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Eddy viscosity of three-dimensional flow

Published online by Cambridge University Press:  26 April 2006

A. Wirth
Affiliation:
CNRS, Observatoire de Nice, B.P. 229, 06304 Nice Cedex 4, France
S. Gama
Affiliation:
CNRS, Observatoire de Nice, B.P. 229, 06304 Nice Cedex 4, France FEUP, Universidade do Porto, R. Bragas, 4099 Porto Codex, Portugal
U. Frisch
Affiliation:
CNRS, Observatoire de Nice, B.P. 229, 06304 Nice Cedex 4, France

Abstract

Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressible flow.

As shown by Dubrulle & Frisch (1991), the eddy viscosity, which is in general a fourth-order anisotropic tensor, is expressible in terms of the solution of auxiliary problems. These are, essentially, three-dimensional linearized Navier–Stokes equations which must be solved numerically.

The dynamics of weak large-scale perturbations of wavevector k is determined by the eigenvalues – called here ‘eddy viscosities’ – of a two by two matrix, obtained by contracting the eddy viscosity tensor with two k-vectors and projecting onto the plane transverse to k to ensure incompressibility. As a consequence, eddy viscosities in three dimensions, but not in two, can become complex. It is shown that this is ruled out for flow with cubic symmetry, the eddy viscosities of which may, however, become negative.

An instance is the equilateral ABC-flow (A = B = C = 1). When the wavevector k is in any of the three coordinate planes, at least one of the eddy viscosities becomes negative for R = 1/v > Rc [bsime ] 1.92. This leads to a large-scale instability occurring for a value of the Reynolds number about seven times smaller than instabilities having the same spatial periodicity as the basic flow.

Type
Research Article
Copyright
© 1995 Cambridge University Press

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References

Arnold, V. I. 1965 Sur la topologie des écoulement stationnaires des fluids parfaits. C.R. Acad. Sci. Paris 261, 1720.Google Scholar
Bayly, B. J. & Yakhot, V. 1986 Positive and negative phenomena in isotropic and anisotropic Beltrami flows. Phys. Rev. A 34, 381391.Google Scholar
Beltrami, E. 1889 Considerazioni idrodinamiche. Rendiconti del Reale Istituto Lombardo serie II, 22, pp. 121130. Also in Opere Mathematische di Beltrami, vol. 4, pp. 300–309, Ulrico Hoepli, Milan 1920.Google Scholar
Bensoussan, A., Lions, J.-L. & Papanicolaou, G. 1978 Asymptotic Analysis for Periodic Structures. North Holland.
Bondarenko, N. F., Gak, M. Z. & Dolzhansky, F. V. 1979 Laboratory and theoretical models of a plane periodic flow. Atmos. Ocean. Phys. 15, 711720.Google Scholar
Boussinesq, J. 1870 Essai théorique sur les lois trouvées expérimentalement par M. Bazin pour l’écoulement uniforme de l'eau dans les canaux découverts. C.R. Acad. Sci. Paris 71, 389393.Google Scholar
Brush, S. G. 1986 The Kind of Motion We Call Heat. North Holland.
Chulaevsky, V. A. 1989 Almost Periodic Operators and Related Nonlinear Integrable Systems. Manchester University Press.
D'Humiéres, D., Lallemand, P. & Frisch, U. 1986 Lattice Gas Models for 3D Hydrodynamics. Europhys. Lett. 2, 291297.Google Scholar
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. M. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353391.Google Scholar
Dubrulle, B. & Frisch, U. 1991 Eddy viscosity of parity-invariant flow. Phys. Rev. A 43, 53555364 (referred to herein as DF).Google Scholar
Frisch, U., She, Z. S. & Sulem, P. L. 1987 Large-scale flow driven by the anisotropic kinetic alpha effect. Physica D 28, 382392.Google Scholar
Frisch, U., She, Z. S. & Thual, O. 1986 Viscoelastic behaviour of cellular solutions to the Kuramoto–Sivashinsky model. J. Fluid Mech. 168, 221240.Google Scholar
Galloway, D. & Frisch, U. 1986 Dynamo action in a family of flows with chaotic streamlines. Geophys. Astrophys. Fluid Dyn. 36, 5383.Google Scholar
Galloway, D. & Frisch, U. 1987 A note on the stability of a family of space-periodic Beltrami flows. J. Fluid Mech. 180, 557564.Google Scholar
Gama, S., Vergassola, M. & Frisch, U. 1994 Negative eddy viscosity in isotropically forced two-dimensional flow: linear and nonlinear dynamics. J. Fluid Mech. 260, 95126.Google Scholar
Gottlieb, D. & Orszag, S. 1977 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM.
Hénon, M. 1966 Sur la topologie des lignes de courant dans un cas particulier. C.R. Acad. Sci. Paris 262, 312314.Google Scholar
Hénon, M. 1992 Implementation of the FCHC lattice gas model on the Connection Machine. J. Stat. Phys. 68, 353377.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 14171423.Google Scholar
Lamb, H. 1619 Hydrodynamics. Cambridge University Press.
Landau, L. D. & Lifshitz, E. M. 1970 Theory of Elasticity, revised edn. Pergamon.
Meshalkin, L. D. & Sinai, Ya. G., 1961 Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. Appl. Math. Mech. 25, 17001705.Google Scholar
Navier, C. L. M. H. 1823 Mémoire sur les lois du mouvement des fluids. Mém. Acad. R. Sci. 6, 389440.Google Scholar
Nepomnyashchy, A. A. 1976 On the stability of the secondary flow of a viscous fluid in an infinite domain. Appl. Math. Mech. 40, 886891.Google Scholar
Pastur, L. & Figotin, A. 1992 Spectra of Random and Almost-Periodic Operators. Springer.
Prandtl, L. 1925 Bericht über Untersuchungen zur ausgebildeten Turbulenz. Z. angew. Math. Mech. 5, 136139.Google Scholar
Rothman, D. H. 1989 Negative-viscosity lattice gases. J. Stat. Phys. 56, 517524.Google Scholar
Saint Venant, A. (Barré) De, 1851 Formules et tables nouvelles pour les eaux courantes. Ann. Mines 20, 49.Google Scholar
She, Z. S. 1987 Metastability and vortex pairing in the Kolmogorov flow. Phys Lett. A 124, 161164.Google Scholar
Sivashinsky, G. I. 1985 Weak turbulence in periodic flows. Physica 17, 243255.Google Scholar
Sulem, P. L., She, Z. S., Scholl, H. & Frisch, U. 1989 Generation of large-scale structures in three-dimensional flow lacking parity-invariance. J. Fluid Mech. 205, 341358.Google Scholar
Taylor, G. I. 1915 Eddy motion in the atmosphere. Phil. Trans. R. Soc. A 215, 126.Google Scholar
Vergassola, M. 1993 Chiral nonlinearities in forced 2-D Navier–Stokes flows. Europhys. Lett. 24, 4145.Google Scholar
Vergassola, M., Gama, S. & Frisch, U. 1994 Proving the existence of negative isotropic eddy viscosity. In NATO-ASI: Solar and Planetary Dynamos (ed. M. R. E. Proctor, P. C. Mathews & A. M. Rucklidge), pp. 321327. Cambridge University Press.
Wirth, A. 1994 Complex eddy viscosity: a three dimensional effect. In NATO-ARW: Chaotic Advection, Tracer Dynamics and Turbulent Dispersion (ed. A. Provenzale), Physica D 76, pp. 312317.
Yakhot, V., Bayly, B. J. & Orszag, S. 1986 Analogy between hyperscale transport and cellular automaton fluid dynamics. Phys. Fluids 29, 20252027.Google Scholar
Zheligovsky, O. & Pouquet, A. 1993 Hydrodynamic stability of the ABC flow. In NATO-ASI: Solar and Planetary Dynamos (ed. M. R. E. Proctor, P. C. Mathews & A. M. Rucklidge), pp. 347354. Cambridge University Press.
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