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Resolvent-based optimal estimation of transitional and turbulent flows

Published online by Cambridge University Press:  31 July 2020

Eduardo Martini Open the ORCID record for Eduardo Martini [Opens in a new window] ,
André V. G. Cavalieri Open the ORCID record for André V. G. Cavalieri [Opens in a new window] ,
Peter Jordan Open the ORCID record for Peter Jordan [Opens in a new window] ,
Aaron Towne Open the ORCID record for Aaron Towne [Opens in a new window]  and
Lutz Lesshafft Open the ORCID record for Lutz Lesshafft [Opens in a new window]
Eduardo Martini*
Affiliation:
Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil
André V. G. Cavalieri
Affiliation:
Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil
Peter Jordan
Affiliation:
Département Fluides, Thermique et Combustion, Institut Pprime, CNRS, Université de Poitiers, ENSMA, 86000Poitiers, France
Aaron Towne
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI48109, USA
Lutz Lesshafft
Affiliation:
Laboratoire d'Hydrodynamique, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128Palaiseau, France
*
†Email address for correspondence: emartini@ita.br
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Abstract

We extend the resolvent-based estimation approach recently introduced by Towne etal. (J. Fluid Mech., vol. 883, 2020, A17) to obtain optimal, non-causal estimates of time-varying flow quantities from low-rank measurements. We derive optimal transfer functions between the measurements and certain nonlinear terms that act as a forcing on the linearised Navier–Stokes equations, and show that the resulting transfer function to the flow state is equivalent to a multiple-input, multiple-output Wiener filter if the colour of the forcing statistics is known. A matrix-free implementation is developed based on integration of the direct and adjoint linearised Navier–Stokes operators, enabling application to the large systems encountered for transitional and turbulent flows without the need for a priori model reduction. Using a linearised Ginzburg–Landau problem, we show that the non-casual resolvent-based method outperforms a casual Kalman filter for general sensor configurations and recovers the Kalman filter transfer function in specific cases, leading to causal estimates at a significantly reduced computational cost. Additionally, our method is shown to be more accurate and robust than popular approaches based on truncation of the resolvent operator to its leading modes. The applicability of the method to transitional and turbulent flows is demonstrated via application to a (linearised) transitional boundary layer and a (nonlinear) turbulent channel flow. Errors on the order of 2 % are achieved for the boundary layer, and the channel flow case highlights the need to account for the forcing colour to achieve accurate flow estimates. In practice, our method can be used as a post-processing tool to reconstruct unmeasured quantities from limited experimental data, and, in cases where the transfer function can be accurately truncated to its causal components, as a low-cost estimator for flow control.

JFM classification

Flow Control: Control theory Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A2
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Abreu, L. I., Cavalieri, A. V. & Wolf, W. 2017 Coherent hydrodynamic waves and trailing-edge noise. In 23rd AIAA/CEAS Aeroacoustics Conference, Denver, Colorado, AIAA Paper 2017-3173.Google Scholar
Ahlfors, L. V. 1979 Complex Analysis, 3rd edn. McGraw-Hill.Google Scholar
Åkervik, E., Hœpffner, J., Ehrenstein, U. W. E. & Henningson, D. S. 2007 Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314.CrossRefGoogle Scholar
Arfken, G. B., Weber, H.-J. & Harris, F. E. 2013 Mathematical Methods for Physicists: A Comprehensive Guide, 7th edn. Elsevier.Google Scholar
Bagheri, S., Henningson, D. S., Hœpffner, 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
Bell, B. M. 1994 The iterated Kalman smoother as a Gauss–Newton method. SIAM J. Optim. 4 (3), 626–636.CrossRefGoogle Scholar
Belson, B. A., Semeraro, O., Rowley, C.W. & Henningson, D. S. 2013 Feedback control of instabilities in the two-dimensional Blasius boundary layer: the role of sensors and actuators. Phys. Fluids 25 (5), 054106.CrossRefGoogle 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
Beneddine, S., Yegavian, R., Sipp, D. & Leclaire, B. 2017 Unsteady flow dynamics reconstruction from mean flow and point sensors: an experimental study. J. Fluid Mech. 824, 174–201.CrossRefGoogle Scholar
Bewley, T. R., Temam, R. & Ziane, M. 2000 A general framework for robust control in fluid mechanics. Physica D 138 (4), 360–392.CrossRefGoogle Scholar
Bode, H. W. & Shannon, C. E. 1950 A simplified derivation of linear least square smoothing and prediction theory. Proc. IRE 38 (4), 417–425.CrossRefGoogle Scholar
Bodony, D. J. 2006 Analysis of sponge zones for computational fluid mechanics. J. Comput. Phys. 212 (2), 681–702.CrossRefGoogle Scholar
Cavalieri, A. V. G., Jordan, P. & Lesshafft, L. 2019 Wave-packet models for jet dynamics and sound radiation. Appl. Mech. Rev. 71 (2), 020802.CrossRefGoogle Scholar
Chevalier, M., Hœpffner, J., Bewley, T. R. & Henningson, D. S. 2006 State estimation in wall-bounded flow systems. Part 2. Turbulent flows. J. Fluid Mech. 552, 167–187.CrossRefGoogle Scholar
Chomaz, J.-M., Huerre, P. & Redekopp, L. G. 1991 A frequency selection criterion in spatially developing flows. Stud. Appl. Maths 84 (2), 119–144.CrossRefGoogle Scholar
Couairon, A. & Chomaz, J.-M. 1999 Fully nonlinear global modes in slowly varying flows. Phys. Fluids 11 (12), 3688–3703.CrossRefGoogle Scholar
Del Alamo, J. C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15 (6), L41–L44.CrossRefGoogle Scholar
Dergham, G., Sipp, D., Robinet, J.-C. & Barbagallo, A. 2011 Model reduction for fluids using frequential snapshots. Phys. Fluids 23 (6), 064101.CrossRefGoogle Scholar
Fabbiane, N., Bagheri, S. & Henningson, D. S. 2015 a Adaptive control of finite-amplitude 3D disturbances in 2D boundary-layer flows. In TSFP Digital Library Online. Begel House Inc.Google Scholar
Fabbiane, N., Simon, B., Fischer, F., Grundmann, S., Bagheri, S. & Henningson, D. S. 2015 b On the role of adaptivity for robust laminar flow control. J. Fluid Mech. 767, R1.CrossRefGoogle Scholar
Fischer, P. F. 1998 Projection techniques for iterative solution of Ax = b with successive right-hand sides. Comput. Methods Appl. Mech. Engng 163 (1–4), 193–204.CrossRefGoogle Scholar
Fischer, P. F. & Patera, A. T. 1989 Parallel spectral element methods for the incompressible Navier–Stokes equations. In Solution of Superlarge Problems in Computational Mechanics (ed. Kane, J. H., Carlson, A. D. & Cox, D. L.), pp.49–65. Springer.CrossRefGoogle Scholar
Fraser, D. & Potter, J. 1969 The optimum linear smoother as a combination of two optimum linear filters. IEEE Trans. Automat. Contr. 14 (4), 387–390.CrossRefGoogle Scholar
Gibson, J. F. 2012 Channelflow: A spectral Navier–Stokes simulator in C++. New Hampshire.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Gillissen, J. J. J., Bouffanais, R. & Yue, D. K. P. 2019 Data assimilation method to de-noise and de-filter particle image velocimetry data. J. Fluid Mech. 877, 196–213.CrossRefGoogle Scholar
Gómez, V. 2007 Wiener–Kolmogorov filtering and smoothing for multivariate series with state-space structure. J. Time Ser. Anal. 28 (3), 361–385.CrossRefGoogle Scholar
Gomez Carrasco, F., Blackburn, H., Rudman, M., Sharma, A. & McKeon, B. 2014 Reconstruction of turbulent pipe flow using resolvent modes: application to low-order models and flow control. In Proceedings of the 19th Australasian Fluid Mechanics Conference 2014 Australasian Fluid Mechanics Society (AFMS 2014), pp. 1–4.Google Scholar
Hervé, A., Sipp, D., Schmid, P. J. & Samuelides, M. 2012 A physics-based approach to flow control using system identification. J. Fluid Mech. 702, 26–58.CrossRefGoogle Scholar
Hespanha, J. P. 2009 Linear Systems Theory. Princeton University Press.Google Scholar
Juang, J.-N. & Pappa, R. S. 1985 An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Control Dyn. 8 (5), 620–627.CrossRefGoogle Scholar
Kailath, T. 1974 A view of three decades of linear filtering theory. IEEE Trans. Inform. Theory 20 (2), 146–181.CrossRefGoogle Scholar
Kailath, T. & Geesey, R. 1971 An innovations approach to least squares estimation–part IV: recursive estimation given lumped covariance functions. IEEE Trans. Automat. Contr. 16 (6), 720–727.CrossRefGoogle Scholar
Kalman, R. E. 1960 A new approach to linear filtering and prediction problems. J. Basic Engng 82 (1), 35–45.CrossRefGoogle Scholar
Kirby, M., Boris, J. P. & Sirovich, L. 1990 A proper orthogonal decomposition of a simulated supersonic shear layer. Intl J. Numer. Methods Fluids 10 (4), 411–428.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 Interpolirovanie i ekstrapolirovanie statsionamykh sluchainykh posledovate1’- nostei (Interpolated and extrapolated stationary random sequences)’. Iivestiya AN SSSR, Seriya Matematicheskaya 5, 119–139.Google Scholar
Lanczos, C. 1997 Linear Differential Operators. SIAM.Google Scholar
Lesshafft, L. 2018 Artificial eigenmodes in truncated flow domains. Theor. Comput. Fluid Dyn. 32 (3), 245–262.CrossRefGoogle Scholar
Lesshafft, L., Semeraro, O., Jaunet, V., Cavalieri, A. V. G. & Jordan, P. 2019 Resolvent-based modeling of coherent wave packets in a turbulent jet. Phys. Rev. Fluids 4 (6), 063901.CrossRefGoogle Scholar
Ma, Z., Ahuja, S. & Rowley, C. W. 2011 Reduced-order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn. 25 (1–4), 233–247.CrossRefGoogle Scholar
Martinelli, F. 2009 Feedback control of turbulent wall flows. PhD thesis, Politecnico di Milano.Google Scholar
Martini, E., Cavalieri, A. V. G., Jordan, P. & Lesshafft, L. 2019 Accurate frequency domain identification of ODEs with arbitrary signals. arXiv:1907.04787.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
Meditch, J. S. 1973 A survey of data smoothing for linear and nonlinear dynamic systems. Automatica 9(2), 151–162.CrossRefGoogle Scholar
Morra, P. 2020 Modelling and Control of Turbulent and Transitional Flows. KTH Royal Institute of Technology.Google Scholar
Morra, P., Nogueira, P. A. S., Cavalieri, A. V. G. & Henningson, D. S. 2020 The colour of forcing statistics in resolvent analyses of turbulent channel flows. arXiv:2004.01565.Google Scholar
Murray, R. M. 2009 Optimization-based control, p. 11. California Institute of Technology.Google Scholar
Nogueira, P. A. S., Morra, P., Martini, E., Cavalieri, A. V. G. & Henningson, D. S. 2020 Forcing statistics in resolvent analysis: application in minimal turbulent Couette flow. arXiv:2001.02576.Google Scholar
Pnevmatikakis, E. A., Rad, K. R., Huggins, J. & Paninski, L. 2014 Fast Kalman filtering and forward–backward smoothing via a low-rank perturbative approach. J. Comput. Graph. Stat. 23 (2), 316–339.CrossRefGoogle Scholar
Sasaki, K., Cavalieri, A. V. G., Jordan, P., Schmidt, O. T., Colonius, T. & Brès, G. A. 2017 a High-frequency wavepackets in turbulent jets. J. Fluid Mech. 830, R2.CrossRefGoogle Scholar
Sasaki, K., Morra, P., Fabbiane, N., Cavalieri, A. V. G., Hanifi, A. & Henningson, D. S. 2018 On the wave-cancelling nature of boundary layer flow control. Theor. Comput. Fluid Dyn. 32(5), 593–616.CrossRefGoogle Scholar
Sasaki, K., Piantanida, S., Cavalieri, A. V. G. & Jordan, P. 2017 b Real-time modelling of wavepackets in turbulent jets. J. Fluid Mech. 821, 458–481.CrossRefGoogle Scholar
Sasaki, K., Tissot, G., Cavalieri, A. V., Silvestre, F. J., Jordan, P. & Biau, D. 2016 Closed-loop control of wavepackets in a free shear-flow. In 22nd AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2016-2758.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
Semeraro, O., Pralits, J. O., Rowley, C. W. & Henningson, D. S. 2013 Riccati-less approach for optimal control and estimation: an application to two-dimensional boundary layers. J. Fluid Mech. 731, 394–417.CrossRefGoogle Scholar
da Silva, A. F. C. & Colonius, T. 2018 Ensemble-based state estimator for aerodynamic flows. AIAA J. 56 (7), 2568–2578.CrossRefGoogle Scholar
Sipp, D. & Schmid, P. J. 2016 Linear closed-loop control of fluid instabilities and noise-induced perturbations: a review of approaches and tools. Appl. Mech. Rev. 68 (2), 020801.CrossRefGoogle Scholar
Symon, S. P. 2018 Reconstruction and estimation of flows using resolvent analysis and data-assimilation. PhD thesis, California Institute of Technology.Google Scholar
Symon, S., Dovetta, N., McKeon, B. J., Sipp, D. & Schmid, P. J. 2017 Data assimilation of mean velocity from 2D PIV measurements of flow over an idealized airfoil. Exp. Fluids 58 (5), 61.CrossRefGoogle Scholar
Tam, C. K. W. & Pastouchenko, N. N. 2002 Noise from fine-scale turbulence of nonaxisymmetric jets. AIAA J. 40 (3), 456–464.CrossRefGoogle 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., Rigas, G. & Colonius, T. 2019 A critical assessment of the parabolized stability equations. Theor. Comput. Fluid Dyn. 33 (3–4), 359–382.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
Trefethen, L. N. 2000 Spectral Methods in MATLAB, vol 10. SIAMCrossRefGoogle Scholar
Weideman, J. A. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465–519.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
Wiener, N. 1942 Extrapolation, Interpolation and Smoothing of Stationary Time Engineering Applications. Massachusettes Insitute of Technology.Google Scholar
Yeh, C.-A. & Taira, K. 2019 Resolvent-analysis-based design of airfoil separation control. J. Fluid Mech. 867, 572–610.CrossRefGoogle Scholar
Zhou, K., Salomon, G. & Wu, E. 1999 Balanced realization and model reduction for unstable systems. Intl J. Robust Nonlinear Control 9 (3), 183–198.3.0.CO;2-E>CrossRefGoogle Scholar