Skip to main content
    • Aa
    • Aa

Asymptotic analysis of the attractors in two-dimensional Kolmogorov flow

  • W. R. SMITH (a1) and J. G. WISSINK (a2)

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] D. Armbruster , B. Nicolaenko , N. Smaoui & P. Chossat (1996) Symmetries and dynamics for 2-D Navier–Stokes flow. Physica D 95, 8193.

[3] V. I. Arnol'd (1991) Kolmogorov's hydrodynamic attractors. Proc. R. Soc. Lond. A 434, 1922.

[5] G. I. Barenblatt (1996) Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, Cambridge.

[6] R. I. Bowles , C. Davies & F. T. Smith (2003) On the spiking stages in deep transition and unsteady separation. J. Eng. Math 45, 227245.

[7] S. J. Chapman (2002) Subcritical transition in channel flows. J. Fluid Mech. 451, 3597.

[9] K. Deguchi & A. G. Walton (2013) A swirling spiral wave solution in pipe flow. J. Fluid Mech. 737, R2.

[10] M. Farazmand (2016) An adjoint-based approach for finding invariant solutions of Navier–Stokes equations. J. Fluid Mech. 795, 278312.

[11] P. Hall & F. T. Smith (1991) On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641666.

[12] Y. Hiruta & S. Toh (2015) Solitary solutions including spatially localized chaos and their interactions in two-dimensional Kolmogorov flow. Phys. Rev. E 92, 063025.

[14] G. R. Joyce & D. Montgomery (1973) Negative temperature states for a two-dimensional guiding-center plasma. J. Plasma Phys. 10, 107121.

[15] S-C. Kim & H. Okamoto (2015) Unimodal patterns appearing in the Kolmogorov flows at large Reynolds numbers. Nonlinearity 28, 32193242.

[19] D. Lucas & R. R. Kerswell (2015) Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow. Phys. Fluids 27, 045106.

[20] J. C. Luke (1966) A perturbation method for nonlinear dispersive wave problems. Proc. Roy. Soc. Lond. A 292, 403412.

[22] D. Montgomery & G. R. Joyce (1974) Statistical mechanics of “negative temperature" states. Phys. Fluids 17, 11391145.

[23] D. Montgomery , W. H. Matthaeus , W. T. Stribling , D. Martinez & S. Oughton (1992) Relaxation in two dimensions and the “sinh-Poisson" equation. Phys. Fluids A 4, 36.

[25] H. Okamoto & M. Shoji (1993) Bifurcation diagrams in Kolmogorov's problem of viscous incompressible fluid on 2-D flat tori. Japan J. Indust. Appl. Math. 10, 191218.

[26] N. Platt , L. Sirovich & N. Fitzmaurice (1991) An investigation of chaotic Kolmogorov flows. Phys. Fluids A 3, 681696.

[27] D. Ruelle & F. Takens (1971) On the nature of turbulence. Comm. Math. Phys. 20, 167192.

[30] F. T. Smith & O. R. Burggraf (1985) On the development of large-sized short-scaled disturbances in boundary layers. Proc. R. Soc. Lond. A 399, 2555.

[31] F. T. Smith , D. J. Doorly & A. P. Rothmayer (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, 255281.

[33] W. R. Smith (2007) Explicit modulation equations, Reynolds averaging and the closure problem for the Korteweg-deVries-Burgers equation. IMA J. Appl. Math. 72, 163179.

[34] W. R. Smith (2007) Modulation equations and Reynolds averaging for finite-amplitude nonlinear waves in an incompressible fluid. IMA J. Appl. Math. 72, 923945.

[35] W. R. Smith (2010) Modulation equations for strongly nonlinear oscillations of an incompressible viscous drop. J. Fluid Mech. 654, 141159.

[36] W. R. Smith , J. R. King , B. Tuck & J. W. Orton (1999) The single-mode rate equations for semiconductor lasers with thermal effects. IMA J. Appl. Math. 63, 136.

[38] W. R. Smith & J. G. Wissink (2014) Parameterization of travelling waves in plane Poiseuille flow. IMA J. Appl. Math. 79, 2232.

[39] W. R. Smith & J. G. Wissink (2015) Travelling waves in two-dimensional plane Poiseuille flow. SIAM J. Appl. Math. 75, 21472169.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 9 *
Loading metrics...

Abstract views

Total abstract views: 47 *
Loading metrics...

* Views captured on Cambridge Core between 24th July 2017 - 21st September 2017. This data will be updated every 24 hours.