Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-18T15:11:41.527Z Has data issue: false hasContentIssue false
  • English
  • Français

Intermittent dynamics in simple models of the turbulent wall layer

Published online by Cambridge University Press:  26 April 2006

Gal Berkooz ,
Philip Holmes  and
J. L. Lumley
Gal Berkooz
Affiliation:
Center for Applied Mathematics, Ithaca, NY 14853-7501, USA
Philip Holmes
Affiliation:
Departments of Theoretical and Applied Mechanics and Mathematics, Ithaca, NY 14853-7501, USA
J. L. Lumley
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Ithaca, NY 14853-7501, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalize the class of models of the wall layer of Aubry et al. (1988), based on the proper orthogonal decomposition, to permit uncoupled evolution of streamwise and cross-stream disturbances. Since the Reynolds stress is no longer constrained, in the absence of streamwise spatial variations all perturbation velocity components eventually decay to zero. However, their transient behaviour is dominated by ’ghosts’ of the non-trivial fixed points and attracting heteroclinic cycles which are characteristic features of those models based on empirical eigenfunctions whose individual velocity components are fixed. This suggests that the intermittent events observed in Aubry et al. do not arise solely because of the effective closure assumption incorporated in those models, but are rooted deeper in the dynamical phenomenon of the wall region.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 230 , September 1991 , pp. 75 - 95
Copyright
© 1991 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

Armbruster, D., Guckenheimeb, J. & Holmes, P. 1988 Heteroclinc cycles and modulated travelling waves in systems with O(2) symmetry. Physica 29D, 257–282.Google Scholar
Armbruster, D., Guckenheimer, J. & Holmes, P. 1989 Kuramoto—Sivashinsky dynamics on the center unstable manifold. SIAM J. Appl. Maths 49, 676–691.Google Scholar
Aubry, N. & Sanghi, S. 1989 Streamwise and spanwise dynamics of the turbulent wall layer. In Forum on Chaotic Flow (ed. X. Ghia), Proc. ASME, New York.
Aubry, N. & Sanghi, S. 1990 Bifurcation and bursting of streaks in the turbulent wall layers. In Turbulence 89: Organized Structures and Turbulence in Fluid Mechanics (ed. M. Lesieur & O. MeAtais). Kluwer.
Aubry, N., Holmes, P. & Lumley, J. L. 1990 The effect of modeled drag reduction on the wall region. In Theoretical and Computational Fluid Dynamics, vol. 1, pp. 229–248. Springer.
Aubry, N., Holmes, P. & Lumley, J. L. & Stone, E.1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.Google Scholar
Bakewell, P. & Lumley, J. L. 1967 Viscous sublayer and adjacent wall region in turbulent pipe flows. Phys. Fluids 10, 1880–1889.Google Scholar
Blackwelder, R. F. & Eckelmann, H. 1979 Streamwise vortices associated with the bursting phenomenon. J. Fluid Mech. 94, 577–594.Google Scholar
Bogard, D. G. & Tiederman, W. G. 1986 Burst detection with single point velocity measurements. J. Fluid Mech. 162, 389–113.Google Scholar
Busse, F. M. 1981 Transition to turbulence in Rayleigh—BeAnard convection. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub), pp. 97–137. Springer.
Busse, F. M. & Heikes, K. E. 1980 Convection in a rotating layer: a simple case of turbulence. Science 208, 173–175.
Corino, E. R. & Bbodkey, R. S. 1969 A visual investigation of the wall region in turbulent flows. J. Fluid Mech. 37, 1–30.Google Scholar
Corrsin, S. 1957 Some current problems in shear flows. In Proc. Naval Hydr. Symp. 24–28 Sept. 1956, (ed. F. S. Sheman).
Guckenheimer, J. & Holmes, P. 1983 Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer. (Corrected 3rd printing, 1990.)
Guckenheimer, J. & Holmes, P. 1988 Structurally stable heteroclinic cycles. Math. Proc. Camb. Phil. Soc. 103, 189–192.Google Scholar
Guckenheimer, J. & Kim, S. 1990 Kaos. MSI Tech. Rep 90–15. Cornell University, Ithaca, NY.Google Scholar
Herzog, S. 1986 The large scale structure in the near-wall region of turbulent pipe flow. PhD thesis, Cornell University, Ithaca, N.Y.
Holmes, P. 1990 Can dynamical systems approach turbulence? In Proc. Whither Turbulence ? urbulence at the Crossroads (ed. J. L. Lumley). Lecture Notes in Applied Physics, vol. 357, pp. 197–249, 306–309. Springer.
Holmesv, P. 1991 Symmetries, heteroclinic cycles and intermittency in fluid flow. In Proc. IMA. Workshop on Dynamical Theories of Turbulence in Fluid Flows (to appear). Springer.
Holmes, P., Berkooz, G. & Lumley, J. L 199 Turbulence, dynamical systems and the unreasonable effectiveness of empirical eigenfunctions. In Proceedings of the Internationa Congress of Mathematicians ICM-90 (Springer (to appear).
Holmes, P., & Stone, E. 1991 Heteroclinic cycles, exponential tails and intermittency in turbulence production. In Proc. Symp. in Honor of J. L. Lumley (to appear). Springer.
Hyman, J. M. & Nicolaenko, B. 1985 The Kuramoto—Sivashinsky equation: a bridge between PDE's and dynamical systems. Los Alamos National Lab Report LA-UR-85–1556Google Scholar
Hyman, J. M., Nicolaenko, B. & Zaleski, S. 1986 Order and complexity in the Kuramoto—Sivashinsky model of weakly turbulent interfaces. Los Alamos National Lab Report LA-UR-86-1947.Google Scholar
Jones, C. & Proctor, M. R. 1987 Strong spatial references and travelling waves in BeAnard convection. Phys. Lett A121, 224–227.Google Scholar
Kim, H. T., Kline, S. J. & Reynolds, W. C. 1971 The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50, 133–160.Google Scholar
Kirby, M., Boris, J. P. & Sirovich, L. 1990 A proper orthogonal decomposition of a simulated supersonic shear layer. Intl J. for Numer. Math. Fluids 10, 411–428.Google Scholar
Kline, S. J. 1967 Observed structure features in turbulent and transitional boundary layers. Fluid Mechanics of Internal Flow (ed. G. Sovran), pp. 27–68. Elsevier.
Kline, S. J. 1978 In Coherent Structure of Turbulent Boundary Layers Proc. AFOSR/Lehigh Workshop (ed. C. R. Smith and D. E. Abbott), pp. 1–26. The role of visualization in the study of the turbulent boundary layer.
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Rundstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech 30, 741–773.Google Scholar
Ladyzhenskaya, 0. A. 1969 The Mathematical Theory of Viscous Incompressible Flows Gordon and Breach.
Lumley, J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. A. M. Yaglom & V. I. Tatarski), pp. 166–178. Moscow: Nauka.
Lumley, J. L. 1970 Stochastic Tools in Turbulence Academic.
Lumley, J. L. (Ed.) 1990 Whither Turbulence? Turbulence at the Crossroads. Springer Lecture Notes in Physics (to appear).
Moffatt, H. K. 1967 The interaction of turbulence with strong wind shear. In Atmospheric Turbulence and Radio Wave Propagations (ed. A. M. Yaglom & V. I. Tatarsky), pp. 139–156. Moscow: Nauka.
Moffatt, H. K. 1990 Fixed points of turbulent dynamical systems and suppression of nonlinearity. In Whither Turbulence ? Turbulence at the Crossroads (ed. J. L. Lumley), pp. 250–257. Lecture Notes in Applied Physics, vol. 357. Springer.
Moin, P. & Moser, R. D. 1989 Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech. 200, 471–509.Google Scholar
Newell, A. C., Rand, D. A. & Russell, D. 1988 Turbulent transport and the random occurrence of coherent events. Physica. 33D, 281–303.Google Scholar
Nicolaenko, B. & She, Z. S. 1990a Temporal intermittency and turbulence production in the Kolomogorov flow. In Topological Dynamics of Turbulence. Cambridge University Press.
Nicolaenko, B. & She, Z. S. 1990b Symmetry-breaking homoclinic chaos in Kolomogorov flows. Preprint, Arizona State University.
Nicolaenko, B., Scheurer, B. & TeAman, R. 1985 Some global dynamical properties of the Kuramotos—Sivashinsky equations: nonlinear stability and attractors. Physics.IbD, 155–183.Google Scholar
Nicolaenko, B., SCheurer, B. & TeAman, R. 1986 Attractors for the Kuramoto—Sivashinsky equations. Los Alamos National Lab. Report LA-UR-85–1630.Google Scholar
Proctor, M. R. & Jones, C. 1988 The interaction of two spatially resonant patterns in thermal convection. J. Fluid Mech. 188, 301–335. Part 1. Exact 1:2 resonance.Google Scholar
Sirovich, L. 1989 Chaotic dynamics of coherent structures. Physica. 37D, 126–145.Google Scholar
Stone, E. 1989 Studies of low dimensional models for the wall region of a turbulent boundary layer. PhD thesis, Cornell University, Ithaca, NY.
Stone, E. & Holmes, P. 1989 Noise induced intermittency in a model of a turbulent boundary Physica. 37D, 20–32.Google Scholar
Stone, E. & Holmes, P. 1990 Random perturbation of heteroclinic attractors S1AM J. Appl. Maths. 50, 726–743.Google Scholar
TeAman, R. 1988 Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer.
Townsend, A. A. 1970 Entrainment and structure of turbulent flow. J. Fluid Mech. 41, 13–46.Google Scholar
Townsend, A. A. 1976 Structure of Turbulent Shear Flows. (2nd edn). Cambridge University Press.
Willmarth, W. W. 1975 Structure of turbulence in boundary layers. Adv. Appl. Mech. 15, 159–254.Google Scholar