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The dynamics of coherent structures in the wall region of a turbulent boundary layer

Published online by Cambridge University Press:  21 April 2006

Nadine Aubry ,
Philip Holmes ,
John L. Lumley  and
Emily Stone
Nadine Aubry
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
Philip Holmes
Affiliation:
Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA Department of Mathematics and Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA
John L. Lumley
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
Emily Stone
Affiliation:
Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
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Abstract

We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier–Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 192 , July 1988 , pp. 115 - 173
Copyright
© 1988 Cambridge University Press

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References

Armbruster D., Guckenheimer, J. & Holmes P. 1988 Heteroclinic cycles and modulated traveling wave systems with O(2) symmetry. Physica D 29, 257–282.Google Scholar
Aubry N. 1987 A dynamical system coherent structure approach to the fully developed turbulent wall layer. Ph.D. thesis, Cornell University.
Aubry N., Holmes P., Lumley, J. L. & Stone E. 1987 The dynamics of coherent structures in the wall region of a turbulent boundary layer. Sibley School of Mechanical and Aerospace Engineering Rep. FDA-86–15. Cornell University.
Bakewell, P. & Lumley J. L. 1967 Viscous sublayer and adjacent wall region in turbulent pipe flow. Phys. Fluids 10, 1880–1889.Google Scholar
Bergé P., Dubois M., Manneville, P. & Pomeau Y. 1980 Intermittency in Rayleigh–Bénard convection. J. Phys. Lett. 41, L341.Google Scholar
Blackwelder, R. F. & Eckelmann H. 1979 Streamwise vortices associated with the bursting phenomenon. J. Fluid Mech. 94, 577–594.Google Scholar
Blackwelder, R. F. & Kaplan R. E. 1976 On the wall structure of the turbulent boundary layer. J. Fluid Mech. 76, 89–112.Google Scholar
Busse F. H. 1981 Transition to turbulence in Rayleigh–Bénard 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.Google Scholar
Campbell, D. & Rose H. 1983 Order in Chaos. North-Holland.
Cantwell B. J. 1981 Organized motion in turbulent flow. Ann. Rev. Fluid Mech. 13, 457–515.Google Scholar
Chapman, D. R. & Kuhn G. D. 1986 The limiting behaviour of turbulence near a wall. J. Fluid Mech. 170, 265–292.Google Scholar
Corino, E. R. & Brodkey R. S. 1969 A visual investigation of the wall region in turbulent flow. J. Fluid Mech. 37, 1–30.Google Scholar
Corrsin S. 1957 Some current problems in turbulent shear flows. In Proc. Naval Hydr. Symp. 24–28 Sept. 1956. (ed. F. S. Shemen): 373. US Office of Naval Research 1956.
Devaney R. L. 1985 An Introduction to Chaotic Dynamical Systems. Benjamin–Cummings.
Doedel, E. J. & Kernevez J. P. 1985 Software for continuation problems in ordinary differential equations with applications. Applied Mathematics Rep. California Institute of Technology.Google Scholar
Drazin, P. G. & Reid W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Dubois M., Rubio, M. A. & Bergé P. 1983 Experimental evidence of intermittencies associated with a subharmonic bifurcation. Phys. Rev. Lett. 51, 1446–1449.Google Scholar
Ersoy, S. & Walker J. D. A. 1985 Viscous flow induced by counter-rotating vortices. Phys. Fluids 28, 2687–2698.Google Scholar
Farabee T. M. 1986 An experimental investigation of wall pressure fluctuations beneath non-equilibrium turbulent flows. David W. Taylor Naval Ship Research and Development Center DTNSRDC-86–047.Google Scholar
Glauser M. N., George, W. K. & Taulbee D. B. 1985b Evaluation of an alternative orthogonal decomposition. 38th Annual American Physical Society. Division of Fluid Dynamics. Tucson, Arizona. Nov. 24–25.Google Scholar
Glauser M. N., Leib, S. J. & George W. K. 1985a Coherent structure in the axisymmetric jet mixing layer. Proc. 5th Symp. of the Turbulent Shear Flow Conf. Cornell University. Springer.
Golubitsky, M. & Guckenheimer, J. (eds) 1986 Multiparameter Bifurcation Theory. A.M.S. Contemporary Mathematics Series, No. 56. American Mathematical Society, Providence, R.I.
Guckenheimer J. 1986 Strange attractors in fluids: another view. Ann. Rev. Fluid Mech. 18, 15–31.Google Scholar
Guckenheimer, J. & Holmes P. J. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer. (Corrected second printing, 1986).
Hatziavramidis, D. T. & Hanratty T. J. 1979 The representation of the viscous wall region by a regular eddy pattern. J. Fluid Mech. 95, 655–679.Google Scholar
Head, M. R. & Bandyopadhyay P. 1981 New aspects of turbulent boundary layer structure. J. Fluid Mech. 107, 297–338.Google Scholar
Herzog S. 1986 The large scale structure in the near-wall region of turbulent pipe flow. Ph.D. thesis, Cornell University.
Hogenes, J. H. A. & Hanratty T. J. 1982 The use of multiple wall probes to identify coherent flow patterns in the viscous wall region. J. Fluid Mech. 124, 363–390.Google Scholar
Hyman, J. M. & Nicolaenko B. 1985 The Kuromoto–Sivashinsky equation: a bridge between PDEs and dynamical systems. Los Alamos Rep. UR-85–1556.Google Scholar
Hyman J. M., Nicolaenko, B. & Zaleski S. 1986 Order and complexity in the Kuromoto–Sivashinsky model of weakly turbulent interfaces. Physica D 23, 265–292.Google Scholar
Jang P. S., Benney, D. J. & Gran R. L. 1986 On the origin of streamwise vortices in a turbulent boundary layer. J. Fluid mech. 169, 109–123.Google Scholar
Keefe L., Moin, P. & Kim J. 1987 The dimension of an attraction in turbulent Poiseuille flow. Bull. Am. Phys. Soc. 32, 2026.Google Scholar
Kim J. 1983 On the structure of wall-bounded turbulent flows. Phys. Fluids 26, 2088–2097.Google Scholar
Kim J. 1985 Turbulence structures associated with the bursting event. Phys. Fluids 28, 52–58.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
Kim, J. & Moin P. 1986 The structure of the vorticity field in turbulent channel flow. Part 2. Study of ensemble-averaged fields. J. Fluid Mech. 162, 339–363.Google Scholar
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
Kubo, I. & Lumley J. L. 1980 A study to assess the potential for using long chain polymers dissolved in water to study turbulence. Ann. Rep. NASA-Ames Grant No. NSG-2382. Cornell University.Google Scholar
Loève M. 1955 Probability Theory. Van Nostrand.
Lorenz E. N. 1963 Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Lu, S. S. & Willmarth W. W. 1973 Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J. Fluid Mech. 60, 481–511.Google Scholar
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. 1981 Coherent structures in turbulence. In Transition and Turbulence (ed. R. E. Meyer), pp. 215–242. Academic.
Lumley, J. L. & Kubo I. 1984 Turbulent drag reduction by polymer additives: a survey. In The Influence of Polymer Additives on Velocity and Temperature Fields. IUTAM Symposium Essen 1984. (ed. B. Gampert), pp. 3–21. Springer.
Marsden, J. E. & McCracken M. 1976 The Hopf Bifurcation and its Applications. Springer.
Maurer, J. & Libchaber A. 1980 Effects of the Prandtl number on the onset of turbulence in liquid helium. J. Phys. Paris Lett. 41, L-515–518.Google Scholar
Moin P. 1984 Probing turbulence via large eddy simulation. AIAA 22nd Aerospace Sciences Meeting.Google Scholar
Monin A. S. 1978 On the nature of turbulence. Sov. Phys. Usp. 21 429–442.Google Scholar
Nicolaenko B. 1986 Lecture delivered at workshop on Computational Aspects of Dynamical Systems, Mathematical Sciences Institute, Cornell University, September 8–10, 1986.
Nikolaides C. 1984 A study of the coherent structures in the viscous wall region. Ph.D. thesis, University of Illinois, Urbana.
Pomeau, Y. & Manneville P. 1980 Intermittent transition to turbulence. Communs Math. Phys. 74, 189–197.Google Scholar
Rand, D. A. & Young, L. S. (eds) 1981 Dynamical Systems and Turbulence. Lecture Notes in Mathematics, vol. 898. Springer.
Richetti P., Argone, F. & Arneodo A. 1985 Type-II intermittency in a periodically driven non-linear oscillator. Preprint, Université de Nice.
Ruelle, D. & Takens F. 1971 On the nature of turbulence. Communs Math. Phys. 82, 137–151.Google Scholar
Silnikov L. P. 1965 A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl 6, 163–166.Google Scholar
Silnikov L. P. 1968 On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type. Math. USSR Sbornik 6, 427–438.Google Scholar
Silnikov L. P. 1970 A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle-focus type. Math USSR Sbornik 10, 91–102.Google Scholar
Sirovich L. 1987a Turbulence and the dynamics of coherent structures: I. Q. Appl. Maths 45, 561–571.Google Scholar
Sirovich L. 1987b Turbulence and the dynamics of coherent structures: II. Q. Appl. Maths 45, 573–582.Google Scholar
Sirovich L. 1987c Turbulence and the dynamics of coherent structures: III. Q. Appl. Maths 45, 583–590.Google Scholar
Sirovich L., Maxey, M. & Tarman H. 1987 Analysis of turbulent thermal convection. Turbulent Shear Flows 6 (ed F. Durst et al.). Springer (in press).
Sirovich, L. & Rodriguez J. D. 1987 Coherent structures and chaos: a model problem Phys. Lett. A 120, 211–214.Google Scholar
Smale S. 1967 Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817.Google Scholar
Smith C. R. 1984 A synthesized model of the near wall behavior in turbulent boundary layers. In Proc. Eighth Symp. on Turbulence (ed. G. K. Patterson & J. L. Zakin), Dept. of Chem. Eng., University of Missouri-Rolla.
Smith, C. R. & Schwarz S. P. 1983 Observation of streamwise rotation in the near-wall region of a turbulent boundary layer. Phys. Fluids 26, 641–652.Google Scholar
Sparrow C. T. 1982 The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer.
Swinney H. L. 1983 Observations of order and chaos in physical systems. Physica 7D, 3–15. Also in Campbell and Rose 1983.Google Scholar
Swinney, H. L. & Gollub, J. P. (eds) 1981 Hydrodynamic Instabilities and the Transition to Turbulence. Springer.
Tennekes, H. & Lumley J. L. 1972 A First Course in Turbulence. M.I.T. Press.
Theodorsen T. 1952 Mechanism of turbulence. Proc. 2nd Midwestern Conf. on Fluid Mech. Ohio State U., Columbus, Ohio.
Tresser C. 1984 About some theorems by L. P. Silnikov. Ann de L'Inst. H. Poincané 40, 441–461.Google Scholar
Tresser C., Coullet, P. & Arneodo A. 1980 On the existence of hysteresis in a transition to chaos after a single bifurcation. J. Phys. Paris Lett. 41, L243–246.Google Scholar
Willmarth W. W. 1975 Structure of turbulence in boundary layers. Adv. Appl. Mech. 15, 159–254.Google Scholar
Willmarth, W. W. & Bogar T. J. 1977 Survey and new measurements of turbulent structure near the wall. Phys. Fluids Suppl. 20, S9–S21.Google Scholar
Willmarth, W. W. & Lu S. S. 1972 Structure of the Reynolds stress near the wall. J. Fluid Mech. 55, 65–92.Google Scholar
Willmarth, W. W. & Tu B. J. 1967 Structure of turbulence in the boundary layer near the wall. Phys. Fluids Suppl. 10, S134–S137.Google Scholar
Zilberman M., Wygnanski, I. & Kaplan R. E. 1977 Transitional boundary layer spot in a fully turbulent environment. Phys. Fluids Suppl. 20, S258–S271.Google Scholar