Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T05:15:37.428Z Has data issue: false hasContentIssue false
  • English
  • Français

Closing the loop: nonlinear Taylor vortex flow through the lens of resolvent analysis

Published online by Cambridge University Press:  04 August 2021

Benedikt Barthel Open the ORCID record for Benedikt Barthel [Opens in a new window] ,
Xiaojue Zhu Open the ORCID record for Xiaojue Zhu [Opens in a new window]  and
Beverley McKeon Open the ORCID record for Beverley McKeon [Opens in a new window]
Benedikt Barthel*
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
Xiaojue Zhu
Affiliation:
Center of Mathematical Sciences and Applications, and School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Beverley McKeon
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present an optimization-based method to efficiently calculate accurate nonlinear models of Taylor vortex flow. Through the resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), we model these Taylor vortex solutions by treating the nonlinearity not as an inherent part of the governing equations but rather as a triadic constraint which must be satisfied by the model solution. We exploit the low-rank linear dynamics to calculate an efficient basis, the coefficients of which are calculated through an optimization problem to minimize the triadic consistency of the solution with itself as well as the input mean flow. Our approach constitutes, what is to our knowledge, the first fully nonlinear and self-sustaining, resolvent-based model in the literature. Our results compare favourably with the direct numerical simulation (DNS) of Taylor–Couette flow at up to five times the critical Reynolds number. Additionally, we find that as the Reynolds number increases, the flow undergoes a fundamental transition from a classical weakly nonlinear regime, where the forcing cascade is strictly down scale, to a fully nonlinear regime characterized by the emergence of an inverse (up scale) forcing cascade. Triadic contributions from the inverse and traditional cascade destructively interfere implying that the accurate modelling of a certain Fourier mode requires knowledge of its immediate harmonic and sub-harmonic. We show analytically that this is a direct consequence of the structure of the quadratic nonlinearity of the Navier–Stokes equations formulated in Fourier space. Finally, using our model as an initial condition to a higher Reynolds number DNS significantly reduces the time to convergence.

JFM classification

Instability: Nonlinear instability Nonlinear Dynamical Systems: Low-dimensional models
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 924 , 10 October 2021 , A9
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

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
Beaume, C., Chini, G.P., Julien, K. & Knobloch, E. 2015 Reduced description of exact coherent states in parallel shear flows. Phys. Rev. E 91 (4), 043010.CrossRefGoogle ScholarPubMed
Bengana, Y. & Tuckerman, L.S. 2021 Frequency prediction from exact or self-consistent meanflows. arXiv:2102.07255CrossRefGoogle Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21 (3), 385–425.CrossRefGoogle Scholar
Dessup, T., Tuckerman, L.S., Wesfreid, J.E., Barkley, D. & Willis, A.P. 2018 Self-sustaining process in Taylor–Couette flow. Phys. Rev. Fluids 3, 123902.CrossRefGoogle Scholar
Gallaire, F., Boujo, E., Mantic-Lugo, V., Arratia, C., Thiria, B. & Meliga, P. 2016 Pushing amplitude equations far from threshold: application to the supercritical Hopf bifurcation in the cylinder wake. Fluid Dyn. Res. 48 (6), 061401.CrossRefGoogle Scholar
Gebhardt, T. & Grossmann, S. 1993 The Taylor–Couette eigenvalue problem with independently rotating cylinders. Z. Phys. B Condens. Matter 90 (4), 475–490.CrossRefGoogle Scholar
van Gils, D.P.M., Bruggert, G., Lathrop, D.P., Sun, C. & Lohse, D. 2011 The Twente turbulent Taylor–Couette (T3C) facility: strongly turbulent (multiphase) flow between two independently rotating cylinders. Rev. Sci. Instrum. 82 (2), 025105.CrossRefGoogle ScholarPubMed
van Gils, D.P.M., Huisman, S.G., Grossmann, S., Sun, C. & Lohse, D. 2012 Optimal Taylor–Couette turbulence. J. Fluid Mech. 706, 118–149.CrossRefGoogle Scholar
Grossmann, S., Lohse, D. & Sun, C. 2016 High-Reynolds number Taylor–Couette turbulence. Annu. Rev. Fluid Mech. 48, 53–80.CrossRefGoogle Scholar
Hamilton, J.M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
Huisman, S.G., van der Veen, R.C.A., Sun, C. & Lohse, D. 2014 Multiple states in highly turbulent Taylor–Couette flow. Nat. Commun. 5, 3820.CrossRefGoogle ScholarPubMed
Illingworth, S.J. 2020 Streamwise-constant large-scale structures in Couette and Poiseuille flows. J. Fluid Mech. 889, A13.CrossRefGoogle Scholar
Jones, C.A. 1981 Nonlinear Taylor vortices and their stability. J. Fluid Mech. 102, 249–261.CrossRefGoogle Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113 (8), 084501.CrossRefGoogle ScholarPubMed
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2015 A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake. Phys. Fluids 27 (7), 074103.CrossRefGoogle Scholar
Marcus, P.S. 1984 Simulation of Taylor–Couette flow. Part 2. Numerical results for wavy-vortex flow with one travelling wave. J. Fluid Mech. 146, 65–113.CrossRefGoogle Scholar
Maretzke, S., Hof, B. & Avila, M. 2014 Transient growth in linearly stable Taylor–Couette flows. J. Fluid Mech. 742, 254–290.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
McMullen, R.M., Rosenberg, K. & McKeon, B.J. 2020 Interaction of forced Orr–Sommerfeld and Squire modes in a low-order representation of turbulent channel flow. Phys. Rev. Fluids 5 (8), 084607.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.CrossRefGoogle Scholar
Morra, P., Nogueira, P.A.S., Cavalieri, A.V.G. & Henningson, D.S. 2021 The colour of forcing statistics in resolvent analyses of turbulent channel flows. J. Fluid Mech. 907, A24.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
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
Ostilla, R., Stevens, R.J.A.M., Grossmann, S., Verzicco, R. & Lohse, D. 2013 Optimal Taylor–Couette flow: direct numerical simulations. J. Fluid Mech. 719, 14–46.CrossRefGoogle Scholar
Ostilla-Mónico, R., van der Poel, E.P., Verzicco, R., Grossmann, S. & Lohse, D. 2014 Exploring the phase diagram of fully turbulent Taylor–Couette flow. J. Fluid Mech. 761, 1–26.CrossRefGoogle Scholar
van der Poel, E.P., Ostilla-Mónico, R., Donners, J. & Verzicco, R. 2015 A pencil distributed finite difference code for strongly turbulent wall-bounded flows. Comput. Fluids 116, 10–16.CrossRefGoogle Scholar
Rand, D. 1982 Dynamics and symmetry. Predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79 (1), 1–37.CrossRefGoogle Scholar
Rigas, G., Sipp, D. & Colonius, T. 2021 Nonlinear input/output analysis: application to boundary layer transition. J. Fluid Mech. 911, A15.CrossRefGoogle Scholar
Rosales, C. & Meneveau, C. 2006 A minimal multiscale Lagrangian map approach to synthesize non-Gaussian turbulent vector fields. Phys. Fluids 18 (7), 075104.CrossRefGoogle Scholar
Rosenberg, K. 2018 Resolvent-based modeling of flows in a channel. PhD thesis, California Institute of Technology.Google Scholar
Rosenberg, K. & McKeon, B.J. 2019 a Computing exact coherent states in channels starting from the laminar profile: a resolvent-based approach. Phys. Rev. E 100 (2), 021101.CrossRefGoogle ScholarPubMed
Rosenberg, K. & McKeon, B.J. 2019 b Efficient representation of exact coherent states of the Navier–stokes equations using resolvent analysis. Fluid Dyn. Res. 51 (1), 011401.CrossRefGoogle Scholar
Rosenberg, K., Symon, S. & McKeon, B.J. 2019 Role of parasitic modes in nonlinear closure via the resolvent feedback loop. Phys. Rev. Fluids 4 (5), 052601.CrossRefGoogle Scholar
Sacco, F., Verzicco, R. & Ostilla-Mónico, R. 2019 Dynamics and evolution of turbulent Taylor rolls. J. Fluid Mech. 870, 970–987.CrossRefGoogle Scholar
Stuart, J.T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9 (3), 353–370.CrossRefGoogle Scholar
Symon, S., Illingworth, S.J. & Marusic, I. 2021 Energy transfer in turbulent channel flows and implications for resolvent modelling. J. Fluid Mech. 911, A3.CrossRefGoogle Scholar
Taylor, G.I. 1923 VIII. Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223 (605–615), 289–343.Google Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123 (2), 402–414.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.CrossRefGoogle Scholar
Yahata, H. 1977 Slowly-varying amplitude of the Taylor vortices near the instability point. II: mode-coupling-theoretical approach. Prog. Theor. Phys. 57 (5), 1490–1496.CrossRefGoogle Scholar
Zhu, X., et al. 2018 AFiD-GPU: a versatile Navier–Stokes solver for wall-bounded turbulent flows on GPU clusters. Comput. Phys. Commun. 229, 199–210.CrossRefGoogle Scholar