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Efficient computation of global resolvent modes

Published online by Cambridge University Press:  20 May 2021

Eduardo Martini Open the ORCID record for Eduardo Martini [Opens in a new window] ,
Daniel Rodríguez Open the ORCID record for Daniel Rodríguez [Opens in a new window] ,
Aaron Towne Open the ORCID record for Aaron Towne [Opens in a new window]  and
André V.G. Cavalieri Open the ORCID record for André V.G. Cavalieri [Opens in a new window]
Eduardo Martini*
Affiliation:
Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil Département Fluides, Thermique et Combustion, Institut Pprime, CNRS, Université de Poitiers, ENSMA, 86000 Poitiers, France
Daniel Rodríguez
Affiliation:
ETSIAE-UPM (School of Aeronautics) – Universidad Politécnica de Madrid, Spain
Aaron Towne
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
André V.G. Cavalieri
Affiliation:
Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil
*
†Email address for correspondence: eduardo.martini@univ-poitiers.fr
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Abstract

Resolvent analysis of the linearized Navier–Stokes equations provides useful insight into the dynamics of transitional and turbulent flows and can provide a model for the dominant coherent structures within the flow, particularly for flows where the linear operator selectively amplifies one particular force component, known as the optimal force mode. Force and response modes are typically obtained from a singular-value decomposition of the resolvent operator. Despite recent progress, the cost of resolvent analysis for complex flows remains considerable, and explicit construction of the resolvent operator is feasible only for simplified problems with a small number of degrees of freedom. In this paper we propose two new matrix-free methods for computing resolvent modes based on the integration of the linearized equations and the corresponding adjoint system in the time domain. Our approach achieves an order of magnitude speedup when compared with previous matrix-free time-stepping methods by enabling all frequencies of interest to be computed simultaneously. Two different methods are presented: one based on analysis of the transient response, providing leading modes with fine frequency discretization; and another based on the steady-state response to periodic forcing, providing optimal and suboptimal modes for a discrete set of frequencies. The methods are validated using a linearized Ginzburg–Landau equation and applied to the three-dimensional flow around a parabolic body.

JFM classification

Instability: Transition to turbulence Mathematical Foundations: Computational methods

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JFM Papers
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Journal of Fluid Mechanics , Volume 919 , 25 July 2021 , A3
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© The Author(s), 2021. 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, p. 3173. AIAA.CrossRefGoogle Scholar
Åkervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D.S. 2008 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. (B/Fluids) 27 (5), 501–513.CrossRefGoogle Scholar
Alizard, F., Cherubini, S. & Robinet, J.-C. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21 (6), 064108.CrossRefGoogle Scholar
Amestoy, P.R., Davis, T.A. & Duff, I.S. 1996 An approximate minimum degree ordering algorithm. SIAM. J. Matrix Anal. Appl. 17 (4), 886–905.CrossRefGoogle Scholar
Arnoldi, W.E. 1951 The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Maths 9 (1), 17–29.CrossRefGoogle 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
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
Brynjell-Rahkola, M., Tuckerman, L.S., Schlatter, P. & Henningson, D.S. 2017 Computing optimal forcing using laplace preconditioning. Commun. Comput. Phys. 22 (5), 1508–1532.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
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
Fischer, P.F. 1998 Projection techniques for iterative solution of $Ax= b$ with successive right-hand sides. Comput. Meth. 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, pp. 49–65. Springer.CrossRefGoogle Scholar
Gennaro, E., Rodríguez, D., Medeiros, M. & Theofilis, V. 2013 Sparse techniques in global flow instability with application to compressible leading-edge flow. AIAA J. 51 (9), 2295–2303.CrossRefGoogle Scholar
Gómez, F., Blackburn, H., Rudman, M., Sharma, A. & McKeon, B. 2016 a A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator. J. Fluid Mech. 798, R2.CrossRefGoogle Scholar
Gómez, F., Sharma, A.S. & Blackburn, H.M. 2016 b Estimation of unsteady aerodynamic forces using pointwise velocity data. J. Fluid Mech. 804, R4.CrossRefGoogle 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
Karypis, G. & Kumar, V. 1998 METIS: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices. University of Minnesota, Department of Computer Science and Engineering, Army HPC Research Center.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. & Jordan, P. 2019 Resolvent-based modeling of coherent wave packets in a turbulent jet. Phys. Rev. Fluids 4 (6), 063901.CrossRefGoogle Scholar
Martini, E., Cavalieri, A.V.G., Jordan, P., Towne, A. & Lesshafft, L. 2020 Resolvent-based optimal estimation of transitional and turbulent flows. J. Fluid Mech. 900, A2.CrossRefGoogle Scholar
McKeon, B.J. & Sharma, A.S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.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
Monokrousos, A., Åkervik, E., Brandt, L. & Henningson, D.S. 2010 Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181.CrossRefGoogle Scholar
Nogueira, P.A.S., Morra, P., Martini, E., Cavalieri, A.V.G. & Henningson, D.S. 2021 Forcing statistics in resolvent analysis: application in minimal turbulent Couette flow. J. Fluid Mech. 908, A32.CrossRefGoogle Scholar
Press, W.H., Teukolsky, S.A., Vetterling, W.T. & Flannery, B.P. 2007 Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge University Press.Google Scholar
Ribeiro, J.H.M., Yeh, C.-A. & Taira, K. 2020 Randomized resolvent analysis. Phys. Rev. Fluids 5 (3), 033902.CrossRefGoogle Scholar
Rodríguez, D. & Gennaro, E.M. 2017 Three-dimensional flow stability analysis based on the matrix-forming approach made affordable. In Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016 (ed. M.L. Bittencourt, N.A. Dumont & J.S. Hesthaven), vol. 119, pp. 639–650. Springer International Publishing.CrossRefGoogle Scholar
Rodríguez, D., Gennaro, E.M. & Souza, L.F. 2021 Self-excited primary and secondary instability of laminar separation bubbles. J. Fluid Mech. 906, A13.CrossRefGoogle Scholar
Rodríguez, D., Tumin, A. & Theofilis, V. 2011 Towards the foundation of a global modes concept. In 6th AIAA Theoretical Fluid Mechanics Conference, p. 3603. AIAA.Google Scholar
Saad, Y. 2003 Iterative Methods for Sparse Linear Systems, vol. 82. SIAM.CrossRefGoogle Scholar
Sasaki, K., Piantanida, S., Cavalieri, A.V.G. & Jordan, P. 2017 Real-time modelling of wavepackets in turbulent jets. J. Fluid Mech. 821, 458–481.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2012 Stability and Transition in Shear Flows, vol. 142. Springer Science & Business Media.Google 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
Shaabani-Ardali, L., Sipp, D. & Lesshafft, L. 2020 Optimal triggering of jet bifurcation: an example of optimal forcing applied to a time-periodic base flow. J. Fluid Mech. 885, A34.CrossRefGoogle Scholar
Sipp, D. & Marquet, O. 2013 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5), 617–635.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
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
Trefethen, L.N. 1997 Pseudospectra of linear operators. SIAM Rev. 39 (3), 383–406.CrossRefGoogle Scholar
Yeh, C.-A. & Taira, K. 2019 Resolvent-analysis-based design of airfoil separation control. J. Fluid Mech. 867, 572–610.CrossRefGoogle Scholar