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Optimizing the control of transition to turbulence using a Bayesian method

Published online by Cambridge University Press:  27 April 2022

Anton Pershin Open the ORCID record for Anton Pershin [Opens in a new window] ,
Cédric Beaume Open the ORCID record for Cédric Beaume [Opens in a new window] ,
Tom S. Eaves Open the ORCID record for Tom S. Eaves [Opens in a new window]  and
Steven M. Tobias Open the ORCID record for Steven M. Tobias [Opens in a new window]
Anton Pershin*
Affiliation:
Department of Physics, University of Oxford, Oxford OX1 3PU, UK School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
Cédric Beaume
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
Tom S. Eaves
Affiliation:
School of Science and Engineering, University of Dundee, Dundee DD1 4HN, UK
Steven M. Tobias
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
*
†Email address for correspondence: [email protected]
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Abstract

The nonlinear robustness of laminar plane Couette flow is considered under the action of in-phase spanwise wall oscillations by computing properties of the edge of chaos, i.e. the boundary of its basins of attraction. Three measures are used to quantify the chosen control strategy on laminar-to-turbulent transition: the kinetic energy of edge states (local attractors on the edge of chaos), the form of the minimal seed (least energetic perturbation on the edge of chaos), and the laminarization probability (the probability that a random perturbation from the laminar flow of given kinetic energy will laminarize). A novel Bayesian approach is introduced to enable the accurate computation of the laminarization probability at a fraction of the cost of previous methods. While the edge state and the minimal seed provide useful information about the dynamics of transition to turbulence, neither measure is particularly useful to judge the effectiveness of the control strategy since they are not representative of the global geometry of the edge. In contrast, the laminarization probability provides global information about the edge and can be used to evaluate the control effectiveness by computing a laminarization score (the expected laminarization probability) and the associated expected dissipation rate of the controlled flow. These two quantities allow for the determination of optimal control parameter values subject to desired constraints. The results discussed in the paper are expected to be applied to a wide range of transitional flows and control strategies aimed at suppressing or triggering transition to turbulence.

JFM classification

Instability: Transition to turbulence Instability: Nonlinear instability Flow Control: Instability control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 941 , 25 June 2022 , A25
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Barkley, D. 2016 Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803, P1.CrossRefGoogle Scholar
Bernardo, J.M. 1979 Reference posterior distributions for Bayesian inference. J. R. Stat. Soc. B 41 (2), 113–128.Google Scholar
Box, G.E.P. & Tiao, G.C. 2011 Bayesian Inference in Statistical Analysis, vol. 40. John Wiley & Sons.Google Scholar
Budanur, N.B., Dogra, A.S. & Hof, B. 2019 Geometry of transient chaos in streamwise-localized pipe flow turbulence. Phys. Rev. Fluids 4 (10), 102401.CrossRefGoogle Scholar
Budanur, N.B., Marensi, E., Willis, A.P. & Hof, B. 2020 Upper edge of chaos and the energetics of transition in pipe flow. Phys. Rev. Fluids 5 (2), 023903.CrossRefGoogle Scholar
Chantry, M. & Schneider, T.M. 2014 Studying edge geometry in transiently turbulent shear flows. J. Fluid Mech. 747, 506–517.CrossRefGoogle Scholar
Chantry, M., Tuckerman, L.S. & Barkley, D. 2017 Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech. 824, R1.CrossRefGoogle Scholar
Cherubini, S. & De Palma, P. 2013 Nonlinear optimal perturbations in a Couette flow: bursting and transition. J. Fluid Mech. 716, 251–279.CrossRefGoogle Scholar
Cherubini, S., Palma, P.D., Robinet, J.-Ch. & Bottaro, A. 2011 Edge states in a boundary layer. Phys. Fluids 23 (5), 051705.CrossRefGoogle Scholar
Chung, S.W. & Freund, J.B. 2022 An optimization method for chaotic turbulent flow. J. Comput. Phys. 457, 111077.CrossRefGoogle Scholar
Couliou, M. & Monchaux, R. 2015 Large-scale flows in transitional plane Couette flow: a key ingredient of the spot growth mechanism. Phys. Fluids 27 (3), 034101.CrossRefGoogle Scholar
Dauchot, O. & Daviaud, F. 1995 Finite amplitude perturbation and spots growth mechanism in plane Couette flow. Phys. Fluids 7 (2), 335–343.CrossRefGoogle Scholar
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, S. 2013 Minimal transition thresholds in plane Couette flow. Phys. Fluids 25, 084103.CrossRefGoogle Scholar
Duguet, Y., Schlatter, P. & Henningson, D.S. 2009 Localized edge states in plane Couette flow. Phys. Fluids 21 (11), 111701.CrossRefGoogle Scholar
Duguet, Y., Schlatter, P. & Henningson, D.S. 2010 Formation of turbulent patterns near the onset of transition in plane Couette flow. J. Fluid Mech. 650, 119–129.CrossRefGoogle Scholar
Eaves, T.S. & Caulfield, C.P. 2015 Disruption of SSP/VWI states by a stable stratification. J. Fluid Mech. 784, 548–564.CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S. & Rubin, D.B. 1995 Bayesian Data Analysis. Chapman and Hall/CRC.CrossRefGoogle Scholar
Gibson, J.F. 2014 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep. U. New Hampshire, Channelflow.org.Google Scholar
Hof, B., De Lozar, A., Avila, M., Tu, X. & Schneider, T.M. 2010 Eliminating turbulence in spatially intermittent flows. Science 327 (5972), 1491–1494.CrossRefGoogle ScholarPubMed
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703–716.CrossRefGoogle Scholar
Jaynes, E.T. 1957 Information theory and statistical mechanics. Phys. Rev. 106 (4), 620.CrossRefGoogle Scholar
Kerswell, R.R. 2018 Nonlinear nonmodal stability theory. Annu. Rev. Fluid Mech. 50, 319–345.CrossRefGoogle Scholar
Khan, H.H., Anwer, S.F., Hasan, N. & Sanghi, S. 2021 Laminar to turbulent transition in a finite length square duct subjected to inlet disturbance. Phys. Fluids 33 (6), 065128.CrossRefGoogle Scholar
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D.S. 2016 Edge states as mediators of bypass transition in boundary-layer flows. J. Fluid Mech. 801, R2.CrossRefGoogle Scholar
Kreilos, T., Khapko, T., Schlatter, P., Duguet, Y., Henningson, D.S. & Eckhardt, B. 2016 Bypass transition and spot nucleation in boundary layers. Phys. Rev. Fluids 1 (4), 043602.CrossRefGoogle Scholar
Kühnen, J., Scarselli, D., Schaner, M. & Hof, B. 2018 a Relaminarization by steady modification of the streamwise velocity profile in a pipe. Flow Turbul. Combust. 100 (4), 919–943.CrossRefGoogle Scholar
Kühnen, J., Song, B., Scarselli, D., Budanur, N.B., Riedl, M., Willis, A.P., Avila, M. & Hof, B. 2018 b Destabilizing turbulence in pipe flow. Nat. Phys. 14 (4), 386.CrossRefGoogle Scholar
Lecoanet, D. & Kerswell, R.R. 2018 Connection between nonlinear energy optimization and instantons. Phys. Rev. E 97, 012212.CrossRefGoogle ScholarPubMed
Lustro, J.R.T., Kawahara, G., van Veen, L., Shimizu, M. & Kokubu, H. 2019 The onset of transient turbulence in minimal plane Couette flow. J. Fluid Mech. 862, R2.CrossRefGoogle Scholar
McMillan, B.F., Pringle, C.C.T. & Teaca, B. 2018 Simple advecting structures and the edge of chaos in subcritical tokamak plasmas. J. Plasma Phys. 84, 905840611.CrossRefGoogle Scholar
Menck, P.J., Heitzig, J., Marwan, N. & Kurths, J. 2013 How basin stability complements the linear-stability paradigm. Nat. Phys. 9 (2), 89–92.CrossRefGoogle Scholar
Meseguer, A. & Trefethen, L.N. 2003 Linearized pipe flow to Reynolds number $10^{7}$. J. Comput. Phys. 186, 178–197.CrossRefGoogle Scholar
Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D.S. 2011 Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106, 134502.CrossRefGoogle ScholarPubMed
Orszag, S.A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.CrossRefGoogle Scholar
Park, S.Y. & Bera, A.K. 2009 Maximum entropy autoregressive conditional heteroskedasticity model. J. Econom. 150 (2), 219–230.CrossRefGoogle Scholar
Pershin, A., Beaume, C. & Tobias, S.M. 2020 A probabilistic protocol for the assessment of transition and control. J. Fluid Mech. 895, A16.CrossRefGoogle Scholar
Pringle, C.C.T., Willis, A.P. & Kerswell, R.R. 2012 Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos. J. Fluid Mech. 702, 415–443.CrossRefGoogle Scholar
Quadrio, M. & Ricco, P. 2004 Critical assessment of turbulent drag reduction through spanwise wall oscillations. J. Fluid Mech. 521, 251–271.CrossRefGoogle Scholar
Quadrio, M. & Sibilla, S. 2000 Numerical simulation of turbulent flow in a pipe oscillating around its axis. J. Fluid Mech. 424, 217–241.CrossRefGoogle Scholar
Rabin, S.M.E., Caulfield, C.P. & Kerswell, R.R. 2012 Triggering turbulence efficiently in plane Couette flow. J. Fluid Mech. 712, 244–272.CrossRefGoogle Scholar
Rabin, S.M.E., Caulfield, C.P. & Kerswell, R.R. 2014 Designing a more nonlinearly stable laminar flow via boundary manipulation. J. Fluid Mech. 738, R1.CrossRefGoogle Scholar
Rinaldi, E., Schlatter, P. & Bagheri, S. 2018 Edge state modulation by mean viscosity gradients. J. Fluid Mech. 838, 379–403.CrossRefGoogle Scholar
Robinson, C. 1998 Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC.CrossRefGoogle Scholar
Romanov, V.A. 1973 Stability of plane-parallel Couette flow. Funct. Anal. Applics. 7 (2), 137–146.CrossRefGoogle Scholar
Scarselli, D., Kühnen, J. & Hof, B. 2019 Relaminarising pipe flow by wall movement. J. Fluid Mech. 867, 934–948.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows, vol. 142. Springer.CrossRefGoogle Scholar
Schneider, T.M., Eckhardt, B. & Yorke, J.A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.CrossRefGoogle ScholarPubMed
Schneider, T.M., Gibson, J.F., Lagha, M., De Lillo, F. & Eckhardt, B. 2008 Laminar–turbulent boundary in plane Couette flow. Phys. Rev. E 78 (3), 037301.CrossRefGoogle ScholarPubMed
Schneider, T.M., Marinc, D. & Eckhardt, B. 2010 Localized edge states nucleate turbulence in extended plane Couette cells. J. Fluid Mech. 646, 441–451.CrossRefGoogle Scholar
Shannon, C.E. 1948 A mathematical theory of communication. Bell Syst. Tech. J. 27 (3), 379–423.CrossRefGoogle Scholar
Shi, L., Avila, M. & Hof, B. 2013 Scale invariance at the onset of turbulence in Couette flow. Phys. Rev. Lett. 110 (20), 204502.CrossRefGoogle ScholarPubMed
Skufca, J.D., Yorke, J.A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.CrossRefGoogle Scholar
Subramanian, P., Mariappan, S., Sujith, R.I. & Wahi, P. 2010 Bifurcation analysis of thermoacoustic instability in a horizontal Rijke tube. Intl J. Spray Combust. Dyn. 2 (4), 325–355.CrossRefGoogle Scholar
van Veen, L. & Kawahara, G. 2011 Homoclinic tangle on the edge of shear turbulence. Phys. Rev. Lett. 107 (11), 114501.CrossRefGoogle ScholarPubMed
Virot, E., Kreilos, T., Schneider, T.M. & Rubinstein, S.M. 2017 Stability landscape of shell buckling. Phys. Rev. Lett. 119 (22), 224101.CrossRefGoogle ScholarPubMed
Vishnampet, R., Bodony, D.J. & Freund, J.B. 2015 A practical discrete-adjoint method for high-fidelity compressible turbulence simulations. J. Comput. Phys. 285, 173–192.CrossRefGoogle Scholar
Watanabe, T., Iima, M. & Nishiura, Y. 2016 A skeleton of collision dynamics: hierarchical network structure among even-symmetric steady pulses in binary fluid convection. SIAM J. Appl. Dyn. Syst. 15 (2), 789–806.CrossRefGoogle Scholar
Wells, D.K., Kath, W.L. & Motter, A.E. 2015 Control of stochastic and induced switching in biophysical networks. Phys. Rev. X 5 (3), 031036.Google ScholarPubMed
Xi, L. & Graham, M.D. 2012 Dynamics on the laminar–turbulent boundary and the origin of the maximum drag reduction asymptote. Phys. Rev. Lett. 108 (2), 028301.CrossRefGoogle ScholarPubMed
Zammert, S. & Eckhardt, B. 2015 Crisis bifurcations in plane Poiseuille flow. Phys. Rev. E 91 (4), 041003.CrossRefGoogle ScholarPubMed