modeling, analysis, and simulation of dynamic wind load effects on structures. J. Engng Mech. 131 (4), 325–339.CrossRefGoogle Scholar
Chu, B.-T. 1965 On the energy transfer to small disturbances in fluid flow (Part I). Acta Mech. 1 (3), 215–234.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
Cordier, L. & Bergmann, M. 2008 Proper 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.Google 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
Dergham, G., Sipp, D. & Robinet, J.-Ch. 2013 Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow. J. Fluid Mech. 719, 406–430.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids 5 (11), 2600–2609.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part I: autonomous operators. J. Atmos. Sci. 53 (14), 2025–2040.Google 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
Garnaud, X., Lesshafft, L., Schmid, P. J. & Huerre, P. 2013 The preferred mode of incompressible jets: linear frequency response analysis. J. Fluid Mech. 716, 189–202.CrossRefGoogle Scholar
George, W. K. 1988 Insight into the dynamics of coherent structures from a proper orthogonal decomposition dy . In International Seminar on Wall Turbulence.Google Scholar
George, W. K. 2017 A 50-year retrospective and the future. In Whither Turbulence and Big Data in the 21st Century? pp. 13–43. Springer.CrossRefGoogle Scholar
George, W. K., Beuther, P. D. & Lumley, J. L. 1978 Processing of random signals. In Proceedings of the Dynamic Flow Conference 1978 on Dynamic Measurements in Unsteady Flows, pp. 757–800. Springer.CrossRefGoogle Scholar
Glauser, M. N., Leib, S. J. & George, W. K. 1987 Coherent Structures in the Axisymmetric Turbulent Jet Mixing Layer, pp. 134–145. Springer.Google Scholar
Gómez, F., Blackburn, H. M., Rudman, M., Sharma, A. S. & McKeon, B. J. 2016a On the coupling of direct numerical simulation and resolvent analysis. In Progress in Turbulence VI, pp. 87–91. Springer.CrossRefGoogle Scholar
Gómez, F., Blackburn, H. M., Rudman, M., Sharma, A. S. & McKeon, B. J. 2016b A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator. J. Fluid Mech. 798, R2.CrossRefGoogle Scholar
Gordeyev, S. V. & Thomas, F. O. 2000 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.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
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).Google Scholar
Hellström, L. H. O. & Smits, A. J. 2014 The energetic motions in turbulent pipe flow. Phys. Fluids 26 (12), 125102.Google Scholar
Hilberg, D., Lazik, W. & Fiedler, H. E. 1994 The application of classical POD and snapshot POD in a turbulent shear layer with periodic structures. Appl. Sci. Res. 53 (3), 283–290.CrossRefGoogle Scholar
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.Google Scholar
Holmes, P. J., Lumley, J. L., Berkooz, G., Mattingly, J. C. & Wittenberg, R. W. 1997 Low-dimensional models of coherent structures in turbulence. Phys. Rep. 287 (4), 337–384.Google Scholar
Hunt, R. E. & Crighton, D. G. 1991 Instability of flows in spatially developing media. Proc. R. Soc. Lond. A 435, 109–128.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.Google Scholar
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.CrossRefGoogle Scholar
Jovanović, M. & Bamieh, B. 2001 Modeling 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.Google Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Landahl, M. T. & Mollo Christensen, E. 1992 Turbulence and Random Processes in Fluid Mechanics. Cambridge University Press.Google Scholar
Liang, Y. C., Lee, H. P., Lim, S. P., Lin, W. Z., Lee, K. H. & Wu, C. G. 2002 Proper orthogonal decomposition and its applications. Part I: theory. J. Sound Vib. 252 (3), 527–544.CrossRefGoogle Scholar
Lumley, J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Propagation (ed. Yaglom, A. M. & Tatarski, V. I.), pp. 166–178. Nauka.Google Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Mezić, I. 2005 Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41 (1), 309–325.CrossRefGoogle Scholar
Moarref, R. & Jovanović, M. R. 2012 Model-based design of transverse wall oscillations for turbulent drag reduction. J. Fluid Mech. 707, 205–240.CrossRefGoogle Scholar
Moarref, R., Jovanović, M. R., Tropp, J. A., Sharma, A. S. & McKeon, B. J. 2014 A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization. Phys. Fluids 26 (5), 051701.Google Scholar
Moin, P. & Moser, R. D. 1989 Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech. 200, 471–509.CrossRefGoogle Scholar
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.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
Noack, B. R., Stankiewicz, W., Morzyński, M. & Schmid, P. J. 2016 Recursive dynamic mode decomposition of transient and post-transient wake flows. J. Fluid Mech. 809, 843–872.Google 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
Pinier, J. T., Ausseur, J. M., Glauser, M. N. & Higuchi, H. 2007 Proportional closed-loop feedback control of flow separation. AIAA J. 45 (1), 181–190.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Rowley, C. W. 2005 Model reduction for fluids, using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (03), 997–1013.CrossRefGoogle Scholar
Rowley, C. W., Colonius, T. & Murray, R. M. 2004 Model reduction for compressible flows using POD and Galerkin projection. Physica D 189 (1), 115–129.Google Scholar
Rowley, C. W. & Dawson, S. T. M. 2017 Model reduction for flow analysis and control. Annu. Rev. Fluid Mech. 49, 387–417.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.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, vol. 142. Springer Science & Business Media.Google Scholar
Schmid, P. J. & Sesterhenn, J. 2008 Dynamic mode decomposition of numerical and experimental data. In Bull. Am. Phys. Soc., 61st APS meeting, p. 208. San Antonio.Google Scholar
Schmid, P. J., Violato, D. & Scarano, F. 2012 Decomposition of time-resolved tomographic PIV. Exp. Fluids 52 (6), 1567–1579.CrossRefGoogle Scholar
Schmidt, O. T.2017 An efficient streaming algorithm for spectral proper orthogonal decomposition. arXiv:1711.04199.Google Scholar
Schmidt, O. T., Towne, A., Colonius, T., Cavalieri, A. V. G., Jordan, P. & Brès, G. A. 2017a Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability. J. Fluid Mech. 825, 1153–1181.Google Scholar
Schmidt, O. T., Towne, A., Rigas, G., Colonius, T. & Brès, G. A.2017b Spectral analysis for jet turbulence. arXiv:1711.06296.Google Scholar
Semeraro, O., Bellani, G. & Lundell, F. 2012 Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes. Exp. Fluids 53 (5), 1203–1220.Google Scholar
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.CrossRefGoogle Scholar
Semeraro, O., Lesshafft, L., Jaunet, V. & Jordan, P. 2016b Modeling of coherent structures in a turbulent jet as global linear instability wavepackets: theory and experiment. Intl J. Heat Fluid Flow 62, 24–32.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
Sharma, A. S., Mezić, I. & McKeon, B. J. 2016 Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier–Stokes equations. Phys. Rev. Fluids 1 (3), 032402.Google Scholar
Sieber, M., Paschereit, C. O. & Oberleithner, K. 2016 Spectral proper orthogonal decomposition. J. Fluid Mech. 792, 798–828.Google Scholar
Sinha, A., Rodriguez, D., Bres, G. A. & Colonius, T. 2014 Wavepacket models for supersonic jet noise. J. Fluid Mech. 742, 71–95.Google 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.Google Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Maths 45 (3), 561–571.Google Scholar
Sirovich, L. 1989 Chaotic dynamics of coherent structures. Physica D 37 (1), 126–145.Google Scholar
Suzuki, T. & Colonius, T. 2006 Instability waves in a subsonic round jet detected using a near-field phased microphone array. J. Fluid Mech. 565, 197–226.Google Scholar
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. 2017 Modal analysis of fluid flows: An overview. AIAA J. 55 (12), 4013–4041.Google 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., Brès, G. A. & Lele, S. K. 2016 Toward a resolvent-based statisitical jet-noise model. In Annual Research Briefs, Center for Turbulence Research, Stanford University.Google Scholar
Towne, A., Brès, G. A. & Lele, S. K.2017 A statistical jet-noise model based on the resolvent framework. AIAA Paper 2017-3406.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Trefethen, L., Trefethen, A., Reddy, S. & Driscoll, T. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.Google Scholar
Tu, J. H., Rowley, C. W., Luchtenburg, D. M., Brunton, S. L. & Kutz, J. N. 2014 On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1 (2), 391–421.Google Scholar
Tutkun, M. & George, W. K. 2017 Lumley decomposition of turbulent boundary layer at high Reynolds numbers. Phys. Fluids 29 (2), 020707.Google Scholar
Tutkun, M., Johansson, P. B. V. & George, W. K. 2008 Three-component vectorial proper orthogonal decomposition of axisymmetric wake behind a disk. AIAA J. 46 (5), 1118.Google 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.Google Scholar
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.CrossRefGoogle Scholar