Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T05:38:59.566Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic analysis of the attractors in two-dimensional Kolmogorov flow

Published online by Cambridge University Press:  24 July 2017

W. R. SMITH  and
J. G. WISSINK
W. R. SMITH
Affiliation:
School of Mathematics, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK email: [email protected]
J. G. WISSINK
Affiliation:
Department of Mechanical, Aerospace and Civil Engineering, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The high Reynolds-number structure of the laminar, chaotic and turbulent attractors is investigated in a two-dimensional Kolmogorov flow. The laminar attractors include the families of multi-phased travelling waves and quasi-periodic standing waves both of which form the backbone of the transition to a turbulent flow. At leading order, each laminar attractor under study is obtained by solving the Euler equations on a manifold subject to the appropriate periodicity and symmetry conditions. The manifold is determined by a finite number of vorticity equations, these being required to suppress the secular terms at the next order. Our results show that, for the multi-phased travelling waves, the first phase velocity can be determined by an integral conservation law for kinetic energy and the subsequent phase velocities can be evaluated by a non-linear eigenvalue problem. The results also reveal that whereas viscosity determines the smallest scales and controls the amplitude of the flow, the inertial terms govern the shape and form of the flow. The comparison of our analytical predictions for evaluating the stable single-phased travelling wave with the direct numerical simulation of the Navier–Stokes equations has been undertaken, the agreement being excellent. For sufficiently high Reynolds number, after the bifurcation to chaotic flow, all of the multi-phased travelling waves and quasi-periodic standing waves become unstable non-wandering sets. Based on the above new findings for these unstable non-wandering sets and other travelling and standing waves of this kind in phase space, necessary conditions for the invariant manifolds of the chaotic and turbulent attractors are obtained, these necessary conditions being conjectured to be also sufficient.

Keywords

singular perturbationswavesNavier–Stokes equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 3 , June 2018 , pp. 393 - 416
Copyright
Copyright © Cambridge University Press 2017 

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

[1] Ablowitz, M. J. & Benney, D. J. (1970) The evolution of multi-phase modes for nonlinear dispersive waves. Stud. Appl. Math. 49, 225–238.Google Scholar
[2] Armbruster, D., Nicolaenko, B., Smaoui, N. & Chossat, P. (1996) Symmetries and dynamics for 2-D Navier–Stokes flow. Physica D 95, 81–93.CrossRefGoogle Scholar
[3] Arnol'd, V. I. (1991) Kolmogorov's hydrodynamic attractors. Proc. R. Soc. Lond. A 434, 19–22.Google Scholar
[4] Arnol'd, V. I. & Meshalkin, L. D. (1960) The seminar of A. N. Kolmogorov on selected topics in analysis (1958–59). Usp. Mat. Nauk 15, 247–250.Google Scholar
[5] Barenblatt, G. I. (1996) Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[6] Bowles, R. I., Davies, C. & Smith, F. T. (2003) On the spiking stages in deep transition and unsteady separation. J. Eng. Math 45, 227–245.CrossRefGoogle Scholar
[7] Chapman, S. J. (2002) Subcritical transition in channel flows. J. Fluid Mech. 451, 35–97.Google Scholar
[8] Craik, A. D. D. (1985) Wave Interactions and Fluid Flows. Cambridge University Press, Cambridge.Google Scholar
[9] Deguchi, K. & Walton, A. G. (2013) A swirling spiral wave solution in pipe flow. J. Fluid Mech. 737, R2.CrossRefGoogle Scholar
[10] Farazmand, M. (2016) An adjoint-based approach for finding invariant solutions of Navier–Stokes equations. J. Fluid Mech. 795, 278–312.CrossRefGoogle Scholar
[11] Hall, P. & Smith, F. T. (1991) On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.CrossRefGoogle Scholar
[12] Hiruta, Y. & Toh, S. (2015) Solitary solutions including spatially localized chaos and their interactions in two-dimensional Kolmogorov flow. Phys. Rev. E 92, 063025.Google Scholar
[13] Hopf, E. (1948) A mathematical example displaying features of turbulence. Comm. Pure Appl. Math. 1, 303–322.CrossRefGoogle Scholar
[14] Joyce, G. R. & Montgomery, D. (1973) Negative temperature states for a two-dimensional guiding-center plasma. J. Plasma Phys. 10, 107–121.CrossRefGoogle Scholar
[15] Kim, S-C. & Okamoto, H. (2015) Unimodal patterns appearing in the Kolmogorov flows at large Reynolds numbers. Nonlinearity 28, 3219–3242.Google Scholar
[16] Kuzmak, G. E. (1959) Asymptotic solutions of nonlinear second order differential equations with variable coefficients. Prikl. Mat. Mekh. 23, 515–526 (In Russian); (1959) J. Appl. Math. Mech. 23, 730–744 (In English).Google Scholar
[17] Ladyzhenskaya, O. A. (1972) A dynamical system generated by the Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI) 27, 91–115 (In Russian); (1975) J. Soviet Math. 3, 458–479 (In English).Google Scholar
[18] Landau, L. D. & Lifshitz, E. M. (1987) Fluid Mechanics, Pergamon Press, Oxford.Google Scholar
[19] Lucas, D. & Kerswell, R. R. (2015) Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow. Phys. Fluids 27, 045106.CrossRefGoogle Scholar
[20] Luke, J. C. (1966) A perturbation method for nonlinear dispersive wave problems. Proc. Roy. Soc. Lond. A 292, 403–412.Google Scholar
[21] Meshalkin, L. D. & Sinai, Ya. G.. (1961) Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. Prikl. Mat. Mekh. 25, 1140–1143 (In Russian); (1961) J. Appl. Math. Mech. 25, 1700–1705 (In English).Google Scholar
[22] Montgomery, D. & Joyce, G. R. (1974) Statistical mechanics of “negative temperature" states. Phys. Fluids 17, 1139–1145.CrossRefGoogle Scholar
[23] Montgomery, D., Matthaeus, W. H., Stribling, W. T., Martinez, D. & Oughton, S. (1992) Relaxation in two dimensions and the “sinh-Poisson" equation. Phys. Fluids A 4, 3–6.Google Scholar
[24] Newhouse, S. E., Ruelle, D. & Takens, F. (1978) Occurrence of strange axiom A attractors near quasi-periodic flows on Tm, m ⩾ 3. Comm. Math. Phys. 64, 35–40.Google Scholar
[25] Okamoto, H. & Shoji, M. (1993) Bifurcation diagrams in Kolmogorov's problem of viscous incompressible fluid on 2-D flat tori. Japan J. Indust. Appl. Math. 10, 191–218.Google Scholar
[26] Platt, N., Sirovich, L. & Fitzmaurice, N. (1991) An investigation of chaotic Kolmogorov flows. Phys. Fluids A 3, 681–696.CrossRefGoogle Scholar
[27] Ruelle, D. & Takens, F. (1971) On the nature of turbulence. Comm. Math. Phys. 20, 167–192.Google Scholar
[28] She, Z. S. (1988) Large-scale dynamics and transition to turbulence in the two-dimensional Kolmogorov flow. In: Branover, H., Mond, M. & Unger, Y. (editors), Proceedings on Current Trends in Turbulence Research, Vol. 117, AIAA, Washington, DC, pp. 374–396.Google Scholar
[29] Smith, F. T. & Bodonyi, R. J. (1982) Amplitude-dependent neutral modes in the Hagen–Poiseuille flow through a circular pipe. Proc. R. Soc. Lond. A 384, 463–489.Google Scholar
[30] Smith, F. T. & Burggraf, O. R. (1985) On the development of large-sized short-scaled disturbances in boundary layers. Proc. R. Soc. Lond. A 399, 25–55.Google Scholar
[31] Smith, F. T., Doorly, D. J. & Rothmayer, A. P. (1990) On displacement-thickness, wall-layer and mid-flow scales in turbulent boundary layers & slugs of vorticity in channel and pipe flows. Proc. R. Soc. Lond. A 428, 255–281.Google Scholar
[32] Smith, W. R. (2005) On the sensitivity of strongly nonlinear autonomous oscillators and oscillatory waves to small perturbations. IMA J. Appl. Math. 70, 359–385.Google Scholar
[33] Smith, W. R. (2007) Explicit modulation equations, Reynolds averaging and the closure problem for the Korteweg-deVries-Burgers equation. IMA J. Appl. Math. 72, 163–179.Google Scholar
[34] Smith, W. R. (2007) Modulation equations and Reynolds averaging for finite-amplitude nonlinear waves in an incompressible fluid. IMA J. Appl. Math. 72, 923–945.CrossRefGoogle Scholar
[35] Smith, W. R. (2010) Modulation equations for strongly nonlinear oscillations of an incompressible viscous drop. J. Fluid Mech. 654, 141–159.CrossRefGoogle Scholar
[36] Smith, W. R., King, J. R., Tuck, B. & Orton, J. W. (1999) The single-mode rate equations for semiconductor lasers with thermal effects. IMA J. Appl. Math. 63, 1–36.CrossRefGoogle Scholar
[37] Smith, W. R. & Wang, Q. X. Viscous decay of nonlinear oscillations of a spherical bubble at large Reynolds number. In preparation.Google Scholar
[38] Smith, W. R. & Wissink, J. G. (2014) Parameterization of travelling waves in plane Poiseuille flow. IMA J. Appl. Math. 79, 22–32.Google Scholar
[39] Smith, W. R. & Wissink, J. G. (2015) Travelling waves in two-dimensional plane Poiseuille flow. SIAM J. Appl. Math. 75, 2147–2169.Google Scholar
[40] Tennekes, H. & Lumley, J. L. (1972) A First Course in Turbulence, MIT Press, London.CrossRefGoogle Scholar
[41] Whitham, G. B. (1974) Linear and Nonlinear Waves, John Wiley & Sons, New York.Google Scholar