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The mixing layer and its coherence examined from the point of view of two-dimensional turbulence

Published online by Cambridge University Press:  21 April 2006

Marcel Lesieur
Affiliation:
Institut de Mécanique, Institut National Polytechnique de Grenoble and Université Scientifique et Médicale, BP 68, 38402 Saint-Martin d'Hères, France
Chantal Staquet
Affiliation:
Institut de Mécanique, Institut National Polytechnique de Grenoble and Université Scientifique et Médicale, BP 68, 38402 Saint-Martin d'Hères, France
Pascal Le Roy
Affiliation:
Institut de Mécanique, Institut National Polytechnique de Grenoble and Université Scientifique et Médicale, BP 68, 38402 Saint-Martin d'Hères, France
Pierre Comte
Affiliation:
Institut de Mécanique, Institut National Polytechnique de Grenoble and Université Scientifique et Médicale, BP 68, 38402 Saint-Martin d'Hères, France

Abstract

A two-dimensional numerical large-eddy simulation of a temporal mixing layer submitted to a white-noise perturbation is performed. It is shown that the first pairing of vortices having the same sign is responsible for the formation of a continuous spatial longitudinal energy spectrum of slope between k−4 and k−3. After two successive pairings this spectral range extends to more than 1 decade. The vorticity thickness, averaged over several calculations differing by the initial white-noise realization, is shown to grow linearly, and eventually saturates. This saturation is associated with the finite size of the computational domain.

We then examine the predictability of the mixing layer, considering the growth of decorrelation between pairs of flows differing slightly at the first roll-up. The inverse cascade of error through the kinetic energy spectrum is displayed. The error rate is shown to grow exponentially, and saturates together with the levelling-off of the vorticity thickness growth. Extrapolation of these results leads to the conclusion that, in an infinite domain, the two fields would become completely decorrelated. It turns out that the two-dimensional mixing layer is an example of flow that is unpredictable and possesses a broadband kinetic energy spectrum, though composed mainly of spatially coherent structures.

It is finally shown how this two-dimensional predictability analysis can be associated with the growth of a particular spanwise perturbation developing on a Kelvin-Helmholtz billow: this is done in the framework of a one-mode spectral truncation in the spanwise direction. Within this analogy, the loss of two-dimensional predictability would correspond to a return to three-dimensionality and a loss of coherence. We indicate also how a new coherent structure could then be recreated, using an eddy-viscosity assumption and the linear instability of the mean inflexional shear.

Type
Research Article
Copyright
© 1988 Cambridge University Press

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References

Arakawa A. 1966 Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. J. Comp. Phys. 1, 119143.Google Scholar
Aref, H. & Siggia E. D. 1980 Vortex dynamics of the two-dimensional turbulent shear layer. J. Fluid Mech. 100, 705737.Google Scholar
Basdevant, C. & Sadourny R. 1983 Modeling of virtual scales in numerical simulation of two-dimensional turbulent flows. J. Méc. Theor. Appl. Special Issue, Two-Dimensional Turbulence (ed. R. Moreau), pp. 243269.
Batchelor G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence Phys. Fluids Suppl. II, 12, 233239.Google Scholar
Bernal L. P. 1981 The coherent structure of turbulent mixing layers. Ph.D. thesis, California Institute of Technology, Pasadena.
Betchov, R. & Szewczyk G. 1963 Stability of a shear layer between parallel streams. Phys. Fluids 6, 13911396.Google Scholar
Browand, F. K. & Ho C. M. 1983 The mixing-layer: an example of quasi two-dimensional turbulence. J. Méc. Theor. Appl., Special Issue, Two-Dimensional Turbulence (ed. R. Moreau), pp. 99–120.
Browand, F. K. & Troutt T. R. 1980 A note on spanwise structure in the two-dimensional mixing layer. J. Fluid Mech. 93, 325336.Google Scholar
Brown, G. L. & Roshko A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775816.Google Scholar
Cain A. B., Reynolds, W. C. & Ferziger J. H. 1981 A three-dimensional simulation of transition and early turbulence in a time-developing mixing layer. Rep. TF-14. Stanford, California.Google Scholar
Chollet, J. P. & Lesieur M. 1981 Parameterization of small scales of three-dimensional isotropic turbulence utilizing spectral closures. J. Atmos. Sci. 38, 27472757.Google Scholar
Corcos, G. M. & Lin S. J. 1984 The mixing layer: deterministic models of a turbulent flow. Part 2. The origin of the three-dimensional motion. J. Fluid Mech. 139, 6795.Google Scholar
Corcos, G. M. & Sherman F. S. 1984 The mixing layer: deterministic models of a turbulent flow. Part 1. Introduction and the two-dimensional flow. J. Fluid Mech. 139, 2965.Google Scholar
Couèt, B. & Leonard A. 1980 Mixing layer simulation by an improved three-dimensional vortex-in-cell algorithm. Proc. 7th Intl Conf. on Numerical Methods in Fluid Dynamics. Stanford-Ames.
Head, M. R. & Bandyopadhyay P. 1981 New aspects of turbulent boundary layer structure. J. Fluid Mech. 107, 297337.Google Scholar
Ho, C. M. & Huang L. S. 1982 Subharmonic and vortex merging in mixing layers. J. Fluid Mech. 119, 443473.Google Scholar
Ho, C. M. & Huerre P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365424.Google Scholar
Holland W. R. 1978 The role of mesoscale eddies in the general circulation of the ocean J. Phys. Oceanogr. 8, 363392.Google Scholar
Hopfinger E. J., Browand, F. K. & Gagne Y. 1982 Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505534.Google Scholar
Hoyer, J. M. & Sadourny R. 1982 Closure modeling of fully developed baroclinc instability. J. Atmos. Sci. 39, 707721.Google Scholar
Jimenez J. 1983 A spanwise structure in the plane shear layer. J. Fluid Mech. 132, 319336.Google Scholar
Jimenez J., Cogollos, M. & Bernal L. P. 1985 A perspective view of the plane mixing layer. J. Fluid Mech. 152, 125143.Google Scholar
Kelly R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27, 657689.Google Scholar
Kline S. J., Reynolds W. C., Schraub, F. A. & Runstadler P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741773.Google Scholar
Kraichnan R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 14171423.Google Scholar
Kraichnan R. H. 1976 Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 15211536.Google Scholar
Lasheras J. C., Cho, J. S. & Maxworthy T. 1986 On the origin and evolution of streamwise vortical structures in a plane, free shear layer. J. Fluid Mech. 172, 231258.Google Scholar
Leith C. E. 1968 Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11, 671673.Google Scholar
Leith, C. E. & Kraichnan R. H. 1972 Predictability of turbulent flows. J. Atmos. Sci. 29, 10411058.Google Scholar
Lesieur M. 1983 Intermittency of coherent structures, an approach using statistical theories of isotropic turbulence. In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi), pp. 339350. North-Holland.
Lesieur M. 1987 Turbulence in Fluids. Nijhoff.
Liepmann, H. W. & Laufer J. 1947 Investigation of free turbulent mixing. NACA Tech. Note, nx 1257.Google Scholar
McWilliams J. C. 1984 The emergence of isolated coherent vortices in turbulent flows. J. Fluid Mech. 146, 2143.Google Scholar
Métais O., Chollet, J. P. & Lesieur M. 1983 Predictability of the large scales of freely-evolving three- and two-dimensional turbulence. In Predictability of Fluid Motions. Proc. A.I.P. Conf., vol. 106, (ed. G. Holloway & B. J. West), pp. 303319.
Métais, O. & Lesieur M. 1986 Statistical predictability of decaying turbulence. J. Atmos. Sci. 43, 857870.Google Scholar
Metcalfe R. W., Orszag S. A., Brachet M. E., Menon, S. & Riley J. 1987 Secondary instability of a temporally growing mixing layer. J. Fluid Mech. 184, 207243.Google Scholar
Michalke A. 1964 On the inviscid instability of the hyperbolic tangent velocity profile. J. Fluid Mech. 19, 543556.Google Scholar
Moffatt H. K. 1986 Geophysical and astrophysical turbulence. In Proc. European Turbulence Conference, Lyon 1986 (ed. G. Comte-Bellot & J. Mathieu), pp. 228244. Springer.
Moin, M. & Kim J. 1982 Numerical investigation of turbulent channel flow. J. Fluid Mech. 118, 341378.Google Scholar
Mory, M. & Hopfinger E. 1986 Structure functions in rotationally dominated turbulent flow. Phys. Fluids 29, 21402146.Google Scholar
Pedlosky J. 1979 Geophysical Fluid Dynamics. Springer.
Perry, A. E. & Chong M. S. 1982 On the mechanism of wall turbulence. J. Fluid Mech. 119, 173217.Google Scholar
Perry A. E., Chong, M. S. & Lim T. T. 1982 Vortices in turbulence. In Vortex motion, Proc. 75th Anniversary of the Aerodynamische Versuchsanstalt, Goettingen (ed. H. G. Hornung & E. A. Muller), pp. 106121. Friedr. Vieweg and Sohn.
Pierrehumbert, R. T. & Widnall S. E. 1982 The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 5982.Google Scholar
Riley, J. J. & Metcalfe R. W. 1980 Direct numerical simulation of a perturbed turbulent mixing layer. AIAA 18th Aerospace Sci. Meeting, Pasadena, 800274.Google Scholar
Schwarztrauber, P. & Sweet R. 1980 Efficient fortran subprograms for the solution of EPD equations. Rep. NCAR. Boulder, Colorado.Google Scholar
Sommeria J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139168.Google Scholar
Staquet C. 1985 Etude numérique de la transition à la turbulence bidimensionnelle dans une couche de mélange. Thèse de I'Université de Grenoble.
Staquet C., Métais, O. & Lesieur M. 1985 Etude de la couche de mélange et de sa cohérence du point de vue de la turbulence bidimensionnelle. C.R. Acad. Sci., Paris 300, II, 833838.Google Scholar
Taneda S. 1981 Large scale periodic motions in turbulent shear layers. J. Phys. Soc. Japan 50, 13981403.Google Scholar
Winant, C. D. & Browand F. K. 1974 Vortex pairing: the mechanism of turbulent mixing layer growth at moderate Reynolds number. J. Fluid Mech. 63, 237255.Google Scholar
Wood, D. H. & Bradshaw P. 1982 A turbulent mixing layer constrained by a solid wall. J. Fluid Mech. 122, 5790.Google Scholar
Wygnanski, I. & Fiedler H. E. 1970 The two-dimensional mixing region. J. Fluid Mech. 41, 327361.Google Scholar
Zabusky, N. J. & Deem G. S. 1971 Dynamical evolution of two-dimensional unstable shear flows. J. Fluid Mech. 47, 353379.Google Scholar