Abstract
The present work studies the isotropic and homogeneous turbulence for incompressible fluids through a specific Lyapunov analysis. The analysis consists in the calculation of the velocity fluctuation through the Lyapunov theory applied to the local deformation using the Navier-Stokes equations, and in the study of the mechanism of energy cascade through the finite scale Lyapunov analysis of the relative motion between two particles. The analysis provides an explanation for the mechanism of energy cascade, leads to the closure of the von Kármán-Howarth equation, and describes the statistics of the velocity difference. Several tests and numerical results are presented.
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References
Ottino J.M.: The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge Texts in Applied Mathematics, New York (1989)
Ottino J.M.: Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech. 22, 207–253 (1990)
Truesdell C.: A First Course in Rational Continuum Mechanics. Academic, New York (1977)
Richardson L.F.: Atmospheric diffusion shown on a distance–neighbour graph. Proc. Roy. Soc. Lond A 110, 709 (1926)
Kolmogorov A.N.: Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32(1), 19–21 (1941)
von Kármán T., Howarth L.: On the statistical theory of isotropic turbulence. Proc. Roy. Soc. A 164(14), 192 (1938)
Batchelor G.K.: The Theory of Homogeneous Turbulence. Cambridge University Press, Cambridge (1953)
Eyink G.L., Sreenivasan K.R.: Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87–135 (2006)
Hasselmann K.: Zur Deutung der dreifachen Geschwindigkeitskorrelationen der isotropen Turbulenz. Dtsch. Hydrogr. Z 11(5), 207–217 (1958)
Millionshtchikov M.: Isotropic turbulence in the field of turbulent viscosity. JETP Lett. 8, 406–411 (1969)
Oberlack M., Peters N.: Closure of the two-point correlation equation as a basis for Reynolds stress models. Appl. Sci. Res. 51, 533–539 (1993)
Skorokhod A.V.: Stochastic Equations for Complex Systems. Springer, Berlin (1988)
Risken H.: The Fokker-Planck Equation: Methods of Solution and Applications. Springer, Berlin (1989)
Domaradzki J.A., Mellor G.L.: A simple turbulence closure hypothesis for the triple-velocity correlation functions in homogeneous isotropic turbulence. Jour. Fluid Mech. 140, 45–61 (1984)
Onufriev, A.: On a model equation for probability density in semi-empirical turbulence transfer theory. In: The Notes on Turbulence. Nauka, Moscow (1994)
Grebenev V.N., Oberlack M.: A Chorin-type formula for solutions to a closure model for the von Kármán-Howarth Equation. J. Nonlinear Math. Phys. 12(1), 19 (2005)
Grebenev V.N., Oberlack M.: A Geometric interpretation of the second-Order structure function arising in turbulence. Math. Phy. Anal. Geom. 12(1), 1–18 (2009)
Toth Z., Kalnay E.: Ensemble forecasting at NMC: the generation of perturbations. Bull. Amer. Meteor. Soc. 74, 2317–2330 (1993)
Kalnay, E., Corazza, M., Cai, M.: Are bred vectors the same as Lyapunov vectors? http://www.atmos.umd.edu/ekalnay/lyapbredamsfinal.htm (2004)
Lu J., Yang G., Oh H., Luo A.C.J.: Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors. Chaos Solitons Fractals 23(5), 1879–1892 (2005)
Lamb H.: Hydrodynamics. Dover Publications, USA (1945)
Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (1990)
Sprott J.C.: Chaos and Time-series Analysis. Oxford University Press, Oxford (2003)
Annan J.D.: On the Orthogonality of Bred Vectors. Mon. Weather Review. 132(3), 843–849 (2004)
Lehmann E.L.: Elements of Large–Sample Theory. Springer, Berlin (1999)
Borisenko A.I., Tarapov I.E.: Vector and Tensor Analysis with Applications. Dover Publication, USA (1990)
Robertson H.P.: The invariant theory of isotropic turbulence. Math. Proc. Camb. Ph. Soc. 36, 209–223 (1940)
Tabeling P., Zocchi G., Belin F., Maurer J., Willaime H.: Probability density functions, skewness, and flatness in large Reynolds number turbulence. Phys. Rev. E 53, 1613 (1996)
Belin F., Maurer J., Willaime H., Tabeling P.: Velocity gradient distributions in fully developed turbulence: an experimental study. Phys. Fluids 9(12), 3843–3850 (1997)
Sreenivasan K.R., Antonia R.A.: The phenomenology of small–scale turbulence. Annu. Rev. Fluid Mech. 29, 435–472 (1997)
Madow W.G.: Limiting distributions of quadratic and bilinear forms. Ann. Appl. Probab. 11(2), 125–146 (1940)
Hildebrand F.B.: Introduction to Numerical Analysis. Dover Publications, USA (1987)
Kolmogorov A.N.: Refinement of previous hypothesis concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 12, 82–85 (1962)
She Z.S., Leveque E.: Universal scaling laws in fully developed turbulence. Phys. Rev. Lett. 72, 336 (1994)
Pullin D., Saffman P.: On the Lundgren Townsend model of turbulent fine structure. Phys. Fluids A 5(1), 126 (1993)
Landau L.D., Lifshitz M.: Fluid Mechanics. Pergamon London, England (1959)
Feigenbaum, M.J.: . J. Stat. Phys. 19, (1978)
Prigogine I.: Time, Chaos and the Laws of Chaos. Progress, Moscow (1994)
Ruelle D., Takens F.: Commun. Math. Phys. 20, 167 (1971)
Pomeau Y., Manneville P.: Commun. Math. Phys. 74, 189 (1980)
Eckmann J.P.: Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643–654 (1981)
Gollub J.P., Swinney H.L.: Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35, 927 (1975)
Giglio M., Musazzi S., Perini U.: Transition to chaotic behavior via a reproducible sequence of period doubling bifurcations. Phys. Rev. Lett. 47, 243 (1981)
Maurer J., Libchaber A.: Rayleigh–Bénard Experiment in liquid helium: frequency Locking and the onset of turbulence. J. de Physique Lett. 40, L419–L423 (1979)
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This work was partially supported by the Italian Ministry for the Universities and Scientific and Technological Research (MIUR).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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de Divitiis, N. Lyapunov analysis for fully developed homogeneous isotropic turbulence. Theor. Comput. Fluid Dyn. 25, 421–445 (2011). https://doi.org/10.1007/s00162-010-0211-9
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DOI: https://doi.org/10.1007/s00162-010-0211-9