Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-16T23:16:01.671Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact coherent structures and shadowing in turbulent Taylor–Couette flow

Published online by Cambridge University Press:  23 July 2021

Michael C. Krygier Open the ORCID record for Michael C. Krygier [Opens in a new window] ,
Joshua L. Pughe-Sanford Open the ORCID record for Joshua L. Pughe-Sanford [Opens in a new window]  and
Roman O. Grigoriev Open the ORCID record for Roman O. Grigoriev [Opens in a new window]
Michael C. Krygier
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA30332, USA
Joshua L. Pughe-Sanford
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA30332, USA
Roman O. Grigoriev*
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA30332, USA
*
†Email address for correspondence: romgrig@gatech.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate a theoretical framework for modelling fluid turbulence based on the formalism of exact coherent structures (ECSs). Although highly promising, existing evidence for the role of ECSs in turbulent flows is largely circumstantial and comes primarily from idealized numerical simulations. In particular, it remains unclear whether three-dimensional turbulent flows in experiment shadow any ECSs. In order to conclusively answer this question, a hierarchy of ECSs should be computed on a domain and with boundary conditions exactly matching experiment. The present study makes the first step in this direction by investigating a small-aspect-ratio Taylor–Couette flow with naturally periodic boundary conditions in the azimuthal direction. We describe the structure of the chaotic set underlying turbulent flow driven by counter-rotating cylinders and present direct numerical evidence for shadowing of a collection of unstable relative periodic orbits and a travelling wave, setting the stage for further experimental tests of the framework.

JFM classification

Nonlinear Dynamical Systems: Chaos Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A7
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aitta, A., Ahlers, G. & Cannell, D.S. 1985 Tricritical phenomena in rotating Couette–Taylor flow. Phys. Rev. Lett. 54 (7), 673.CrossRefGoogle ScholarPubMed
Allgower, E.L. & Georg, K. 2003 Introduction to Numerical Continuation Methods. SIAM.CrossRefGoogle Scholar
Altmeyer, S., Do, Y., Marques, F. & Lopez, J.M. 2012 Symmetry-breaking Hopf bifurcations to 1-, 2-, and 3-tori in small-aspect-ratio counterrotating Taylor–Couette flow. Phys. Rev. E 86 (4), 046316.CrossRefGoogle ScholarPubMed
Andereck, C.D., Liu, S.S. & Swinney, H.L. 1986 Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183.CrossRefGoogle Scholar
Arnold, V.I. & Avez, A. 1968 Ergodic Problems of Classical Mechanics. W. A. Benjamin.Google Scholar
Auerbach, D., Cvitanović, P., Eckmann, J.-P., Gunaratne, G. & Procaccia, I. 1987 Exploring chaotic motion through periodic orbits. Phys. Rev. Lett. 58 (23), 2387.CrossRefGoogle ScholarPubMed
Avila, M. 2012 Stability and angular-momentum transport of fluid flows between corotating cylinders. Phys. Rev. Lett. 108 (12), 124501.CrossRefGoogle ScholarPubMed
Avila, M., Grimes, M., Lopez, J.M. & Marques, F. 2008 Global endwall effects on centrifugally stable flows. Phys. Fluids 20 (10), 104104.CrossRefGoogle Scholar
Benjamin, T.B. & Mullin, T. 1981 Anomalous modes in the Taylor experiment. Proc. R. Soc. Lond. A 377 (1770), 221–249.Google Scholar
Budanur, N.B., Short, K.Y., Farazmand, M., Willis, A.P. & Cvitanović, P. 2017 Relative periodic orbits form the backbone of turbulent pipe flow. J. Fluid Mech. 833, 274–301.CrossRefGoogle Scholar
Buzug, T., Von Stamm, J. & Pfister, G. 1992 Fractal dimensions of strange attractors obtained from the Taylor–Couette experiment. Physica A 191 (1–4), 559–563.CrossRefGoogle Scholar
Buzug, T., von Stamm, J. & Pfister, G. 1993 Characterization of period-doubling scenarios in Taylor–Couette flow. Phys. Rev. E 47 (2), 1054.CrossRefGoogle ScholarPubMed
Chandler, G.J. & Kerswell, R.R. 2013 Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech. 722, 554–595.CrossRefGoogle Scholar
Chaté, H. & Manneville, P. 1987 Transition to turbulence via spatio-temporal intermittency. Phys. Rev. Lett. 58 (2), 112.CrossRefGoogle ScholarPubMed
Cliffe, K.A. 1983 Numerical calculations of two-cell and single-cell Taylor flows. J. Fluid Mech. 135, 219–233.CrossRefGoogle Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21 (3), 385–425.CrossRefGoogle Scholar
Cvitanović, P. 1988 Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett. 61 (24), 2729.CrossRefGoogle ScholarPubMed
Cvitanović, P. 2013 Recurrent flows: the clockwork behind turbulence. J. Fluid Mech. 726, 1–4.CrossRefGoogle Scholar
Cvitanović, P. & Gibson, J.F. 2010 Geometry of the turbulence in wall-bounded shear flows: periodic orbits. Phys. Scr. 2010 (T142), 014007.CrossRefGoogle Scholar
De Lozar, A., Mellibovsky, F., Avila, M. & Hof, B. 2012 Edge state in pipe flow experiments. Phys. Rev. Lett. 108 (21), 214502.CrossRefGoogle Scholar
Deguchi, K. & Altmeyer, S. 2013 Fully nonlinear mode competitions of nearly bicritical spiral or Taylor vortices in Taylor–Couette flow. Phys. Rev. E 87 (4), 043017.CrossRefGoogle ScholarPubMed
Deguchi, K., Meseguer, A. & Mellibovsky, F. 2014 Subcritical equilibria in Taylor–Couette flow. Phys. Rev. Lett. 112 (18), 184502.CrossRefGoogle ScholarPubMed
Dennis, D.J.C. & Sogaro, F.M. 2014 Distinct organizational states of fully developed turbulent pipe flow. Phys. Rev. Lett. 113, 234501.CrossRefGoogle ScholarPubMed
Eckhardt, B., Faisst, H., Schmiegel, A. & Schneider, T.M. 2008 Dynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. R. Soc. A 366 (1868), 1297–1315.CrossRefGoogle ScholarPubMed
Eckhardt, B., Schneider, T.M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39 (1), 447–468.CrossRefGoogle Scholar
Fiedler, B., Sandstede, B., Scheel, A. & Wulff, C. 1996 Bifurcation from relative equilibria of noncompact group actions: skew products, meanders, and drifts. Doc. Math. 141, 479–505.Google Scholar
Furukawa, H., Watanabe, T., Toya, Y. & Nakamura, I. 2002 Flow pattern exchange in the Taylor–Couette system with a very small aspect ratio. Phys. Rev. E 65 (3), 036306.CrossRefGoogle ScholarPubMed
Gaspard, P. 2005 Chaos, Scattering and Statistical Mechanics, vol. 9. Cambridge University Press.Google Scholar
Heise, M., Hoffmann, C., Will, C., Altmeyer, S., Abshagen, J. & Pfister, G. 2013 Co-rotating Taylor–Couette flow enclosed by stationary disks. J. Fluid Mech. 716, R4.CrossRefGoogle Scholar
Hochstrate, K., Abshagen, J., Avila, M., Will, C. & Pfister, G. 2010 Decay of turbulent bursting in enclosed flows. In 7th IUTAM Symposium on Laminar-Turbulent Transition (ed. P. Schlatter & D.S. Henningson), pp. 195–200. Springer.CrossRefGoogle Scholar
Hof, B., van Doorne, C.W.H., Westerweel, J., Nieuwstadt, F.T.M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R.R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305 (5690), 1594–1598.CrossRefGoogle ScholarPubMed
Hoffmann, C., Altmeyer, S., Heise, M., Abshagen, J. & Pfister, G. 2013 Axisymmetric propagating vortices in centrifugally stable Taylor–Couette flow. J. Fluid Mech. 728, 458–470.CrossRefGoogle Scholar
Hopf, E. 1942 Abzweigung einer periodischen lösung von einer stationären lösung eines differentialsystems. Ber. Math.-Phys. Kl Sächs. Akad. Wiss. Leipzig 94, 1–22.Google Scholar
Hopf, E. 1948 A mathematical example displaying features of turbulence. Commun. Pure Appl. Maths 1 (4), 303–322.CrossRefGoogle Scholar
Hugues, S. & Randriamampianina, A. 1998 An improved projection scheme applied to pseudospectral methods for the incompressible Navier–Stokes equations. Intl J. Numer. Meth. Fluids 28 (3), 501–521.3.0.CO;2-S>CrossRefGoogle Scholar
Hussain, A.K.M.F. 1983 Coherent structures–reality and myth. Phys. Fluids 26 (10), 2816–2850.CrossRefGoogle Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.CrossRefGoogle Scholar
Kawahara, G., Uhlmann, M. & Van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.CrossRefGoogle Scholar
Kazantsev, E. 1998 Unstable periodic orbits and attractor of the barotropic ocean model. Nonlinear Process. Geophys. 5, 193–208.CrossRefGoogle Scholar
Kerswell, R.R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18 (6), R17.CrossRefGoogle Scholar
Kerswell, R.R. & Tutty, O.R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301–305.Google Scholar
Kostelich, E.J., Kan, I., Grebogi, C., Ott, E. & Yorke, J.A. 1997 Unstable dimension variability: a source of nonhyperbolicity in chaotic systems. Physica D 109 (1–2), 81–90.CrossRefGoogle Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22 (4), 047505.CrossRefGoogle ScholarPubMed
Lan, Y. 2010 Cycle expansions: from maps to turbulence. Commun. Nonlinear Sci. Numer. Simul. 15 (3), 502–526.CrossRefGoogle Scholar
Lorenz, E.N. 1963 Deterministic non-periodic flow. J. Atmos. Sci. 20 (1), 30–41.2.0.CO;2>CrossRefGoogle Scholar
Lorenzen, A., Pfister, G. & Mullin, T. 1983 End effects on the transition to time-dependent motion in the Taylor experiment. Phys. Fluids 26 (1), 10–13.CrossRefGoogle Scholar
Lucas, D. & Kerswell, R.R. 2015 Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow. Phys. Fluids 27 (4), 045106.CrossRefGoogle Scholar
Mandelbrot, B. 1967 How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156 (3775), 636–638.CrossRefGoogle Scholar
Marques, F. & Lopéz, J.M. 2006 Onset of three-dimensional unsteady states in small-aspect-ratio Taylor–Couette flow. J. Fluid Mech. 561, 255–277.CrossRefGoogle Scholar
Mercader, I., Batiste, O. & Alonso, A. 2010 An efficient spectral code for incompressible flows in cylindrical geometries. Comput. Fluids 39 (2), 215–224.CrossRefGoogle Scholar
Meseguer, A., Mellibovsky, F., Avila, M. & Marques, F. 2009 Families of subcritical spirals in highly counter-rotating Taylor–Couette flow. Phys. Rev. E 79 (3), 036309.CrossRefGoogle ScholarPubMed
Moody, L.F. 1944 Friction factors for pipe flow. Trans. ASME 66, 671–684.Google Scholar
Mullin, T. 1982 Mutations of steady cellular flows in the Taylor experiment. J. Fluid Mech. 121, 207–218.CrossRefGoogle Scholar
Mullin, T., Toya, Y. & Tavener, S.J. 2002 Symmetry breaking and multiplicity of states in small aspect ratio Taylor–Couette flow. Phys. Fluids 14 (8), 2778–2787.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
Nikuradse, J. 1932 Gesetzmassigkeiten der turbulenten stromung in glatten rohren. Ver. Deutsch. Ing. Forsch. 356.Google Scholar
Orszag, S.A. & Patera, A.T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.CrossRefGoogle Scholar
Page, J. & Kerswell, R.R. 2020 Searching turbulence for periodic orbits with dynamic mode decomposition. J. Fluid Mech. 886, A28.CrossRefGoogle Scholar
Park, J.S. & Graham, M.D. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430–454.CrossRefGoogle Scholar
Pfister, G., Buzug, T. & Enge, N. 1992 Characterization of experimental time series from Taylor–Couette flow. Physica D 58 (1–4), 441–454.CrossRefGoogle Scholar
Pfister, G., Schmidt, H., Cliffe, K.A. & Mullin, T. 1988 Bifurcation phenomena in Taylor–Couette flow in a very short annulus. J. Fluid Mech. 191, 1–18.CrossRefGoogle Scholar
Pfister, G., Schulz, A. & Lensch, B. 1991 Bifurcations and a route to chaos of an one-vortex-state in Taylor–Couette flow. Eur. J. Mech. B/Fluids 10 (2), 247–252.Google Scholar
Sandstede, B., Scheel, A. & Wulff, C. 1999 Dynamical behavior of patterns with Euclidean symmetry. In Pattern Formation in Continuous and Coupled Systems (ed. M. Golubitsky, D. Luss & S.H. Strogatz), pp. 249–264. Springer.CrossRefGoogle Scholar
Schneider, T.M., Eckhardt, B. & Vollmer, J. 2007 Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75 (6), 066313.CrossRefGoogle ScholarPubMed
Schulz, A., Pfister, G. & Tavener, S.J. 2003 The effect of outer cylinder rotation on Taylor–Couette flow at small aspect ratio. Phys. Fluids 15 (2), 417–425.CrossRefGoogle Scholar
Streett, C.L. & Hussaini, M.Y. 1991 A numerical simulation of the appearance of chaos in finite-length Taylor–Couette flow. Appl. Numer. Maths 7 (1), 41–71.CrossRefGoogle Scholar
Suri, B., Kageorge, L., Grigoriev, R.O. & Schatz, M.F. 2020 Capturing turbulent dynamics and statistics in experiments with unstable periodic orbits. Phys. Rev. Lett. 125 (6), 064501.CrossRefGoogle ScholarPubMed
Suri, B., Tithof, J., Grigoriev, R.O. & Schatz, M.F. 2017 Forecasting fluid flows using the geometry of turbulence. Phys. Rev. Lett. 118 (11), 114501.CrossRefGoogle ScholarPubMed
Suri, B., Tithof, J., Grigoriev, R.O. & Schatz, M.F. 2018 Unstable equilibria and invariant manifolds in quasi-two-dimensional Kolmogorov-like flow. Phys. Rev. E 98 (2), 023105.CrossRefGoogle ScholarPubMed
Tsameret, A. & Steinberg, V. 1994 Competing states in a Couette–Taylor system with an axial flow. Phys. Rev. E 49 (5), 4077.CrossRefGoogle Scholar
van Veen, L., Kida, S. & Kawahara, G. 2006 Periodic motion representing isotropic turbulence. Fluid Dyn. Res. 38 (1), 19–46.CrossRefGoogle Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.CrossRefGoogle Scholar
Viswanath, D. 2009 The critical layer in pipe flow at high Reynolds number. Phil. Trans. R. Soc. Lond. A 367, 561–576.Google ScholarPubMed
Von Kármán, T. 1930 Mechanische ahnlichkeit und turbulenz. Math. Phys. Klasse. 58–76.Google Scholar
Waleffe, F. & Wang, J. 2005 Transition threshold and the self-sustaining process. In IUTAM Symposium on Laminar-Turbulent Transition and Finite Amplitude Solutions (ed. T. Mullin & R. Kerswell), pp. 85–106. Springer.CrossRefGoogle Scholar
Willis, A.P., Cvitanović, P. & Avila, M. 2013 Revealing the state space of turbulent pipe flow by symmetry reduction. J. Fluid Mech. 721, 514–540.CrossRefGoogle Scholar
Yalnız, G., Hof, B. & Budanur, N.B. 2020 Coarse graining the state space of a turbulent flow using periodic orbits. Phys. Rev. Lett. 126, 244502.Google Scholar

Krygier et al. Supplementary Movie 1

Shadowing event for RPO05 in lobe 1.
Download Krygier et al. Supplementary Movie 1(Video)
Video 9.8 MB

Krygier et al. Supplementary Movie 2

Shadowing event for reflected copy of RPO05 in lobe 1.

Download Krygier et al. Supplementary Movie 2(Video)
Video 12.8 MB

Krygier et al. Supplementary Movie 3

Shadowing event for RPO01 in lobe 1.

Download Krygier et al. Supplementary Movie 3(Video)
Video 14.4 MB

Krygier et al. Supplementary movie 4

Shadowing event for TW01 in lobe 1.

Download Krygier et al. Supplementary movie 4(Video)
Video 11.3 MB

Krygier et al. Supplementary Movie 5

Shadowing event for RPO15 in lobe 3.

Download Krygier et al. Supplementary Movie 5(Video)
Video 3.3 MB

Krygier et al. Supplementary Movie 6

RPO07 and RPO12 shadowing each other.

Download Krygier et al. Supplementary Movie 6(Video)
Video 22.9 MB