Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-18T09:51:08.861Z Has data issue: false hasContentIssue false
  • English
  • Français

A model for large-scale structures in turbulent shear flows

Published online by Cambridge University Press:  26 April 2006

Andrew C. Poje  and
J. L. Lumley
Andrew C. Poje
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14855, USA Present address: IGPP, University of California, Los Alamos National Laboratory, Los Alamos, NM 87544 USA.
J. L. Lumley
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14855, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A procedure based on energy stability arguments is presented as a method for extracting large-scale, coherent structures from fully turbulent shear flows. By means of two distinct averaging operators, the instantaneous flow field is decomposed into three components: a spatial mean, coherent field and random background fluctuations. The evolution equations for the coherent velocity, derived from the Navier–Stokes equations, are examined to determine the mode that maximizes the growth rate of volume-averaged coherent kinetic energy. Using a simple closure scheme to model the effects of the background turbulence, we find that the spatial form of the maximum energy growth modes compares well with the shape of the empirical eigenfunctions given by the proper orthogonal decomposition. The discrepancy between the eigenspectrum of the stability problem and the empirical eigenspectrum is explained by examining the role of the mean velocity field. A simple dynamic model which captures the energy exchange mechanisms between the different scales of motion is proposed. Analysis of this model shows that the modes which attain the maximum amplitude of coherent energy density in the model correspond to the empirical modes which possess the largest percentage of turbulent kinetic energy. The proposed method provides a means for extracting coherent structures which are similar to those produced by the proper orthogonal decomposition but which requires only modest statistical input.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 285 , 25 February 1995 , pp. 349 - 369
Copyright
© 1995 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

Alper, A. & Liu, J. T. C. 1977 On the interaction between large scale structure and fine-grained turbulence in a free shear flow. II. The development of spatial interactions in the mean. Proc. R. Soc. Lond. A 359, 497–523.Google Scholar
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, 115–197.Google Scholar
Aubrey, N. & Sanghi, S. 1990 Bifurcation and bursting of streaks in the turbulent wall layer. In Organized Structures and Turbulence in Fluid Mechanics (ed. M. Lesieur and O. Métais). Kluwer.CrossRefGoogle Scholar
Berkooz, G. 1992 Feedback control by boundary deformation of models of the turbulent wall layer. Tech. Rep. FDA-92–16. Sibley School of Mechanical and Aerospace Engineering, Cornell University.Google Scholar
Berkooz, G., Holmes, P. & Lumley, J. L. 1991 Intermittent dynamics in simple models of the turbulent wall layer. J. Fluid Mech. 230, 75–95.Google Scholar
Blackwelder, R. F. & Kaplan, R. E. 1976 On the wall structure of the turbulent boundary layers. J. Fluid Mech. 76, 89–112.Google Scholar
Brereton, G. J. & Kodal, A. 1992 A frequency-domain filtering technique for triple decomposition of unsteady turbulent flow. J. Fluids Engng 114, 45–51.Google Scholar
Butler, K. M. & Farrell, B. F. 1992a Optimal perturbations in wall-bounded turbulent shear flow. Bull. Am. Phys. Soc. 37, 1738.Google Scholar
Butler, K. M. & Farrell, B. F. 1992b Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A, 4, 1637–1650.Google Scholar
Cantwell, B. J. 1981 Organized motions in turbulent flow. Ann. Rev. Fluid Mech. 13, 457–515.Google Scholar
Elswick, R. C. 1967 Investigation of a theory for the structure of the viscous sublayer in wall turbulence. Master's thesis, The Pennsylvania State University.Google Scholar
Gtski, T. B. & Liu, J. T. C. 1979 On the interaction between large scale structure and fine-grained turbulence in a free shear flow. III. A numerical solution. Phil. Trans. R. Soc. Lond. A 293, 473–509.Google Scholar
Hussain, A. K. M. F. & Reynolds, W. C. 1970 The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41, 241–258.Google Scholar
Hussain, A. K. M. F. & Reynolds, W. C. 1972 The mechanics of an organized wave in turbulent shear flow. Part 2. Experimental results. J. Fluid Mech. 54, 241–262.Google Scholar
Joseph, D. D. 1966 Nonlinear stability of the Boussinesq equations by the method of energy. Arc. for Rat. Mech. Anal. 22, 163.Google Scholar
Joseph, D. D. 1973 Stability of Fluid Flow. Springer.Google Scholar
Liu, J. T. C. 1988 Contributions to the understanding of large-scale coherent structures in developing free turbulent shear flows. Adv. Appl. Mech. 26, 183–309.Google Scholar
Liu, J. T. C. & Merkine, L. 1976 On the interaction between large scale structure and fine-grained turbulence in a free shear flow. I. The development of temporal interactions in the mean. Proc. R. Soc. Lond. A 352, 213–247.Google Scholar
Lumley, J. L. 1967 The structure of inhomgeneous turbulence. In Atmospheric Turbulence and Tave Propagation. Nauka.Google Scholar
Lumley, J. L. 1970 Towards a turbulent constitutive relation. J. Fluid Mech., 41, 413–434.Google Scholar
Lumley, J. L. 1971 Some comments on the energy method. In Developments in Mechanics 6 (ed. L. Lee & A. Szewczyk). Notre Dame Press.Google Scholar
Moin, P. & Moser, R. 1989 Characteristic eddy decomposition of turbulence in a channel. J. Fluid Mech. 200, 471–500.Google Scholar
Naot, D., Shivit, A. & Wolfshtein, M. 1970 Interactions between componenents of the turbulent velocity correlation tensor. Israel J. Tech. 8, 259.Google Scholar
Orr, W. M. 1907 The stability or instability of steady motions in a liquid. Part II: A viscous liquid. Proc. R. Irish Acad. A 28, 122.Google Scholar
Payne, F. R. 1992 Lumley's PODT definition of large eddies and a trio of numerical procedures. In Studies in Turbulence (ed. T. B. Gatski, S. Sarkar & C. G. Speziale). Springer-Verlag.CrossRefGoogle Scholar
Phillips, W. R. C. 1987 The wall region of a turbulent boundary layer. Phys. Fluids 30, 2354–2361.Google Scholar
Poje, A. C. 1993 An energy method stability model for large scale structures in turbulent shear flows. PhD thesis, Cornell University.Google Scholar
Pope, S. B. 1975 A more general effective–viscosity hypothesis. J. Fluid Mech. 72, 331–340.Google Scholar
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiment. J. Fluid Mech. 54, 263–287.Google Scholar
Reynolds, W. C. & Tiederman, W. G. 1967 Stability of turbulent channel flow with application to Malkus's theory. J. Fluid Mech. 27, 253–272.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Ann. Rev. Fluid Mech. 23, 601–639.Google Scholar
Serrin, J. 1959 On the stability of viscous fluid motions. Arch. Rat. Mech. Anal. 3, 1.Google Scholar
Stone, E. & Holmes, P. 1989 Noise induced intermittency in a model of a turbulent boundary layer. Physica 37D, 20–32.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 1–21.Google Scholar